CN103004160B - Select wave beam group and the device of beam subset, method and computer program product in a communications system - Google Patents

Abstract

Disclose a kind of apparatus and method for selecting wave beam group and beam subset in a communications system.The method includes: measure the channel condition information (CSI) (920) about the downlink from base station；Broadband properties according to CSI identifies the selected wave beam group (930) from wave beam group set；In selected wave beam group, selected beam subset (940) is identified according at least one subband.Wherein, the characteristic of wave beam group set depends on that the number of the wave beam in transmission rank, and selected beam subset is equal to transmission rank.The method farther includes: generates with this form of dicode and identifies selected wave beam group and the encoder feedback information (950) of the selected beam subset for each subband, transmits encoder feedback information to base station.Also disclose a kind of computer program, including being configured so that device realizes the program code of above operation.

Description

Apparatus, method and computer program product for selecting beam sets and beam subsets in a communication system

related application

The present application claims the priority of PCT application PCT/CN2010/073411 (Attorney Docket No. EIE100124PCT) entitled "Apparatus and Method for Selection of Beam Groups and Subset of Beamsina Communication System" filed with SIPO on June 1, 2010, the entire contents of which are hereby incorporated by reference.

technical field

The present invention relates generally to communication systems, and in particular, to apparatus, methods and systems for selecting beam sets and beam subsets in a communication system.

Background technique

3rd Generation Partnership Project ("3GPP") Long Term Evolution ("LTE") (also referred to as 3GPP LTE) refers to research and development involving 3GPP LTE Release 8 and beyond, which is commonly used in the industry to describe the The name of an ongoing effort to identify the technologies and capabilities of systems such as the Universal Mobile Telecommunications System ("UMTS"). The designation "LTE-A" is commonly used in the industry to refer to a further development of LTE. The goals of this broad-based initiative include improving communication efficiency, reducing costs, improving services, using new spectrum resources, and achieving better integration with other open standards.

The Evolved Universal Terrestrial Radio Access Network ("E-UTRAN") in 3GPP includes the provision of a user plane (including Packet Data Convergence Protocol/Radio Link Control/Media Access Control/Physical ( "PDCP/RLC/MAC/PHY") sublayer) and control plane (including radio resource control ("RRC") sublayer) protocol termination base stations. A wireless communication device or terminal is commonly referred to as User Equipment (also "UE"). A base station is an entity of a communication network commonly referred to as a NodeB or NB. Specifically, in E-UTRAN, an "evolved" base station is called an eNodeB or eNB. See 3GPP Technical Specification ("TS") 36.300 v8.7.0 (2008-12), which is hereby incorporated by reference, for details on the overall architecture of E-UTRAN. For details of radio resource control management, see 3GPP TS 25.331 v.9.1.0 (2009-12) and 3GPP TS 36.331 v.9.1.0 (2009-12), which are hereby incorporated by reference.

As wireless communication systems such as cellular telephony, satellite, and microwave communication systems become widely deployed and continue to attract an ever-increasing number of users, there is an urgent need to accommodate the transmission of increasingly large Data volume, large and variable number of communication devices. The increased data volume is a result of wireless communication devices communicating video information and surfing the Internet as well as performing ordinary voice communications. To address these continuing needs, a topic of current general interest in 3GPP is the efficient use of spatially multiplexed cellular transmissions. Efficient use of spatially multiplexed transmissions can support higher data rates per Hertz ("Hz") of bandwidth at limited transmit power levels, thereby enabling a wireless communication device to transmit a greater amount of data in a shorter period of time. The data, or equivalently, accommodates substantially simultaneous operation of a larger number of wireless communication devices.

To meet peak spectral efficiency requirements (up to 30 bits/Hz), support for up to 8 transmit ("Tx") antennas in the downlink ("DL") will be standardized in 3GPP LTE Release 10, enabling the use of Up to 8 spatial layers for downlink spatial multiplexing transmission. Both 8-transmit downlink multiple-input/multiple-output ("MIMO") and enhanced multi-user multiple-input/multiple-output ("MU-MIMO") are now agreed as Release 10 in relation to enhanced downlink MIMO transmission part of a work item. Such a procedure would enable the transmission of higher data rates at limited transmitter power levels per Hertz of bandwidth.

However, the process of enabling a wireless communication device to communicate channel status and other relevant information back to a base station so that the base station can efficiently perform spatially multiplexed transmissions in the downlink introduces many challenges. One of the many problems is how to address the increased degrees of freedom and communication channel dimensions associated with downlink antenna beamforming (also known as transmit precoding) without the reporting of channel state information, which Reporting places a burden on the uplink communication channel for the wireless communication device. Another problem is achieving improved Single User Multiple Input/Multiple Output ("SU-MIMO") performance where the transmit antenna array has a large azimuth spread in the wireless communication channel. It is generally believed that under current configurations, coverage for wireless communication devices located at beam intersections in the antenna beam space may be poor.

In view of the continued increasing deployment of communication systems such as cellular communication systems and these unresolved issues, it would be beneficial to employ an improved codebook format to enable wireless communication devices to efficiently determine channel conditions and antenna beam characteristics, and to The channel state and antenna beam characteristics are communicated to the base station, which avoids the drawbacks of current communication systems.

Contents of the invention

These and other problems are generally solved or avoided, and technical advantages are generally realized, by embodiments of the present invention, including apparatus, methods for selecting beam sets and beam subsets in a communication system and system. In one embodiment, an apparatus includes a processor and a memory including computer program code. The memory and computer program code are configured, with the processor, to cause the apparatus to measure channel state information on a downlink from the base station and identify a selected beam set from the set of beam sets based on a wideband property of the channel state information. The characteristics of the set of beam groups depend on the transmission rank. The memory and computer program code are further configured, with the processor, to cause the apparatus to identify a selected subset of beams in the selected beam set based on at least one subband. The number of beams in the selected beam subset is equal to the transmission rank.

The foregoing has outlined rather broadly the features and technical advantages of the invention in order that the following detailed description of the invention may be better understood. Additional features and advantages of the invention will be described hereinafter which form the subject of the claims of the invention. It should be appreciated by those skilled in the art that the conception and specific embodiment disclosed may be readily utilized as a basis for modifying or designing other structures or processes for carrying out the same purposes of the present invention. It should also be realized by those skilled in the art that such equivalent constructions do not depart from the spirit and scope of the invention as set forth in the appended claims.

Description of drawings

For a fuller understanding of the present invention and its advantages, reference is now made to the following detailed description taken in conjunction with the accompanying drawings in which:

Figures 1 and 2 illustrate system level diagrams of an embodiment of a communication system including a base station and a wireless communication device providing an application environment for the principles of the present invention;

3 and 4 illustrate system-level diagrams of embodiments of communication systems, including wireless communication systems, that provide an environment for application of the principles of the invention;

Figure 5 illustrates a system level diagram of an embodiment of a communication unit of a communication system for application of the principles of the invention;

Figures 6A, 6B, 7A and 7B illustrate graphical representations of embodiments of forming beamgroups in accordance with the principles of the present invention;

Figure 8 illustrates a graphical representation of an embodiment of a beamset in accordance with the principles of the present invention; and

Figure 9 illustrates a flowchart of an embodiment of a method of operating a communication system in accordance with the principles of the present invention.

detailed description

The making and using of the presently preferred embodiments are discussed in detail below. It should be appreciated, however, that the present invention provides many applicable inventive concepts that can be embodied in a wide variety of specific contexts. The specific embodiments discussed are merely illustrative of specific ways to make and use the invention, and do not limit the scope of the invention. In light of the foregoing, reference will be made to examples in the specific context of apparatuses, methods, and systems for determining and communicating channel states and antenna beam characteristics from a wireless communication device, such as a user equipment, to a base station in a communication system The present invention will be described by way of example. The apparatus, methods and systems are applicable to, but not limited to, any communication system including existing and future 3GPP technologies (ie, UMTS, LTE and future variations thereof, such as fourth generation ("4G") communication systems).

Turning now to FIG. 1 , illustrated is a system level diagram of an embodiment of a communication system including a base station 115 and wireless communication devices (eg, user equipment) 135 , 140 , 145 providing an environment for application of the principles of the present invention. Base station 115 is coupled to a public switched telephone network (not shown). Base station 115 is configured with multiple antennas for transmitting and receiving signals in multiple sectors, including first sector 120, second sector 125, and third sector 130, each of which Sectors typically span 120 degrees. Although FIG. 1 illustrates one wireless communication device (e.g., wireless communication device 140) in each sector (e.g., first sector 120), a sector (e.g., first sector 120) may typically contain multiple a wireless communication device. In an alternative embodiment, base station 115 may be formed of only one sector (eg, first sector 120), and multiple base stations may be configured to operate according to the cooperative (co-operative) multiple-input/multiple-output ("C-MIMO ”) operation etc. to transmit.

Sectors (eg, first sector 120 ) are formed by focusing and phase-adjusting radiated signals from base station antennas, and each sector (eg, first sector 120 ) may employ separate antennas. Multiple sectors 120, 125, 130 can increase the number of subscriber stations (e.g., wireless communication devices 135, 140, 145) that can communicate with base station 115 simultaneously by reducing interference resulting from focusing and phase adjustment of the base station antenna , without increasing the utilized bandwidth. While the wireless communication devices 135, 140, 145 are part of the primary communication system, the wireless communication devices 135, 140, 145 and other devices such as machines (not shown) may be part of a secondary communication system to participate in, without limitation, Device-to-device and machine-to-machine or other communications.

Turning now to FIG. 2 , illustrated is a system level diagram of an embodiment of a communication system comprising a base station 210 and wireless communication devices (eg, user equipment) 260 , 270 providing an environment for application of the principles of the present invention. The communication system includes a base station 210 coupled to a core telecommunications network, such as a public switched telephone network ("PSTN") 230, by a communication path or link 220 (eg, by a fiber optic communication path). The base station 210 is coupled to wireless communication devices 260, 270 within its cellular area 290 by wireless communication paths or links 240, 250, respectively.

In operation of the communication system illustrated in FIG. 2, the base station 210 communicates with each wireless communication device 260, 270 over communication paths 240, 250, respectively, through control and data communication resources allocated by the base station 210. In frequency division duplex ("FDD") and/or time division duplex ("TDD") communication modes, the control and data communication resources may include frequency and time slot communication resources. While the wireless communication devices 260, 270 are part of the primary communication system, the wireless communication devices 260, 270 and other devices such as machines (not shown) may be part of a secondary communication system to participate in, but not limited to, device-to-device and Machine-to-Machine or Other Communications.

Turning now to FIG. 3 , illustrated is a system level diagram of an embodiment of a communication system, including a wireless communication system, providing an environment for application of the principles of the present invention. The wireless communication system may be configured to provide an Evolved UMTS Terrestrial Radio Access Network ("E-UTRAN") Universal Mobile Telecommunications Service. Mobility Management Entity/System Architecture Evolution Gateway (“MME/SAEGW”, one of which is indicated with 310) provides for E-UTRAN Node Bs (indicated is the control function of an "eNB", "evolved Node B", also known as a "base station", one of which is indicated with 320). Base stations 320 communicate via X2 communication links (some of which are indicated with "X2 links"). The various communication links are typically fiber optic, microwave or other high frequency metallic communication paths, such as coaxial links, or combinations thereof.

Base station 320 communicates with wireless communication devices such as user equipment ("UE", some of which are indicated with 330), which is typically a mobile transceiver carried by a user. Accordingly, the communication links (indicated by "Uu" communication links, some of which are indicated by "Uu links") coupling the base station 320 to the user equipment 330 are implemented using methods such as, for example, Orthogonal Frequency Division Multiplexing ("OFDM") ) signal such as the air link for wireless communication signals. While wireless communication device 330 is part of a primary communication system, user equipment 330 and other devices such as machines (not shown) may be part of a secondary communication system to participate in, but not limited to, device-to-device and machine-to-machine communications or other communications.

Turning now to FIG. 4 , illustrated is a system level diagram of an embodiment of a communication system, including a wireless communication system, providing an environment for application of the principles of the present invention. The wireless communication system provides an E-UTRAN architecture, including providing an E-UTRAN user plane (Packet Data Convergence Protocol/ The base stations (one of which is indicated with 410) where the Radio Link Control/Medium Access Control/Physical) and Control Plane (Radio Resource Control) protocols terminate. Base stations 410 are interconnected with X2 interfaces or communication links (indicated with "X2"). The base station 410 is also connected to an Evolved Packet Core ("EPC") comprising a Mobility Management Entity/System Architecture Evolved Gateway ("MME/SAEGW", one of which is indicated at 430) through an S1 interface or communication link (indicated by "S1") ). The S1 interface supports a multi-entity relationship between the MME/SEG 430 and the base station 410 . For applications supporting public terrestrial inter-mobile handover, inter-eNB active mode mobility is supported by the MME/SIA Evolution Gateway 430 relocation via the S1 interface.

Base station 410 may host functions such as radio resource management. For example, base station 410 may perform various functions such as Internet Protocol ("IP") header compression and encryption of user data streams, decoding of user data streams, radio bearer control, radio admission control, connection mobility control, Dynamic allocation of communication resources to user equipment in link and downlink, selection of mobility management entity at user equipment attachment, routing of user plane data towards user plane entity, (from mobility management entity source Scheduling and transmission of paging messages, scheduling and transmission of broadcast information (from mobility management entities or operations and maintenance sources), and measurement and reporting configuration for mobility and scheduling. The mobility management entity/system architecture evolution gateway 430 can host various functions, such as distribution of paging messages to the base station 410, security control, termination of U-plane packets for paging reasons, for user equipment mobility Supported U-plane handover, idle state mobility control, and system architecture evolution bearer control. The user equipment 420 and the machine 425 receive from the base station 410 an assignment of information block groups.

Furthermore, some of the base stations 410 are coupled home base stations 440 (devices) that are coupled to devices such as user equipment 450 and/or machines (not shown) for the secondary communication system. Base station 410 may allocate secondary communication system resources directly to user equipment 420 and machine 425 or to home base station 440 for communication (eg, local communication) within the secondary communication system. For a better understanding of the home base station (indicated with "HeNB"), see 3GPP TS 32.871 v.9.1.0 (2010-03), which is hereby incorporated by reference. Although user equipment 420, machine 425 is part of the primary communication system, user equipment 420, machine 425 and home base station 440 (communicating with other user equipment 450 and machines (not shown)) may be part of a secondary communication system to participate in , but not limited to device-to-device and machine-to-machine communications or other communications.

Turning now to FIG. 5 , illustrated is a system level diagram of an embodiment of a communication unit 510 of a communication system for application of the principles of the present invention. Communication unit or device 510 may represent, but is not limited to, a base station, a wireless communication device (eg, subscriber station, terminal, mobile station, user equipment, machine), network control unit, communication node, and the like. The communication unit 510 includes at least a processor 520, a memory 550 storing programs and data of a temporary or more permanent nature, an antenna 560, and a radio frequency transceiver 570 coupled to the antenna 560 and the processor 520 for two-way wireless communication. The communication unit 510 may provide point-to-point and/or point-to-multipoint communication services.

A communication unit 510, such as a base station in a cellular network, may be coupled to a communication network unit, such as a network control unit 580 of a public switched telecommunications network ("PSTN"). The network control unit 580 may in turn be formed by a processor, memory and other electronic components (not shown). Network control unit 580 typically provides access to a telecommunications network such as the PSTN. Access may be provided using fiber optic, coaxial, twisted pair, microwave communication, or similar links coupled to appropriate link terminating units. The communication unit 510 formed as a wireless communication device is typically a self-contained device to be carried by the end user.

The processor 520 in the communication unit 510, which may be implemented with one or more processing devices, performs functions associated with its operation, including but not limited to precoding of antenna gain/phase parameters (precoder 521), forming individual components of the communication message. Encoding and decoding of bits (encoder/decoder 523), formatting of information, and overall control of the communication unit (controller 525), including processes related to management of communication resources (resource manager 528). Exemplary functions related to management of communication resources include, but are not limited to, hardware installation, traffic management, performance data analysis, end user and device tracking, configuration management, end user management, management of wireless communication devices, billing management, subscriptions, security , billing, etc. For example, in accordance with memory 550, resource manager 528 is configured to allocate primary and secondary communication resources (e.g., time and frequency communication resources) to communicate voice communications and data to/from communication unit 510, and thus communicate between primary and secondary communication systems Format messages that include communication resources in .

Execution of all or part of specific functions or processes related to management of communication resources may be performed in a device separate from and/or coupled to the communication unit 510 , and results of such functions or processes are communicated to the communication unit 510 regarding the execution. The processor 520 of the communication unit 510 may be of any type suitable for the local application environment, and may include, by way of non-limiting examples, a general purpose computer, a special purpose computer, a microprocessor, a digital signal processor ("DSP"), a field programmable One or more of gate arrays ("FPGAs"), application specific integrated circuits ("ASICs"), and processors based on multi-core processor architectures.

Transceiver 570 of communication unit 510 modulates information onto a carrier waveform for transmission by communication unit 510 to another communication unit via antenna 560 . Transceiver 570 demodulates information received via antenna 560 for further processing by other communication units. The transceiver 570 is capable of supporting duplex operation of the communication unit 510 .

As noted above, the memory 550 of the communication unit 510 may be one or more memories, and be of any type suitable for the local application environment, and may be implemented using any suitable volatile or non-volatile data storage technology, such as Semiconductor-based memory devices, magnetic memory devices and systems, optical memory devices and systems, fixed memory and removable memory. Programs stored in memory 550 may include program instructions or computer program code that, when executed by an associated processor, enable communications unit 510 to perform the tasks described herein. Of course, the memory 550 may form a data buffer for data communicated to or from the communication unit 510 . Exemplary embodiments of the systems, subsystems and modules described herein may be implemented at least in part by computer software executable by processors such as wireless communication devices and base stations, or by hardware or combinations thereof. As will become more apparent, systems, subsystems and modules may be implemented in communications unit 510 as illustrated and described herein.

Discussions in 3GPP have recently focused on codebook design for 8 base station transmit antennas and associated transmit precoding, which is not present in the new LTE Release 10 standard. In the RAN1No.59 meeting, it has been agreed to extend the Release 8 implicit feedback framework to LTE Release 10. This is based on a modular (or multi-granularity) design combining two feedback components from separate codebooks representing different properties of the channel state information. One feedback component is for wideband communication channel properties (also referred to as broadband properties) and/or long-term communication channel properties (also referred to as long-term properties), while the other is for frequency-selective communication channel properties (also referred to as frequency-selective attributes) and/or short-term communication channel attributes (also referred to as short-term attributes). An example of a long-term property is the directional structure of an optimal transmit beam. For example, the location of a user equipment does not change rapidly, and thus, its azimuth direction may be substantially fixed. Thus, the directional structure of the transmit beam can be represented by long-term properties that happen to be broadband in nature, especially in the presence of strong spatial correlation at the transmit antenna array, which is very likely in closely spaced antenna elements (e.g., separated by half a wavelength ) was observed under the assumption. Examples of short-term properties are fast amplitude and phase fluctuations in communication paths over the air. Such rapid fluctuations can be represented by short-term properties that are generally frequency selective in nature (ie, vary according to frequency subbands).

This communication channel feedback structure is also referred to herein as a dual codebook structure. Although the missing LTE Release 10 standard specification is related to one of the 8-layer transmissions in a base station with 8 transmit antenna configurations, the principle of the dual codebook structure can still be generalized to any number of transmit antennas. As described herein, support for downlink SU-/MU-MIMO operation for dual codebook based channel state information (“CSI”) feedback is described for applications not limited to LTE Release 10 and later. New codebook design and structure.

3GPP LTE downlink MIMO operation is one of several work items under LTE Release-10 consideration. Two new improvements to LTE Release 8/9 for downlink MIMO are being considered. One improvement lies in the optimization of MU-MIMO operation, which benefits from the use of precoded user equipment-specific reference symbols (called UE-RS in the 3GPP organization, or "Dedicated Reference Symbol "DM-RS") and periodic channel New Reference Signature ("RS") design package for Status Information Reference Signature ("CSI-RS"). The second improvement is to extend the downlink transmission operation to 8-layer downlink SU-MIMO.

These improvements serve as support for an enhanced user equipment feedback mode following the implicit feedback principle from LTE Release-8. Accurate channel state information feedback plays an important role for reliable, interference-free (or substantially interference-free) communication, especially for MU-MIMO. Furthermore, signaling aspects and codebook sizes are important when considering extensions to 8-transmit SU/MU-MIMO operation due to the increased degrees of freedom and communication channel dimension therein.

Similar to LTE Release 8, LTE Release 10 UE feedback design is built on the implicit feedback principle (Channel Quality Indicator/Precoding Matrix Indicator/Rank Indicator) with the following differences: namely, dual-code This format rather than a single codebook format. However, single codebook feedback can still be viewed as a special case of setting one of the codebook entries as the identity matrix. Decisions on the Release 10 design date back to 3GPP Working Group RAN1 No. 59, where the slide presentation presented in 3GPP document R1-101683 titled "WayForwardforRel-10 FeedbackFramework", which is hereby incorporated by reference, describes The two matrices of this form the precoder for the subbands. One codebook addresses wideband communication channel properties and/or long-term properties, so the matrix is here denoted "W ₁ ". Another codebook addresses frequency selectivity and/or short-term properties, so here the matrix is _denoted "W2". The resulting precoder for each subband may eg be structured as a matrix multiplication of two matrices.

In the recent 3GPP RAN1 meeting, a way of utilizing long-term/short-term properties and several codebook design proposals were proposed. Several key design aspects are included in these proposals: the feedback concept is conceived to operate with array-type base station antenna setups of cross-polarized (“XP”) and uniform linear array (“ULA”), and as such, must Design and optimize the codebook accordingly. Long-term and short-term attributes can be sampled at the same or different time periods and reported accordingly (at the same or different times). While taking into account a relatively fixed total feedback rate budget (i.e., a fixed total number of bits in a given time interval), one can try to find the best way to invest feedback bits between codebooks characterized by long-term and short-term properties. good balance. The final precoder is the output of an operation (eg, matrix multiplication) between the long-term precoder and the short-term precoder.

In such products, the order of wideband/long _- term precoder matrix W1 and short _- term precoder matrix W2 can further differentiate this concept. If the broadband/long-term properties are processed after the communication channel (i.e., the channel matrix "H" is right-multiplied as H*W ₁ ), it can be seen as steering the main antenna beam towards the user equipment signal space, while further refinement Co-phase (transmission rank -1) or orthogonality (transmission rank > 1) between beams/precoders can be enhanced at the subband level. This can be viewed as a W ₁ *W ₂ matrix multiplication operation. On the other hand, a larger beam space can be created for matrix W ₁ , which can be fine-tuned by left-multiplying matrix W ₂ . The final precoder matrix W is the output of the W ₂ *W ₁ matrix multiplication. It can be argued that these two ways of forming the product of two codebooks are almost the same. The main difference is how to define the beams and fine _- tuning _of the matrices W2 and W1. A common denominator is that the matrix W ₁ is constructed using (oversampled) discrete Fourier transform ("DFT") vectors or matrices.

_The method of choosing the matrices W2 and W1 at the user equipment _of the communication channel is also important from a complexity point of view, and this may affect the performance of the scheme itself. For example, under the assumption of an exhaustive search among all possible combinations of wideband/long-term and short-term precoders (matrices W ₂ and W ₁ ), a feedback proposal can perform optimally, whereas in the more realistic and less complex Degradation under precoder selection. When considering up to 8 spatial layers (or streams), the dual codebook concept is attractive mainly for the lower transmission ranks, i.e. transmission ranks 1-2 and possibly also for transmission ranks 3-4, and more A high transmission rank may rely _on only a single feedback component (e.g., a matrix W2 of size Nt times R, where Nt and R are the number of transmit antennas at the base station and the transmission rank, respectively) to operate, while other components in It is theoretically set as an identity matrix (for example, a matrix W ₁ =I of size Nt by Nt).

As noted above, the dual codebook concept can be reduced to a definition of beams and beam selection/combination as vectors or precoders that are part of two codebooks. Since matrix W can be constructed using oversampled DFT matrices/vectors, the design effectively builds on some form _of the well-known beam grid concept, where the user equipment effectively chooses to provide the best transmission A beam of performance (a column of the discrete Fourier transform matrix).

The Rel-10 feedback concept should support SU-MIMO and MU-MIMO, where SU-MIMO is typical and provides most of the performance gain for less relevant scenarios with higher communication channel azimuth (angle) spread, whereas For highly correlated scenarios with small azimuth spread, MU-MIMO is typical and provides most performance gains. In the case of higher azimuth spread and SU-MIMO operation, it is usually not sufficient to select only one beam (or multiple beams in case of transmission rank > 1) for the entire frequency band, as in conventional beam grids A good communication performance is achieved since it is known that frequency selective precoding and beam selection at the subband level perform best in this case. On the other hand, for very low azimuth spread and MU-MIMO operation, the beam grid is known to perform well, since wideband and frequency selective precoding achieve very close to the same performance in that case, and wideband precoding It is more attractive due to the much lower associated channel state information feedback overhead. As described here, the conventional beam grid concept is improved such that low azimuth spread cases are still well supported, while improving the performance of SU-/MU-MIMO operation in cases with high azimuth spread.

The main codebook structure was presented during the RAN1 No. 61 meeting, May 10-14, 2010 in Montreal, Canada. However, some concepts are even older. A codebook structure is described in 3GPP TSG-RANWG1 #61 document R1-102630 entitled "Refinements of Feedback and Codebook Design" (Montreal, Canada, 10-14 May 2010), which is hereby incorporated by reference.

In the first proposal for the codebook structure, the precoder matrix W ₁ is mainly treated in the wideband sense, compressing the communication channel in the spatial dimension such that the resulting equivalent communication channel matrix H*W ₁ has a higher ratio than the physical transmission communication The channel matrix H (eg, of size Nr by Nt, where Nt and Nr are the number of transmit and receive antennas, respectively) is of lower dimensionality. Further combination (co-phasing) of the _two remaining dimensions (or beams) (for cross-polarized antennas) or orthogonalization between the streams is handled by the matrix W2, which applies in a frequency-selective manner at the subband level . The precoder matrix W ₁ is a block diagonal matrix, where each block comprises columns of an oversampled discrete Fourier transform matrix. Careful design _of the _two matrices W2 and W1 may allow support of cross-polarized and uniform linear array antenna configurations utilizing the same codebook. This is achieved by using 4 bits for matrix W ₁ and two bits for matrix W ₂ . In the case of four-antenna transmission (e.g. corresponding to four transmit uniform linear array configurations or to each block of four co-polarized transmit antenna elements in the case of an 8 transmit antenna configuration assuming cross-polarization) , the ₄ bits for matrix W1 essentially translate into oversampling coefficients for the four discrete Fourier transform matrices. Entries in the codebook for matrix W ₂ include the 2 transmit antenna codebook of Release 8, while the concept also addresses the operation of transmission rank 1-2 in Release 10. After the W ₁ matrix structure has been adapted to the communication channel, the purpose of the precoder matrix W ₂ is to handle cross-polarization combinations (co-phase and orthogonalization), and also provide support for uniform linear array operations, all based on W ₂ The operation of the matrix is done in a frequency selective manner at the subband level.

Additionally/alternatively, under the constraint of very low associated feedback overhead, some user equipment feedback reporting modes may be devised (eg, on the Physical Uplink Control Channel ("PUCCH")). In the latter case, it makes sense, for example, to consider doing _both W2 and W1 matrix selection and reporting in _a broadband manner. The main difference compared to conventional beam grids is that the feedback is based on a dual codebook format or structure, where this format is used to provide similar beam grids for uniform linear arrays and arrays of cross-polarization type with the same feedback grid operation. The problem with this scheme is exactly the one mentioned in the foregoing (ie, it does not support improved SU-MIMO performance when the azimuth spread is high).

In a second proposal, the precoder matrix W ₁ selects a set of column vectors from the oversampled discrete Fourier transform matrix. For the case of cross-polarized antennas, four antenna beams are created according to polarity, while 8 beams are used for uniform linear arrays. Using 1 bit for matrix W ₁ signaling, the codebook divides the codeword (beam) space into _two non-overlapping parts and selects one of them for further Fine tuning or proper beam adjustment. Note that in a bit selection space, beams are predefined for further processing; thus, in the case of 8 transmit antennas, since there are 8 beams defined by _a W matrix, the size of the matrix W is ₈ × A matrix of 16. The codebook structure is described in 3GPP TSG-RANWG1 #61 document R1-102823 entitled "8TxCodebookDesign" (Montreal, Canada, May 10-14, 2010), which is hereby incorporated by reference.

Note also that the main difference from previous proposals is that the communication channel is compressed in the spatial dimension with the matrix _W1 . Allowing further processing of larger spaces (i.e., with full communication channel dimensions ₎ at the subband level using the matrix W2 can be seen as one of the advantages of the concept, especially in cases with larger azimuth spread and higher transmission rank In the case of SU-MIMO transmission. The precoder matrix W2 is signaled by _four bits, and the associated codebook includes combiners and beam selectors.

Especially in the SU-MIMO case described above, the freedom to choose multiple beams in each subband due to the primary partitioning operated by the matrix W ₁ can be seen as an advantage. However, this flexibility brings its own disadvantages because the beam space defined by one matrix W ₁ is so large that matrix W ₁ does not work well on its own. Furthermore, due to the two spaces created with the matrix W ₁ there is poor coverage for user equipments located at the intersection of the spaces.

In a third proposal, multiple beams are supported per subband by matrix W ₁ . The second precoder matrix W ₂ redefines W ₁ by left multiplication (ie, W ₂ *W ₁ ), thereby performing the rotation of the initial beam. The first codebook (i.e., the codebook associated with matrix W ₁ ) also contains combiners along the main beams and has a larger size of 32 beams in total, while the second codebook (i.e., associated with W ₂ codebook) may only have 2-3 bits, including several rotation matrices. In fact, there are only two interpole combiners available for cross-polarized operation, which can be seen as a disadvantage. A codebook subset restriction for matrix W ₁ is also possible. The codebook structure is described in 3GPP TSG-RANWG1 #61 document R1-103026 entitled "Viewson the Feedback Framework for Rel. 10" (Montreal, Canada, May 10-14, 2010), which is hereby incorporated by reference.

An overview of the main proposed concepts is illustrated in Table 1 below:

Table I

Consider reporting at the same time for matrices W ₁ and W ₂ (which affects Physical Uplink Shared Channel (“PUSCH”) utilization) and 6 PRB granularity and 10 millisecond (“ms”) periodicity with matrix W ₂ Based on the reported 50 physical resource blocks ("PRBs"), Table 2 below illustrates the feedback rates of the main proposed concepts.

Table II

As described herein, a codebook structure is described that provides improved support for situations with low spatial correlation/high azimuth spread, where improved frequency selective feedback is required for good performance. The codebook format or structure also provides improved support for higher transmission ranks, especially for the case of uniform linear arrays.

As mentioned before, the codebook structure is based on similar beam grouping as in the second proposal (i.e., the first part of the feedback signal performs beam group selection and the second part of the feedback signal performs beam selection from the selected beam group ). The first part of the feedback applies to the entire frequency band or broadband properties, while the second part of the feedback is subband-specific. As a special case, the second part of the feedback can also be applied in a broadband manner. In this particular case, it should be understood that there is a single subband for channel state information feedback, where the width is equal to the wideband system bandwidth. To support beam group selection with a dual codebook format or structure and beam selection from the selected beam group, there is a precoder matrix W containing e.g. _a discrete Fourier transform based sub-matrix and a column selection vector and phase The shifted precoder matrix W ₂ is such that the actual beams in the beamset are determined by matrix multiplication of the form:

W=W ₁ *W ₂ .

As described herein, beamsets can overlap. The overlapping beamsets are intended to cover situations with higher azimuth spread where the enhanced broadband/long-term transmission direction is at the "edge" or in other words at the border of two beamsets. In such cases where there is no beamset overlap, a portion of the azimuth spread may not be captured with precoding because it falls into adjacent beamsets. Overlap generally means that the total angular domain (or angular domains) spanned by the beamsets is overlapping. The angular domain may refer to the angular range from the direction of maximum array gain of the first beam in a beam group to the direction of maximum array gain of the last beam, or equivalently, the maximum array gain between any two beams within a beam group Maximum possible angular range in terms of direction of the gain. A special case of such beamset overlap is when beamsets partially contain the same beam. In this case, different matrices W ₁ can be partially constructed from the same column vectors.

Second, the number of beamsets and/or beamset size (number of beams in the set) and/or beamset total angular domain (angular range spanned by a beamset) depends on the transmission rank. Typically an 8 transmit antenna array has antennas that are physically very closely spaced. Because higher transmission ranks employ relatively uncorrelated communication channels, the communication channel azimuth spread must be large to support higher rank transmissions. In the case of a dual codebook format or structure, such a high azimuth spread can be properly captured if the beamset size and overall angular domain is large enough that it covers a large range of azimuth angles. Therefore, for higher transmission ranks, the beamset size and overall angular domain should be made larger. Therefore, the two aspects presented can be combined. Overlapping beamsets are constructed, where the size, number and angular domain of the overlapping beamsets can vary according to the transmission rank.

As introduced herein in the exemplary embodiments, this description is provided in terms of beamsets selected from a set of beamsets and beam subset selection from the selected beamsets and possibly overlapping beamsets. This is equivalently convertible in terms of matrix/vector description and codebook format or structure. The set of beamsets specific to the transmission rank is transformed into W A collection of ₁ matrices. The number of beamgroups within a beamgroup set depends on the transmission rank means that the number of W ₁ matrices in the matrix set associated with a given rank depends on the rank itself. The beams within a given beam group are transformed into specific column vectors within the W ₁ matrix associated with the beam group or equivalently, specific columns of the resulting precoder matrix W=W ₁ *W ₂ . Thus, the number of beams within a beam group translates into the number of columns within an associated precoder. The overlap of beamsets within a set of beamsets can be described as _a subset of the column vectors with column vectors found in other precoders (e.g., other W1 matrix or matrices) associated with other beamsets within the set of beamsets. The precoder (eg, W ₁ matrix) associated with the beamgroup of the set. Selecting a subset of beams from a selected beam set eg translates to a column selection vector/matrix (eg W2 matrix) that selects a subset of columns from _a matrix (eg W1 matrix ₎ associated with the selected beam set. Furthermore, other vector or matrix precoder components can be utilized to handle beam co-phasing or orthogonalization at the subband level (eg, W ₂ includes a phase shifter in addition to column selection elements).

Therefore, user equipment feedback is adopted, wherein the user equipment firstly measures the channel state information, and then selects a beamset from the set of beamsets according to the wideband or long-term channel state information, wherein the size of each beamset (i.e., the size of the beamset set The number of beams in each beam group) and/or the number of beam groups (i.e., the number of beam groups in a set of beam groups) and/or the total angular domain spanned by each beam group (i.e., in The total angular domain spanned from the direction of maximum array gain of the first beam to the direction of maximum array gain of the last beam in each beam group in the set of beam groups) depends on the transmission rank, and/or where different beams Groups overlap. In other words, the characteristics of the set of beam sets include at least one of the following: the number of beams in each beam set of the set of beam sets, the number of beam sets in the set of beam sets, and the number of beam sets in each beam set of the set of beam sets. The total angular domain spanned from the direction of maximum array gain of the first beam to the direction of maximum array gain of the last beam. For each subband, the user equipment selects a subset of beams in the selected beam set, where the size of the subset is equal to the transmission rank. Furthermore, the selected subset of beams may include beams that are orthogonal to each other. The user equipment encodes the feedback information into _{a dual} codebook or precoder format using matrices W1 and W2 for transmission on the uplink communication channel and transmits the dual codebook format to the base station.

The base station receives the feedback transmitted on the uplink communication channel, decodes the feedback information, and converts it into a dual codebook or precoder format or structure (ie, into matrices W ₁ and W ₂ ). The base station calculates the final precoders (i.e. antenna weights ₎ to be used for the transmission to the user equipment by frequency subband based _on the matrices W1 and W2, e.g. by matrix multiplication as follows:

W=W ₁ *W ₂ .

When transmitting data to the user equipment, the base station performs antenna weighting on the scheduled subbands according to the weights in the matrix W.

In order to calculate the feedback to the base station, the user equipment first measures channel state information. In case of an LTE based communication system, this measurement may be made using reference signals such as CSI-RS with 8 transmit antennas or CSI-RS with 4 transmit antennas (or less). Then, the user equipment obtains channel state information for the entire system bandwidth.

Once the user equipment has obtained the channel state information, the feedback can be calculated (ie, the user equipment determines the beam set and selects the beam for each subband). In the case of a dual codebook format or structure, the user equipment selects the wideband/long-term precoder matrix W ₁ and, conditional on the selected precoder matrix W ₁ , selects the precoder matrix for each subband W ₂ . In one aspect, matrix W ₁ is utilized to select beam groups, and precoder matrices W ₁ and W ₂ together form the final beam (precoder) within each beam group. Then, matrix multiplication is employed to construct the final precoder:

W=W ₁ *W ₂ .

Turning now to FIGS. 6A and 6B , pictorial representations of embodiments of forming beamgroups in accordance with the principles of the present invention are illustrated. It should be understood that the azimuthal gain characteristic of a particular transmit beam is not flat and is precisely confined in the angular domain as represented in the figure. In Fig. 6A, 4 beam groups are illustrated for the case of transmission rank 1-2, each beam group comprising eg 4 beams. As shown in Figure 6B for transmission ranks 3-4, there are two beam groups, but with a larger beam group angular (azimuth) domain, each group comprising, for example, 8 beams. The size of the beamset and/or the number of beamsets and/or the total angular domain spanned by the beamset depends on the transmission rank. Exemplary codebook entries for precoder matrices W ₁ , W ₂ supporting this format or structure are presented below.

Turning now to FIGS. 7A and 7B , there are illustrated graphical representations of embodiments of forming beamgroups in accordance with the principles of the present invention. In the illustrated embodiment, the beam sets overlap. Note that the number, size, and angular domain spanned by each beamset are dependent on the transmission rank. Exemplary codebook entries for such formats or structures are presented below.

Both base station and user equipment know the codebook constructed using these principles. The user equipment may select codebook entries using, for example, the following exemplary matrix selection scheme. First, the user equipment calculates the wideband communication channel spatial covariance matrix R:

R=E{H ^H H},

where E is the expectation operator, and the superscript operator "H" denotes the Hermitian operator (ie, conjugate and transpose the corresponding matrix). Again, (non-superscript) H denotes the communication channel matrix. Then, the user equipment uses, for example but not limited to, one of the following ways to select the matrix W ₁ .

_In an exemplary method, the user equipment forms all matrix combinations by scanning (searching ₎ among all possible choices for matrices W1 and W2,

W＝W ₁ *W ₂

This search can be done _over a wide bandwidth for matrix W2 (ie, over the entire system bandwidth) to limit computational complexity. Then, the user equipment finds a precoder matrix W that reduces (eg, minimizes) the chord, Fubini-Shi graph, or projection two-norm distance with respect to V, where the matrix S,

R = USV ^H

is the singular value decomposition of the matrix R. Matrices U and V are custom unitary matrices associated with transforming matrix R into diagonal matrix S. The precoder matrix W1 is selected as the _one corresponding to the enhanced (eg optimal) W.

In another exemplary method, the user equipment forms all combinations of matrices by scanning (eg, in a broadband sense) all possible choices of matrices W ₁ and W ₂ .

W=W ₁ *W ₂ .

Then, the user equipment finds the precoder matrix W that enhances (for example, maximizes) the capability (for example, maximizes the following expression):

$\mathrm{<span\; class="notranslate">\; det</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; I</span>}<span\; class="notranslate">\; +</span>\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{{\mathrm{<span\; class="notranslate">\; \&sigma;</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}}{\mathrm{<span\; class="notranslate">\; W</span>}}^{\mathrm{<span\; class="notranslate">\; h</span>}}\mathrm{<span\; class="notranslate">\; RW</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; .</span>$

The precoder matrix W1 is selected as the _one corresponding to the enhanced (eg optimal) W.

In order to reduce the complexity of the above method, the combination

W＝W ₁ *W ₂

can be sampled such that, for example, only one beam from each beam group is selected as representative of the beam group, and the selection is done based on a subset of the matrix W, choosing to enhance (e.g., maximize) $\det (I+\frac{1}{{\mathrm{\&sigma;}}^2}{W_1}^h{\mathrm{RW}}_1)$ The matrix W ₁ .

A second option consists in choosing the matrix W ₁ that enhances (eg maximizes) trace(W ₁ ^H RW ₁ ). Once the wideband/long-term precoder matrix W ₁ is selected, conditional _on the choice of matrix W ₁ , e.g. Frequency subband selection precoder matrix W ₂ .

Once the user equipment has determined the beam group and determined the beam selection within the beam group by subband, the user equipment encodes the selection for transmission on the uplink communication channel. Both matrices W ₁ and W ₂ are encoded as indices into a codebook known to both the base station and the user equipment. The user equipment transmits the selected index of matrix W ₁ and the selected index of matrix W ₂ to the base station (eg, dual codebook format). In the case of an LTE-based communication system, the uplink communication channel used for feedback transmission may be a Physical Uplink Control Channel ("PUCCH") or PUSCH. In case of PUCCH, at least two alternative signaling solutions are considered.

In one signaling solution, the index of matrix W ₁ may be encoded in the first PUCCH report together with the transmission rank indicator, and the index of matrix W ₂ may be encoded together with the channel quality indicator (“CQI”) in In another PUCCH report. In another signaling solution, the transmission rank indicator is transmitted separately in the first PUCCH report, and the index of matrix W ₁ and the index of matrix W ₂ and the CQI are transmitted in another PUCCH report. In an exemplary embodiment, the data for matrix W ₁ and the data for matrix W ₂ may be reported with the same frequency, or the data for matrix W 1 may be reported with a higher frequency than the data for matrix W _{1 2} data. In case of PUSCH, all information including transmission rank indicator, two codebook indexes (eg, dual codebook format) and CQI can be transmitted in one PUSCH report. In contrast to PUCCH, the transmission of PUSCH feedback reports is usually triggered by the base station, which happens periodically according to some semi-static configuration.

The base station obtains the final precoder matrix according to the result of matrix multiplication,

W=W ₁ *W ₂ .

Neither of the codebook matrices are self-contained, it is advantageous to receive both matrices in one report. However, as matrix W ₁ exploits the broadband/long-term properties of the communication channel, reporting to matrix W ₁ can be done with a lower temporal periodicity, while matrix W ₂ fine-tuning can be sent with a higher temporal periodicity. Such decoupled signaling can be employed assuming that matrix W1 is fairly stable _over time and that error-free transmissions have been performed.

An exemplary codebook design for transmission rank dependent beamsets with transmission rank 1-4 operation is now described. In this exemplary design, there are 4 beamsets with transmission ranks 1 and 2, and two beamsets across a larger beam space with transmission ranks 3 and 4. The 4 beam groups in the case of transmission ranks 1 and 2 are described with a discrete Fourier transform based block diagonal matrix as follows.

${\mathrm{<span\; class="notranslate">\; W</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1,2</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 0</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}}\left[\begin{array}{cccccccc}\mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}\\ \mathrm{<span\; class="notranslate">\; 1</span>}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 8</span>}}}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 4</span>}}}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; 3</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 8</span>}}}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}\\ \mathrm{<span\; class="notranslate">\; 1</span>}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 4</span>}}}& \mathrm{<span\; class="notranslate">\; i</span>}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; 3</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 4</span>}}}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}\\ \mathrm{<span\; class="notranslate">\; 1</span>}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; 3</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 8</span>}}}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; 3</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 4</span>}}}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; 9</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 8</span>}}}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}\\ \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}\\ \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 8</span>}}}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 4</span>}}}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; 3</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 8</span>}}}\\ \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 4</span>}}}& \mathrm{<span\; class="notranslate">\; i</span>}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; 3</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 4</span>}}}\\ \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; 3</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 8</span>}}}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; 3</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 4</span>}}}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; 9</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 8</span>}}}\end{array}\right]$

${\mathrm{<span\; class="notranslate">\; W</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1,2</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}}\left[\begin{array}{cccccccc}\mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}\\ \mathrm{<span\; class="notranslate">\; i</span>}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; 5</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 8</span>}}}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; 3</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 4</span>}}}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; 7</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 8</span>}}}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}\\ <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; 5</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 4</span>}}}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; i</span>}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; 7</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 4</span>}}}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}\\ <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; i</span>}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; 15</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 8</span>}}}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 4</span>}}}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; 5</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 8</span>}}}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}\\ \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}\\ \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; i</span>}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; 5</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 8</span>}}}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; 3</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 4</span>}}}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; 7</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 8</span>}}}\\ \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; 5</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 4</span>}}}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; i</span>}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; 7</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 4</span>}}}\\ \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; i</span>}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; 15</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 8</span>}}}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 4</span>}}}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; 5</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 8</span>}}}\end{array}\right]$

${\mathrm{<span\; class="notranslate">\; W</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1,2</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 2</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}}\left[\begin{array}{cccccccc}\mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}\\ <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; 9</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 8</span>}}}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; 5</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 4</span>}}}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; 11</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 8</span>}}}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}\\ \mathrm{<span\; class="notranslate">\; 1</span>}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 4</span>}}}& \mathrm{<span\; class="notranslate">\; i</span>}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; 3</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 4</span>}}}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}\\ <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; 11</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 8</span>}}}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; 7</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 4</span>}}}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 8</span>}}}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}\\ \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}\\ \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; 9</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 8</span>}}}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; 5</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 4</span>}}}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; 11</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 8</span>}}}\\ \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 4</span>}}}& \mathrm{<span\; class="notranslate">\; i</span>}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; 3</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 4</span>}}}\\ \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; 11</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 8</span>}}}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; 7</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 4</span>}}}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 8</span>}}}\end{array}\right]$

${\mathrm{<span\; class="notranslate">\; W</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1,2</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 3</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}}\left[\begin{array}{cccccccc}\mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}\\ <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; i</span>}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; 13</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 8</span>}}}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; 7</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 4</span>}}}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; 15</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 8</span>}}}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}\\ <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; 5</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 4</span>}}}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; i</span>}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; 7</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 4</span>}}}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}\\ \mathrm{<span\; class="notranslate">\; i</span>}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; 7</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 8</span>}}}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; 5</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 4</span>}}}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; 13</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 8</span>}}}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}\\ \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}\\ \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; i</span>}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; 13</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 8</span>}}}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; 7</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 4</span>}}}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; 15</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 8</span>}}}\\ \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; 5</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 4</span>}}}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; i</span>}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; 7</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 4</span>}}}\\ \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; i</span>}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; 7</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 8</span>}}}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; 5</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 4</span>}}}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; 13</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 8</span>}}}\end{array}\right]$

The two beamsets in the case of transmission rank 3-4 can be described as follows:

${\mathrm{<span\; class="notranslate">\; W</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 3,4</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; \&Center\; Dot;</span>\left[\begin{array}{cc}\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; )</span>& \mathrm{<span\; class="notranslate">\; 0</span>}\\ \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; )</span>\end{array}\right]<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 0,1</span>}$

in,

$\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\left[\begin{array}{cccc}\mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}\\ \mathrm{<span\; class="notranslate">\; 0</span>}& {<span\; class="notranslate">\; (</span><span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; no</span>}}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}\\ \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}\\ \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& {<span\; class="notranslate">\; (</span><span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; no</span>}}\end{array}\right]\left[\begin{array}{ccccc}\mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& <span\; class="notranslate">\; .</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; .</span>& \mathrm{<span\; class="notranslate">\; 1</span>}\\ \mathrm{<span\; class="notranslate">\; 1</span>}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; j</span>}\frac{\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 8</span>}}}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 2</span>}<span\; class="notranslate">\; )</span>\frac{\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 8</span>}}}& <span\; class="notranslate">\; .</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; .</span>& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 7</span>}<span\; class="notranslate">\; )</span>\frac{\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 8</span>}}}\\ \mathrm{<span\; class="notranslate">\; 1</span>}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 2</span>}<span\; class="notranslate">\; )</span>\frac{\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 8</span>}}}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 2</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 2</span>}<span\; class="notranslate">\; )</span>\frac{\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 8</span>}}}& <span\; class="notranslate">\; .</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; .</span>& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 7</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 2</span>}<span\; class="notranslate">\; )</span>\frac{\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 8</span>}}}\\ \mathrm{<span\; class="notranslate">\; 1</span>}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 3</span>}<span\; class="notranslate">\; )</span>\frac{\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 8</span>}}}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 2</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 3</span>}<span\; class="notranslate">\; )</span>\frac{\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 8</span>}}}& <span\; class="notranslate">\; .</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; .</span>& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 7</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 3</span>}<span\; class="notranslate">\; )</span>\frac{\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 8</span>}}}\end{array}\right]$

_The corresponding W2 vector/matrix includes the beam selection vector, multiplied by a complex number to shift the phase of the vector. Only those for transmission rank-1 are listed in Table 3, while the designs for transmission rank 2-4 follow similarly by making appropriate phase shifts to the transmission rank 2-4 beam selection vectors. In Table III below, e _i represents a 4×1 vector for selecting the i-th beam, e.g.,

e ₂ =[0100] ^T .

Table III

An exemplary codebook design for overlapping beamsets for transmission rank 1-2 operation is now described. In this exemplary design, there are 8 beam sets that partially overlap. This overlap is made visible in the _W1 matrix by setting some columns in the two adjacent matrices to be the same.

${\mathrm{<span\; class="notranslate">\; W</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1,2</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 0</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}}\left[\begin{array}{cccccccc}\mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}\\ \mathrm{<span\; class="notranslate">\; 1</span>}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 8</span>}}}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 4</span>}}}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; 3</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 8</span>}}}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}\\ \mathrm{<span\; class="notranslate">\; 1</span>}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 4</span>}}}& \mathrm{<span\; class="notranslate">\; i</span>}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; 3</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 4</span>}}}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}\\ \mathrm{<span\; class="notranslate">\; 1</span>}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; 3</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 8</span>}}}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; 3</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 4</span>}}}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; 9</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 8</span>}}}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}\\ \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}\\ \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 8</span>}}}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 4</span>}}}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; 3</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 8</span>}}}\\ \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 4</span>}}}& \mathrm{<span\; class="notranslate">\; i</span>}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; 3</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 4</span>}}}\\ \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; 3</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 8</span>}}}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; 3</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 4</span>}}}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; 9</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 8</span>}}}\end{array}\right]$

${\mathrm{<span\; class="notranslate">\; W</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1,2</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}}\left[\begin{array}{cccccccc}\mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}\\ {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 4</span>}}}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; 3</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 8</span>}}}& \mathrm{<span\; class="notranslate">\; i</span>}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; 5</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 8</span>}}}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}\\ \mathrm{<span\; class="notranslate">\; i</span>}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; 3</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 4</span>}}}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; 5</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 4</span>}}}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}\\ {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; 3</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 4</span>}}}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; 9</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 8</span>}}}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; i</span>}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; 15</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 8</span>}}}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}\\ \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}\\ \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 4</span>}}}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; 3</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 8</span>}}}& \mathrm{<span\; class="notranslate">\; i</span>}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; 5</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 8</span>}}}\\ \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; i</span>}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; 3</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 4</span>}}}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; 5</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 4</span>}}}\\ \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; 3</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 4</span>}}}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; 9</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 8</span>}}}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; i</span>}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; 15</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 8</span>}}}\end{array}\right]$

${\mathrm{<span\; class="notranslate">\; W</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1,2</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 2</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}}\left[\begin{array}{cccccccc}\mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}\\ \mathrm{<span\; class="notranslate">\; i</span>}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; 5</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 8</span>}}}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; 3</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 4</span>}}}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; 7</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 8</span>}}}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}\\ <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; 5</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 4</span>}}}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; i</span>}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; 7</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 4</span>}}}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}\\ <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; i</span>}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; 15</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 8</span>}}}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 4</span>}}}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; 5</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 8</span>}}}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}\\ \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}\\ \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; i</span>}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; 5</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 8</span>}}}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; 3</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 4</span>}}}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; 7</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 8</span>}}}\\ \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; 5</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 4</span>}}}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; i</span>}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; 7</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 4</span>}}}\\ \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; i</span>}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; 15</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 8</span>}}}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 4</span>}}}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; 5</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 8</span>}}}\end{array}\right]$

${\mathrm{<span\; class="notranslate">\; W</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1,2</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 3</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}}\left[\begin{array}{cccccccc}\mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}\\ {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; 3</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 4</span>}}}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; 7</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 8</span>}}}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; 9</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 8</span>}}}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}\\ <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; i</span>}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; 7</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 4</span>}}}& \mathrm{<span\; class="notranslate">\; 1</span>}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 4</span>}}}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}\\ {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 4</span>}}}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; 5</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 8</span>}}}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; 11</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 8</span>}}}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}\\ \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}\\ \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; 3</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 4</span>}}}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; 7</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 8</span>}}}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; 9</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 8</span>}}}\\ \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; i</span>}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; 7</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 4</span>}}}& \mathrm{<span\; class="notranslate">\; 1</span>}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 4</span>}}}\\ \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 4</span>}}}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; 5</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 8</span>}}}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; 11</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 8</span>}}}\end{array}\right]$

${\mathrm{<span\; class="notranslate">\; W</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1,2</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 4</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}}\left[\begin{array}{cccccccc}\mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}\\ <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; 9</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 8</span>}}}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; 5</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 4</span>}}}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; 11</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 8</span>}}}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}\\ \mathrm{<span\; class="notranslate">\; 1</span>}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 4</span>}}}& \mathrm{<span\; class="notranslate">\; i</span>}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; 3</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 4</span>}}}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}\\ <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; 11</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 8</span>}}}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; 7</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 4</span>}}}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 8</span>}}}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}\\ \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}\\ \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; 9</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 8</span>}}}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; 5</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 4</span>}}}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; 11</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 8</span>}}}\\ \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 4</span>}}}& \mathrm{<span\; class="notranslate">\; i</span>}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; 3</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 4</span>}}}\\ \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; 11</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 8</span>}}}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; 7</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 4</span>}}}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 8</span>}}}\end{array}\right]$

${\mathrm{<span\; class="notranslate">\; W</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1,2</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 5</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}}\left[\begin{array}{cccccccc}\mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}\\ {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; 5</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 4</span>}}}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; 11</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 8</span>}}}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; i</span>}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; 13</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 8</span>}}}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}\\ \mathrm{<span\; class="notranslate">\; i</span>}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; 3</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 4</span>}}}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; 5</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 4</span>}}}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}\\ {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; 7</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 4</span>}}}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 8</span>}}}& \mathrm{<span\; class="notranslate">\; i</span>}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; 7</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 8</span>}}}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}\\ \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}\\ \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; 5</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 4</span>}}}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; 11</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 8</span>}}}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; i</span>}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; 13</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 8</span>}}}\\ \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; i</span>}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; 3</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 4</span>}}}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; 5</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 4</span>}}}\\ \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; 7</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 4</span>}}}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 8</span>}}}& \mathrm{<span\; class="notranslate">\; i</span>}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; 7</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 8</span>}}}\end{array}\right]$

${\mathrm{<span\; class="notranslate">\; W</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1,2</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 6</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}}\left[\begin{array}{cccccccc}\mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}\\ <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; i</span>}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; 13</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 8</span>}}}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; 7</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 4</span>}}}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; 15</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 8</span>}}}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}\\ <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; 5</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 4</span>}}}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; i</span>}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; 7</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 4</span>}}}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}\\ \mathrm{<span\; class="notranslate">\; i</span>}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; 7</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 8</span>}}}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; 5</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 4</span>}}}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; 13</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 8</span>}}}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}\\ \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}\\ \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; i</span>}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; 13</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 8</span>}}}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; 7</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 4</span>}}}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; 15</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 8</span>}}}\\ \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; 5</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 4</span>}}}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; i</span>}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; 7</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 4</span>}}}\\ \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; i</span>}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; 7</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 8</span>}}}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; 5</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 4</span>}}}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; 13</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 8</span>}}}\end{array}\right]$

${\mathrm{<span\; class="notranslate">\; W</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1,2</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 7</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}}\left[\begin{array}{cccccccc}\mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}\\ {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; 7</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 4</span>}}}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; 15</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 8</span>}}}& \mathrm{<span\; class="notranslate">\; 1</span>}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 8</span>}}}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}\\ <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; i</span>}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; 7</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 4</span>}}}& \mathrm{<span\; class="notranslate">\; 1</span>}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 4</span>}}}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}\\ {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; 5</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 4</span>}}}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; 13</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 8</span>}}}& \mathrm{<span\; class="notranslate">\; 1</span>}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; 3</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 8</span>}}}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}\\ \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}\\ \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; 7</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 4</span>}}}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; 15</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 8</span>}}}& \mathrm{<span\; class="notranslate">\; 1</span>}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 8</span>}}}\\ \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; i</span>}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; 7</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 4</span>}}}& \mathrm{<span\; class="notranslate">\; 1</span>}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 4</span>}}}\\ \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; 5</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 4</span>}}}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; 13</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 8</span>}}}& \mathrm{<span\; class="notranslate">\; 1</span>}& {\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\frac{\mathrm{<span\; class="notranslate">\; 3</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 8</span>}}}\end{array}\right]$

_The corresponding W2 vectors/matrices are similar to those in the first exemplary codebook design (ie, these matrices include beam selection column vectors).

Thus, improved support for higher azimuth spread cases can be achieved in the case of beam-type feedback grids. When there is less spatial correlation in the communication channel (ie, when there is a higher azimuth spread), the entire signal space can be better captured in a frequency-selective manner. The benefit of the beam grid for MU-MIMO in highly correlated cases is also preserved.

Turning to Figure 8, there is illustrated a graphical representation of an embodiment of a beamset for the case of transmission rank 1-2 in accordance with the principles of the present invention. The first beam set (indicated as Beam Set 1) includes 4 beams, where the first beam 810 has a maximum gain of 6 decibels (“dB”) in the azimuth direction of zero degrees, and the fourth beam 820 has a gain of -22 degrees has a maximum gain of 6 decibels (“dB”) in the azimuth direction. The second beam set (indicated as beam set 2) includes 4 beams, where the first beam 830 has a maximum gain of 6 decibels (“dB”) in the azimuth direction of 18 degrees, and the 4th beam has a gain of -4 degrees has a maximum gain of 6 decibels (“dB”) in the azimuth direction. The angular range 850 from the direction of maximum array gain for the first beam 810 in the first beam set to the direction of maximum array gain for the fourth beam (the last beam) 820 is 22 degrees and is identical in the angular domain to that from the second The angle range 860 from the direction of maximum array gain of the first beam 830 to the direction of maximum array gain of the fourth beam (last beam) 840 in the beam groups (adjacent beam groups) overlaps.

Turning now to FIG. 9 , illustrated is a flowchart of an embodiment of a method of operating a communication system in accordance with the principles of the present invention. After starting step or module 910, elements of the communication system (eg user equipment and/or base stations therein) measure in step or module 920 channel state information on the downlink from the base station. In step or module 930, a beamset selected from a set of beamsets is identified based on the wideband properties of the channel state information, wherein the characteristics of the set of beamsets depend on the transmission rank. The characteristic of the set of beam sets may be the number of beams in each beam set of the set of beam sets, the number of beam sets in the set of beam sets, and/or the maximum The total angular domain spanned from the direction of array gain to the direction of maximum array gain of the last beam. In step or module 940, a selected subset of beams is identified in the selected beam set according to at least one subband, wherein the number of beams in the selected subset of beams is equal to the transmission rank.

In step or module 950, encoded feedback information is generated in a dual codebook format to identify the selected set of beams and the selected subset of beams for each subband. The dual codebook format is constructed as a first matrix representing the selected beamset and a second matrix representing the selected subset of beams for each subband. The first matrix may be formed using a set of columns obtained from an oversampled discrete Fourier transform matrix. Therefore, the angular range from the direction of maximum array gain of the first beam in the selected beam group to the direction of maximum array gain of the last beam may be different in the angular domain from the maximum of the first beam in an adjacent beam group. The angular range from the direction of array gain to the direction of maximum array gain of the last beam overlaps. The selected beam set can be used to drive the 8 transmit antennas of the base station. Based on the coding feedback information, a precoder is formed using a dual codebook format (eg, in a base station) in step or module 960 to transmit a signal in a communication system. The method ends at step or module 970 .

Accordingly, described herein are apparatuses, methods and systems for selecting beam sets and beam subsets in a communication system. In one embodiment, an apparatus (eg implemented in user equipment) comprises a processor and a memory comprising computer program code. The memory and computer program code are configured, with the processor, to cause the apparatus to measure channel state information on a downlink from a base station and identify a beam set selected from the set of beam sets based on a wideband property of the channel state information. The properties of the beamset set depend on the transmission rank. The memory and computer program code are further configured to, with the processor, cause the apparatus to identify a selected subset of beams in the selected beam set from at least one subband. The number of beams in the selected beam subset is equal to the transmission rank.

Additionally, the memory and computer program code are further configured to, with the processor, cause the apparatus to generate encoded feedback information in a dual codebook format, the encoded feedback information identifying the selected beam set and the selected beam subbands for each subband set, and transmit coded feedback information to the base station. The dual codebook format includes a first matrix representing the selected set of beams and a second matrix representing the selected subset of beams for each subband. A first matrix is formed using a set of columns obtained from the oversampled discrete Fourier transform matrix. Moreover, the characteristics of the beam group set include at least one of the following: the number of beams in each beam group of the beam group set, the number of beam groups in the beam group set, and the number of beam groups in each beam group of the beam group set. The total angular domain spanned from the direction of maximum array gain of one beam to the direction of maximum array gain of the last beam. Furthermore, the selected beam set may overlap adjacent beam sets in the overall angular domain (or angle domain). For example, the angular range from the direction of maximum array gain of the first beam in the selected beam group to the direction of maximum array gain of the last beam may be different in the angular domain from the maximum of the first beam in the adjacent beam group The angular range from the direction of array gain to the direction of maximum array gain of the last beam overlaps. Also, the selected beam set may feature 8 transmit antennas.

In another embodiment, an apparatus (eg implemented in a base station) comprises a processor and a memory comprising computer program code. The memory and computer program code are configured to, with the processor, cause the apparatus to receive from a user equipment encoded feedback information in a dual codebook format for identifying the selected beam set and the selected beam set for the at least one subband. The selected subset of beams. The selected beam group represents one of the beam group sets that conforms to the broadband property of the channel state information measured by the user equipment, and the characteristics of the beam group set are based on the transmission rank. Also, the selected beam subset is selected in the selected beam group according to at least one subband, and the number of beams in the selected beam subset is equal to the transmission rank. The memory and computer program code are further configured to, with the processor, cause the apparatus to use a dual codebook format to form a precoder for transmitting a signal to a user equipment. Although the devices, methods and systems described herein have been described with respect to cellular-based communication systems, the devices and methods are equally applicable to other types of communication systems, such as WiMax Communication Systems.

Programs or code segments constituting various embodiments of the present invention may be stored in a computer-readable medium or transmitted through a transmission medium by a computer data signal embodied in a carrier wave, or a signal modulated by a carrier wave. For example, a computer program product comprising program code stored on a computer readable medium may form various embodiments of the present invention. "Computer-readable medium" may include any medium that can store or transmit information. Examples of computer-readable media include circuits, semiconductor memory devices, read-only memory ("ROM"), flash memory, erasable ROM ("EPROM"), floppy disk, compact disk ("CD")-ROM, optical disk, Hard drives, fiber optic media, radio frequency ("RF") links, etc. A computer data signal may include any signal that can propagate over a transmission medium such as an electronic communication network communication channel, fiber optics, air, electromagnetic links, RF links, or the like. Code segments may be downloaded via computer networks such as the Internet, Intranet, and the like.

As described above, the exemplary embodiments provide methods and corresponding apparatuses including various modules providing functions for performing method steps. A module may be implemented as hardware (in one or more chips, including integrated circuits such as application specific integrated circuits), or as software or firmware for execution by a computer processor. Specifically, in the case of firmware or software, the exemplary embodiments may be provided as a computer program product comprising a computer readable storage structure on which is implemented computer program code (i.e., software or firmware) to be executed by a computer processor implement.

Although the present invention and its advantages have been described in detail herein, it should be understood that various changes, substitutions and alterations can be made herein without departing from the spirit and scope of the invention as defined by the appended claims. For example, many of the features and functions described above can be implemented in software, hardware or firmware, or a combination thereof. Moreover, many features, functions, and steps for operating them may be reordered, omitted, added, etc., and still fall within the broad scope of the present invention.

Furthermore, the scope of the present invention is not intended to be limited to the specific embodiments of the process, machine, manufacture, composition of matter, components, methods or steps described in the specification. Those of ordinary skill in the art will readily appreciate from this disclosure that currently existing or later developed devices that perform substantially the same function or achieve substantially the same results as the corresponding embodiments described herein can be utilized in accordance with the present invention. A process, machine, manufacture, composition, part, method or step of a thing. Accordingly, the appended claims are intended to include within their scope such processes, machines, manufacture, compositions of matter, means, methods, or steps.

Claims (22)

1. An apparatus for communication, comprising:

an apparatus for measuring channel state information on a downlink from a base station,

means for identifying a selected beam group from a set of beam groups according to a broadband property of the channel state information, wherein a characteristic of the set of beam groups depends on a transmission rank, an

Means for identifying a selected subset of beams in the selected beam group according to at least one subband, wherein the number of beams in the selected subset of beams is equal to the transmission rank.

2. The apparatus of claim 1, further comprising means for performing at least the following:

generating coded feedback information in a dual codebook format, the coded feedback information identifying the selected beam group and the selected subset of beams for each subband, an

Transmitting the encoded feedback information to the base station.

3. The apparatus of claim 2, wherein the dual codebook format comprises a first matrix representing the selected beam group and a second matrix representing the selected beam subset for each subband.

4. The apparatus of claim 3, wherein the first matrix is formed using a set of columns obtained from an oversampled Discrete Fourier Transform (DFT) matrix.

5. The apparatus of claim 1, wherein the characteristics of the set of beam groups comprise at least one of: a number of beams in each beam group of the set of beam groups, a number of beam groups in the set of beam groups, and a total angular domain spanned in each beam group of the set of beam groups from a direction of maximum array gain for a first beam to a direction of maximum array gain for a last beam.

6. The apparatus of any of claims 1-5, wherein an angular range from a direction of maximum array gain for a first beam to a direction of maximum array gain for a last beam in the selected beam group overlaps in an angular domain with an angular range from a direction of maximum array gain for a first beam to a direction of maximum array gain for a last beam in an adjacent beam group.

7. A method for communication, comprising:

measuring channel state information on a downlink from a base station;

identifying a selected beam group from a set of beam groups according to a wideband property of the channel state information, wherein a characteristic of the set of beam groups depends on a transmission rank; and

identifying a selected subset of beams in the selected beam group according to at least one subband, wherein a number of beams in the selected subset of beams is equal to the transmission rank.

8. The method of claim 7, further comprising:

generating coded feedback information in a dual codebook format, the coded feedback information identifying the selected beam group and the selected subset of beams for each subband; and

transmitting the encoded feedback information to the base station.

9. The method of claim 8, wherein the dual codebook format includes a first matrix representing the selected beam group and a second matrix representing the selected beam subset for each subband.

10. The method of claim 8, wherein the characteristics of the set of beam groups include at least one of: a number of beams in each beam group of the set of beam groups, a number of beam groups in the set of beam groups, and a total angular domain spanned in each beam group of the set of beam groups from a direction of maximum array gain for a first beam to a direction of maximum array gain for a last beam.

11. The method according to any of claims 7-10, wherein an angular range from a direction of maximum array gain of a first beam to a direction of maximum array gain of a last beam in the selected beam group overlaps in an angular domain with an angular range from a direction of maximum array gain of a first beam to a direction of maximum array gain of a last beam in an adjacent beam group.

12. An apparatus for communication, comprising:

a device for a user to receive from a user equipment encoded feedback information having a dual codebook format, the encoded feedback information identifying a selected beam group and a selected subset of beams for at least one subband, wherein the selected beam group represents one of a set of beam groups that is consistent with a wideband property of channel state information measured by the user equipment and a characteristic of the set of beam groups is based on a transmission rank, and wherein the selected subset of beams is selected in the selected beam group according to the at least one subband and a number of beams in the selected subset of beams is equal to the transmission rank; and

means for forming a precoder for transmitting a signal to the user equipment using the dual codebook format.

13. The apparatus of claim 12, wherein the dual codebook format comprises a first matrix representing the selected beam group and a second matrix representing the selected beam subset for each subband.

14. The apparatus of claim 13, wherein the first matrix is formed using a set of columns obtained from an oversampled Discrete Fourier Transform (DFT) matrix.

15. The apparatus of claim 12, wherein the characteristics of the set of beam groups comprise at least one of: a number of beams in each beam group of the set of beam groups, a number of beam groups in the set of beam groups, and a total angular domain spanned in each beam group of the set of beam groups from a direction of maximum array gain for a first beam to a direction of maximum array gain for a last beam.

16. The apparatus of claim 12, wherein an angular range from a direction of maximum array gain for a first beam to a direction of maximum array gain for a last beam in the selected beam group overlaps in an angular domain with an angular range from a direction of maximum array gain for a first beam to a direction of maximum array gain for a last beam in an adjacent beam group.

17. The apparatus of any of claims 12-16, wherein the selected beam group is characterized by 8 transmit antennas.

18. A method for communication, comprising:

receiving encoded feedback information from a user equipment having a dual codebook format, the encoded feedback information identifying a selected beam group and a selected subset of beams for at least one subband, wherein the selected beam group represents one of a set of beam groups that is consistent with a wideband property of channel state information measured by the user equipment and a characteristic of the set of beam groups is based on a transmission rank, and wherein the selected subset of beams is selected in the selected beam group according to the at least one subband and a number of beams in the selected subset of beams is equal to the transmission rank; and

forming a precoder for transmitting a signal to the user equipment using the dual codebook format.

19. The method of claim 18, wherein the dual codebook format includes a first matrix representing the selected beam group and a second matrix representing the selected beam subset for each subband.

20. The method of claim 19, wherein the first matrix is formed using a set of columns obtained from an oversampled Discrete Fourier Transform (DFT) matrix.

21. The method of claim 18, wherein the characteristics of the set of beam groups include at least one of: a number of beams in each beam group of the set of beam groups, a number of beam groups in the set of beam groups, and a total angular domain spanned in each beam group of the set of beam groups from a direction of maximum array gain for a first beam to a direction of maximum array gain for a last beam.

22. The method of any of claims 18-21, wherein an angular range from a direction of maximum array gain for a first beam to a direction of maximum array gain for a last beam in the selected beam group overlaps in an angular domain with an angular range from a direction of maximum array gain for a first beam to a direction of maximum array gain for a last beam in an adjacent beam group.