WO2010005997A2

WO2010005997A2 - Closed form singular value decomposition

Info

Publication number: WO2010005997A2
Application number: PCT/US2009/049851
Authority: WO
Inventors: Yi Jiang; Ron Porat
Original assignee: Wi-Lan, Inc.
Priority date: 2008-07-07
Filing date: 2009-07-07
Publication date: 2010-01-14
Also published as: WO2010005997A3

Abstract

The subject matter disclosed herein provides methods and apparatus for determining a singular value decomposition. In one aspect, there is provided a method. The method may include receiving a plurality of signals transmitted by a base station implementing a plurality of antennas configured for a multiple-input and multiple-output transmission. Moreover, one or more singular vectors may be determined using a closed form singular value decomposition. The one or more determined singular vectors may be provided to a precoder at the base station as feedback. Related systems, apparatus, methods, and/or articles are also described.

Description

CLOSED FORM SINGULAR VALUE DECOMPOSITION

CROSS-REFERENCE TO RELATED APPLICATION [0001] This application claims priority of U.S. Patent Application Serial No. 61/078,766, filed on July 7, 2008, entitled "CLOSED FORM SINGULAR VALUE DECOMPOSITION," the disclosure of which is hereby incorporated by reference in its entirety.

FIELD

[0002] The subject matter described herein relates to wireless communications.

BACKGROUND [0003] In signal processing associated with wireless devices, the singular value decomposition is frequently used to process signals. The singular value decomposition is a decomposition that may include determining a pseudoinverse, a least squares fitting of data, a matrix approximation, and determining the rank, the range and/or the null space of a matrix. As such, the singular value decomposition is a computationally intensive operation, which in the case of a wireless device, may be problematic. Moreover, in many implementations, the singular value decomposition is an iterative solution, not of a closed form.

SUMMARY [0004] The subject matter disclosed herein provides methods and apparatus for determining a singular value decomposition. [0005] In one aspect, there is provided a method. The method may include receiving a plurality of signals transmitted at a base station implementing a plurality of antennas configured for a multiple-input and multiple-output transmission. Moreover, one or more singular vectors may be determined using a closed form singular value decomposition. The one or more determined singular vectors may be provided to a precoder at the base station as feedback.

[0006] In another aspect, there is provided a method. The method may include receiving, at a base station, a singular vector determined, at a client station, using a closed form singular value decomposition. The received singular vector may be used when precoding a plurality of signals for transmission using a multiple-input and multiple-output transmission. Moreover, the precoded plurality of signals may be transmitted as the multiple-input and multiple-output transmission.

[0007] The details of one or more variations of the subject matter described herein are set forth in the accompanying drawings and the description below. Features and advantages of the subject matter described herein will be apparent from the description and drawings, and from the claims.

DESCRIPTION OF DRAWINGS [0008] In the drawings,

[0009] FIG. 1 depicts a block diagram of a network including client stations and base stations;

[0010] FIG. 2 A depicts a block diagram of a client station including a channel estimator configured to perform the closed form singular value decomposition described herein; [0011] FIG. 2B depicts a process for determining the closed form singular value decomposition;

[0012] FIG. 3A depicts a block diagram of a base station including a precoder configured to use singular vectors determined at the client station using the closed form singular value decomposition described herein; and

[0013] FIG. 3B depicts a process, at the base station, for using the singular vectors determined using the closed form singular value decomposition.

[0014] Like labels are used to refer to same or similar items in the drawings.

DETAILED DESCRIPTION [0015] FIG. 1 is a simplified functional block diagram of an embodiment of a wireless communication system 100. The wireless communication system 100 includes a plurality of base stations HOA and HOB, each supporting a corresponding service or coverage area 112A and 112B. The base stations are capable of communicating with wireless devices within their coverage areas. For example, the first base station 11OA is capable of wirelessly communicating with a first client station 114A and a second client station 114B within the coverage area 112A. The first client station 114A is also within the coverage area 112B and is capable of communicating with the second base station HOB. In this description, the communication path from the base station to the client station is referred to as a downlink 116A and the communication path from the client station to the base station is referred to as an uplink 116B.

[0016] Although for simplicity only two base stations are shown in FIG. 1, a typical wireless communication system 100 includes a much larger number of base stations. The base stations 11 OA and HOB can be configured as cellular base station transceiver subsystems, gateways, access points, radio frequency (RF) repeaters, frame repeaters, nodes, or any wireless network entry point.

[0017] The base stations HOA and HOB can be configured to support an omni-directional coverage area or a sectored coverage area. For example, the second base station HOB is depicted as supporting the sectored coverage area 112B. The coverage area 112B is depicted as having three sectors, 118A, 118B, and 118C. In typical embodiments, the second base station HOB treats each sector 118 as effectively a distinct coverage area.

[0018] Although only two client stations 114A and 114B are shown in the wireless communication system 100, typical systems are configured to support a large number of client stations. The client stations 114A and 114B can be mobile, nomadic, or stationary units. The client stations 114A and 114B are often referred to as, for example, mobile stations, mobile units, subscriber stations, wireless terminals, or the like. A client station can be, for example, a wireless handheld device, a vehicle mounted device, a portable device, client premise equipment, a fixed location device, a wireless plug-in accessory or the like. In some cases, a client station can take the form of a handheld computer, notebook computer, wireless telephone, personal digital assistant, wireless email device, personal media player, meter reading equipment or the like and may include a display mechanism, microphone, speaker and memory.

[0019] In a typical system, the base stations 11 OA and 11 OB also communicate with each other and a network control module 124 over backhaul links 122 A and 122B. The backhaul links 122A and 122B may include wired and wireless communication links. The network control module 124 provides network administration and coordination as well as other overhead, coupling, and supervisory functions for the wireless communication system 100.

[0020] In some embodiments, the wireless communication system 100 can be configured to support both bidirectional communication and unidirectional communication. In a bidirectional network, the client station is capable of both receiving information from and providing information to the wireless communications network. Applications operating over the bidirectional communications channel include traditional voice and data applications. In a unidirectional network, the client station is capable of receiving information from the wireless communications network but may have limited or no ability to provide information to the network. Applications operating over the unidirectional communications channel include broadcast and multicast applications. In one embodiment, the wireless system 100 supports both bidirectional and unidirectional communications. In such an embodiment, the network control module 124 is also coupled to external entities via, for example, content link 126 (e.g., a source of digital video and/or multimedia) and two-way traffic link 128.

[0021] The wireless communication system 100 can be configured to use Orthogonal Frequency Division Multiple Access (OFDMA) communication techniques. For example, the wireless communication system 100 can be configured to substantially comply with a standard system specification, such as IEEE 802.16 and its progeny or some other wireless standard such as, for example, WiBro, WiFi, Long Term Evolution (LTE), or it may be a proprietary system. The subject matter described herein is not limited to application to OFDMA systems or to the noted standards and specifications. The description in the context of an OFDMA system is offered for the purposes of providing a particular example only.

[0022] As used herein, IEEE 802.16 refers to one or more Institute of Electrical and Electronic Engineers (IEEE) Standard for Local and metropolitan area networks, Part 16: Air Interface for Fixed Broadband Wireless Access Systems, 1 October 2004, IEEE Standard for Local and metropolitan area networks, Part 16: Air Interface for Fixed and Mobile Broadband Wireless Access Systems, 26 February 2006, and any subsequent additions or revisions to the IEEE 802.16 series of standards.

[0023] In some embodiments, downlinks 116A-B and uplink 116C each represent a radio frequency (RF) signal. The RF signal may include data, such as voice, video, images, Internet Protocol (IP) packets, control information, and any other type of information. When IEEE-802.16 is used, the RF signal may use OFDMA. OFDMA is a multi-user version of orthogonal frequency division multiplexing (OFDM). In OFDMA, multiple access is achieved by assigning to individual users groups of subcarriers (also referred to as subchannels or tones). The subcarriers are modulated using BPSK (binary phase shift keying), QPSK (quadrature phase shift keying), QAM (quadrature amplitude modulation), and carry symbols (also referred to as OFDMA symbols) including data coded using a forward error-correction code.

[0024] In some embodiments, a base station is implemented using multiple- input and multiple-output (MIMO). When MIMO is used, a base station may include a plurality of antennas. For example, base station HOA may be configured for MIMO and include a precoder coupled to two antennas for transmitting the MIMO transmission via downlinks 116A-B. A client station may include a plurality of antennas to receive the MIMO transmission sent via downlinks 116A-B. The client station may also combine the received signals, which may result in fewer errors and/or enhanced data transfer. Although the examples given herein are made in the context of MIMO, other smart antenna techniques may be used as well including MISO (multiple input, single output) and SIMO (single input, multiple output).

[0025] Moreover, when MIMO is used, the base station may perform precoding, which may use additional information, such as channel estimation information, to code, for each antenna, a stream of symbols for transmission over the corresponding antenna. In a closed loop feedback-based approach, the channel estimation information is provided by the client station. For example, a client station may receive each of the downlinks 116A-B transmitted by the antennas of the base station, decode the received downlink signals, determine channel estimation information for the decoded channels (e.g., subcarriers) in each of the received downlink signals, and then provide to the base station the determined channel estimation information. The channel estimation information provided by the client station may include the singular vectors determined by the client station using a singular value decomposition (SVD). Although the channel estimation information is described as including singular vectors, the channel estimation information may also include other channel information, such as the signal to noise ratio of a subcarrier, carrier to noise ratio, and the like.

[0026] To determine the singular vectors, the client station may perform a singular value decomposition. However, a typical singular value decomposition is an iterative calculation, which is computationally intensive for a client station. As such, the subject matter described herein relates to a closed form singular value decomposition, which is less computationally intensive when compared to typical, iterative singular value decomposition algorithms.

[0027] FIG. 2A depicts an exemplary client station, such as client station 114B. The client station 114B includes a plurality of antennas 220A-B for receiving the downlinks 116A-B, each transmitted by a base station, such as base station HOA, which implements MIMO as described further below. Although the examples described herein refer to two antennas at the base station and two antennas at the client station, other quantities of antennas can be used at the base station and the client station. The client station 114B also includes a radio interface 240, which may include other components, such as filters, converters (e.g., digital-to-analog converters and the like), symbol demappers, an Inverse Fast Fourier Transform (IFFT) module, and the like, to process the received MIMO transmission sent via downlinks 116A-B, to determine channel estimation information, and to decode any data carried by the downlinks. In some implementations, the client station 114B is also compatible with IEEE 802.16 and MIMO transmissions (which are sent via downlinks 116A-B). The client station 114B further includes a channel estimator 260 (described further below), a processor 220 for controlling client station 114B and for accessing and executing program code stored in memory 225.

[0028] For each of the MIMO transmissions sent via downlinks 116A-B and received at each of antennas 220A-B, the channel estimator 260 may determine channel estimation information. In some embodiments, rather than perform a traditional, iterative singular value decomposition to determine the singular vectors, the channel estimator 260 performs a closed form singular value decomposition. The determined the singular vectors are then provided as feedback by the client station to a precoder at the base station.

[0029] In some embodiments, channel estimator 260 determines the singular vector Vi based on the following:

and determines the singular vector V₂ based on the following:

[0030] The determined singular vectors vj and V₂ are then provided as feedback to the base station (e.g., as a management message transmitted via uplink 116C).

[0031] The following provides example implementations for determining the singular vectors vy and v_∑ , although other approaches may be used as well. Generally, analog feedback may provide advantages, such as low complexity at the client station (e.g., a mobile station) and unbounded feedback accuracy which may be particularly important for multi-user MIMO with uncorrelated antennas or multi-base station MIMO.

[0032] In any case, there are several options for feedback, such as using a full channel matrix H, a covariance matrix H^H H , a strongest (e.g., large) right singular vector, and two strongest right singular vectors. While feeding back the full channel matrix or channel covariance provides sufficient information for single user and/or multi-user MIMO, feeding back just the first or two strongest (e.g., largest) eigenvectors may reduce feedback overhead.

[0033] For example, a 4 antenna base station requires 8 complex values for feedback consisting of a full channel matrix or a covariance matrix, and only 3 complex values for the strongest right singular vector option and 5 complex values for two strongest right singular vectors option. An 8 antenna base station or two 4 antenna base stations in multi-base station MIMO will require 16, 7, and 13 values for the full channel matrix option, the strongest right singular vector option, and the two strongest right singular vectors option respectively. In addition, a typical operation of closed loop MIMO in frequency division duplex may require the client station (e.g., a mobile station) to send an initial estimate of the post precoding carrier-to-interference-plus-noise ratio (CINR).

[0034] To further illustrate, the following provides an example of how the singular values of the channel is calculated as these will determine the post precoding SNR (signal to noise ratio).

[0035] Assume the channel matrix H is of dimension 2 by N (2xN), where N is greater than or equal to 2 (N>=2). The singular values can be found from the eigenvalues of HH which is of a 2x2 dimension matrix, which can be solved based on

the following two equations:

[0036] where Tr is trace and Tr is trace and σi and σ₂ are the two singular values of H.

[0037] To calculate the singular value decomposition, the focus is on the matrix H which is of size Nx2, and then find the correct singular vectors that are of size 2 and use those singular vectors to find the left singular vectors which are the right

singular vectors of Η.

[0038] First, the form of Η^Η is of the following form: H^H = V∑U^H , where

U can be generally expressed as follows:

[0039] By the definition of the singular value decomposition, the first column of matrix U is calculated according to the following equation:

where ||x|| represents a Euclidean norm. The maximum Euclidean norm is the maximal singular value <x, .

[0040] Denoting by ' the i^th row of matrix H and developing the above expression, the following equation is formed:

which can be expressed (e.g., via substitution) as follows:

6> = argmax | A₁ |² cos² θ+ \ h₂ \² sin²0 + 2 | A₂ ⁴A₁ I sinøcosø .

[0041] Differentiating and equating to zero, the following equation is formed:

after which a CORDIC rotation is used to calculate cos θ and sin θ . Using CZH^ = V ∑ , the strongest singular vector is determined by normalizing the following equation:

[0042] At this point, the strongest right singular vector has been finalized. To finalize the two right strongest singular vectors, a solution to the following equation is found: , which may be normalized.

[0043] To illustrate further, the singular value decomposition (SVD) is generally done through iterative procedures. But for the special case of N x 2 matrix, the decomposition may be done via a closed-form solution for singular value decomposition. For example, given a matrix of the following form:

, and then calculating the singular value decomposition as follows:

H = UΣV^* where

^xZ is a orthonormal matrix, and ∑ is of the following form:

and V is of the following form:

[0044] By the definition of the singular value decomposition, the first column of V, which is parameterized by θ and φ , is calculated according to the following equation:

where represents for Euclidean norm, and this maximum Euclidean norm is the

maximal singular value . Denoting a_h b, as the fth element of h 1 and h2, respectively,

the solution of the θ and φ equation above can be derived based on the following equation:

[0045] Thus, the optimal , which can be expressed as

follows:

[0046] Further simplification of the θ and φ equation yields the following:

and further constraining

causes no loss of optimality, and equating the derivative to zero yields the following:

[0047] Hence, the following:

[0048] The value of θ may be calculated based on the above equation, which has only one solution in the interval 0 to π/2. However, since the solution of interest is cos θ and sin θ rather than θ itself, the following may be used to find a solution:

[0049] When α is denoted as follows:

[0050] The following equation results:

which leads to the unique solution in θ over the interval of 0 to π/2 (i.e., as foHows-

and

and

[0051] Given the above cos θ and sin θ, V may be formed as follows:

and given the following:

UΣ = HV

[0052] and, given the following:

which is the first column of UΣ, the largest singular value has the following form.

wherein || represents the Euclidean norm and the second equality follows from the θ symbol above. As such, the corresponding singular vector is

[0053] Given the following equation,

which is the second column of UΣ, and given

, where

represents the Frobenius norm, the smaller singular value is obtained as follows:

[0054] Thus, the corresponding singular vector is

[0055] Table 1 depicts example Matlab pseudo-code for determining the singular vectors as described above.

5 Once the singular vectors (e.g., V₁ and v₂ which may also be referred to as singular vectors ui and U₂) are determined by channel estimator 260, channel estimator 260 may provide the singular vectors to a precoder at the base station. For example, channel estimator 260 may provide the determined singular vectors V₁ and v₂ to processor 220, which forwards to the base station the determined singular vectors V₁ and V₂ in a management message via radio interface 240, one of the antennas 220A or 220B, and uplink 116C.

[0057] FIG. 2B depicts a process 200 for determining the closed form singular value decomposition. The description of process 200 will also refer to FIG. 2A. At 292, a client station receives a plurality of signals. For example, client station 114B may receive a plurality of RF signals transmitted by base station HOB as a multiple-input and multiple-output transmission. At 294, a singular vector may be determined using a closed form singular value decomposition. For example, channel estimator 260 may determine singular vectors vi and v₂ using the closed form singular value decomposition as described herein. At 296, the determined singular vector is provided (e.g., sent) to the base station to configure a precoder. In some implementations, process 200 is stored as program code stored at memory 225.

[0058] FIG. 3A depicts a base station, such as base station HOA. The base station 11OA includes antennas 320A-B configured to transmit via downlinks 116A-B and configured to receive uplink 116C via at least one of antennas 320A-B. The base station 11OA further includes a radio interface 340 coupled to the antennas 320A-B, a precoder 360 (described further below), a processor 330 for controlling base station 11OA and for accessing and executing program code stored in memory 335. The radio interface 340 further includes other components, such as filters, converters (e.g., digital-to-analog converters and the like), mappers, a Fast Fourier Transform (FFT) module, and the like, to generate a MIMO transmission via downlinks 116A-B and to receive the channel estimation information provided via uplink 116C. In some implementations, the base station HOA is also compatible with IEEE 802.16 and the RF signals of the MIMO downlinks 116A-B and uplink 116C are configured in accordance with OFDMA.

[0059] The radio interface 340 decodes the uplink 116C carrying the singular vectors vj and V₂ and then provides the singular vectors V₁ and V₂ to the precoder 360. The precoder 360 uses the singular vectors V₁ and V₂ (as well as any other channel estimation information provided as feedback by the client station to the base station) to precode symbols for MIMO transmission via each antennas 320A-B and downlinks 116A-B. The term "precoding" refers to beamforming to support MIMO transmission at each of the antennas (e.g., using singular vectors to weight orthogonal "eigen-beams" transmitted via each of the antennas).

[0060] FIG. 3B depicts a process 300, at the base station, for using the singular vectors determined using the closed form singular value decomposition. The description of process 300 will refer to FIGs. 2A and 3A as well.

[0061] At 393, a base station, such as base station 11OA, receives one or more singular vectors determined at a client station, such a client station 114B, using the closed form singular value decomposition described herein. At 394, the received singular vectors may be used at a precoder, such as precoder 360, when precoding signals for transmission using multiple-input and multiple-output transmission. At 396, the base station transmits, as the multiple-input and multiple-output transmission, the precoded plurality of signals (e.g., symbols carried via downlink 116A-B). In some implementations, process 300 is stored as program code stored at memory 335.

[0062] Moreover, in some implementations, the client station feeds back the largest singular vector of the collaborating base station antennas. This is referred to as joint antenna processing multi-basestation MIMO in 802.16m or coordinated multi-point transmission (COMP) in LTE. For example, if each base station has 4 antennas and 3 base stations are collaborating, the singular vector has L= 12 complex numbers. The computation of the singular vector(s) for a client station with 2 antennas may be performed as noted above. For example, the largest singular vector for any subcarrier is given by v = where h_t i= 1 ,2 is the ith row of H, and the 2xL

global downlink channel between the client station and all of the collaborating base

stations is as follows: and tan The phase aligned

singular vectors of several subcarriers is averaged within a band to get the average

where j denotes one of

the subcarriers.

[0063] The subject matter described herein may be embodied in systems, apparatus, methods, and/or articles depending on the desired configuration. Base station HOA (or one or more components therein) can be implemented using one or more of the following: a processor executing program code, an application-specific integrated circuit (ASIC), a digital signal processor (DSP), an embedded processor, a field programmable gate array (FPGA), and/or combinations thereof. Client station 114B (or one or more components therein) can be implemented using one or more of the following: a processor executing program code, an application-specific integrated circuit (ASIC), a digital signal processor (DSP), an embedded processor, a field programmable gate array (FPGA), and/or combinations thereof. These various implementations may include implementation in one or more computer programs that are executable and/or interpretable on a programmable system including at least one programmable processor, which may be special or general purpose, coupled to receive data and instructions from, and to transmit data and instructions to, a storage system, at least one input device, and at least one output device. These computer programs (also known as programs, software, software applications, applications, components, program code, or code) include machine instructions for a programmable processor, and may be implemented in a high-level procedural and/or object-oriented programming language, and/or in assembly/machine language. As used herein, the term "machine-readable medium" refers to any computer program product, computer-readable medium, apparatus and/or device (e.g., magnetic discs, optical disks, memory, Programmable Logic Devices (PLDs)) used to provide machine instructions and/or data to a programmable processor, including a machine- readable medium that receives machine instructions as a machine-readable signal. Similarly, systems are also described herein that may include a processor and a memory coupled to the processor. The memory may include one or more programs that cause the processor to perform one or more of the operations described herein.

[0064] Although a few variations have been described in detail above, other modifications or additions are possible. In particular, further features and/or variations may be provided in addition to those set forth herein. For example, the above closed form singular value decomposition may be used in other applications, such as at the base station to determine singular vectors for a sounder transmitted by the client station to the base station. Moreover, the implementations described above may be directed to various combinations and subcombinations of the disclosed features and/or combinations and subcombinations of several further features disclosed above. In addition, the logic flow depicted in the accompanying figures and/or described herein does not require the particular order shown, or sequential order, to achieve desirable results. Other embodiments may be within the scope of the following claims.

Claims

WHAT IS CLAIMED IS:

1. A method comprising: receiving a plurality of signals transmitted by a base station implementing a plurality of antennas configured for a multiple-input and multiple-output transmission; determining, using a closed form singular value decomposition, one or more singular vectors; and providing, as feedback, the one or more determined singular vectors to a precoder at the base station.

2. The method of claim 1, wherein determining further comprises: determining, using the closed form singular value decomposition, the one or more singular vectors, the closed form singular value decomposition determined without iteration.

3. The method of claim 1, wherein determining further comprises: determining, using the closed form singular value decomposition, a first singular vector based on the following equation:

₁ ; and and a second singular vector based on the following equation:

4. A method comprising: receiving, at a base station, a singular vector determined, at a client station, using a closed form singular value decomposition; using the received singular vector when precoding a plurality of signals for transmission using multiple-input and multiple-output transmission; and transmitting, as the multiple-input and multiple-output transmission, the precoded plurality of signals.