Patent Publication Number: US-10320454-B2

Title: Technique for precoder determination

Description:
TECHNICAL FIELD 
     The present disclosure relates to a technique for determining a precoder for a radio transmission. More specifically, and without limitation, a method and an apparatus for determining a precoder for a radio transmission via a channel including two or more transmit antennas is provided. 
     BACKGROUND 
     Data rates achievable in a mobile telecommunications network have been increased using base stations with two or more transmit antennas that provide one or multiple spatial streams to one or more mobile devices. The base station applies precoding weights to a signal to be transmitted on the different transmit antennas. For example, Long Term Evolution (LTE) introduced a further Transmission Mode, TM9, in Release 10, which supports up to 8 transmit antennas for a downlink channel with up to 8 spatial streams (which are also referred to as layers) from the base station to a User Equipment (UE). 
     The base station relies upon Channel State Information (CSI) reported by the UE for specifying a precoder, i.e., specifying the precoding weights for each transmit antenna and each layer. Codebook-based precoding uses a common codebook comprising a set of precoders. The codebook is known at both the transmitting base station and the receiving UE, so that one precoder is specified in the CSI report by reference to the codebook. 
     As the number of transmit antennas is increased, the size of the codebook (i.e., the number of codebook entries) increases. Furthermore, the size of each codebook entry (i.e., the number of precoding weights) increases with the number of layers. Signal processing for determining the precoder to be reported in the CSI requires memory resources and computational resources at the UE. The signal processing resources for a conventional implementation approximately scale with the size of the codebook and the size of the entries. For example in an LTE implementation supporting 8 transmit antennas and 2 layers, the codebook according to Table 7.2.4-2 in standard document 3GPP TS 36.213 (Version 11.4.0) includes 256 precoders, each precoder being represented by a matrix of dimension 8×2. In other words, 256 sets of precoding weights for the different transmit antennas have to be processed for each of the 2 layers. In the case of 4 transmit antennas, 16 sets of precoding weights for the different transmit antennas have to be processed for each of the 2 layers according to Table 6.3.4.2.3-2 in standard document 3GPP TS 36.211 (Version 12.0.0). The increase in codebook size by a factor of 16 for doubling the number of transmit antennas indicates a strong non-linear increase in resource consumption caused by increasing the number of transmit antennas. 
     SUMMARY 
     Accordingly, there is a need for a technique that determines a precoder more efficiently in at least some situations. 
     According to one aspect, a method of determining a precoder for a radio transmission via a channel including two or more transmit antennas is provided. The method comprises a step of measuring at least one of a channel state of the channel and a channel noise characteristic of the channel; a step of selecting, based on a result of the measurement, a subset of precoders out of a codebook including a plurality of precoders; and a step of determining a precoder for precoding the transmission out of the subset. 
     By dividing the precoder determination in the selecting step and the determining step resource requirements for determining the precoder can be reduced at least in some situations. At least some implementations of the technique avoid costly hypothesis testing over all codebook entries. For example, the selected subset can be significantly reduced compared to the codebook, so that the resource requirements of the determining step are reduced compared to a conventional precoder determination based on the entire codebook. 
     When the result of the measurement is indicative of channel noise, the result of the measurement may include a characteristic of the channel noise. The channel noise characteristic may include noise power of the channel noise, a covariance of the channel noise and/or at least some elements of a noise covariance matrix of the channel noise. 
     When the result of the measurement is indicative of the channel state, the result of the measurement may include at least some elements of a channel state matrix. E.g., each element of the channel state matrix may represent a combination of transmit antenna and receive antenna. Alternatively or in addition, each element may be a function of frequency (e.g., a subcarrier) and time (e.g., a symbol). E.g., the measuring step may be implemented by estimating a channel state. 
     Each entry in the codebook may correspond to, or may indicate, a different one of the plurality of precoders. The precoder may specify, or may be represented by, a set of precoding weights (also referred to as precoding coefficients). The set of precoding weights may include a precoding weight for each transmit antenna. If the channel provides multiple spatial streams (also referred to as layers), the set of precoding weights may include a precoding weight for each combination of transmit antenna and spatial stream of the channel. Alternatively or in addition, the precoder may be represented by a vector of precoding weights, or by a matrix of precoding weights. The matrix may include one column for each spatial stream. Each precoding weight may specify a phase shift, e.g., represented by a complex number of magnitude  1  or another predefined common magnitude. The number of precoding weights specified by each precoder may be proportional to the product of the number of transmit antennas and the number of spatial streams. The number of transmit antennas may be equal to 2, 4 or 8. The number of spatial streams may be equal to 1, 2, 3, 4, 5, 6, 7 or 8. 
     The plurality of precoders included in the codebook may be parameterized. For example, one or more precoders may be determined by a combination of a first parameter and a second parameter. Each combination of a first parameter and a second parameter may uniquely identify one precoder in the codebook. Alternatively or in addition, each precoder in the codebook may correspond to a unique first parameter and a unique second parameter. 
     The first parameter may specify the subset of precoders. The selecting step may specify the first parameter. Alternatively or in addition, the second parameter may specify one precoder in the subset. The determining step may specify the second parameter. 
     The subset, e.g., the first parameter, may be selected based on a first objective function. The first objective function may be a numerical approximation of signal power on the channel, Mutual Information (MI) on the channel, a channel capacity of the channel, a Signal-to-Noise Ratio (SNR) and/or a Signal-to-Noise and Interference Ratio (SNIR). The first objective function may be a function of the result of the measurement and at least the first parameter. The result of the measurement may include at least one of the channel state and the channel noise. The subset, e.g., the first parameter, may be selected so as to maximize the first objective function. Maximizing the first objective function may be an approximation of maximizing MI on the channel, channel capacity, SNR and/or SNIR. The first objective function may ensure, e.g., with a predetermined level of confidence, that the precoder, which maximizes MI on the channel, channel capacity, SNR and/or SNIR, is included in the subset. The first objective function may be a quadratic function of the channel state. 
     The first objective function may be independent of a quantity indicative of the channel noise. The first objective function may depend on the measured quantity indicative of the channel state. E.g., the first objective function may depend on a relation between transmit antennas or groups of transmit antennas. E.g., the first objective function may depend on the channel states of different transmit antennas (which may also be referred to as transmit antenna correlation). The transmit antenna correlation may be quadratic in the channel state. 
     The first parameter may be a real number. The first parameter may correspond to a time delay between the different transmit antennas. The first objective function may include a sum over correlations that are phase-shifted according to the first parameter. The first objective function may be computed consistent with g(β,R)=Σ m,n  r m,n  e j(m-n)β , wherein the first parameter is denoted by β, and wherein a relation, R=(r m,n ), between the transmit antennas m and n is denoted by r m,n . The relation may depend on the measurement result. 
     Alternatively or in combination, a first objective function used for the selection may depend on a quantity indicative of both the channel state and the channel noise. The quantity may be representable by a matrix R. The first objective function may include a determinant. The determinant may be computed for a matrix that is quadratic in a candidate precoder W out of the codebook as a candidate for the subset and/or linear in the quantity R. The first objective function may be computed consistent with g(W,R)=det (I+W H RW), wherein the candidate precoder is denoted by W, and wherein the relation between the transmit antennas is denoted by R and depends on the measurement result. Herein, a unity matrix is denoted by I. 
     The relation R may be computed according to R=H H  H, wherein the channel state is denoted by H. Alternatively or in addition, the relation R may be computed according to R=H H  N −1  H, wherein the channel state is denoted by H and noise covariance is denoted by N. Latter computation may, at least in certain situation, eliminate or reduce an influence of noise on the relation R and/or the selecting step. 
     A predefined number of the candidate precoders that correspond to the highest values of the first objective function may be selected for the subset. The selecting step may include sorting at least some precoders of the codebook according to an evaluation of the first objective function. 
     The first parameter may represent the subset of precoders. For example, the first parameter may indicate one precoder in the subset and the subset may be defined to include all precoders of the codebook that deviate from the indicated precoder up to a deviation limit. 
     The precoder may be determined based on the result of the measurement or another second result of the measurement. The precoder may be determined based on a second objective function. The second objective function may be different from the first objective function. The second objective function may be a function of the result of the measurement (or the second result) and the second parameter. The second objective function may be a numerical representation or estimation of MI on the channel, channel capacity, SNR and/or SNIR. The first objective function may be an approximation of the second objective function. The computational complexity for evaluating the second objective function may be greater than the computational complexity for evaluating the first objective function. The approximation may allow computing the first objective function numerically efficient compared to the second objective function. The first objective function may ensure, e.g., with a predetermined level of confidence, that the precoder maximizing the second objective function is included in the subset. 
     The precoder may be determined with a higher temporal resolution than the selection of the subset. The second objective function may depend on a plurality of measurements, e.g., measurements for individual Resource Blocks. The first objective function may depend on a single measurement in time. 
     The precoder may be determined by evaluating the second objective function for each of the precoders in the selected subset. The determining step may ignore precoders in the codebook that are not in the subset. The determined precoder may correspond to an extremum, e.g., a minimum or a maximum, of the evaluations. 
     The term objective function as used herein may encompass a metric, a merit function (e.g., to be maximized) or a cost function (e.g., to be minimized) for candidates of the precoders. The term objective function as used herein may encompass a function that allows assessing the suitability of a candidate precoder, e.g., given the measurement result. The first objective function may be a function that allows comparing and/or discriminating between different subsets. The second objective function may be a function that allows comparing and/or discriminating between different precoders in the subset. 
     The spatial streams of the channel may also be referred to as layers of spatial multiplexing. A number of layers or a rank of the channel may be equal to one. Each of the precoders may include a precoding vector. Alternatively or in addition, a number of layers or a rank of the channel may be equal to at least two. Each of the precoders may include a precoding matrix. The number of layers of the channel may be dynamically adapted to channel conditions, e.g., the rank of the channel. 
     The radio transmission may be a transmission from a base station to a mobile device. At least one of the measuring step, the selecting step and the determining step may be performed by the mobile device. The base station may include the two or more antennas. The precoding according to the determined precoder may be performed by the base station. For example, the base station may apply precoding weights according to the determined precoder for the different at least two transmit antennas. 
     The determined precoder may be reported from the mobile device to the base station. The determined precoder may be reported to the base station, e.g., by means of a Channel State Information (CSI) report. The measuring step may be based on reference signals. The reference signals may be received on the channel by the mobile device. The reference signals may be spread in space and time. The reference signals may include CSI Reference Signals (CSI-RS) from the base station. 
     The base station may use the codebook for codebook-based precoding, e.g., according to the determined precoder. The CSI report may include a Precoding Matrix Indicator (PMI), a Rank Indicator (RI) and/or a Precoding Type Indicator (PTI). The PMI, the RI and/or the PTI may indicate the determined precoder, e.g., by reference to the codebook. 
     The selected subset may include a fraction of the plurality of precoders included in the codebook. For example, the selected subset may include less than half of the plurality of precoders. The codebook is optionally restricted by the base station. An indicator of a restriction of the codebook may be received from the base station. The selecting step may select the subset of precoders out of the restricted codebook. 
     The channel noise may be indicative of noise covariance and/or noise power on the channel. The channel state may be indicative of a response function of the channel. The channel may be a Multiple Input Single Output (MISO) channel or a Multiple Input Multiple Output (MIMO) channel. The radio transmission via the MISO channel may include beamforming. Alternatively or in addition, the radio transmission via the MIMO channel may include spatial diversity and/or spatial multiplexing. 
     The precoder may increase, e.g., maximize, a Poynting vector of the radio transmission at the base station towards the mobile device. The selecting step may specify a coarse direction for the maximum Poynting vector of the radio transmission. The determining step may adjust the coarse direction, e.g., within a predetermined solid angle centered about the coarse direction. 
     According to another aspect, a computer program product is provided. The computer program product comprises program code portions for performing one or more of the steps of the method described herein when the computer program product is executed on one or more computing devices. The computer program product may be stored on a computer-readable recording medium such as a permanent or re-writable memory. The computer program product may also be provided for download via one or more computer networks, such as the Internet, a cellular telecommunications network, or a wireless or wired Local Area Network (LAN). 
     According to a further aspect, an apparatus for determining a precoder for a radio transmission via a channel including two or more transmit antennas is provided. The apparatus comprises a measuring unit adapted to measure at least one of a channel state of the channel and a channel noise characteristic of the channel; a selecting unit adapted to select, based on a result of the measurement, a subset of precoders out of a codebook including a plurality of precoders; and a determining unit adapted to determine a precoder for precoding the transmission out of the subset. 
     The apparatus may further be adapted to, or may further include one or more units adapted to, perform any one of the steps disclosed in the context of the method aspect. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       In the following, the present disclosure is described in more detail with reference to exemplary embodiments illustrated in the drawings, wherein 
         FIG. 1  schematically illustrates a base station and a mobile device of a mobile telecommunications network; 
         FIG. 2  schematically illustrates reference signals, particularly Channel State Information-Reference Signals, CSI-RS, for measuring a channel state and channel noise of a channel between the base station and the mobile device of  FIG. 1 ; 
         FIG. 3  shows a flowchart for a method of determining a precoder for a radio transmission between the base station and the mobile device of  FIG. 1 ; 
         FIG. 4  schematically illustrates a block diagram of an apparatus for determining a precoder in the mobile device of  FIG. 1 ; 
         FIG. 5A  schematically illustrate a first parameter of a codebook used for the precoding determination; 
         FIG. 5B  schematically illustrate a second parameter of a codebook used for the precoding determination; 
         FIG. 6  schematically illustrates a linear array of 4 transmit antennas; and 
         FIG. 7  schematically illustrates an arrangement of 4 pairs of cross-polarized transmit antennas. 
     
    
    
     DETAILED DESCRIPTION 
     In the following description, for purposes of explanation and not limitation, specific details are set forth, such as specific network environments, apparatus configurations to and transmission channels in order to provide a thorough understanding of the technique disclosed herein. It is apparent to one skilled in the art that the technique may be practiced in other embodiments that depart from these specific details. Moreover, while the following embodiments are primarily described in relation to Long Term Evolution (LTE), it is readily apparent that the technique may also be practiced in context with other mobile telecommunications networks including Universal Mobile Telecommunications System (UMTS) and Global System for Mobile Communications (GSM), as well as other wireless networks, e.g., a Wireless Local Area Network (W-LAN) according to the family of standards IEEE 802.11. Moreover, while embodiments are described for a linear array of transmit antennas, the technique is also applicable to two-dimensional arrays, such as 2×2, 2×4 or 4×4 antenna arrays. 
     Those skilled in the art will further appreciate that the methods, steps and functions explained herein may be implemented using individual hardware circuitry, using software functioning in conjunction with a programmed microprocessor or general purpose computer, using a Field-Programmable Gate Array (FPGA), an Application Specific Integrated Circuit (ASIC) and/or using one or more Digital Signal Processors (DSPs). It will also be appreciated that while the following embodiments are primarily described in the form of methods and apparatuses, the technique disclosed herein may also be embodied in a computer processor and a memory coupled to the processor, wherein the memory stores one or more programs that perform the steps discussed herein when executed by the processor. 
       FIG. 1  schematically illustrates a mobile telecommunications system including a base station  102  and a mobile device  104 . In the exemplary scenario shown in  FIG. 1 , the base station  102  includes 4 transmit antennas  106  and the mobile device  104  includes 4 receive antennas  108  for a downlink transmission on a channel  110  including two spatial layers. For a closed-loop Single User Multiple Input Multiple Output (SU-MIMO) transmission on the downlink channel  110 , precoding is applied to the data carried on a Physical Downlink Shared Channel (PDSCH) in order to increase the received Signal-to-Interference plus Noise power Ratio (SINR). 
     The basis station includes a unit  112  for generating precoding weights  114 . For each transmission layer (also referred to as spatial stream), precoding weights  114  are applied to the transmit antennas  106  based on Channel State Information (CSI) reported by the mobile device  104 . A set of precoding weights is also referred to as a precoder. 
     An ideal precoder can be computed, e.g., from eigenvectors of a covariance matrix, H H  H, of a channel state represented by the matrix H. Herein, the superscript “H” indicates transposition and complex conjugation. However, reporting the eigenvectors from the mobile device  104  to the base station  102  is not practical in terms of the required control signaling overhead. Long Term Evolution (LTE) introduced in Release 8 codebook-based precoding, in which the best precoder among a plurality of predetermined precoders is reported from the mobile device  104  to the base station  102 . The plurality of predetermined precoders is collectively referred to as a codebook  116 . To minimize the overhead for control signaling, the CSI includes a Precoding Matrix Indicator (PMI)  118 . The PMI  118  is an index that identifies the precoder in the codebook  116 . The codebook  116  is also known to the unit  112  in the base station  104  applying the precoding weights  114  according to the PMI  118 . 
     The mobile device  104  includes an apparatus  120  for precoder determination. The apparatus includes a measuring unit  122  and a selecting and determining unit  124 . Based on an estimation for the channel state H and, optionally, channel noise of the channel  110  provided by the measuring unit  122 , the selecting and determining unit  124  specifies the PMI  118 . 
     Frequency-selective precoding is adapted in LTE since Release 8.  FIG. 2  shows a Resource Block (RB)  200  with time on the horizontal axis and frequency on the vertical axis. The RB  200  encompasses on the time axis one subframe, i.e., one millisecond. The subframe includes 14 Orthogonal Frequency Division Multiplexing (OFDM) symbols (one of which is shown at reference sign  202 ). The RB  200  encompasses 12 subcarriers (one of which is shown at reference sign  204 ) on the frequency axis. Each OFDM symbol within the RB  200  includes 12 Resource Elements (REs). The RB  200  thus includes 12×14 REs (each of which is indicated by a square in the RB  200  shown in  FIG. 2 ). 
     The channel  110  includes a Physical Downlink Control Channel (PDCCH) and the PDSCH. The first two symbols of the RB  200  are allocated to the PDCCH. The further 12 symbols are allocated to the PDSCH. The basis station  102  notifies the mobile device  104  of the precoding weight information actually used for the PDSCH through the PDCCH. The mobile device  104  uses the precoding weight information for demodulation. 
     Reference Signals  206  are transmitted by the base station  102  in REs distributed in the RB  200  according to a predetermined pattern. The measuring unit  122  estimates a channel state and channel noise by receiving the predetermined reference signals  206 . 
     While the mobile telecommunications network  100  has been exemplarily described with reference to  FIG. 1  for a closed-loop SU-MIMO transmission with rank 2 via a 4×4 MIMO channel  110 , the technique is also applicable to a 2×2 MIMO channel or a Multiple Input Single Output (MISO) channel. 
       FIG. 3  shows a flowchart for a method  300  of determining a precoder for a radio transmission via a channel including two or more transmit antennas. The method  300  may be performed at the mobile device  104 . The mobile device  104  may receive the channel  110  from the transmit antennas  106 . 
     The method comprises a step  302  of measuring a channel state of the channel and, optionally, channel noise of the channel. Based on a result of the measurement, a subset of two or more precoders out of a codebook including a plurality of precoders is selected in a step  304 . A precoder for precoding the transmission is determined in a step  306  out of the selected subset. 
     The step  302  may be performed by the measuring unit  122 . The steps  304  and  306  may be performed by the selecting and determining unit  124 . The unit  124  further reports the PMI  118  to the base station  102  of the radio transmission. Optionally, a further functional unit  124  determines a Channel Quality Indicator (CQI) and Rank Indication (RI), e.g., for rank adaption according to channel conditions. In this case, the CSI report to the base station  102  includes the PMI  118 , the CQI and the RI. 
       FIG. 4  schematically illustrates a block diagram of an embodiment for the apparatus  120 . The apparatus  120  includes a selecting unit  400  and a determining unit  402 . The determining unit  402  is also referred to as core signal processing for precoder determination or Channel Quality Indicator Entity (CQIE). The determining unit  402  is downstream of the selecting unit  400 . The selecting unit  400  is adapted to select, based on a results of the measurement provided by the measuring unit  122 , a subset to of precoders out of the codebook  116  according to the step  304 . The determining unit  402  is adapted to determine the precoder (e.g., to be used for precoding the transmission) out of the selected subset according to the step  306 . The PMI  118  corresponding to the determined precoder is reported to the base station  102  by the apparatus  120 . 
     The selecting unit  400  reduces the complexity of signal processing downstream of the selecting unit  400 . E.g., the signal processing effort in the determining unit  402  is reduced. The selecting unit  400  reduces entries in a Codebook Subset Restriction (CSR) bit field in front of the core signal processing for precoder determination according to the selecting step  304 . The complexity for the signal processing according to the selecting step  304  performed by the selecting unit  400  is more than compensated by the reduction of complexity for the signal processing according to the determining step  306  downstream of the selecting unit  400 . 
     Optionally, the selecting unit  400  accounts for an additional restriction signaled by the base station  102 . In an LTE implementation, a codebook subset restriction by the base station  102  is supported in Transmission Modes (TM)  3 ,  4 ,  5 ,  6 ,  8  and  9 , if the mobile device  104  is configured to include the PMI  118  and/or the RI in the CSI report. 
     The selecting unit  400  evaluates a first objective function. The selected subset of precoders maximizes the first objective function. In one embodiment for the selecting unit  400 , the subset is selected based on the channel state provided by the measuring unit  122 . In an advanced embodiment of the selecting unit  400 , the subset is selected based on the channel state and the channel noise provided by the measuring unit  122  (which is indicated by a dotted arrow in  FIG. 1 ). The evaluation of the first objective function is a substep of the selecting step  304 . 
     In the embodiment for the determining unit  402  shown in  FIG. 4 , the determining unit  402  includes a Channel Quality Metric Estimator (CQME)  404  and a Channel Quality Report Generator (CQRG)  406 . The CQME  404  evaluates (e.g., by numerically computation) a second objective function based on the channel state and the channel noise provided by the measuring unit  122 . The evaluation of the second objective function is a substep of the determining step  306 . 
     The computational complexity for evaluating the second objective function is significantly greater than the computational complexity for evaluating the first objective function. Typically, the computational complexity of the second objective function is at least 2 times (e.g., about 4 times or about 10 times) the computational complexity of the first objective function. 
     The CQME  404  evaluates the second objective function for each precoder in the subset (e.g., represented by the reduced CSR) given the measurement provided by the measuring unit  122 . The second objective function estimates a metric for the corresponding one of the precoders in the subset. The CQRG  406  determines, based on the metrics provided by the CQME  404 , which precoder out of the subset is determined according to the step  306 . The determining unit  402  evaluates and compares the second objective function only for those precoders included in the subset selected by the selecting unit  400 . 
     The second objective function yields a metric that is more accurate than a metric provided by the first objective function. E.g., the metric provided by the first objective function is accurate enough for distinguishing between different subset in the codebook. The metric provided by the second objective function is accurate enough for distinguishing between different precoders in the subset. The metric provided by the first objective function is not accurate enough for distinguishing between different precoders in the subset. In one embodiment for the CQME  404 , the second objective function estimates the mutual information on the channel  110 . In a variant of the embodiment, the second objective function estimates the SNIR of the channel  110 . 
     A first use case of the technique is described with reference to  FIGS. 5A and 5B . Each of  FIGS. 5A and 5B  schematically illustrates an azimuthal cross-section of an emission characteristic for an exemplary array of the transmit antennas  106  in a plane  500 . 
     Each subset corresponds to a coarse direction  502  of emitted power centered about a first angle  504  relative to a normal of the plane  500 . The emission characteristic  503  results from a hypothetical superposition of the precoders in the subset. For the exemplary array of transmit antennas  106  underlying  FIG. 5A , the selection step  304  specifies the coarse direction  502 . 
     The determining step  306  specifies an emission direction  506  by determining one precoder out of the subset. As schematically illustrated in  FIG. 5B , the determined precoder corresponds to a narrower emission characteristic  507  centered about the emission direction  506 . The emission direction  506  includes a second angle  508  with the coarse direction  502 . The technical effect schematically illustrated in  FIGS. 5A and 5B  for the selecting step  304  and the determining  306 , respectively, shows that the selecting step  306  corresponds to selecting a coarse direction  502  according to the first angle  504 , which is refined or adjusted in the determining step  306  according to the second angle  508  resulting in the emission direction  506 . 
     The codebook  116  for an LTE implementation using 2 transmit antennas or 4 transmit antennas  106  is specified in standard document 3GPP TS 36.211 (Version 12.0.0) in Sect. 6.3.4.2.3. Loc. cit., Table 6.3.4.2.3-1 specifies the codebook  116  for 2 transmit antennas  106 , and Table 6.3.4.2.3-2 specifies the codebook for 4 transmit antennas  106 . 
     An exemplary LTE implementation according to standard document 3GPP TS 36.213 (Version 11.4.0), Sect. 7.2.4 is described. Loc. cit., Tables 7.2.4-1 to 7.2.4-8 specify codebooks  116  for 8 transmit antennas  106 , wherein the channel  110  provides 1 to 8 layers, respectively. For example, the number of precoders in the codebook is 16×16=256 according to Table 7.2.4-1. Each precoder specifies 8 precoding weights, i.e., one precoding weight for each transmit antenna. For two or more layers on the channel  110 , each precoder specifies 8 precoding weights, w=[w 0 , . . . , w 7 ] T , for each of the layers. 
     The precoding vectors, w, to be considered for PMI reporting when CSI-RS is transmitted over 8 transmit antennas  106  are composed in the following way. For each layer, a complex vector w n,m  of length 8 is given by 
     
       
         
           
             
               
                 
                   
                     
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     A first index m of the codebook ranges from 0 to 31. The complex column vector v m  of length 4 in Eq. (1) is defined as
 
 v   m =[1 e   jβ   e   j2β   e   j3β ] T ,  (2)
 
wherein a first phase angle
 
             β   =       2   ⁢           ⁢   π   ⁢           ⁢   m     32           
is an alternative representation of the first parameter.
 
     A second index n of the codebook ranges from 0 to 3. The phase factor φ n  is defined as
 
φ n ϵ{1, e   jα   ,e   j2α   ,e   j3α },  (3)
 
wherein a second phase angle α=2πn/4 is an alternative representation of the second parameter.
 
     The precoding vector is fully determined by two parameters, the phase angles α and β. A second use case of the technique is described with reference to  FIGS. 6 and 7 . A physical interpretation of the first phase angle β is illustrated in  FIG. 6  for a linear array  500  of the transmit antennas  106  spaced apart by a distance λ/2 between two adjacent transmit antennas  106 . For clarity, 4 transmit antennas  106  are shown in  FIG. 6 . The first partition, v m , of the precoding vector, w n,m , is sequentially associated to the transmit antennas  106  shown in  FIG. 6 . An emission angle γ relative to the normal direction of the linear array  500  is directly related to the first phase angle β. Hence, the first phase angle β specifies the emission direction  506 . 
       FIG. 7  shows a linear array of 4 pairs of cross-polarized transmit antennas  106 .  FIG. 7  shows the plane including the transmit antennas  106 , i.e., antenna dipoles are parallel to the plane of the drawing sheet. The 4 transmit antennas with cross-polarization direction “up” are sequentially associated to the first partition, v m , of the precoding vector, w n,m . The second partition, φ n v m , of the precoding vector, w n,m , is sequentially associated to the cross-polarization direction “down”. Consequentially, the emission direction is specified by the first phase angle β. The polarization directions “up” and “down” have the same emission characteristic in the azimuthal cross-section shown in  FIG. 6 . The phase angle α controls the combination of the two cross-polarization components. 
     For an LTE implementation with 8 transmit antennas  106  and two layers, the codebook  116  includes 256 different precoders according to 3GPP TS 36.213 (Version 11.4.0), Table 7.2.4-2. Each precoder specifies two precoding vectors w n,m , i.e., one precoding vector for each layer. The two precoding vectors can be represented as columns of one precoding matrix. Hence, each precoder is representable by one precoding matrix W(i 1 ,i 2 ). 
     The codebook entries, i.e., the plurality of precoders in the codebook  116 , are parameterized by the two indices i 1  and i 2 , each index ranging from 0 to 15. For periodic CSI reporting on a Physical Uplink Control Channel (PUCCH), the two indices are sequentially reported to the base station  102 , e.g., within different subframes  200 . 
     The codebook  116  obeys the following structure according to afore-mentioned Tables 7.2.4-1 and 7.2.4-2 for transmission ranks 1 and 2, respectively. The index i 1  determines the coarse angle of radio emission, e.g., in terms of the first phase angle β. For each given index i 1 , the index i 2  spans codebook entries for fine adjustment of the phase angles α and β. The variation of β about β mean  for a given index i 1  is within a narrow range ±Δβ. Below table includes the range ±Δβ of the first phase angle β for different i 2  and a given index i 1 . 
     
       
         
           
               
               
               
               
               
               
               
             
               
                   
               
               
                   
                 i 1   
                 β mean   
                 ±Δβ 
                 i 1   
                 β mean   
                 ±Δβ 
               
               
                   
               
             
            
               
                   
                 0 
                 
                   
                     
                       
                         2 
                         ⁢ 
                         
                           π 
                           · 
                           
                             3 
                             64 
                           
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         2 
                         ⁢ 
                         
                           π 
                           · 
                           
                             3 
                             64 
                           
                         
                       
                     
                   
                 
                  8 
                 
                   
                     
                       
                         2 
                         ⁢ 
                         
                           π 
                           · 
                           
                             3 
                             64 
                           
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         2 
                         ⁢ 
                         
                           π 
                           · 
                           
                             3 
                             64 
                           
                         
                       
                     
                   
                 
               
               
                   
               
               
                   
                 1 
                 
                   
                     
                       
                         2 
                         ⁢ 
                         
                           π 
                           · 
                           
                             7 
                             64 
                           
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         2 
                         ⁢ 
                         
                           π 
                           · 
                           
                             3 
                             64 
                           
                         
                       
                     
                   
                 
                  9 
                 
                   
                     
                       
                         2 
                         ⁢ 
                         
                           π 
                           · 
                           
                             39 
                             64 
                           
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         2 
                         ⁢ 
                         
                           π 
                           · 
                           
                             3 
                             64 
                           
                         
                       
                     
                   
                 
               
               
                   
               
               
                   
                 2 
                 
                   
                     
                       
                         2 
                         ⁢ 
                         
                           π 
                           · 
                           
                             11 
                             64 
                           
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         2 
                         ⁢ 
                         
                           π 
                           · 
                           
                             3 
                             64 
                           
                         
                       
                     
                   
                 
                 10 
                 
                   
                     
                       
                         2 
                         ⁢ 
                         
                           π 
                           · 
                           
                             43 
                             64 
                           
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         2 
                         ⁢ 
                         
                           π 
                           · 
                           
                             3 
                             64 
                           
                         
                       
                     
                   
                 
               
               
                   
               
               
                   
                 3 
                 
                   
                     
                       
                         2 
                         ⁢ 
                         
                           π 
                           · 
                           
                             15 
                             64 
                           
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         2 
                         ⁢ 
                         
                           π 
                           · 
                           
                             3 
                             64 
                           
                         
                       
                     
                   
                 
                 11 
                 
                   
                     
                       
                         2 
                         ⁢ 
                         
                           π 
                           · 
                           
                             47 
                             64 
                           
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         2 
                         ⁢ 
                         
                           π 
                           · 
                           
                             3 
                             64 
                           
                         
                       
                     
                   
                 
               
               
                   
               
               
                   
                 4 
                 
                   
                     
                       
                         2 
                         ⁢ 
                         
                           π 
                           · 
                           
                             19 
                             64 
                           
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         2 
                         ⁢ 
                         
                           π 
                           · 
                           
                             3 
                             64 
                           
                         
                       
                     
                   
                 
                 12 
                 
                   
                     
                       
                         2 
                         ⁢ 
                         
                           π 
                           · 
                           
                             51 
                             64 
                           
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         2 
                         ⁢ 
                         
                           π 
                           · 
                           
                             3 
                             64 
                           
                         
                       
                     
                   
                 
               
               
                   
               
               
                   
                 5 
                 
                   
                     
                       
                         2 
                         ⁢ 
                         
                           π 
                           · 
                           
                             23 
                             64 
                           
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         2 
                         ⁢ 
                         
                           π 
                           · 
                           
                             3 
                             64 
                           
                         
                       
                     
                   
                 
                 13 
                 
                   
                     
                       
                         2 
                         ⁢ 
                         
                           π 
                           · 
                           
                             55 
                             64 
                           
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         2 
                         ⁢ 
                         
                           π 
                           · 
                           
                             3 
                             64 
                           
                         
                       
                     
                   
                 
               
               
                   
               
               
                   
                 6 
                 
                   
                     
                       
                         2 
                         ⁢ 
                         
                           π 
                           · 
                           
                             27 
                             64 
                           
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         2 
                         ⁢ 
                         
                           π 
                           · 
                           
                             3 
                             64 
                           
                         
                       
                     
                   
                 
                 14 
                 
                   
                     
                       
                         2 
                         ⁢ 
                         
                           π 
                           · 
                           
                             59 
                             64 
                           
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         2 
                         ⁢ 
                         
                           π 
                           · 
                           
                             3 
                             64 
                           
                         
                       
                     
                   
                 
               
               
                   
               
               
                   
                 7 
                 
                   
                     
                       
                         2 
                         ⁢ 
                         
                           π 
                           · 
                           
                             31 
                             64 
                           
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         2 
                         ⁢ 
                         
                           π 
                           · 
                           
                             3 
                             64 
                           
                         
                       
                     
                   
                 
                 15 
                 
                   
                     
                       
                         2 
                         ⁢ 
                         
                           π 
                           · 
                           
                             63 
                             64 
                           
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         2 
                         ⁢ 
                         
                           π 
                           · 
                           
                             3 
                             64 
                           
                         
                       
                     
                   
                 
               
               
                   
               
            
           
         
       
     
     The first and second uses cases are combined in an embodiment. According to the parameterization structure of the codebook  116 , the first index i 1  specifies a coarse beam direction (as schematically illustrated in  FIG. 5A  at reference sign  502 ) and the second index i 2  allows for fine adjustment of α and (as schematically illustrated in  FIG. 5B  at reference sign  508 ) for β about β mean . The selection  304  of the subset specifies i 1 . The determination  306  (which is preferably dealt in the CQIE  402  only) specifies the index i 2 . As compared to a conventional implementation performing the precoder determination by evaluating each of the 256 different possible precoders each including one precoding vector per transmission rank, the embodiment reduces the number of evaluations to 16. The technique thus achieves a level of computational complexity for 8 transmit antennas  106 , which is similar to a conventional precoder determination for 4 transmit antennas. 
     Embodiments for the first objective function are described below. The embodiments for the first objective function can be combined with afore-mentioned embodiments for the second objection function. While below considerations provide a clear understanding of the first objective function to the skilled person, steps described below, e.g., for justifying examples of the first objective function, are not essential for an implementation of the first objective function. For example, a result of the consideration can be implemented, e.g., using a FPGA, a program executed on an Advanced RISC Machine (ARM) processor and/or an Embedded Vector Processor (EVP). 
     A first embodiment for the first objective function relates to a single-layer transmission on the channel  110 . The CSI-RSs, s(k), are received by the mobile device  104  at Resource Element k. A time index l for the OFDM symbol may be omitted, since the CSI-RSs have only one allocation in time direction per subframe  200 . 
     For 8 transmit antennas  106  and N Rx  receive antennas  108 , the CSI-RSs s(k) are received over the N Rx ×8 MIMO channel  110 , the channel state of which is represented by H(k), and corrupted by noise n(k). The UE observation is given as 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           y 
                           ⁡ 
                           
                             ( 
                             k 
                             ) 
                           
                         
                         = 
                           
                         ⁢ 
                         
                           
                             
                               H 
                               ⁡ 
                               
                                 ( 
                                 k 
                                 ) 
                               
                             
                             ⁢ 
                             
                               w 
                               
                                 n 
                                 , 
                                 m 
                               
                             
                             ⁢ 
                             
                               s 
                               ⁡ 
                               
                                 ( 
                                 k 
                                 ) 
                               
                             
                           
                           + 
                           
                             n 
                             ⁡ 
                             
                               ( 
                               k 
                               ) 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                         ⁢ 
                         
                           
                             
                               
                                 H 
                                 1 
                               
                               ⁡ 
                               
                                 ( 
                                 k 
                                 ) 
                               
                             
                             ⁢ 
                             
                               v 
                               m 
                             
                             ⁢ 
                             
                               s 
                               ⁡ 
                               
                                 ( 
                                 k 
                                 ) 
                               
                             
                           
                           + 
                           
                             
                               
                                 H 
                                 2 
                               
                               ⁡ 
                               
                                 ( 
                                 k 
                                 ) 
                               
                             
                             ⁢ 
                             
                               φ 
                               n 
                             
                             ⁢ 
                             
                               v 
                               m 
                             
                             ⁢ 
                             
                               s 
                               ⁡ 
                               
                                 ( 
                                 k 
                                 ) 
                               
                             
                           
                           + 
                           
                             n 
                             ⁡ 
                             
                               ( 
                               k 
                               ) 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   4 
                   ) 
                 
               
             
           
         
       
     
     In the first embodiment, an objective of the first objective function is to find the first phase angle β such that the signal energy is maximized (irrespectively of the choice of φ n ), i.e. 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           β 
                           max 
                         
                         = 
                           
                         ⁢ 
                         
                           
                             
                               arg 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               max 
                             
                             β 
                           
                           ⁢ 
                           E 
                           ⁢ 
                           
                             { 
                             
                               
                                  
                                 
                                   
                                     
                                       
                                         H 
                                         1 
                                       
                                       ⁡ 
                                       
                                         ( 
                                         k 
                                         ) 
                                       
                                     
                                     ⁢ 
                                     
                                       v 
                                       m 
                                     
                                     ⁢ 
                                     
                                       s 
                                       ⁡ 
                                       
                                         ( 
                                         k 
                                         ) 
                                       
                                     
                                   
                                   + 
                                   
                                     
                                       
                                         H 
                                         2 
                                       
                                       ⁡ 
                                       
                                         ( 
                                         k 
                                         ) 
                                       
                                     
                                     ⁢ 
                                     
                                       φ 
                                       n 
                                     
                                     ⁢ 
                                     
                                       v 
                                       m 
                                     
                                     ⁢ 
                                     
                                       s 
                                       ⁡ 
                                       
                                         ( 
                                         k 
                                         ) 
                                       
                                     
                                   
                                 
                                  
                               
                               2 
                               2 
                             
                             } 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                         ⁢ 
                         
                           
                             
                               arg 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               max 
                             
                             β 
                           
                           ⁢ 
                           
                             v 
                             m 
                             H 
                           
                           ⁢ 
                           
                             
                               E 
                               ⁢ 
                               
                                 { 
                                 
                                   
                                     
                                       
                                         H 
                                         1 
                                         H 
                                       
                                       ⁡ 
                                       
                                         ( 
                                         k 
                                         ) 
                                       
                                     
                                     ⁢ 
                                     
                                       
                                         H 
                                         1 
                                       
                                       ⁡ 
                                       
                                         ( 
                                         k 
                                         ) 
                                       
                                     
                                   
                                   + 
                                   
                                     
                                       
                                         H 
                                         2 
                                         H 
                                       
                                       ⁡ 
                                       
                                         ( 
                                         k 
                                         ) 
                                       
                                     
                                     ⁢ 
                                     
                                       
                                         H 
                                         2 
                                       
                                       ⁡ 
                                       
                                         ( 
                                         k 
                                         ) 
                                       
                                     
                                   
                                 
                                 } 
                               
                             
                             
                               ︸ 
                               R 
                             
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             v 
                             m 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   5 
                   ) 
                 
               
             
           
         
       
     
     The intermediate matrix R of dimension 4×4 can be estimated by 
     
       
         
           
             
               
                 
                   
                     R 
                     = 
                     
                       
                         
                           ∑ 
                           
                             k 
                             = 
                             0 
                           
                           
                             
                               N 
                               RB 
                               DL 
                             
                             - 
                             1 
                           
                         
                         ⁢ 
                         
                           
                             
                               H 
                               1 
                               H 
                             
                             ⁡ 
                             
                               ( 
                               k 
                               ) 
                             
                           
                           ⁢ 
                           
                             
                               H 
                               1 
                             
                             ⁡ 
                             
                               ( 
                               k 
                               ) 
                             
                           
                         
                       
                       + 
                       
                         
                           
                             H 
                             2 
                             H 
                           
                           ⁡ 
                           
                             ( 
                             k 
                             ) 
                           
                         
                         ⁢ 
                         
                           
                             H 
                             2 
                           
                           ⁡ 
                           
                             ( 
                             k 
                             ) 
                           
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   6 
                   ) 
                 
               
             
           
         
       
     
     wherein the numerical complexity includes a number of (4+3+2+1)·N RB   DL ·2 complex multiplications and a number of (4+3+2+1)·(N RB   DL −1)·2 complex additions. The intermediate matrix R may be interpreted as a relation between the 4 pairs of transmit antennas, e.g., a transmit antenna correlation that is averaged over the different polarizations. 
     Inserting Eq. (2) into Eq. (5), the first objective function can be rewritten as 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           g 
                           ⁡ 
                           
                             ( 
                             β 
                             ) 
                           
                         
                         = 
                           
                         ⁢ 
                         
                           
                             ∑ 
                             
                               m 
                               = 
                               0 
                             
                             
                               N 
                               - 
                               1 
                             
                           
                           ⁢ 
                           
                             
                               ∑ 
                               
                                 n 
                                 = 
                                 0 
                               
                               
                                 N 
                                 - 
                                 1 
                               
                             
                             ⁢ 
                             
                               
                                 r 
                                 
                                   m 
                                   , 
                                   n 
                                 
                               
                               ⁢ 
                               
                                 e 
                                 
                                   
                                     j 
                                     ⁡ 
                                     
                                       ( 
                                       
                                         m 
                                         - 
                                         n 
                                       
                                       ) 
                                     
                                   
                                   ⁢ 
                                   β 
                                 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                         ⁢ 
                         
                           
                             
                               ∑ 
                               
                                 m 
                                 = 
                                 0 
                               
                               
                                 N 
                                 - 
                                 1 
                               
                             
                             ⁢ 
                             
                               r 
                               
                                 m 
                                 , 
                                 m 
                               
                             
                           
                           + 
                           
                             
                               ∑ 
                               
                                 v 
                                 = 
                                 1 
                               
                               
                                 N 
                                 - 
                                 1 
                               
                             
                             ⁢ 
                             
                               
                                 ∑ 
                                 
                                   m 
                                   = 
                                   0 
                                 
                                 
                                   N 
                                   - 
                                   1 
                                   - 
                                   v 
                                 
                               
                               ⁢ 
                               
                                 
                                   r 
                                   
                                     m 
                                     , 
                                     
                                       m 
                                       + 
                                       v 
                                     
                                   
                                 
                                 ⁢ 
                                 
                                   e 
                                   
                                     jv 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     β 
                                   
                                 
                               
                             
                           
                           + 
                           
                             
                               r 
                               
                                 m 
                                 , 
                                 
                                   m 
                                   + 
                                   v 
                                 
                               
                               * 
                             
                             ⁢ 
                             
                               e 
                               
                                 
                                   - 
                                   jv 
                                 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 β 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                         ⁢ 
                         
                           
                             
                               ∑ 
                               
                                 m 
                                 = 
                                 0 
                               
                               
                                 N 
                                 - 
                                 1 
                               
                             
                             ⁢ 
                             
                               r 
                               
                                 m 
                                 , 
                                 m 
                               
                             
                           
                           + 
                         
                       
                     
                   
                   
                     
                       
                           
                         ⁢ 
                         
                           2 
                           ⁢ 
                           
                             
                               ∑ 
                               
                                 v 
                                 = 
                                 1 
                               
                               
                                 N 
                                 - 
                                 1 
                               
                             
                             ⁢ 
                             
                               ( 
                               
                                 
                                   
                                     cos 
                                     ⁡ 
                                     
                                       ( 
                                       
                                         v 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         β 
                                       
                                       ) 
                                     
                                   
                                   ⁢ 
                                   
                                     
                                       
                                         ∑ 
                                         
                                           m 
                                           = 
                                           0 
                                         
                                         
                                           N 
                                           - 
                                           1 
                                           - 
                                           v 
                                         
                                       
                                       ⁢ 
                                       
                                         ℜ 
                                         ⁢ 
                                         
                                           { 
                                           
                                             r 
                                             
                                               m 
                                               , 
                                               
                                                 m 
                                                 + 
                                                 v 
                                               
                                             
                                           
                                           } 
                                         
                                       
                                     
                                     
                                       ︸ 
                                       
                                         
                                           r 
                                           _ 
                                         
                                         v 
                                         ′ 
                                       
                                     
                                   
                                 
                                 - 
                                 
                                   
                                     sin 
                                     ⁡ 
                                     
                                       ( 
                                       
                                         v 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         β 
                                       
                                       ) 
                                     
                                   
                                   ⁢ 
                                   
                                     
                                       
                                         ∑ 
                                         
                                           m 
                                           = 
                                           0 
                                         
                                         
                                           N 
                                           - 
                                           1 
                                           - 
                                           v 
                                         
                                       
                                       ⁢ 
                                       
                                         𝔍 
                                         ⁢ 
                                         
                                           { 
                                           
                                             r 
                                             
                                               m 
                                               , 
                                               
                                                 m 
                                                 + 
                                                 v 
                                               
                                             
                                           
                                           } 
                                         
                                       
                                     
                                     
                                       ︸ 
                                       
                                         
                                           r 
                                           _ 
                                         
                                         v 
                                         ″ 
                                       
                                     
                                   
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   7 
                   ) 
                 
               
             
           
         
       
     
     The numerical effort for summing up  r ′ v  and  r ″ v  includes 2×(3+2+1) real additions. 
     Two variants of the first embodiment that approximate β max  are disclosed. The first variant has the lower numerical complexity compared to the second variant at the cost of less accuracy. 
     The first variant is described. At β=β max , the derivative of g(β) with respect to β vanishes: 
     
       
         
           
             
               
                 
                   
                     
                       ∂ 
                       
                         g 
                         ⁡ 
                         
                           ( 
                           β 
                           ) 
                         
                       
                     
                     
                       ∂ 
                       β 
                     
                   
                   = 
                   
                     
                       
                         
                           ∑ 
                           
                             v 
                             = 
                             1 
                           
                           
                             N 
                             - 
                             1 
                           
                         
                         ⁢ 
                         
                           
                             - 
                             v 
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             
                               r 
                               _ 
                             
                             v 
                             ′ 
                           
                           ⁢ 
                           
                             sin 
                             ⁡ 
                             
                               ( 
                               
                                 v 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 β 
                               
                               ) 
                             
                           
                         
                       
                       + 
                       
                         v 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           
                             r 
                             _ 
                           
                           v 
                           ″ 
                         
                         ⁢ 
                         
                           cos 
                           ⁡ 
                           
                             ( 
                             
                               v 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               β 
                             
                             ) 
                           
                         
                       
                     
                     = 
                     0. 
                   
                 
               
               
                 
                   ( 
                   8 
                   ) 
                 
               
             
           
         
       
     
     For N transmit antennas  106 , N≤4, a closed-form solution of Eq. (8) may be found, e.g., using Chebyshev polynomials. In general and/or for numerical efficiency, the derivative of the first objective function is approximated by neglecting the higher order terms ν&gt;1 of the sum. Hence, Eq. (8) becomes 
     
       
         
           
             
               
                 
                   
                     
                       
                         ∂ 
                         
                           g 
                           ⁡ 
                           
                             ( 
                             β 
                             ) 
                           
                         
                       
                       
                         ∂ 
                         β 
                       
                     
                     ≈ 
                     
                       
                         
                           - 
                           
                             
                               r 
                               _ 
                             
                             1 
                             ′ 
                           
                         
                         ⁢ 
                         
                           sin 
                           ⁡ 
                           
                             ( 
                             β 
                             ) 
                           
                         
                       
                       + 
                       
                         
                           
                             r 
                             _ 
                           
                           1 
                           ″ 
                         
                         ⁢ 
                         
                           cos 
                           ⁡ 
                           
                             ( 
                             β 
                             ) 
                           
                         
                       
                     
                   
                   = 
                   0 
                 
               
               
                 
                   ( 
                   9 
                   ) 
                 
               
             
           
         
       
     
     An approximation of β max  is obtained by calculating 
     
       
         
           
             
               
                 
                   
                     β 
                     max 
                   
                   = 
                   
                     arctan 
                     ⁡ 
                     
                       ( 
                       
                         
                           
                             r 
                             _ 
                           
                           1 
                           ″ 
                         
                         
                           
                             r 
                             _ 
                           
                           1 
                           ′ 
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   10 
                   ) 
                 
               
             
           
         
       
     
     Eq. (10) specifies the first index i 1  according to the first variant in the selecting step  304 . The first index is specified, e.g., by rounding. For example, the entry in above table for β mean  that is closest to β max  specifies the first index i 1 . 
     The numerical complexity of the first variant is summarized in below table. 
     
       
         
           
               
               
               
               
             
               
                   
               
               
                 Function 
                 Multiplications 
                 Additions 
                 Specific Operations 
               
               
                   
               
             
            
               
                 Covariance 
                 3 × N RB   DL  × 2 
                 3 × (N RB   DL  − 1) × 2 
                   
               
               
                 matrix 
                 complex- 
                 complex-valued 
               
               
                 estimation 
                 valued 
               
               
                 Summing up 
                   
                 2 × 3 
               
               
                 in Eq. (7) 
                   
                 real-valued 
               
               
                 Arctan 
                   
                   
                 Division and 
               
               
                 computation 
                   
                   
                 Look-Up Table 
               
               
                   
                   
                   
                 (LUT) for arctan 
               
               
                   
               
            
           
         
       
     
     Subsequently, the second index i 2  is determined according to the determining step  306 . The first index i 1  specifies β mean . The precoder is determined out of the subset of precoders associated to the specified first index i 1 . 
     The second variant is described. The second variant is of higher numerical complexity and more accurate, since all summands of in the objective function in Eq. (7) are taken into account according to:
 
 g (β)=Σ ν=1   N-1 (cos(νβ)   r ′   ν +sin(νβ)   r ″   ν ).  (11)
 
     Moreover, the numerical complexity is decreased, if the Codebook Subset Restriction field (e.g., as specified by the base station  102 ) already excludes some candidates with respect to the first index i 1 . 
     The first phase angle β is discretized into L=32 elements, i.e., g(β) is sampled within the range of β=0, . . . , 2π at a rate π/16. Matrices C and S of dimension L×(N−2) are defined to have elements given by 
     
       
         
           
             
               
                 
                   
                     
                       C 
                       
                         k 
                         , 
                         
                           v 
                           - 
                           1 
                         
                       
                     
                     = 
                     
                       cos 
                       ⁡ 
                       
                         ( 
                         
                           
                             vl 
                             L 
                           
                           ⁢ 
                           π 
                         
                         ) 
                       
                     
                   
                   , 
                   and 
                 
               
               
                 
                   ( 
                   12 
                   ) 
                 
               
             
             
               
                 
                   
                     S 
                     
                       k 
                       , 
                       
                         v 
                         - 
                         1 
                       
                     
                   
                   = 
                   
                     
                       sin 
                       ⁡ 
                       
                         ( 
                         
                           
                             vl 
                             L 
                           
                           ⁢ 
                           π 
                         
                         ) 
                       
                     
                     . 
                   
                 
               
               
                 
                     
                 
               
             
           
         
       
     
     Two column vectors  r ′ and  r ″ of length N−1 are defined as
 
   r ′=[ r ′   1    . . .  r ′   N-1 ] T , and
 
   r ″=[ r ″   1    . . .  r ″   N-1 ] T .  (13)
 
     Hence, the discretized function g(β) can be expressed as a vector:
 
 g=C r ′+S r ″.   (14)
 
     The first index i 1  associated to β max  corresponds to the index of the maximum value in the vector g. 
     The numerical complexity of the second variant is summarized in below table. 
     
       
         
           
               
               
               
               
             
               
                   
               
               
                   
                   
                   
                 Specific 
               
               
                 Function 
                 Multiplications 
                 Additions 
                 Operations 
               
               
                   
               
             
            
               
                 Covariance 
                 (4 + 3 + 2 + 1) × 
                 (4 + 3 + 2 + 1) × 
                   
               
               
                 matrix 
                 N RB   DL  × 2 
                 (N RB   DL  − 1) × 2 
               
               
                 estimation 
                 complex-valued 
                 complex-valued 
               
               
                 Summing up 
                   
                 2 × (3 + 2 + 1) 
               
               
                 in Eq. (7) 
                   
                 real-valued 
               
               
                 Matrix vector 
                 32 × 3 × 2 
                 32 × 2 × 2 
               
               
                 multiplication 
                 real-valued 
               
               
                 according to 
               
               
                 Eq. (14) 
               
               
                 Search for 
                   
                   
                 Tree search with 
               
               
                 maximum in 
                   
                   
                 complexity 
               
               
                 vector of 
                   
                   
                 O(log 2 (16)) 
               
               
                 size 32 
               
               
                   
               
            
           
         
       
     
     A second embodiment for the first objective function relates to a transmission on the channel  110  providing an arbitrary number of layers. While the second embodiment is described for 2 receive antennas  108 , the technique can readily be applied to any other number of receive antennas  108 . 
     The measuring unit  122  provides estimations for the coarse channel state, or filtered coarse channel state, Ĥ(k). The coarse channel states are computed by de-rotation, e.g., by multiplying the received signal with the complex-conjugate of the CSI-RS as sent. The filtering is applied to the coarse channel state for improving noise suppression. 
     The measuring unit  122  further provides a noise covariance, 
     
       
         
           
             
               N 
               = 
               
                 [ 
                 
                   
                     
                       
                         σ 
                         00 
                       
                     
                     
                       
                         σ 
                         01 
                       
                     
                   
                   
                     
                       
                         σ 
                         01 
                         * 
                       
                     
                     
                       
                         σ 
                         11 
                       
                     
                   
                 
                 ] 
               
             
             , 
           
         
       
     
     of the noise n between the receive antennas  108 . 
     The correlation matrix {circumflex over (R)} is estimated according to 
     
       
         
           
             
               
                 
                   
                     
                       R 
                       ^ 
                     
                     ⁡ 
                     
                       ( 
                       
                         H 
                         ^ 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       1 
                       
                         N 
                         RB 
                         DL 
                       
                     
                     ⁢ 
                     
                       
                         ∑ 
                         
                           k 
                           = 
                           0 
                         
                         
                           
                             N 
                             RB 
                             DL 
                           
                           - 
                           1 
                         
                       
                       ⁢ 
                       
                         
                           
                             
                               H 
                               ^ 
                             
                             H 
                           
                           ⁡ 
                           
                             ( 
                             k 
                             ) 
                           
                         
                         ⁢ 
                         
                           N 
                           
                             - 
                             1 
                           
                         
                         ⁢ 
                         
                           
                             
                               H 
                               ^ 
                             
                             ⁡ 
                             
                               ( 
                               k 
                               ) 
                             
                           
                           . 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   15 
                   ) 
                 
               
             
           
         
       
     
     The noise is present in the estimation of the coarse channel state, Ĥ(k), and is thus also present in the correlation matrix {circumflex over (R)}. The noise is removed from in the correlation matrix, R, e.g., according to: 
     
       
         
           
             
               R 
               
                 i 
                 , 
                 j 
               
             
             = 
             
               { 
               
                 
                   
                     
                       
                         
                           
                             R 
                             ^ 
                           
                           
                             i 
                             , 
                             j 
                           
                         
                         ⁡ 
                         
                           ( 
                           
                             H 
                             ^ 
                           
                           ) 
                         
                       
                       - 
                       
                         
                           2 
                           ⁢ 
                           
                             g 
                             ⁡ 
                             
                               ( 
                               
                                 
                                   
                                     σ 
                                     00 
                                   
                                   ⁢ 
                                   
                                     σ 
                                     11 
                                   
                                 
                                 - 
                                 
                                   ℜ 
                                   ⁢ 
                                   
                                     { 
                                     
                                       σ 
                                       01 
                                       2 
                                     
                                     } 
                                   
                                 
                               
                               ) 
                             
                           
                         
                         
                           
                             
                               σ 
                               00 
                             
                             ⁢ 
                             
                               σ 
                               11 
                             
                           
                           - 
                           
                             
                                
                               
                                 σ 
                                 01 
                               
                                
                             
                             2 
                           
                         
                       
                     
                   
                   
                     
                       
                         for 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         i 
                       
                       = 
                       j 
                     
                   
                 
                 
                   
                     
                       
                         
                           R 
                           ^ 
                         
                         
                           i 
                           , 
                           j 
                         
                       
                       ⁡ 
                       
                         ( 
                         
                           H 
                           ^ 
                         
                         ) 
                       
                     
                   
                   
                     
                       
                         for 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         i 
                       
                       ≠ 
                       j 
                     
                   
                 
               
             
           
         
       
     
     Herein, gϵ[0,1] is the ratio of residual noise variance in the estimation of the coarse channel state. 
     The objective for the second embodiment of the objective function is to maximize the Shannon Capacity,
 
log 2  [det( I+W   H    R W )].
 
     Since the logarithm is strictly monotonically increasing, maximizing the Shannon Capacity is equal to maximizing the first object function
 
 g ( W,R )=det( I+W   H   R W ).  (16)
 
     Herein, w is a matrix representation of one of the precoders in the codebook  116 . 
     Maximizing the first objective function in Eq. (16) according to the selecting step  304  yields a precoder 
     
       
         
           
             
               W 
               * 
             
             = 
             
               
                 
                   
                     arg 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     max 
                   
                   β 
                 
                 ⁡ 
                 
                   [ 
                   
                     g 
                     ⁡ 
                     
                       ( 
                       
                         W 
                         , 
                         R 
                       
                       ) 
                     
                   
                   ] 
                 
               
               = 
               
                 
                   
                     W 
                     * 
                   
                   ⁡ 
                   
                     ( 
                     
                       
                         H 
                         ^ 
                       
                       , 
                       N 
                     
                     ) 
                   
                 
                 . 
               
             
           
         
       
     
     The precoder W* defines a subset including a predefined number of precoders (e.g., 6 neighboring precoders) out of the codebook. For the determination  306 , the CQME  404  evaluates the second objective function with a higher temporal resolution compared to the evaluation of the first objective function for the selection  304 . E.g., the CQME  404  evaluates the second objective function for each Resource Block  200  and for each of the selected precoders in the subset. 
     As has become apparent from above exemplary embodiments, at least some embodiments allow efficiently determining the precoder, e.g., even for a large codebook. The determined precoder can be indicated in a CSI report. The numerical effort for determining a precoder that is applicable for a radio transmission via a channel including two or more transmit antennas may be similar to the numerical effort necessary for determining a precoder for a radio transmission via a channel including one or two transmit antennas. 
     In the foregoing, principles, preferred embodiments and various modes of implementing the technique disclosed herein have exemplarily been described. However, the present invention should not be construed as being limited to the particular principles, embodiments and modes discussed above. Rather, it will be appreciated that variations and modifications may be made by a person skilled in the art without departing from the scope of the present invention as defined in the following claims.