WO2023233006A1

WO2023233006A1 - Beam determination with reduced phase quantization levels

Info

Publication number: WO2023233006A1
Application number: PCT/EP2023/064846
Authority: WO
Inventors: Fredrik RUSEK; Chaitanya TUMULA; Erik Bengtsson
Original assignee: Sony Group Corporation; Sony Europe B.V.
Priority date: 2022-06-02
Filing date: 2023-06-02
Publication date: 2023-12-07

Abstract

Various techniques are disclosed for determining multiple beams pointing towards different directions such that the multiple beams are defined by a comparatively small number of phase quantization levels of the associated antenna weights of the antenna elements. A beamforming codebook can be populated using these multiple beams. Such beamforming codebook can be used for beamforming of a wireless device.

Description

D E S C R I P T I O N BEAM DETERMINATION WITH REDUCED PHASE QUANTIZATION LEVELS TECHNICAL FIELD Various examples of the disclosure generally pertain to beamforming using beams defined by antenna weights stored in a codebook. Various examples specifically relate to determining the antenna weights. BACKGROUND Beamforming is a multi-antenna technique. Beamforming pertains to a technique that ena- bles directing radio energy towards a specific direction in space, e.g., to a specific receiver. This is achieved by using an antenna array including multiple antenna elements and adjust- ing the phase and amplitude applied at the signal of each antenna element. The phase and amplitude adjustments of a given antenna element is referred to as the antenna weights of this antenna element; the set of antenna weights for all antennas of the antenna array is re- ferred to as a beamformer. This results in constructive addition of the corresponding signals along the specific direction; while the signals cancel out in other directions. Typically, beams are defined that have a certain beamwidth and are centered around the direction of maxi- mum signal strength. Beyond such transmit (TX) beamforming, also receive (RX) beamform- ing is possible. Here, energy is selectively collected from a certain direction, i.e., from a re- spective receive beam. Antenna weights can define a TX beam or an RX beam. SUMMARY There is a need for advanced beamforming techniques. There is a need for beams having a wide beamwidth. There is a need for compact codebooks. This need is met by the features of the independent claims. The dependent claims define embodiments. A computer-implemented method of determining respective antenna weights for each of a plurality of beams is disclosed. The method includes determining the respective antenna weights of a first beam of the plurality of first beams. The first beam points towards a given direction. The method also includes determining the respective antenna weights of multiple second beams of the plurality of beams. Each one of the multiple second beam points to- wards a respective one of multiple directions. The determining of the antenna weights of the multiple second beams includes replicating the first beam to point to each one of the multiple directions. Also, the method may include determining the multiple directions based on a predetermined rule that defines allowed inclinations between the multiple directions and the given direction. The allowed inclinations may, e.g., be represented by grid points of a square grid defined in the space of directional cosines of azimuth and elevation angles. For example, it would be possible to populate a beamforming codebook using the plurality of beams. I.e., a beamforming codebook can be generated based on such techniques. A computer program includes program code that can be executed by at least one processor. The processor, upon executing the program code, is configured to execute a method of de- termining respective antenna weights for each of a plurality of beams. The method includes determining the respective antenna weights of a first beam of the plurality of first beams. The first beam points towards a given direction. The method also includes determining the re- spective antenna weights of multiple second beams of the plurality of beams. Each one of the multiple second beam points towards a respective one of multiple directions. The deter- mining of the antenna weights of the multiple second beams includes replicating the first beam to point to each one of the multiple directions. A computing device is disclosed which includes such at least one processor. It is then possible that the beamforming codebook is used by a wireless device for beam- forming. For instance, a computer program can include program code that is executable by at least one processor of a wireless device, e.g., a base station or a terminal. The wireless device can include a memory and an antenna array. The execution of the program code causes the at least one processor to load, from the memory, the beamforming codebook that is deter- mined according to such computer-implemented method. Then, the antenna array can be controlled to pre-code signals using the antenna weights of a selected beam of the plurality of beams retrieved from the beamforming codebook. A method of operating a wireless device is disclosed. The wireless device includes an an- tenna array and a memory. The memory stores a beamforming codebook. The beamforming codebook includes, for each of a plurality of beams, respective antenna weights for the an- tenna array. The method includes retrieving, from the beamforming codebook, antenna weights of the selected beam of the plurality of beams. For instance, the beam can be se- lected based on the management procedure executed by the wireless device. The method also includes controlling the antenna array to pre-code signals using the antenna weights of the selected beam. The plurality of beams point towards multiple directions that are arranged at grid points of a square grid defined in a space of directional cosines of azimuth and eleva- tion angles. A computer program includes program code that can be executed by at least one processor. The processor, upon executing the program code, is configured to execute a method of oper- ating a wireless device. The wireless device includes an antenna array and a memory. The memory stores a beamforming codebook. The beamforming codebook includes, for each of a plurality of beams, respective antenna weights for the antenna array. The method includes retrieving, from the beamforming codebook, antenna weights of a selected beam of the plu- rality of beams. For instance, the beam can be selected based on a beam management pro- cedure executed by the wireless device. The method also includes controlling the antenna array to pre-code signals using the antenna weights of the selected beam. The plurality of beams point towards multiple directions that are arranged at grid points of a square grid de- fined in a space of directional cosines of azimuth and elevation angles. A wireless device includes an antenna array and a memory and a processor. The memory stores a beamforming codebook. The beamforming codebook includes, for each of a plurality of beams, respective antenna weights for the antenna array. The processor is configured to retrieve, from the beamforming codebook, antenna weights of a selected beam of the plural- ity of beams. The processor is also configured to control the antenna array to pre-code signal using the antenna weights of the selected beam. The plurality of beams point towards multi- ple directions that are arranged at grid points of a square grid defined in a space of direc- tional cosines of azimuth and elevation angles. It is to be understood that the features mentioned above and those yet to be explained below may be used not only in the respective combinations indicated, but also in other combina- tions or in isolation without departing from the scope of the invention. BRIEF DESCRIPTION OF THE DRAWINGS FIG.1 schematically illustrates a beam pattern in the space of directional cosines according to various examples. FIG.2 schematically illustrates multiple beams pointing towards different directions. FIG.3 schematically illustrates a communication system. FIG.4 schematically illustrates details of the communication system. FIG.5 is a flowchart of a method according to various examples. FIG.6 schematically illustrates a processing device according to various examples. FIG.7 is a flowchart of a method according to various examples. DETAILED DESCRIPTION Some examples of the present disclosure generally provide for a plurality of circuits or other electrical devices. All references to the circuits and other electrical devices and the function- ality provided by each are not intended to be limited to encompassing only what is illustrated and described herein. While particular labels may be assigned to the various circuits or other electrical devices disclosed, such labels are not intended to limit the scope of operation for the circuits and the other electrical devices. Such circuits and other electrical devices may be combined with each other and/or separated in any manner based on the particular type of electrical implementation that is desired. It is recognized that any circuit or other electrical device disclosed herein may include any number of microcontrollers, a graphics processor unit (GPU), integrated circuits, memory devices (e.g., FLASH, random access memory (RAM), read only memory (ROM), electrically programmable read only memory (EPROM), electrically erasable programmable read only memory (EEPROM), or other suitable variants thereof), and software which co-act with one another to perform operation(s) disclosed herein. In addition, any one or more of the electrical devices may be configured to execute a program code that is embodied in a non-transitory computer readable medium programmed to perform any number of the functions as disclosed. In the following, embodiments of the invention will be described in detail with reference to the accompanying drawings. It is to be understood that the following description of embodiments is not to be taken in a limiting sense. The scope of the invention is not intended to be limited by the embodiments described hereinafter or by the drawings, which are taken to be illustra- tive only. Hereinafter, techniques of determining beamformers at antenna arrays are disclosed. For instance, the antenna arrays could be part of communication devices such as access nodes to cellular networks, e.g., base stations, or mobile terminals. It would also be possible that the antenna arrays are part of a coverage enhancing device, e.g., a reconfigurable re- flective array that has an antenna array for reflecting incident signals. Another example of coverage enhancing devices would be a repeater device or a relay device. Techniques are disclosed which enable selecting antenna weights {x_mn} applied at antenna element (m, n) of a two-dimensional array so that the radiated signal has desired properties. One such property is the design of, so-called, wide beams. This means that the radiated sig- nal should be sent towards a large part of the surrounding space; or in other words, in many spatial directions simultaneously. This is a long-standing problem and if the antenna weights {x_mn} have no restrictions (e.g., with respect to quantization) many type of beams exist. However, according to examples, the beamformers (i.e., a set of antenna weights for all an- tenna elements of an antenna array, thereby defining a TX or RX beam) are determined tak- ing into consideration antenna weight quantization. Consideration of antenna weight quanti- zation has the following reason: In practice, antenna weights without restrictions (e.g., an- tenna weights that can take any arbitrary value, e.g., in an analog fashion) are not easy to implement. Instead, the antenna weights are chosen from a finite set of possible values (quantization levels); this is typically referred to as “quantization”. Let the set of possible val- ues taken by each antenna weight be denoted Q. A standard construction for Q is to quantize amplitude of phase independenly, so that the set is given as

Thus, there are |A| possible amplitude levels / amplitude quantization levels, and |P| possible phase levels / phase quantization levels for each antenna weight (of any given antenna ele- ment in the array). In general, it is desired to keep both these cardinalities small. For instance, a small cardinal- ity can provide for a compact codebook that is accessed and used by the participating com- munication devices or communication devices. Generally, the size of the codebook (i.e., data amount required to store the codebook) is af- fected by the number of codebook entries. The number of codebook entries defines the num- ber of available beams from which the respective wireless device can choose during opera- tion. Typically, the number of codebook entries would be defined by requirements of the beam management implemented by the node that uses the codebook. For instance, in some scenarios it may be desirable to have a larger count of beams to choose from, e.g., to in- crease spatial selectivity of any transmission. The size of the codebook is also affected by the size of each codebook entry. The size of each codebook entry depends on a number of bits used to define the antenna weights for each antenna element. This depends on the num- ber of required quantization levels. Techniques are disclosed which facilitate reducing the number of quantization levels, thereby reducing the codebook size – even for a fixed number of codebook entries. For a given set of antenna weights {x_mn}, where x_mn ∈ Q, the radiation pattern towards spherical coordinates (azimuth and elevation) is

The radiation pattern defines the geometrical shape of the beam. More specifically, for a TX beam, the radiation pattern refers to the directional (angular) dependence of the strength of the radio waves transmitted; for an RX beam the radiation pattern refers to the directional de- pendence of the sensitivity of the radio waves received. For instance, widebeam design is to select the antenna weights {x_mn | x_mn ∈ Q} such that the radiation pattern looks as follows

where F is some arbitrary set with measure (for sets, “measure” is conventionally deas

) above a predetermined threshold defining where in space the wide beam points. “High” means above a threshold; “low” means below the threshold. Note that for a codebook multiple beams are needed, which implies that it is necessary to se- lect {x_mn | x_mn ∈ Q} for several different sets F to cover the whole space. Various techniques disclosed herein are based on the finding that selecting the appropriate antenna weights for a given quantization {x_mn | x_mn ∈ Q} can be challenging when the num- ber of quantization levels |^| and |^| are small. For instance, a beam may have a subopti- mum shape in the sense that it does not offer constant beam response within the beam and also suffers from significant sidelobes in the stopband. Techniques are disclosed hereinafter that mitigate such restrictions or drawbacks. Some preliminaries are needed before the solution is explained. In array processing, it is convenient to abandon spherical coordinates in favor of directional cosines. These are de- fined as

so that the radiation pattern of Equation 2 becomes

The domain of the directional cosines depends on the setting. For the purpose of this disclo- sure it is sufficient to consider

for BS beamforming and −1 ≤ k_x, k_y ≤ 1 for CED beamforming. As the latter includes the former, we assume the latter case in what fol- lows. To span the space -1 ≤ k_x, k_y ≤ 1 several beams are designed, each one distinctly located in the directional cosine space (i.e., pointing to a specific direction).

Now it is assumed that L beams have been designed, each one centered at (c_{x ,l}, c_{y, l}). This defines the direction into which each beam points. For each beam, there is a set F_l, located around (c_{x ,l}, c_{y, l}), for which the radiation pattern of beam I has large magnitude. Further, each beam is charactierized from its set of beamforming coefficients {x_{mn, l} }. This is illus- trated in FIG. 1.

FIG. 1 illustrates the center positions 321, 322 along the k_x, k_y axes for a given beam 201, i.e., c_x, c_y. Also illustrated is the footprint F denoted as 311. The sidelength of the footprint defines the beamwidth.

Clearly, the beam 201 has an ideal shape and cannot be obtained in practice. However, if the beamforming coefficients are unrestricted (i.e., no quantization), a reasonable approxi- mation of the pattern shown in FIG. 1 can be achieved - especially if the number of antennas is large.

For quantized antenna weights, large distortions tend to occur where reference techniques are employed, unless the values |A| and |P| are large.

The techniques disclosed herein mitigate such drawbacks.

Hereinafter techniques are disclosed that facilitate finding L sets {x_mn,l } with each coefficient taken from the alphabet Q so that each beam pattern is well concentrated in, and reasonably uniform over, the sets {F_l}, and, further, so that ∪ F_l = [- 1,1]² , for small values of |A| and |P|

Various techniques disclosed herein are based on the finding that the key to have a small value of |P | is to appropriately select the centers (c_{x ,l}, c_{y, l}) , i.e., to appropriately select the directions at which the beams of the codebook are pointing. It has been found that using cer- tain allowed inclinations between different directions into which the beams included in the codebook are pointing is an important aspect of quantized wide-beam design.

For illustration, FIG. 2 illustrates a set of three beams 200-202 and respective center direc- tions 341-343 (in real space coordinates, for illustrative purposes).

First, the situation is discussed without quantization; quantization will be introduced later.

A “center” beam located at (c_x,0, c_{y, 0}) = (0,0) is considered. Let h_x and h_y be two low pass FIR filters with real-valued coefficients. Then, the unrestricted 2-D antenna weights can be chosen as

which generates the beam pattern

where H_x(∙) is the discrete-time Fourier transform of ℎ_^, defined on [−1,1]. This initial pattern should be seen as a starting point from which further beams are replicated. A beam pattern B₀(k_x, k_y) satisfying Equation 7 is said to be separable. In other words, initially, the respective antenna weights of a first beam are determined. That first beam points towards a given direction, e.g., is the center beam as described above at k_x = 0, k_y = 0. Then, antenna weights for multiple second beams are determined, wherein each one of the multiple second beams points towards a respective one of multiple direc- tions. Such determining of the antenna weights of the multiple second beams includes repli- cating the first beam to point to each one of the multiple directions. For instance, initially beam 200 could be determined; and then beams 201, 202 – having the same beam pattern, but directed towards a different direction 342, 343 – could be obtained by replicating the beam 200. In fact, the beam pattern does not need to be separable - any pattern B₀(k_x, k_y) stemming from real-valued coefficients can be used. A beam located at (c_{x ,l}, c_{y, l}) can be replicated from the initial beam at {x_mn,0 } as

This means, in other words, that replicating the first beam to point to each one of the multiple directions includes offsetting each antenna weight of the first beam by a respective phase shift that is defined by the inclination of the respective one of the multiple directions with re- spect to the initial direction. Thereby, a respective antenna weight of the respective one of the second beams is obtained. The resulting beam pattern B_l(k_x, k_y) becomes a copy of B₀(k_x, k_y) but shifted to (c_{x ,l}, c_{y, l}), i.e.,

An exact copy will be obtained if the antennas are isotropic; else the copy will be scaled by the pattern of the antenna elements (e.g., for a patch antenna, it is only possible to have beams covering a half sphere so that the antenna design affects the beam pattern of the copies).

Next, quantization of the antenna weights is introduced. Quantization refers to restricting the set of allowable values assumed by the antenna weights to certain discrete numbers.

For instance, from Equation 9 certain antenna weights for a given beam are obtained. Typi- cally, the values of these antenna weights will deviate from the quantization levels. Accord- ingly, the values of the antenna weights are shifted to the nearest element in Q, i.e. , the near- est allowed quantization level is selected as the antenna weight.

First beam: For x_mn,0 , the coefficients can be entirely real-valued (by appropriately designing the initial beam using prior-art techniques known to the skilled person). In other words, it is possible to restrict the beam pattern to have symmetric magnitude response |B₀(k_x,k_y) |, which implies that the coefficients xmn,0 are entirely real-valued. Thus, for phase quantization it is sufficient to create ±1, or in previous notation P = {0, π }.

The beam pattern B₀{k_x,k_y) is a Fourier transformation of the coefficients xmnfi. Selecting x_mn,0 as real-valued the following impact on the beam pattern: The real-valued sequences x_mn,0 have Fourier transforms that are symmetric (in magnitude) around 0. So the conse- quence of real-valued x_mn,0 is that the beams are symmetric around the centers. As the beam pattern is assumed separable, the symmetry implies that |B₀(k_x,k_y)|=|B₀(±k_x,±k_y)| . Various techniques for design of the first beam are conceivable. One option is that all an- tenna elements have the same phase value (any value); then the beam will be perpendicular to the array toward the far-field (i.e., non-focused). The beam width will be defined by the number of elements (referred to as pencil beam). Then, with 1 bit quantization of the phase, wider beams can be configured without issues of sidelobes or performance loss.

For this initial beam having the beam pattern B₀, there is no advantage of using a larger P. I.e., two quantization levels for the phase are sufficient. Thus, quantized beamforming design for the center beam becomes a problem of amplitude quantization only, provided that P = {0, π }. It has been found empirically that the number of amplitude levels |A| can be main- tained reasonably small, 3-5, without any major degradation to the beam pattern. I.e., the number of amplitude quantization levels used by the codebook for indicating amplitudes of the antenna weights of the initial beam can be not larger than 10, optionally not larger than 5. Second beam, i.e., x_mn,l for I ≠ 0: Due to the phase shift according to Equation 9, the an- tenna weights are no longer real-valued, but complex. This implies that an expanded set P is needed, beyond P = {0, π}. For beam directions (c_{x ,l}, c_{y, l}) that are determined based on a predetermined rule (which, in other words, defines certain allowed inclinations between the directions of the beams; and other inclinations are then forbidden; while not all allowed inclinations must be necessarily represented in the codebook), the terms can only take finitely many values for

any triplet m, n, l. In particular, the number of values can be small. This leads to a to a less complex design of the codebook with fewer quantization levels. Suppose that no matter the values m, n, l, always

is one of 2T distinct values. This implies that with |P| = 2T, it is possible to phase quantize the unrestricted values {x_mn,l } at no loss. Further, this transforms the beamforming problem for any (c_{x ,l}, c_{y, l}) into that for (c_{x ,0}, c_{y, 0}) = (0,0). This is so since after the phase quantization, the quantized version of {x_mn,l } has phases that exactly coincide with that of the unrestricted version. Thus, only am- plitude remains to be quantized. It remains to discuss how center locations (c_{x ,l}, c_{y, l}) can be chosen such that only a few phase values can occur. The predetermined rule for the selection of the center locations can specify that the allowed inclinations are represented by grid points of a rectangular or specifi- cally square grid (in the space of directional cosines) with side length ∆, i.e.,

Then, for some integers (s_l, r_l) one obtains

Note now that (ms_l + nr_l) is integer valued. If one chooses the side-length of the grid as:

for two relatively prime integers Y and T, then it follows that is one out of 2T distinct values.

This means, in other words, that the side length of the grid is a rational number that equals the fraction of two co-prime integers. At the same time, this fraction has a denominator T that is comparatively small, e.g., not larger than 10, optionally not larger than 5. Specifically, it would be possible that the denominator is selected based on a predetermined size threshold that is associated with the beamforming codebook. I.e., for larger sizes of codebooks, larger values of T may be acceptable. FIG.3 schematically illustrates a communication system 100. The communication system 100 includes two CDs 101, 102 that are configured to communicate with each other via a ra- dio link 112. In the example of FIG.1, the device 101 is implemented by a BS 101 and the CD 102 is implemented by a UE 102. The UE 102 could be, e.g., a smartphone, a tablet PC, a smartwatch, a smart TV, a smart meter, to give just a few examples. As a general rule, the techniques described herein could be used for various types of com- munication systems, e.g., also for peer-to-peer communication, etc. For the sake of simplic- ity, however, hereinafter, various techniques will be described in the context of a communica- tion system that is implemented by an BS 101 of a cellular NW and a UE 102. As illustrated in FIG.3, there can be DL communication, as well as UL communication. Ex- amples described herein particularly focus on the DL communication, but similar techniques may be applied to UL communication and/or sidelink communication. FIG.4 illustrates details with respect to the BS 101. The BS 101 implements an access node of a cellular network, e.g., a 3GPP-specified cellular network. The BS 101 includes control circuitry that is implemented by a processor 1011 and a non-volatile memory 1015. The pro- cessor 1011 can load program code that is stored in the memory 1015. The processor 1011 can then execute the program code. Executing the program code causes the processor to perform techniques as described herein, e.g.: transmitting and/or receiving (communicating) payload data on the radio link 112 on the data carrier 111; loading a beamforming codebook 190 from the memory 1015; applying RX or TX beamforming based on a codebook 190 of beams by controlling antenna elements 1014 of an array 1013 to pre-code respective signals using antenna weights (amplitude and phase) retrieved from the codebook; etc. FIG.4 also illustrates details with respect to the UE 102. The UE 102 includes control cir- cuitry that is implemented by a processor 1021 and a non-volatile memory 1025. The proces- sor 1021 can load program code that is stored in the memory 1025. The processor can exe- cute the program code. Executing the program code causes the processor to perform tech- niques as described herein, e.g.: transmitting and/or receiving (communicating) payload data on the radio link 112 on the data carrier 111; loading a beamforming codebook from the memory and control the antenna array to pre-code signals using antenna weights retrieved from the beamforming codebook; etc. FIG.4 also illustrates details with respect to communication between the BS 101 and the UE 102 on the data carrier 111. The BS 101 includes an interface 1012 that can access and con- trol multiple antennas 1014. Likewise, the UE 102 includes an interface 1022 that can access and control multiple antennas 1024. While the scenario of FIG.4 illustrates the antennas 1014 being coupled to the BS 101, as a general rule, it would be possible to employ transmit-receive points (TRPs) that are spaced apart from the BS. The interfaces 1012, 1022 can each include one or more TX chains and one or more RX chains. For instance, such RX chains can include low noise amplifiers, analogue to digital converters, mixers, etc. Analog and/or digital beamforming would be possible. Thereby, phase-coherent transmitting and/or receiving (communicating) can be implemented across the multiple antennas 1014, 1024. Multi-antenna techniques can be implemented. Specifi- cally, beamforming can be employed. By using a TX beam, the direction of signals transmitted by a transmitter of the communica- tion system is controlled. Energy is focused into a respective direction or even multiple direc- tions, by phase-coherent superposition of the individual signals originating from each an- tenna 1014, 1024. Thereby, a spatial data stream can be directed. The spatial data streams transmitted on multiple beams can be independent, resulting in spatial multiplexing multi-an- tenna transmission; or dependent on each other, e.g., redundant, resulting in diversity multi- input multi-output (MIMO) transmission. As a general rule, alternatively or additionally to such TX beams, it is possible to employ re- ceive (RX) beams. In the scenario of FIG.3, the BS 101 employs beamforming based on the codebook 190. The codebook 190 includes multiple entries, each entry being associated with a respective beam. Each entry is associated with a certain set of antenna weights specifying the ampli- tude and phase for each antenna element of the antenna array 1013. The antenna weights specify the amplitude and phase for each antenna element by being indicative of an ampli- tude quantization level and a phase quantization level. For instance, if there are 10 phase quantization levels, then a 4-bit indicator would suffice. Thus, for each beam and for each an- tenna element, a 4-bit indicator for phase is required. Since there can be a large count of beams (typically more than 100 beams) and a large count of antenna elements (e.g., more than 100 or even 1000 antenna elements), it is desirable to size-limit the codebook by size- limiting the quantization levels and thereby the data structure indicating the quantization lev- els. FIG.5 is a flowchart of a method according to various examples. The method of FIG.5 can be executed by a communication device such as a base station, e.g., by the BS 101. More specifically, it would be possible that the method of FIG.5 is executed by the processor 1011 of the BS 101, upon loading and executing program code from the memory 1015. For instance, the method of FIG.5 can be executed when the BS operates in a beamforming mode. The method of FIG.5 can facilitate transmitting or receiving data to or from another communication device such as a UE such as the UE 102. At box 6105, the BS retrieves antenna weights of a selected beam from a beam codebook. For instance, the selected beam could be determined based on a beam sweeping measure- ment. The selected beam could be predetermined. The selected beam could be determined based on a relative position of the other communication device with which communication is intended via the selected beam with respect to the BS. The antenna weights specify the amplitude and phase of each antenna element of the an- tenna array of the BS. The antenna weights specify the amplitude and phase using indicators indicative of amplitude quantization levels and phase quantization levels. Depending on the number of quantization levels, larger or smaller indicators are required. The codebook can be loaded from a local memory to retrieve the antenna weights. At box 6110, it is then possible to control the antenna elements of the array to pre-code sig- nals using the antenna weights retrieved from the codebook. The respective beamforming is then applied. According to various examples, a special type of codebook is used. Specifically, a codebook is used that includes beams that point towards multiple directions, wherein those multiple di- rections are arranged at grid points of a square grid that is defined in the space of directional cosines of azimuthal and elevation angles. The plurality of beams can be replicates of each other. The number of phase quantization levels used by the codebook for indicating phases of antenna weights of the plurality of beams can be equal or smaller than 10, optionally equal or smaller than 5. Next, techniques with respect to constructing such codebook are disclosed. FIG.6 schematically illustrates a processing device 901. The processing device 901 can be configured for determining a codebook that includes multiple beams that are arranged at grid points of a square grid defined in the space of directional cosines. A codebook can be con- structed that has only a small number of phase quantization levels; while still obtaining beam patterns of the beams that have a wide beamwidth and small amplitude variations within the beam footprint. The processing device 901 includes a processor 912 and a memory 913. The processor 912 can load program code from the memory 913. The processor 912 can then execute the pro- gram code. Upon loading and executing the program code, the processor can determine beams of a codebook and then populate the codebook with those beams. The codebook could then be provided to a communication device that implements the beamforming based on the codebook, via the interface 911. For instance, upon providing the beamforming code- book, the communication device could execute the method according to FIG.5 relying on the codebook. Techniques with respect to construction of the codebook will be explained next in connection with FIG.7. FIG.7 is a flowchart of a method according to various examples. FIG.7 pertains to a method of determining a beamforming codebook. FIG.7 could be executed by the processor 912 upon loading program code from the memory 913 and executing the program code. At box 6010, antenna weights of a first beam of a plurality of beams to be included in the codebook is determined. The first beam points towards a given direction. For instance, the beam could be determined in accordance with Eq.7. The first beam could be centered in the space of directional cosines (cf. FIG.1). At box 6010, it is thus possible to choose a set of real-valued beamforming coefficients (an- tenna weights) corresponding to a beam pattern that is located around k_x, k_y = (0,0). It is helpful to choose the coefficients carefully so that they can be amplitude quantized using a small |A|. From experiments, |A| = 3 − 5 suffices. As a general rule, the number of amplitude quantization levels used by the codebook may not be larger than 10, optionally not larger than 5. Said coefficients should be selected such that the one-sided width of F is ∆ where ∆= Y/T for relatively prime Y and T. Then, the beamforming coefficients are quantized and the result is denoted as {x_mn,0 }. This is already one entry of the codebook. Next, at box 6015, antenna weights of multiple second beams of the plurality of beams to be included in the codebook are determined. Each one of the multiple second beam points to- wards a respective one of multiple directions. The determination of the antenna weights of the multiple second beams includes replicating the first beam to point to each one of the mul- tiple directions. For instance, the replication in accordance with Eq.8 could be applied. The multiple directions are selected in accordance with a predetermined rule. In particular, the predetermined rule specifies allowed positions in the space of directional cosines, i.e., specifies allowed inclinations between the given direction of the first beam in the multiple di- rections of the second beams. Specifically, beams are allowed to be positioned (but do not have to be positioned) at grid points of a square grid defined in the space of directional co- sines, wherein the side length of the square grid is a rational number that equals a fraction of to co-prime integers. This has been explained in connection with Eq.12: The parameter T in the denominator of this fraction controls the side length of the grid as well as the number of phase quantization levels. Accordingly, the parameter T controls the size of the codebook. For instance, it would be possible to select the denominator T based on a predetermined size threshold that is associated with the beamforming codebook. The denominator T could be selected based on a predetermined phase quantization level threshold that is associated with the beamforming codebook. For instance, there can be a tendency that a larger T results in higher quality beams, i.e., beams that have a beam pattern that is homogeneous and does not show a large number of local variations within the beamwidth. At the same time, a larger T will increase the size of the codebook. Accordingly, a sweet spot for the size of T can be chosen by considering the above. Typically, a number of phase quantization levels used by the codebook for indicating phases of the antenna weights can be selected to not be larger than 10, optionally not larger than 5. In other words, at box 6015 choose center locations for all other beams as

(c_{x ,l}, c_{y, l} ) =

) with integer valued s_l, r_l. The phase quantization levels are then set as: Set P = ; i.e., depend on the denominator T.

The beamwidth of the first beam can be chosen so as to be not smaller than the side length of the square grid. This ensures that neighboring beams are overlapping (cf. FIG.2); thereby, it is possible to address any direction by an appropriate beam. Summarizing, techniques have been disclosed that are based on the finding that if the center beam located around (0,0) is formed from coefficients x_mn,0 then an identical beam, but lo- cated around (c_{x ,l}, c_{y, l} ) has coefficients

_, For arbitrarily selected values (c_{x ,l}, c_{y, l} ) these coefficients are complex valued and essentially all possible phases occur; in effect, the number of phase quantization levels must be gigantic. But if

can only take on finitely many values, then the same is true also for the phase of . values.

If the center is chosen as (c_{x ,l}, c_{y, l} ) = (∆s_l, ∆r_l), then

_{, ,} where (ms_l + nr_l) is an integer by definition. Thus, for some integer q. If the

sidelength of this grid of allowed directions is constructed as a fraction of two coprime inte- gers, ∆= Y/T then one obtains

where a_mn,0 = 0 for x_mn,0 > 0 and a_mn,0 = 1 otherwise. By inspection, this can take on pre- cisely 2T distinct values. Thus, a small codebook is possible (for a given number of code- book entries) due to a limited number of required phase quantization levels. Although the invention has been shown and described with respect to certain preferred em- bodiments, equivalents and modifications will occur to others skilled in the art upon the read- ing and understanding of the specification. The present invention includes all such equiva- lents and modifications and is limited only by the scope of the appended claims. For illustration, above, various techniques have been disclosed in connection with beamform- ing at a base station. Similarly, beamforming could be applied at a UE or a repeater device or a relay device according to the examples disclosed herein. For still further illustration, techniques have been disclosed where a codebook includes a first beam and replicates of the first beam at certain directions. In some scenarios, it would be possible that a codebook includes multiple sub-codebooks, wherein each one of the multiple subcode books includes a respective first beam and replicates of that respective first beam. Thereby, a wider coverage of directions would be possible and different beam patterns could be provided for. For still further illustration, techniques have been disclosed where a codebook includes a first beam and replicates of the first beam at certain directions. In some scenarios, it would be possible that a codebook includes multiple subcode books, wherein each one of the multiple subcode books includes a respective first beam and replicates of that respective first beam. Thereby, a wider coverage of directions would be possible and different beam patterns could be provided for.

Claims

C L A I M S 1. A computer-implemented method of determining, for each of a plurality of beams (200, 201, 202), respective antenna weights, the method comprising: - determining (6010) the respective antenna weights of a first beam (200) of the plu- rality of beams, the first beam pointing towards a given direction (341), - determining (6015) the respective antenna weights of multiple second beams (201, 202) of the plurality of beams, each one of the multiple second beams pointing towards a re- spective one of multiple directions (342, 343), said determining of the antenna weights of the multiple second beams comprising replicating the first beam to point to each one of the multi- ple directions, and - determining the multiple directions based on a predetermined rule that defines al- lowed inclinations between the multiple directions and the given direction, wherein the allowed inclinations are represented by grid points of a square grid de- fined in a space of directional cosines of azimuth and elevation angles. 2. The computer-implemented method of claim 1, wherein a side length of the square grid is a rational number that equals a fraction of two coprime integers, wherein the denominator of the fraction is not larger than ten, optionally not larger than five. 3. The computer-implemented method of claim 2, further comprising: - selecting the denominator of the fraction based on a predetermined size threshold associated with a beamforming codebook comprising the plurality of beams. 4. The computer-implemented method of claim 2 or 3, further comprising: - selecting the denominator of the fraction based on a predetermined phase quantiza- tion level threshold associated with a beamforming codebook comprising the plurality of beams. 5. The computer-implemented method of any one of claims 2 to 4, wherein a beamwidth of the first beam is not smaller than a side length of the square grid. 6. The computer-implemented method of any one of the preceding claims, wherein a number of phase quantization levels used by a beamforming codebook for indicating phases of the antenna weights of the plurality of beams is not larger than ten, op- tionally not larger than five. 7. The computer-implemented method of any one of the preceding claims, wherein a number of phase quantization levels of the antenna weights of the first beam is two. 8. The computer-implemented method of any one of the preceding claims, wherein a number of amplitude quantization levels used by a beamforming codebook for indicating amplitudes the antenna weights of the plurality of beams is not larger than ten, optionally not larger than five. 9. The computer-implemented method of any one of the preceding claims, wherein said replicating of the first beam to point to each one of the multiple direc- tions comprises offsetting, each antenna weight of the first beam, by a respective phase shift defined by an inclination of a respective one of the multiple directions with respect to the given direction, to thereby obtain a respective antenna weight of the respective one of the multiple second beams. 10. A computer program comprising program code executable by at least one processor of a wireless device comprising a memory and an antenna array, wherein execution of the program code causes at least one processor to load, from the memory, a beamforming code- book determined according to the computer-implemented method of any one of the preced- ing claims, and to control the antenna array to pre-code signals using the antenna weights of a selected beam of the plurality of beams retrieved from the beamforming codebook. 11. A method of operating a wireless device (101, 102) comprising an antenna array (1014, 1024) and a memory storing a beamforming codebook (190) comprising, for each of a plurality of beams (200, 201, 202), respective antenna weights for the antenna array, the method comprising: - retrieving (6105), from the beamforming codebook, antenna weights of a selected beam of the plurality of beams, and - controlling (6110) the antenna array to pre-code signals using the antenna weights of the selected beam, wherein the plurality of beams point towards multiple directions that are arranged at grid points of a square grid defined in a space of directional cosines of azimuth and elevation angles. 12. The method of claim 11, wherein the plurality of beams are replicates of each other. 13. The method of claim 11 or 12, wherein a number of phase quantization levels used by the codebook for indicating phases of the antenna weights of the plurality of beams is not larger than ten, optionally not larger than five.