Patent Publication Number: US-11387877-B2

Title: Device and method using adaptive codebook for dual beamforming feedback and wireless communication system including the same

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
     This application claims priority from U.S. Provisional Application No. 63/083,411 filed on Sep. 25, 2020 in the U.S. Patent and Trademark Office, and Korean Patent Application No. 10-2020-0174101 filed on Dec. 14, 2020 in the Korean Intellectual Property Office, the contents of which are herein incorporated by reference in their entireties. 
    
    
     BACKGROUND 
     1. Field 
     Methods, apparatuses and systems consistent with example embodiments relate to semiconductor integrated circuits, and more particularly to an adaptive codebook for dual beamforming feedback. 
     2. Related Art 
     In a wireless communication system, it is important to have a strong communication signal. In particular, the greatest possible signal-to-noise ratio (SNR) is desired at a receiver end. Similarly, for a wireless fidelity (Wi-Fi) system, an increased SNR at the receiving device increases the probability that frames are correctly received and reduces the amount of retransmissions necessary from the source. SNR at the receiver end may be increased by increasing transmit power, decreasing distance between the source and the receiver, and increasing antenna gain. 
     A signal can be transmitted based on a beamforming method in a wireless local area network (WLAN) system such as a Wi-Fi system. In a wireless communication system, a beamforming is a technique of a smart antenna, and is a technique for directing a beam to a corresponding terminal. A beamforming feedback should be preceded for a beamforming transmission, and various methods for providing efficient beamforming feedback have been researched. 
     SUMMARY 
     At least one example embodiment provides a beamformee device capable of adaptively utilizing a codebook while performing a dual beamforming feedback to reduce the overhead of beamforming feedback. 
     At least one example embodiment provides a wireless communication system including the beamformee device. 
     At least one example embodiment provides a beamforming feedback method performed by the beamformee device. 
     According to an aspect of an example embodiment, a beamformee device includes a processor configured to implement: a channel estimator configured to receive a null data packet (NDP) from a beamformer device through a channel, and to obtain a plurality of channel information associated with a plurality of subcarriers of the channel based on the NDP; a first beamforming matrix provider configured to provide a plurality of wideband beamforming matrices based on the plurality of channel information; a dimension reduction unit configured to generate a plurality of equivalent channel information corresponding to the plurality of channel information based on the plurality of wideband beamforming matrices; and a second beamforming matrix provider configured to provide a plurality of subcarrier beamforming matrices based on the plurality of equivalent channel information. The processor is further configured to feed the plurality of wideband beamforming matrices and the plurality of subcarrier beamforming matrices back to the beamformer device, and any one or any combination of the plurality of wideband beamforming matrices and the plurality of subcarrier beamforming matrices are selected from a plurality of codebooks that are stored in the beamformee device. 
     According to an aspect of an example embodiment, a wireless communication system includes: a beamformer device configured to transmit an NDP through a channel; and a beamformee device configured to receive the NDP through the channel, to estimate the channel based on the NDP, and to feed a result of estimating the channel back to the beamformer device. The beamformee device includes a processor configured to implement: a channel estimator configured to obtain a plurality of channel information associated with a plurality of subcarriers based on the NDP; a first beamforming matrix provider configured to provide a plurality of wideband beamforming matrices based on the plurality of channel information; a dimension reduction unit configured to generate a plurality of equivalent channel information corresponding to the plurality of channel information based on the plurality of wideband beamforming matrices; and a second beamforming matrix provider configured to provide a plurality of subcarrier beamforming matrices based on the plurality of equivalent channel information. The processor is further configured to feed the plurality of wideband beamforming matrices and the plurality of subcarrier beamforming matrices back to the beamformer device as the result of estimating the channel, and any one or any combination of the plurality of wideband beamforming matrices and the plurality of subcarrier beamforming matrices are selected from a plurality of codebooks that are stored in the beamformee device. 
     According to an aspect of an example embodiment, a beamformee device includes: a processor configured to implement: a channel estimator configured to receive a null data packet (NDP) from a beamformer device through a channel, and to obtain a plurality of channel information associated with a plurality of subcarriers of the channel based on the NDP; a feedback mode selector configured to select one of a first feedback mode, a second feedback mode or a third feedback mode as a selected feedback mode based on a characteristic of the channel; a first beamforming matrix provider configured to generate a plurality of wideband beamforming matrices by performing a singular value decomposition (SVD) and compressing the plurality of channel information based on the selected feedback mode being the second feedback mode, and to select and output one of a plurality of first codebooks as a selected first codebook based on the plurality of channel information as one of the plurality of wideband beamforming matrices based on the selected feedback mode being the first feedback mode or the third feedback mode; a dimension reduction unit configured to generate a plurality of equivalent channel information corresponding to the plurality of channel information based on the plurality of wideband beamforming matrices; and a second beamforming matrix provider configured to generate a plurality of subcarrier beamforming matrices by performing an SVD and compressing the plurality of equivalent channel information based on the selected feedback mode being the first feedback mode, and to select and output one of a plurality of second codebooks as a selected second codebook based on the plurality of equivalent channel information as one of the plurality of subcarrier beamforming matrices based on the selected feedback mode being the second feedback mode or the third feedback mode. The processor is further configured to feed the plurality of wideband beamforming matrices and the plurality of subcarrier beamforming matrices back to the beamformer device, and a codebook index corresponding to the selected first codebook and the selected second codebook is fed back to the beamformer device. 
     According to example embodiments, a beamforming feedback method includes: obtaining a plurality of channel information associated with a plurality of subcarriers of a channel based on an NDP that is received from a beamformer device through the channel; providing a plurality of wideband beamforming matrices based on the plurality of channel information; generating a plurality of equivalent channel information corresponding to the plurality of channel information based on the plurality of wideband beamforming matrices; providing a plurality of subcarrier beamforming matrices based on the plurality of equivalent channel information; and feeding the plurality of wideband beamforming matrices and the plurality of subcarrier beamforming matrices back to the beamformer device. Any one or any combination of the plurality of wideband beamforming matrices and the plurality of subcarrier beamforming matrices are selected from a plurality of codebooks that are stored in memory. 
     According to example embodiments, a beamformee device includes a processor configured to: estimate a channel based on an NDP that is received from a beamformer device through the channel, provide a plurality of wideband beamforming matrices and a plurality of subcarrier beamforming matrices as a result of estimating the channel, select any one or any combination of the plurality of wideband beamforming matrices and the plurality of subcarrier beamforming matrices from a plurality of codebooks that are stored in the beamformee device, generate a beamforming feedback report based on the plurality of wideband beamforming matrices and the plurality of subcarrier beamforming matrices, and feed the beamforming feedback report back to the beamformer device. The beamforming feedback report includes media access control (MAC) header information, category information, multiple-input multiple-output (MIMO) control information, codebook index information, compressed beamforming report (CBR) information, and feedback mode information. 
     In the beamformee device, the wireless communication system and the beamforming feedback method according to example embodiments, the dual beamforming feedback may be used when the beamforming feedback is performed in the feedback mode, and the codebook utilization scheme may be applied to at least one of the plurality of wideband beamforming matrices and the plurality of subcarrier beamforming matrices. In addition, the adaptive codebook utilization scheme in which the feedback mode is selected and/or changed depending on the condition and/or environment of the channel may be implemented. Accordingly, the feedback overhead of the beamforming feedback may be efficiently reduced, and beamforming feedback may be performed with improved efficiency. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       The above and other aspects will be more apparent from the following description of example embodiments taken in conjunction with the accompanying drawings, in which: 
         FIG. 1  is a block diagram illustrating a beamformee device and a wireless communication system including the beamformee device according to example embodiments. 
         FIGS. 2 and 3  are diagrams for describing a dual beamforming feedback according to example embodiments. 
         FIGS. 4A, 4B, 4C, 4D and 5  are diagrams for describing a process of designing a codebook used in a beamformee device according to example embodiments. 
         FIG. 6  is a block diagram illustrating a beamformee device and a wireless communication system according to example embodiments. 
         FIG. 7  is a diagram illustrating a beamforming feedback report generated by a beamformee device according to example embodiments. 
         FIG. 8  is a diagram illustrating performance of a beamformee device according to example embodiments. 
         FIG. 9  is a block diagram illustrating a beamformee device and a wireless communication system according to example embodiments. 
         FIG. 10  is a diagram illustrating a beamforming feedback report generated by a beamformee device according to example embodiments. 
         FIGS. 11 and 12  are block diagrams illustrating beamformee devices and wireless communication systems according to example embodiments. 
         FIG. 13  is a block diagram illustrating a feedback mode selector included in a beamformee device according to example embodiments. 
         FIG. 14  is a diagram illustrating a look-up table included in a feedback mode selector according to example embodiments. 
         FIG. 15  is a block diagram illustrating a beamformee device and a wireless communication system according to example embodiments. 
         FIGS. 16, 17 and 18  are flowcharts illustrating beamforming feedback methods according to example embodiments. 
         FIG. 19  is a block diagram illustrating an electronic device in a network environment according to example embodiments. 
     
    
    
     DETAILED DESCRIPTION 
     Example embodiments will be described with reference to the accompanying drawings. The present disclosure may, however, be embodied in many different forms and should not be construed as limited to the example embodiments set forth herein. Like reference numerals refer to like elements throughout this application. Expressions such as “at least one of,” when preceding a list of elements, modify the entire list of elements and do not modify the individual elements of the list. For example, the expression, “at least one of a, b, and c,” should be understood as including only a, only b, only c, both a and b, both a and c, both b and c, or all of a, b, and c. 
       FIG. 1  is a block diagram illustrating a beamformee device and a wireless communication system including the beamformee device according to example embodiments. 
     Referring to  FIG. 1 , a wireless communication system  10  includes a beamformer device  100  and a beamformee device  200 . 
     In some example embodiments, the wireless communication system  10  may be a wireless communication system that is implemented or formed based on a wireless local area network (WLAN). For example, the wireless communication system  10  may be a wireless communication system that is implemented or formed based on Wi-Fi. For example, the WLAN system may be implemented based on the Institute of Electrical and Electronics Engineers (IEEE) 802.11ac standard or the IEEE 802.11ax standard, or may be implemented based on the IEEE 802.11be standard that is a next generation standard. 
     In the WLAN system, a beamforming transmission may be performed between the beamformer device  100  and the beamformee device  200 . The beamforming transmission is a technology in which a beam of an antenna is directed and limited to a specific terminal in a multi-antenna orthogonal frequency division modulation (OFDM) system. The beamforming transmission may be divided into a technique of forming a transmission beam to increase a reception data rate of a single user (e.g., a single-user beamforming) and a technique of forming a transmission beam for mutual interference cancellation during simultaneous transmission between multiple users (e.g., a multi-user beamforming). The beamformee device  200  may decode a packet transmitted for channel measurement from the beamformer device  100 , may compress channel information based on a technique specified in the WLAN standard, and may feed the compressed channel information back to the beamformer device  100 . The above-described operation of feeding back the channel information may be referred to as a beamforming feedback, and the beamforming feedback may be performed in a feedback mode. 
     According to example embodiments, the beamforming feedback that is performed in the feedback mode may be a dual beamforming feedback or a two-step beamforming feedback. The dual beamforming feedback may be used to reduce the feedback overhead. For example, one matrix may be used with wideband, and another matrix may be used with a sub-band (e.g., with a subcarrier). The dual beamforming feedback will be described in detail with reference to  FIGS. 2 and 3 . 
     Hereinafter, operations of the wireless communication system  10  and the beamformee device  200  according to example embodiments will be described based on operations for reducing the feedback overhead when the dual beamforming feedback is performed in the feedback mode. However, example embodiments are not limited thereto, and the wireless communication system  10  may perform a normal beamforming transmission based on the channel information obtained by the dual beamforming feedback in a normal operation mode after the feedback mode. 
     The beamformer device  100  transmits a null data packet (NDP) used for channel measurement. The beamformer device  100  may be referred to as a transmitter or an access point (AP). The NDP may be referred to as a sounding packet. 
     The beamformer device  100  may include a plurality of antennas (e.g., transmission antennas)  101 . The beamformer device  100  may transmit or output the NDP using the plurality of antennas  101 . For example, the number of the plurality of antennas  101  may be N t , where N t  is a natural number greater than or equal to two. 
     The beamformee device  200  receives the NDP from the beamformer device  100  through the channel, estimates the channel based on the NDP, and feeds back a result of estimating the channel to the beamformer device  100 . The beamformee device  200  performs the dual beamforming feedback to reduce the feedback overhead of the beamforming feedback, and thus feeds back only a plurality of wideband beamforming matrices WBM and a plurality of subcarrier beamforming matrices SBM, instead of the entire matrix for the channel. For example, as will be described later, the plurality of wideband beamforming matrices WBM and the plurality of subcarrier beamforming matrices SBM may be fed back in the form of a beamforming feedback report. The beamformee device  200  may be referred to as a receiver or a station (STA). 
     The beamformee device  200  includes a channel estimator  210 , a first beamforming matrix provider  220 , a dimension reduction unit  230  and a second beamforming matrix provider  240 . The beamformee device  200  may further include a plurality of antennas (e.g., reception antennas)  201 . 
     The beamformee device  200  may receive the NDP from the beamformer device  100  through the channel using the plurality of antennas  201 . For example, the number of the plurality of antennas  201  may be N r , where N r  is a natural number greater than or equal to two. The channel (e.g., a wireless channel) may be formed between the plurality of antennas  101  of the beamformer device  100  and the plurality of antennas  201  of the beamformee device  200 . 
     The channel estimator  210  obtains a plurality of channel information ECI associated with or related to a plurality of subcarriers by estimating the channel based on the NDP. For example, one channel information may be estimated and obtained for one subcarrier. For example, each channel information may be obtained in the form of a channel matrix representing a frequency response. 
     The first beamforming matrix provider  220  provides the plurality of wideband beamforming matrices WBM based on the plurality of channel information ECI. The first beamforming matrix provider  220  may be referred to as a wideband beamforming matrix (BM) provider. 
     The dimension reduction unit  230  generates a plurality of equivalent channel information EECI corresponding to the plurality of channel information ECI based on the plurality of wideband beamforming matrices WBM. For example, each equivalent channel information may be generated in the form of an equivalent channel matrix whose size is reduced from that of corresponding channel information (e.g., that of corresponding channel matrix). 
     The second beamforming matrix provider  240  provides a plurality of subcarrier beamforming matrices SBM based on the plurality of equivalent channel information EECI. The second beamforming matrix provider  240  may be referred to as a subcarrier beamforming matrix provider. 
     According to example embodiments, at least one of the plurality of wideband beamforming matrices WBM and the plurality of subcarrier beamforming matrices SBM is selected from among a plurality of codebooks that are pre-designed or are designed in advance. In other words, a codebook utilization scheme may be applied or employed to at least one of the plurality of wideband beamforming matrices WBM and the plurality of subcarrier beamforming matrices SBM. 
     For example, the first beamforming matrix provider  220  may include a plurality of first codebooks  221  that are pre-designed and correspond to the plurality of wideband beamforming matrices WBM. The plurality of first codebooks  221  may be stored in a memory. The second beamforming matrix provider  240  may include a plurality of second codebooks  241  that are pre-designed and correspond to the plurality of subcarrier beamforming matrices SBM. The plurality of second codebooks  241  may be stored in a memory. The plurality of first codebooks  221  may be referred to as a plurality of wideband codebooks (WCB), and the plurality of second codebooks  241  may be referred to as a plurality of subcarrier codebooks (SCB). 
     In some example embodiments, as will be described with reference to  FIG. 6 , when the codebook utilization scheme is applied to the plurality of wideband beamforming matrices WBM, the first beamforming matrix provider  220  may select one of the plurality of first codebooks  221  based on the plurality of channel information ECI, and may output the selected first codebook as one of the plurality of wideband beamforming matrices WBM. 
     In other example embodiments, as will be described with reference to  FIG. 9 , when the codebook utilization scheme is applied to the plurality of subcarrier beamforming matrices SBM, the second beamforming matrix provider  240  may select one of the plurality of second codebooks  241  based on the plurality of equivalent channel information EECI, and may output the selected second codebook as one of the plurality of subcarrier beamforming matrices SBM. 
     In still other example embodiments, as will be described with reference to  FIG. 11 , the codebook utilization technique may be applied to both the plurality of wideband beamforming matrices WBM and the plurality of subcarrier beamforming matrices SBM. 
     In some example embodiments, as will be described with reference to  FIG. 12 , the beamformee device  200  may operate in one of a first feedback mode, a second feedback mode and a third feedback mode based on a characteristic of the channel. The first feedback mode may be a feedback mode in which the codebook utilization scheme is applied to the plurality of wideband beamforming matrices WBM. The second feedback mode may be a feedback mode in which the codebook utilization scheme is applied to the plurality of subcarrier beamforming matrices SBM. The third feedback mode may be a feedback mode in which the codebook utilization scheme is applied to both the plurality of wideband beamforming matrices WBM and the plurality of subcarrier beamforming matrices SBM. In other words, an adaptive codebook utilization scheme in which a feedback mode is selected depending on a condition and/or an environment of the channel may be implemented. 
     In some example embodiments, as will be described with reference to  FIGS. 4A, 4B, 4C, 4D, 5 and 15 , the plurality of codebooks (e.g., at least one of the first codebooks  221  and the second codebooks  241 ) may be designed and stored in the beamformee device  200  at the time of manufacture and/or at an initial operation time. 
     In the WLAN system, the beamforming may increase antenna gain while maintaining omnidirectional coverage, which results in increased SNR and more stable, higher bandwidth WLAN connections by focusing transmissions to the recipient. These benefits may be achieved by transmitting a signal via an array of antennas and slightly altering the phase of the signal at each antenna in the array. For example, the IEEE 802.11ac standard (e.g., a wireless networking standard in the 802.11 family, developed by the IEEE Standards Association) may provide a protocol for calibrating an array of antennas to direct a signal to any point covered under omnidirectional propagation. The beamformer device  100  may be a device that augments the phase shift of antennas to produce a gain in a desired direction. The beamformee device  200  may be a device that is a target of the beamformer device  100 . The beamformee device  200  may participate in the establishment of the beam, but may not augment timings of its antennas. 
     For the beamforming transmission in the WLAN system, the beamformer device  100  should first obtain the channel information (or channel state information) from the beamformee device  200 . For obtaining the channel information, the beamformer device  100  may transmit the NDP (or sounding packet) through the plurality of antennas  101 . The beamformee device  200  may receive the NDP through the plurality of antennas  201  and may obtain the channel information based on the NDP. The received signals may represent a result of undergoing the channel, and a relationship between the transmitted signals and the received signals may satisfy Equation 1.
 
 Y [ k ]= H [ k ] S [ k ]+ N [ k ]  [Equation 1]
 
     In Equation 1, Y[k] denotes a vector of the received signals, S[k] denotes a vector of the transmitted signals (e.g., a vector of sounding signals), H[k] denotes a channel matrix representing the frequency response, N[k] denotes a noise vector, and k denotes a subcarrier index. Each of Y[k] and N[k] may be an N r *1 vector, S[k] may be an N t *1 vector, and H[k] may be an N r *N t  matrix. N t  denotes the number of the plurality of antennas  101  of the beamformer device  100 , and N r  denotes the number of the plurality of antennas  201  of the beamformee device  200 . 
     The beamformee device  200  may estimate channel information H for each subcarrier using the channel estimator  210 , and the channel information H may correspond to one of the plurality of channel information ECI. In addition, when the beamformee device  200  operates in the feedback mode, the beamformee device  200  may perform a singular value decomposition (SVD) on the channel information H based on Equation 2. 
     
       
         
           
             
               
                 
                   
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     In Equation 2, each of U[k] and V[k] denotes an unitary matrix, Σ[k] is a diagonal matrix including channel singular values, and V[k]H denotes a conjugate transpose matrix of V[k]. 
     In the WLAN system, to reduce the feedback overhead, the beamformee device  200  may not directly feed back V[k]. Instead, the beamformee device  200  may first obtain Q[k] by multiplying a diagonal matrix 
               D   ~     ⁡     (       e       -   j     ⁢           ⁢     ϕ       N   t     ,   1           ,   …   ⁢           ,     e       -   j     ⁢           ⁢     ϕ       N     t   -   1       ,     N     t   -   1                 )           
for performing a common-phase shift by V[k] based on Equation 3, where e −jϕ     i,j    is a phase value corresponding to an element in an i-th row and a j-th column of V[k].
 
 Q [ k ]= V [ k ] {tilde over (D)} [Equation 3]
 
     Subsequently, the beamformee device  200  may compress and feed angle values φ and ψ corresponding back to each element of Q[k] using Givens rotation. In other words, the beamformee device  200  may quantize the angle values φ and ψ obtained based on Equation 4, and may feed the quantized angle values back to the beamformer device  100 . 
     
       
         
           
             
               
                 
                   
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                               I 
                               
                                 Nt 
                                 - 
                                 1 
                               
                             
                           
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   [ 
                   
                     Equation 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     6 
                   
                   ] 
                 
               
             
           
         
       
     
     In other words, the plurality of wideband beamforming matrices WBM and the plurality of subcarrier beamforming matrices SBM that are fed back from the beamformee device  200  to the beamformer device  100  may include the angle values φ and ψ that are obtained based on Equation 4. 
       FIGS. 2 and 3  are diagrams for describing a dual beamforming feedback. 
       FIG. 2  illustrates a plurality of subcarriers that are used for the signal transmission between a beamformer device and a beamformee device in a wireless communication system. In  FIG. 2 , one arrow may represent one subcarrier SC.  FIG. 3  illustrates an example of a wireless communication system that performs the dual beamforming feedback, e.g., a dimension reduction scheme for reducing a size of a beamforming matrix. 
     Referring to  FIGS. 2 and 3 , a wireless communication system  20  may include a beamformer device  105  and a beamformee device  205 . The beamformer device  105  may include a plurality of antennas  101 , a wideband precoder unit  120  and a subcarrier precoder unit  140 . The beamformee device  205  may include a plurality of antennas  201 , a channel estimator  210 , a wideband precoder calculator  250 , a dimension reduction unit  230  and a subcarrier precoder calculator  260 . The wideband precoder calculator  250  may include a covariance calculator  252 , an SVD unit  254  and a compression unit  256 . The subcarrier precoder calculator  260  may include an SVD unit  262  and a compression unit  264 . The plurality of antennas  101  and  201 , the channel estimator  210  and the dimension reduction unit  230  may be substantially the same as the plurality of antennas  101  and  201 , the channel estimator  210  and the dimension reduction unit  230  in  FIG. 1 , respectively. 
     The beamformee device  205  may receive a NDP or sounding packet, and may obtain channel information (e.g., a channel matrix) H[k] by estimating a channel using the channel estimator  210 , where k=0, 1, . . . , N fft −1. Here, N fft  may represent the number of the plurality of subcarriers SC in one OFDM symbol. 
     The beamformee device  205  may divide the plurality of subcarriers SC into M groups SCG, where M is a natural number greater than or equal to two, and thus each of the groups SCG may include Ng (=N fft /M) subcarriers. The beamformee device  205  may form a wideband beam (e.g., wideband beamforming information or a wideband beamforming matrix) having the same value for all of the groups SCG. The beamformee device  205  may obtain the wideband beam based on Equation 7 using the covariance calculator  252 . 
     
       
         
           
             
               
                 
                   
                     
                       Cov 
                       ⁡ 
                       
                         [ 
                         m 
                         ] 
                       
                     
                     = 
                     
                       
                         1 
                         Ng 
                       
                       ⁢ 
                       
                         
                           ∑ 
                           
                             k 
                             = 
                             0 
                           
                           
                             Ng 
                             - 
                             1 
                           
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           
                             
                               ( 
                               
                                 
                                   H 
                                   
                                     
                                       Ng 
                                       * 
                                       m 
                                     
                                     + 
                                     k 
                                   
                                 
                                 ⁡ 
                                 
                                   [ 
                                   
                                     
                                       Ng 
                                       * 
                                       m 
                                     
                                     + 
                                     k 
                                   
                                   ] 
                                 
                               
                               ) 
                             
                             H 
                           
                           ⁢ 
                           
                             ( 
                             
                               
                                 H 
                                 
                                   
                                     Ng 
                                     * 
                                     m 
                                   
                                   + 
                                   k 
                                 
                               
                               ⁡ 
                               
                                 [ 
                                 
                                   
                                     Ng 
                                     * 
                                     m 
                                   
                                   + 
                                   k 
                                 
                                 ] 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                   , 
                   
                     m 
                     = 
                     0 
                   
                   , 
                   1 
                   , 
                   … 
                   ⁢ 
                   
                       
                   
                   , 
                   
                     M 
                     - 
                     1 
                   
                 
               
               
                 
                   [ 
                   
                     Equation 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     7 
                   
                   ] 
                 
               
             
           
         
       
     
     The beamformee device  205  may perform an SVD on the covariance matrix (e.g., Cov[m]) obtained by Equation 7 using the SVD unit  254 , and may obtain a unitary matrix W WB [m] of the wideband beam as in Equation 8.
 
 W   WB [ m ]= SVD (Cov[ m ])  [Equation 8]
 
     The beamformee device  205  may take first to K-th columns among a plurality of columns of the unitary matrix W WB [m], may perform compression on the unitary matrix W WB [m] based on the Givens rotation according to Equations 3 to 6 using the compression unit  256 , and may obtain angle values φ and ψ associated with the wideband beam. Here, K may be a design parameter and may be an adjustable parameter. 
     Next, the beamformee device  205  may form a per-subcarrier beam (e.g., subcarrier beamforming information or a subcarrier beamforming matrix) for each subcarrier SC. 
     First, the beamformee device  205  may multiply the channel matrix H[k] by the unitary matrix W WB [m] of the wideband beam using the dimension reduction unit  230 , and may obtain equivalent channel information (e.g., an equivalent channel matrix) as in Equation 9.
 
 {tilde over (H)} [ k ]= H [ k ] W   WB [ m ]  [Equation 9]
 
     The equivalent channel matrix {tilde over (H)}[k] may be reduced in size compared to the channel matrix H[k]. For example, H[k] may have a size of N r *N t , and W WB [m] may have a size of N t *K. Thus, {tilde over (H)}[k] may have a size of Nr*K, and the size of the matrix may be thereby reduced (e.g., dimension reduction). For example, when N t =16, N r =2 and K=8, {tilde over (H)}[k] having a size of 2*8 may be generated based on H[k] having a size of 2*16. As 2*8 is less than 2*16, the size of the matrix may be reduced. 
     The beamformee device  205  may perform an SVD on the equivalent channel matrix {tilde over (H)}[k] obtained by Equation 9 using the SVD unit  262 , and may obtain a unitary matrix W SC [k] of the subcarrier beam as in Equation 10.
 
 W   SC [ k ]= SVD ( {tilde over (H)} [ k ])  [Equation 10]
 
     The beamformee device  205  may perform compression on the unitary matrix W SC [k] based on the Givens rotation according to Equations 3 to 6 using the compression unit  264 , and may obtain angle values φ and ψ associated with the subcarrier beam. 
     When feeding back the channel information for the beamforming transmission, the beamformee device  205  may feed back the angle values φ and ψ for M wideband beams obtained by the compression unit  256 , and may feed back the angle values φ and ψ for Nm subcarrier beams obtained by the compression unit  264 . In  FIG. 3 , an arrow from the wideband precoder calculator  250  to the wideband precoder unit  120  may represent the angle values φ and ψ for the M wideband beams, and an arrow from the subcarrier precoder calculator  260  to the subcarrier precoder unit  140  may represent the angle values φ and ψ for the N fft  subcarrier beams. 
     The wideband precoder unit  120  may restore the M wideband beams based on the angle values φ and ψ fed back from the wideband precoder calculator  250 , the subcarrier precoder unit  140  may restore the N fft  subcarrier beams based on the angle values φ and ψ fed back from the subcarrier precoder calculator  260 , and the channel information may be obtained based on the restored M wideband beams and the restored N fft  subcarrier beams. 
     As described above, the wideband beamforming matrix may be determined by the beamformee device  205  from the covariance matrix of the wideband channel. The wideband bandwidth may depend on a channel condition and a MIMO mode. For the single-user beamforming, the wideband bandwidth may be about 80 MHz. For the multi-user beamforming, less than about 80 MHz may be required (e.g., about 5 or 10 MHz per user). After calculating the covariance matrix, the wideband beamforming matrix may be obtained by a method used in compressed beamforming, e.g., an SVD. The design parameter K may be determined by the beamformee device  205  or the beamformer device  105 . For example, in a case of the single-user beamforming, K may be determined by the beamformee device  205 . In a case of the multi-user beamforming or trigger-based feedback, K may be determined by the beamformer device  105 . The tradeoff between complexity and performance affects the choice of K. Various factors include timing, delay spread, side of bandwidth, spatial correlation between antennas, etc. 
     According to example embodiments, the wideband precoder calculator  250  and the subcarrier precoder calculator  260  may be replaced with the first beamforming matrix provider  220  and the second beamforming matrix provider  240 , respectively. In other words, instead of calculating and feeding back both the plurality of wideband beamforming matrices WBM and the plurality of subcarrier beamforming matrices SBM based on the operations described with reference to  FIGS. 2 and 3 , the codebook utilization scheme may be applied to at least one of the plurality of wideband beamforming matrices WBM and the plurality of subcarrier beamforming matrices SBM. Accordingly, the amount of computation and the amount of data to be fed back may be reduced, and thus the feedback overhead may be reduced. 
       FIGS. 4A, 4B, 4C, 4D and 5  are diagrams for describing a process of designing a codebook used in a beamformee device according to example embodiments. 
     Referring to  FIGS. 4A, 4B, 4C, 4D and 5 , when generating the plurality of codebooks, a Lloyd algorithm may be performed using a discrete-time Fourier transform (DFT) codebook as an initial value. 
     For example, an initial DFT codebook w(n) may be obtained based on Equation 11. 
     
       
         
           
             
               
                 
                   
                     w 
                     ⁡ 
                     
                       ( 
                       n 
                       ) 
                     
                   
                   = 
                   
                     
                       
                         1 
                         
                           
                             N 
                             t 
                           
                         
                       
                       ⁡ 
                       
                         [ 
                         
                           1 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             exp 
                             ⁡ 
                             
                               ( 
                               
                                 
                                   - 
                                   j 
                                 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 π 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 1 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 n 
                               
                               ) 
                             
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             exp 
                             ⁡ 
                             
                               ( 
                               
                                 
                                   - 
                                   j 
                                 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 π 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 n 
                               
                               ) 
                             
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           … 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             exp 
                             ⁡ 
                             
                               ( 
                               
                                 
                                   - 
                                   j 
                                 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 π 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   ( 
                                   
                                     
                                       N 
                                       t 
                                     
                                     - 
                                     1 
                                   
                                   ) 
                                 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 n 
                               
                               ) 
                             
                           
                         
                         ] 
                       
                     
                     T 
                   
                 
               
               
                 
                   [ 
                   
                     Equation 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     11 
                   
                   ] 
                 
               
             
           
         
       
     
     In Equation 11, 
               n   =   0     ,     1   NB     ,     2   NB     ,   …   ⁢           ,       NB   -   1     NB     ,         
N denotes the number of codebooks, and B denotes the number of columns of each codebook. A j-th vector of an i-th codebook may be determined by
 
     
       
         
           
             n 
             = 
             
               
                 
                   
                     N 
                     * 
                     j 
                   
                   + 
                   i 
                 
                 NB 
               
               . 
             
           
         
       
     
     According to example embodiments, N DFT codebooks W n   (0)  (n=0, 1, . . . , N−1) may be generated using Equation 11, and the Lloyd algorithm may be performed using the N DFT codebooks as initial values. 
     In some example embodiments, the Lloyd algorithm may be performed based on the following order (1), (2), (3) and (4). First, the Euclidean distance function ED(⋅) for performing the Lloyd algorithm may be defined as in Equation 12.
 
 ED ( A,B )=∥ AD−B ∥ where  D =diag( e   −jϕ     1     ,e   −jϕ     2     , . . . ,e   −jϕ     N   ), e   jϕ     N     =b   H   a/|b   H   a|   [Equation 12]
 
     In Equation 12, diag denotes a function for extracting diagonal elements of a matrix, capital letters denote the matrix, and small letters denote a column of the matrix. 
     (1) Based on the IEEE D-channel model, multiple (e.g., about 10000) samples of the channel matrix H may be generated. V∈  may be generated by performing an SVD for each sample of the channel matrix H. 
     (2) Next, for a given codebook W n   (i-1)  (n=0, 1, . . . , N−1) (an initial codebook is a DFT codebook, i=1), V samples may be divided into N regions based on the Euclidean distance as in Equation 13.
 
 X   n   ={V∈     |ED ( V,W   n   (i-1) )&lt; ED ( V,W   m   (i-1) ),  n≠m, n,m= 0,1, . . . , N− 1}  [Equation 13]
 
     (3) Next, for X n  (n=0, 1, . . . , N−1), W at which an average Euclidean distance is the minimum may be calculated as in Equation 13 and may be set as the i-th codebook.
 
 W   n   (i) =arg min W   E{ED ( V,W )} V∈X   n   ,W   H   W=I   [Equation 14]
 
     (4) Next, the processes of (2) and (3) may be repeatedly performed until the convergence condition J (i)  is satisfied. In Equation 15, V→W n   (i)  may be a set of V, which is ED(V, W n   (i) )&lt;ED(V, W m   (i) ), n≠m.
 
If  J   (i) &gt;∈, then  i=i+ 1
 
Else, Stop
 
where  J   (i) =Σ n=0   N-1 Σ V→W     n       (i)     ED ( V,W   n   (i) )  [Equation 15]
 
       FIGS. 4A, 4B, 4C and 4D  illustrate a process of detecting a centroid or a geometric center using the Lloyd algorithm. In  FIGS. 4A, 4B, 4C and 4D , a portion marked with a dot (⋅) may represent a centroid of each region, and a portion marked with a cross (+) may represent a codebook corresponding to each region. 
       FIGS. 4A, 4B, 4C and 4D  illustrate results of iteration of the above-described algorithm once, twice, three times and fifteen times, respectively. As the algorithm is repeated, five regions R 11 , R 21 , R 31 , R 41  and R 51  in  FIG. 4A  may be changed to five regions R 12 , R 22 , R 32 , R 42  and R 52  in  FIG. 4B , five regions R 13 , R 23 , R 33 , R 43  and R 53  in  FIG. 4C , and five regions R 14 , R 24 , R 34 , R 44  and R 54  in  FIG. 4D . In addition, it can be seen that the centroid of each region and the corresponding codebook are matched and the convergence condition of Equation 15 is satisfied. 
       FIG. 5  illustrates a process of convergence under the convergence condition of Equation 15 as the Lloyd algorithm is repeatedly performed. It can be seen that the corresponding codebook is converged to a desired optimal codebook when the Lloyd algorithm is repeated about ten or more times. 
       FIG. 6  is a block diagram illustrating an example of a beamformee device and a wireless communication system of  FIG. 1 . Repeated description will be omitted. 
     Referring to  FIG. 6 , a wireless communication system  10   a  includes a beamformer device  100  and a beamformee device  200   a . The beamformer device  100  may include a plurality of antennas  101 . The beamformee device  200   a  may include a plurality of antennas  201 , a channel estimator  210 , a first beamforming matrix provider  220   a , a dimension reduction unit  230  and a second beamforming matrix provider  240   a.    
       FIG. 6  illustrates an example where the codebook utilization scheme is applied only to the plurality of wideband beamforming matrices WBM, and thus a configuration of the first beamforming matrix provider  220   a  in the example of  FIG. 6  may be changed. The beamformer device  100 , the plurality of antennas  101  and  201 , the channel estimator  210  and the dimension reduction unit  230  may be substantially the same as those described with reference to  FIGS. 1 and 3 . 
     The first beamforming matrix provider  220   a  may include a first storage unit (STG)  222  and a first selector (SEL)  224 . 
     The first storage unit  222  may store a plurality of first codebooks WCB_1, WCB_2, . . . , WCB_N. For example, the plurality of first codebooks WCB_1 to WCB_N may be pre-designed and pre-stored based on the Lloyd algorithm described with reference to  FIGS. 4A, 4B, 4C, 4D and 5 . For example, the first storage unit  222  may include a volatile memory, such as a dynamic random access memory (DRAM), a static random access memory (SRAM), a mobile DRAM, or the like, and/or a nonvolatile memory, such as an erasable programmable read-only memory (EPROM), an electrically erasable programmable read-only memory (EEPROM), a flash memory, a phase change random access memory (PRAM), a resistance random access memory (RRAM), a nano floating gate memory (NFGM), a polymer random access memory (PoRAM), a magnetic random access memory (MRAM), a ferroelectric random access memory (FRAM), or the like. 
     The first selector  224  may select the plurality of wideband beamforming matrices WBM based on the plurality of first codebooks WCB_1 to WCB_N. For example, the first selector  224  may select a codebook that is capable of obtaining the largest beamforming gain. For example, based on the plurality of channel information ECI, the first selector  224  may select one of the plurality of first codebooks WCB_1 to WCB_N as one of the plurality of wideband beamforming matrices WBM, and the selected first codebook may be used to maximize the power of the channel. 
     In some example embodiments, the first selector  224  may select one of the plurality of wideband beamforming matrices WBM based on Equation 16. 
     
       
         
           
             
               
                 
                   
                     W 
                     w 
                   
                   = 
                   
                     arg 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         max 
                         
                           
                             W 
                             i 
                           
                           ∈ 
                           ℂ 
                         
                       
                       ⁢ 
                       
                         
                           1 
                           
                             N 
                             g 
                           
                         
                         ⁢ 
                         
                           
                             ∑ 
                             
                               k 
                               = 
                               0 
                             
                             
                               
                                 N 
                                 g 
                               
                               - 
                               1 
                             
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                              
                             
                               
                                 
                                   H 
                                   H 
                                 
                                 ⁡ 
                                 
                                   [ 
                                   k 
                                   ] 
                                 
                               
                               ⁢ 
                               
                                 H 
                                 ⁡ 
                                 
                                   [ 
                                   k 
                                   ] 
                                 
                               
                               ⁢ 
                               
                                 W 
                                 i 
                               
                             
                              
                           
                         
                       
                     
                   
                 
               
               
                 
                   [ 
                   
                     Equation 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     16 
                   
                   ] 
                 
               
             
           
         
       
     
     In Equation 16, W w  denotes the selected wideband beamforming matrix, N g  denotes the number of subcarriers corresponding to one wideband beam, H[k] denotes a channel matrix corresponding to one of the plurality of channel information ECI, H H [k] denotes a conjugate transpose matrix of H[k], W, denotes an i-th codebook of a codebook set  , e.g., an i-th codebook among the plurality of first codebooks WCB_1 to WCB_N, and k denotes an index of the plurality of subcarriers. A function argmax represents arguments of max, e.g., a function that returns a value that maximizes the function. 
     The second beamforming matrix provider  240   a  may include an SVD unit  242  and a compression unit (COMP)  244 . 
     The SVD unit  242  may perform an SVD on the plurality of equivalent channel information EECI. The compression unit  244  may compress an output of the SVD unit  242 . The SVD unit  242  and the compression unit  244  may be substantially the same as the SVD unit  262  and the compression unit  264  in  FIG. 3 , respectively. In other words, when the codebook utilization scheme is applied only to the plurality of wideband beamforming matrices WBM, the second beamforming matrix provider  240   a  may be implemented substantially the same as the subcarrier precoder calculator  260 . 
     The plurality of wideband beamforming matrices WBM provided from the first beamforming matrix provider  220   a  and the plurality of subcarrier beamforming matrices SBM provided from the second beamforming matrix provider  240   a  may be fed back to the beamformer device  100 . For example, a beamforming feedback report BFR generated based on the plurality of wideband beamforming matrices WBM and the plurality of subcarrier beamforming matrices SBM may be fed back to the beamformer device  100 . 
       FIG. 7  is a diagram illustrating an example of a beamforming feedback report generated by a beamformee device of  FIG. 6 . 
     Referring to  FIG. 7 , a beamforming feedback report BFR may include media access control (MAC) header information  271 , category information  272 , multiple-input multiple-output (MIMO) control information  273 , codebook index information  274   a , compressed beamforming report (CBR) information  275   a , additional common phase information  276   a  and feedback mode information  277   a . The beamforming feedback report BFR of  FIG. 7  may further include the codebook index information  274   a , the additional common phase information  276   a  and the feedback mode information  277   a.    
     The codebook index information  274   a  may correspond to the plurality of wideband beamforming matrices WBM, and the CBR information  275   a  may correspond to the plurality of subcarrier beamforming matrices SBM. The feedback mode information  277   a  may represent the first feedback mode to which the codebook utilization scheme is applied to the plurality of wideband beamforming matrices WBM. 
     The beamforming feedback report BFR of  FIG. 7  may not include all information of the plurality of wideband beamforming matrices WBM and the plurality of subcarrier beamforming matrices SBM. Instead, the beamforming feedback report BFR of  FIG. 7  may include only some or partial information of the plurality of wideband beamforming matrices WBM and the plurality of subcarrier beamforming matrices SBM. For example, for the plurality of subcarrier beamforming matrices SBM, the CBR information  275   a  generated by compressing the angle values φ and ψ of the N fft  subcarrier beams, which are described with reference to  FIGS. 2 and 3 , may be fed back. For the plurality of wideband beamforming matrices WBM to which the codebook utilization scheme is applied, the codebook index information  274   a  representing codebooks corresponding to the M wideband beams may be fed back. Accordingly, the amount of computation and the amount of data to be fed back may be reduced, and thus the feedback overhead may be reduced. 
     In addition, the additional common phase information  276   a  may include common phase information φ common  constituting 
               D   ~     ⁡     (       e       -   j     ⁢           ⁢     ϕ       N   t     ,   1           ,   …   ⁢           ,     e       -   j     ⁢           ⁢     ϕ       N     t   -   1       ,     N     t   -   1                 )           
in Equation 3. For the accuracy of computation, the common phase information φ common  for the subcarrier beams may be additionally fed back together with the angle values φ and ψ.
 
       FIG. 8  is a diagram illustrating performance of a beamformee device of  FIG. 6 . 
     Referring to  FIG. 8 , CASE1 represents a beamformee device in which the dual beamforming feedback is not performed, CASE2 represents a beamformee device in which only the dual beamforming feedback is performed as illustrated in  FIGS. 2 and 3 , and CASE3 represents the beamformee device  200   a  in which the dual beamforming feedback is performed and the codebook utilization scheme is applied to the plurality of wideband beamforming matrices WBM as illustrated in  FIG. 6 . It can be seen that the feedback overhead is reduced and the amount of data transmission increases in the beamformee device  200   a  according to example embodiments. 
       FIG. 9  is a block diagram illustrating another example of a beamformee device and a wireless communication system of  FIG. 1 . Repeated description will be omitted. 
     Referring to  FIG. 9 , a wireless communication system  10   b  includes a beamformer device  100  and a beamformee device  200   b . The beamformer device  100  may include a plurality of antennas  101 . The beamformee device  200   b  may include a plurality of antennas  201 , a channel estimator  210 , a first beamforming matrix provider  220   b , a dimension reduction unit  230  and a second beamforming matrix provider  240   b.    
       FIG. 9  illustrates an example where the codebook utilization scheme is applied only to the plurality of subcarrier beamforming matrices SBM, and thus a configuration of the second beamforming matrix provider  240   b  in the example of  FIG. 9  may be changed. The beamformer device  100 , the plurality of antennas  101  and  201 , the channel estimator  210  and the dimension reduction unit  230  may be substantially the same as those described with reference to  FIGS. 1 and 3 . 
     The first beamforming matrix provider  220   b  may include a covariance calculator (COV)  225 , an SVD unit  226  and a compression unit  228 . 
     The covariance calculator  225  may calculate a covariance matrix based on the plurality of channel information ECI. The SVD unit  226  may perform an SVD on an output of the covariance calculator  225 . The compression unit  228  may compress an output of the SVD unit  226 . The covariance calculator  225 , the SVD unit  226  and the compression unit  228  may be substantially the same as the covariance calculator  252 , the SVD unit  254  and the compression unit  256  in  FIG. 3 , respectively. In other words, when the codebook utilization scheme is applied only to the plurality of subcarrier beamforming matrices SBM, the first beamforming matrix provider  220   b  may be implemented substantially the same as the wideband precoder calculator  250 . 
     The second beamforming matrix provider  240   b  may include a second storage unit  246  and a second selector  248 . 
     The second storage unit  246  may store a plurality of second codebooks SCB_1, SCB_2, . . . , SCB_N. For example, the plurality of second codebooks SCB_1 to SCB_N may be pre-designed and pre-stored based on the Lloyd algorithm described with reference to  FIGS. 4A, 4B, 4C, 4D and 5 . For example, the second storage unit  246  may include a volatile memory and/or a nonvolatile memory. 
     The second selector  248  may select the plurality of subcarrier beamforming matrices SBM based on the plurality of second codebooks SCB_1 to SCB_N. For example, the second selector  248  may select a codebook that is capable of obtaining the largest beamforming gain. For example, based on the plurality of equivalent channel information EECI, the second selector  248  may select one of the plurality of second codebooks SCB_1 to SCB_N as one of the plurality of subcarrier beamforming matrices SBM, and the selected second codebook may be closest to the plurality of equivalent channel information EECI. 
     In some example embodiments, the second selector  248  may select one of the plurality of subcarrier beamforming matrices SBM based on Equation 17. 
     
       
         
           
             
               
                 
                   
                     W 
                     s 
                   
                   = 
                   
                     arg 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         min 
                         
                           
                             W 
                             i 
                           
                           ∈ 
                           ℂ 
                         
                       
                       ⁢ 
                       
                         
                           1 
                           
                             N 
                             g 
                           
                         
                         ⁢ 
                         
                           
                             ∑ 
                             
                               k 
                               = 
                               0 
                             
                             
                               
                                 N 
                                 g 
                               
                               - 
                               1 
                             
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             ED 
                             ⁡ 
                             
                               ( 
                               
                                 
                                   
                                     V 
                                     sc 
                                   
                                   ⁡ 
                                   
                                     [ 
                                     k 
                                     ] 
                                   
                                 
                                 , 
                                 
                                   W 
                                   i 
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   [ 
                   
                     Equation 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     17 
                   
                   ] 
                 
               
             
           
         
       
     
     In Equation 17, W s  denotes the selected subcarrier beamforming matrix, N g  denotes the number of subcarriers corresponding to one wideband beam, ED(⋅) denotes the Euclidean distance function, V sc [k] denotes a unitary matrix for a channel matrix corresponding to one of the plurality of equivalent channel information EECI, W i  denotes an i-th codebook of a codebook set  , e.g., an i-th codebook among the plurality of second codebooks SCB_1 to SCB_N, and k denotes an index of the plurality of subcarriers. A function argmin represents arguments of min, e.g., a function that returns the value that minimizes the function. 
     The plurality of wideband beamforming matrices WBM provided from the first beamforming matrix provider  220   b  and the plurality of subcarrier beamforming matrices SBM provided from the second beamforming matrix provider  240   b  may be fed back to the beamformer device  100 . For example, a beamforming feedback report BFR generated based on the plurality of wideband beamforming matrices WBM and the plurality of subcarrier beamforming matrices SBM may be fed back to the beamformer device  100 . 
       FIG. 10  is a diagram illustrating an example of a beamforming feedback report generated by a beamformee device of  FIG. 9 . 
     Referring to  FIG. 10 , a beamforming feedback report BFR may include MAC header information  271 , category information  272 , MIMO control information  273 , codebook index information  274   b , CBR information  275   b  and feedback mode information  277   b . The beamforming feedback report BFR of  FIG. 10  may further include the codebook index information  274   b  and the feedback mode information  277   b.    
     The codebook index information  274   b  may correspond to the plurality of subcarrier beamforming matrices SBM, and the CBR information  275   b  may correspond to the plurality of wideband beamforming matrices WBM. The feedback mode information  277   b  may represent the second feedback mode to which the codebook utilization scheme is applied to the plurality of subcarrier beamforming matrices SBM. 
     The beamforming feedback report BFR of  FIG. 10  may include only partial information of the plurality of wideband beamforming matrices WBM and the plurality of subcarrier beamforming matrices SBM. For example, for the plurality of wideband beamforming matrices WBM, the CBR information  275   b  generated by compressing the angle values φ and ψ for the M wideband beams, which are described with reference to  FIGS. 2 and 3 , may be fed back. For the plurality of subcarrier beamforming matrices SBM to which the codebook utilization scheme is applied, the codebook index information  274   b  representing codebooks corresponding to the Nm subcarrier beams may be fed back. Accordingly, the amount of computation and the amount of data to be fed back may be reduced, and thus the feedback overhead may be reduced. 
       FIGS. 11 and 12  are block diagrams illustrating other examples of a beamformee device and a wireless communication system of  FIG. 1 . Repeated description will be omitted. 
     Referring to  FIG. 11 , a wireless communication system  10   c  includes a beamformer device  100  and a beamformee device  200   c . The beamformer device  100  may include a plurality of antennas  101 . The beamformee device  200   c  may include a plurality of antennas  201 , a channel estimator  210 , a first beamforming matrix provider  220   c , a dimension reduction unit  230  and a second beamforming matrix provider  240   c.    
       FIG. 11  illustrates an example where the codebook utilization scheme is applied to both the plurality of wideband beamforming matrices WBM and the plurality of subcarrier beamforming matrices SBM, and thus configurations of the first and second beamforming matrix providers  220   c  and  240   c  may be changed. The beamformer device  100 , the plurality of antennas  101  and  201 , the channel estimator  210  and the dimension reduction unit  230  may be substantially the same as those described with reference to  FIGS. 1 and 3 . 
     The first beamforming matrix provider  220   c  may be substantially the same as the first beamforming matrix provider  220   a  in  FIG. 6 , and may include a first storage unit  222  and a first selector  224 . The second beamforming matrix provider  240   c  may be substantially the same as the second beamforming matrix provider  240   b  in  FIG. 9 , and may include a second storage unit  246  and a second selector  248 . 
     A beamforming feedback report that is generated based on the plurality of wideband beamforming matrices WBM provided from the first beamforming matrix provider  220   c  and the plurality of subcarrier beamforming matrices SBM provided from the second beamforming matrix provider  240   c  may include MAC header information, category information, MIMO control information, first codebook index information corresponding to the codebook index information  274   a  in  FIG. 7 , second codebook index information corresponding to the codebook index information  274   b  in  FIG. 10 , and feedback mode information. 
     Referring to  FIG. 12 , a wireless communication system  10   d  includes a beamformer device  100  and a beamformee device  200   d . The beamformer device  100  may include a plurality of antennas  101 . The beamformee device  200   d  may include a plurality of antennas  201 , a channel estimator  210 , a first beamforming matrix provider  220   d , a dimension reduction unit  230 , a second beamforming matrix provider  240   d  and a feedback mode selector  280 . 
       FIG. 12  illustrates an example where the codebook utilization scheme is selectively and/or adaptively applied to at least one of the plurality of wideband beamforming matrices WBM and the plurality of subcarrier beamforming matrices SBM, and thus the beamformee device  200   d  may further include the feedback mode selector  280  and configurations of the first and second beamforming matrix providers  220   d  and  240   d  may be changed. The beamformer device  100 , the plurality of antennas  101  and  201 , the channel estimator  210  and the dimension reduction unit  230  may be substantially the same as those described with reference to  FIGS. 1 and 3 . 
     The beamformee device  200   d  may operate in one of the first feedback mode, the second feedback mode and the third feedback mode. In the first feedback mode, one of the plurality of first codebooks  221  may be selected, and the selected first codebook may be output as one of the plurality of wideband beamforming matrices WBM. In the second feedback mode, one of the plurality of second codebooks  241  may be selected, and the selected second codebook may be output as one of the plurality of subcarrier beamforming matrices SBM. In the third feedback mode, one of the plurality of first codebooks  221  may be selected, one of the plurality of second codebooks  241  may be selected, and the selected first codebook and the selected second codebook may be output as one of the plurality of wideband beamforming matrices WBM and as one of the plurality of subcarrier beamforming matrices SBM, respectively. 
     For selecting and/or changing the feedback mode, the first beamforming matrix provider  220   d  may include a first processing unit (PU 1 )  2202  and a second processing unit (PU 2 )  2204 , and the second beamforming matrix provider  240   d  may include a third processing unit (PU 3 )  2402  and a fourth processing unit (PU 4 )  2404 . The first processing unit  2202  and the second processing unit  2204  may have configurations substantially the same as those of the first beamforming matrix provider  220   a  in  FIG. 6  and the first beamforming matrix provider  220   b  in  FIG. 9 , respectively. The third processing unit  2402  and the fourth processing unit  2404  may have configurations substantially the same as those of the second beamforming matrix provider  240   b  in  FIG. 9  and the second beamforming matrix provider  240   a  in  FIG. 6 , respectively. 
     The feedback mode selector  280  may select one of the first feedback mode, the second feedback mode and the third feedback mode based on the characteristics of the channel, and may generate a mode signal MD representing the selected feedback mode. For example, the selected feedback mode may be a feedback mode that is predicted to have the largest amount of data transmission. 
     In some example embodiments, one of the first and second processing units  2202  and  2204  may be enabled or activated, and one of the third and fourth processing units  2402  and  2404  may be enabled or activated, depending on the feedback mode (e.g., based on the mode signal MD). For example, the first processing unit  2202  may be enabled in the first and third feedback modes, and the second processing unit  2204  may be enabled in the second feedback mode. In addition, the third processing unit  2402  may be enabled in the second and third feedback modes, and the fourth processing unit  2404  may be enabled in the first feedback mode. 
       FIG. 13  is a block diagram illustrating an example of a feedback mode selector included in a beamformee device of  FIG. 12 . 
     Referring to  FIG. 13 , a feedback mode selector  280   a  may include an estimator  282  and a selector  284 . 
     The estimator  282  may estimate the characteristic of the channel, and may generate a characteristic signal CCHA representing the estimated characteristic of the channel. For example, the characteristic of the channel may include at least one of a signal-to-noise ratio (SNR), a modulation and coding scheme (MCS), a physical data unit (PDU) length, a bandwidth (BW), the number of spatial stream (Nss), and an encoding scheme including a binary convolution coding (BCC) and a low density parity check (LDPC). 
     The selector  284  may select one of the feedback modes based on the characteristic of the channel (e.g., based on the characteristic signal CCHA), and may generate the mode signal MD. 
     In some example embodiments, the selector  284  may include a look-up table (LUT)  286  that is preset or predetermined. For example, the look-up table  286  may represent a relationship between the estimated characteristic of the channel and the feedback mode. 
       FIG. 14  is a diagram illustrating an example of a look-up table that may be included in the feedback mode selector of  FIG. 13 . 
     Referring to  FIG. 14 , an example of the look-up table  286  in which the SNR and the MCS are used as the characteristics of the channel is illustrated. 
     In an example of  FIG. 14 , the look-up table  286  may include M different MCS cases for each SNR and data rates in the first, second and third feedback modes for each MCS. 
     For example, for a first SNR SNR_1 and first through M-th MCSs MCS_1, MCS_2, . . . , MCS_M, the look-up table  286  may include data rates RATE_1_1_1, RATE_1_2_1, . . . , RATE_1_M_1 in the first feedback mode, data rates RATE_1_1_2, RATE_1_2_2, . . . , RATE_1_M_2 in the second feedback mode, and data rates RATE_1_1_3, RATE_1_2_3, . . . , RATE_1_M_3 in the third feedback mode. Similarly, for a second SNR SNR_2 and the first through M-th MCSs MCS_1, MCS_2, . . . , MCS_M, the look-up table  286  may include data rates RATE_2_1_1, RATE_2_2_1, . . . , RATE_2_M_1 in the first feedback mode, data rates RATE_2_1_2, RATE_2_2_2, . . . , RATE_2_M_2 in the second feedback mode, and data rates RATE_2_1_3, RATE_2_2_3, . . . , RATE_2_M_3 in the third feedback mode. For a K-th SNR SNR_K and the first through M-th MCSs MCS_1, MCS_2, . . . , MCS_M, the look-up table  286  may include data rates RATE_K_1_1, RATE_K_2_1, . . . , RATE_K_M_1 in the first feedback mode, data rates RATE_K_1_2, RATE_K_2_2, . . . , RATE_K_M_2 in the second feedback mode, and data rates RATE_K_1_3, RATE_K_2_3, . . . , RATE_K_M_3 in the third feedback mode. 
     When a SNR estimated by the estimator  282  is greater than SNR_m and less than or equal to SNR (m+1) among the SNRs SNR_1 to SNR_K, the SNR_m may be selected, and a corresponding MCS may be selected from among the MCSs MCS_1 to MCS_M. Among the data rates corresponding to the selected SNR and the selected MCS, a feedback mode corresponding to the largest data rate (e.g., the amount of data transmission amount is predicted to be the largest) may be selected. 
       FIG. 15  is a block diagram illustrating another example of a beamformee device and a wireless communication system of  FIG. 1 . Repeated description will be omitted. 
     Referring to  FIG. 15 , a wireless communication system  10   e  includes a beamformer device  100  and a beamformee device  200   e . The beamformer device  100  may include a plurality of antennas  101 . The beamformee device  200   e  may include a plurality of antennas  201 , a channel estimator  210 , a first beamforming matrix provider  220 , a dimension reduction unit  230 , a second beamforming matrix provider  240  and a codebook generator  290 . 
       FIG. 15  illustrates an example where the beamformee device  200   e  further includes the codebook generator  290 . The beamformer device  100 , the plurality of antennas  101  and  201 , the channel estimator  210 , the first beamforming matrix provider  220 , the dimension reduction unit  230  and the second beamforming matrix provider  240  may be substantially the same as those described with reference to  FIGS. 1 and 3 . 
     The codebook generator  290  may design the plurality of first codebooks  221  and/or the plurality of second codebooks  241 , and may generate codebook information CBI representing the designed codebooks. For example, the codebook generator  290  may design the codebooks based on the Lloyd algorithm described with reference to  FIGS. 4A, 4B, 4C, 4D and 5 . The codebook information CBI may be provided to the first and second beamforming matrix providers  220  and  240 . 
     The first and second beamforming matrix providers  220  and  240  may be implemented as described with reference to  FIGS. 6, 9, 11 and 12 . 
     In some example embodiments, at least a part of the elements or components included in the beamformee device according to example embodiments may be implemented as hardware. For example, at least a part of the elements or components included in the beamformee device may be included in a computer-based electronic system. In other example embodiments, at least a part of the elements or components included in the beamformee device according to example embodiments may be implemented as instruction codes or program routines (e.g., a software program). For example, the instruction codes or the program routines may be executed by a computer-based electronic system, and may be stored in any storage device located inside or outside the computer-based electronic system. 
     In the beamformee device and the wireless communication system according to example embodiments, the dual beamforming feedback may be used when the beamforming feedback is performed in the feedback mode, and the codebook utilization scheme may be applied to at least one of the plurality of wideband beamforming matrices WBM and the plurality of subcarrier beamforming matrices SBM. In addition, the adaptive codebook utilization scheme in which the feedback mode is selected and/or changed depending on the condition and/or environment of the channel may be implemented. Accordingly, the feedback overhead of the beamforming feedback may be efficiently reduced, and beamforming feedback may be performed with improved efficiency. 
       FIGS. 16, 17 and 18  are flowcharts illustrating a beamforming feedback method according to example embodiments. 
     Referring to  FIG. 16 , a beamforming feedback method according to example embodiments may be performed by a wireless communication system including a beamformer device and a beamformee device. For example, the beamforming feedback method according to example embodiments may be performed by the beamformee device. The wireless communication system and the beamformee device may be implemented as described with reference to  FIGS. 1 through 15 . 
     In the beamforming feedback method according to example embodiments, a plurality of channel information associated with a plurality of subcarriers are obtained by estimating a channel based on a NDP that is received from the beamformer device through the channel (operation S 100 ). For example, operation S 100  may be performed by the channel estimator  210  in  FIG. 1 . 
     A plurality of wideband beamforming matrices are provided based on the plurality of channel information (operation S 200 ). For example, operation S 200  may be performed by the first beamforming matrix provider  220  in  FIG. 1 . 
     In some example embodiments, when performing operation S 200 , one of a plurality of first codebooks that are pre-designed may be selected based on the plurality of channel information, and the selected first codebook may be output as one of the plurality of wideband beamforming matrices. For example, based on the plurality of channel information, one of the plurality of first codebooks may be selected as one of the plurality of wideband beamforming matrices, and the selected first codebook may be used to maximize the power of the channel. For example, one of the plurality of wideband beamforming matrices may be selected based on Equation 16 described with reference to  FIG. 6 . 
     A plurality of equivalent channel information corresponding to the plurality of channel information are generated based on the plurality of wideband beamforming matrices (operation S 300 ). For example, operation S 300  may be performed by the dimension reduction unit  230  in  FIG. 1 . 
     A plurality of subcarrier beamforming matrices are provided based on the plurality of equivalent channel information (operation S 400 ). For example, operation S 400  may be performed by the second beamforming matrix provider  240  in  FIG. 1 . 
     In some example embodiments, when performing operation S 400 , one of a plurality of second codebooks that are pre-designed may be selected based on the plurality of equivalent channel information, and the selected second codebook may be output as one of the plurality of subcarrier beamforming matrices. For example, based on the plurality of equivalent channel information, one of the plurality of second codebooks may be selected as one of the plurality of subcarrier beamforming matrices, and the selected second codebook may be selected because it is closest to the plurality of equivalent channel information. For example, one of the plurality of subcarrier beamforming matrices may be selected based on Equation 17 described with reference to  FIG. 9 . 
     The plurality of wideband beamforming matrices and the plurality of subcarrier beamforming matrices are fed back to the beamformer device. For example, the plurality of wideband beamforming matrices and the plurality of subcarrier beamforming matrices may be fed back to the beamformer device in the form of a beamforming feedback report. 
     At least one of the plurality of wideband beamforming matrices and the plurality of subcarrier beamforming matrices are selected from a plurality of codebooks that are pre-designed. For example, as described with reference to  FIGS. 6, 9 and 11 , the codebook utilization scheme may be applied only to the plurality of wideband beamforming matrices WBM, the codebook utilization scheme may be applied only to the plurality of subcarrier beamforming matrices SBM, or the codebook utilization scheme may be applied to both the plurality of wideband beamforming matrices WBM and the plurality of subcarrier beamforming matrices SBM. 
     Referring to  FIG. 17 , in a beamforming feedback method according to example embodiments. Repeated description will be omitted. 
     A feedback mode of the beamformee device is selected based on a characteristic of the channel (operation S 500 ). For example, as described with reference to  FIG. 12 , one of the first feedback mode, the second feedback mode and the third feedback mode may be selected based on the characteristic of the channel, and the selected feedback mode may be a feedback mode that is predicted to have the largest amount of data transmission. For example, operation S 500  may be performed by the feedback mode selector  280  in  FIG. 12 . 
     Operations S 100 , S 200 , S 300  and S 400  subsequent to operation S 500  may be substantially the same as those described with reference to  FIG. 16 . 
     Referring to  FIG. 18 , in a beamforming feedback method according to example embodiments. Repeated description will be omitted. 
     The plurality of codebooks are designed (operation S 600 ). For example, as described with reference to  FIGS. 4A, 4B, 4C, 4D and 5 , the plurality of codebooks may be designed based on the Lloyd algorithm. For example, operation S 600  may be performed once at the time of manufacture and/or at the initial operation time of the beamforming device, and may be omitted thereafter. For example, operation S 600  may be performed by the codebook generator  290  in  FIG. 15 . 
     Operations S 100 , S 200 , S 300  and S 400  subsequent to operation S 600  may be substantially the same as those described with reference to  FIG. 16 . 
     In some example embodiments, operation S 600  may be additionally performed in the example of  FIG. 17 . 
     As will be appreciated by those skilled in the art, example embodiments may be implemented as a system, method, computer program product, and/or a computer program product embodied in one or more computer readable medium(s) having computer readable program code embodied thereon. The computer readable program code may be provided to a processor of a general purpose computer, special purpose computer, or other programmable data processing apparatus. The computer readable medium may be a computer readable signal medium or a computer readable storage medium. The computer readable storage medium may be any tangible medium that can contain or store a program for use by or in connection with an instruction execution system, apparatus, or device. For example, the computer readable medium may be a non-transitory computer readable medium. 
       FIG. 19  is a block diagram illustrating an electronic device in a network environment according to example embodiments. 
     Referring to  FIG. 19 , an electronic device  301  in a network environment  300  may communicate with an electronic device  302  via a first network  398  (e.g., a short-range wireless communication network), or an electronic device  304  or a server  308  via a second network  399  (e.g., a long-range wireless communication network). In some example embodiments, the electronic device  301  may communicate with the electronic device  304  via the server  308 . In some example embodiments, the electronic device  301  may include a processor  320 , memory  330 , an input device  350 , a sound output device  355 , a display device  360 , an audio module  370 , a sensor module  376 , an interface  377 , a haptic module  379 , a camera module  380 , a power management module  388 , a battery  389 , a communication module  390 , a subscriber identification module (SIM)  396 , and/or an antenna module  397 . 
     The processor  320  may execute, for example, software (e.g., a program  340 ) to control at least one other component (e.g., a hardware or software component) of the electronic device  301  coupled with the processor  320 , and may perform various data processing or computation. In some example embodiments, the processor  320  may include a main processor  321  (e.g., a central processing unit (CPU) or an application processor (AP)), and an auxiliary processor  323  (e.g., a graphics processing unit (GPU), an image signal processor (ISP), a sensor hub processor, or a communication processor (CP)) that is operable independently from, or in conjunction with, the main processor  321 . 
     The memory  330  may store various data used by at least one component (e.g., the processor  320  or the sensor module  376 ) of the electronic device  301 . The various data may include, for example, software (e.g., the program  340 ) and input data or output data for a command related thereto. The memory  330  may include a volatile memory  332  or a nonvolatile memory  334 . 
     The program  340  may be stored in the memory  330  as software, and may include, for example, an operating system (OS)  342 , middleware  344 , and/or an application  346 . 
     The input device  350  may receive a command or data to be used by another component (e.g., the processor  320 ) of the electronic device  301 , from the outside (e.g., a user) of the electronic device  301 . The sound output device  355  may output sound signals to the outside of the electronic device  301 . The display device  360  may visually provide information to the outside (e.g., to a user) of the electronic device  301 . 
     The audio module  370  may convert a sound into an electrical signal and vice versa. The sensor module  376  may detect an operational state (e.g., power or temperature) of the electronic device  301  or an environmental state (e.g., a state of a user) external to the electronic device  301 , and then generate an electrical signal or data value corresponding to the detected state. The interface  377  may support one or more specified protocols to be used to couple the electronic device  301  with the external electronic device (e.g., the electronic device  302 ) directly (e.g., wired) or wirelessly. 
     A connecting terminal  378  may include a connector via which the electronic device  301  may be physically connected with the external electronic device (e.g., the electronic device  302 ). The haptic module  379  may convert an electrical signal into a mechanical stimulus (e.g., a vibration or a movement) or electrical stimulus which may be recognized by a user via his tactile sensation or kinesthetic sensation. The camera module  380  may capture a still image or moving images. 
     The power management module  388  may manage power supplied to other components of the electronic device  301 . In some example embodiments, the power management module  388  may be implemented as at least part of, for example, a power management integrated circuit (PMIC). The battery  389  may supply power to at least one component of the electronic device  301 . 
     The communication module  390  may support establishing a direct (e.g., wired) communication channel or a wireless communication channel between the electronic device  301  and the external electronic device (e.g., the electronic device  302 , the electronic device  304 , or the server  308 ) and performing communication via the established communication channel. In some example embodiments, the communication module  390  may include a wireless communication module  392  (e.g., a cellular communication module, a short-range wireless communication module, or a global navigation satellite system (GNSS) communication module) or a wired communication module  394  (e.g., a local area network (LAN) communication module or a power line communication (PLC) module). 
     In some example embodiments, the wireless communication module  392  included in the communication module  390  may include the beamformer device and the beamformee device according to example embodiments, and may be implemented to perform the beamforming feedback method according to example embodiments. For example, the wireless communication module  392  included in the electronic device  301  may include the beamformee device (e.g., the beamformee device  200  in  FIG. 1 ) according to example embodiments, a wireless communication module included in the electronic device  304  may include the beamformer device (e.g., the beamformer device  100  in  FIG. 1 ), and the second network  399  formed between the electronic devices  301  and  304  may correspond to the channel between the beamformee device and the beamformer device. The beamformee device included in the electronic device  301  may communicate with the beamformer device included in the electronic device  304 , and may operate based on the codebook utilization scheme and/or the adaptive codebook utilization scheme according to example embodiments while performing the dual beamforming feedback. Similarly, the beamformee device included in the electronic device  304  may communicate with the beamformer device included in the electronic device  301 , and may operate based on the codebook utilization scheme and/or the adaptive codebook utilization scheme according to example embodiments while performing the dual beamforming feedback. 
     The antenna module  397  may transmit or receive a signal or power to or from the outside (e.g., the external electronic device) of the electronic device  301 . 
     Example embodiments may be applied to various communication devices and systems that perform the beamforming and various electronic devices and systems that include the communication devices and systems. For example, example embodiments may be applied to systems such as a personal computer (PC), a workstation, a mobile phone, a smart phone, a tablet computer, a laptop computer, a personal digital assistant (PDA), a portable multimedia player (PMP), a digital camera, a portable game console, a music player, a camcorder, a video player, a navigation device, a wearable device, an internet of things (IoT) device, an internet of everything (IoE) device, an e-book reader, a virtual reality (VR) device, an augmented reality (AR) device, a robotic device, a drone, etc. 
     The foregoing is illustrative of example embodiments and is not to be construed as limiting thereof. Although some example embodiments have been described, those skilled in the art will readily appreciate that many modifications are possible without materially departing from the novel teachings and advantages of the example embodiments. Accordingly, all such modifications are intended to be included within the scope of the example embodiments as defined in the claims. Therefore, it is to be understood that the foregoing is illustrative of various example embodiments and is not to be construed as limited to the specific example embodiments disclosed, and that modifications to the disclosed example embodiments, as well as other example embodiments, are intended to be included within the scope of the appended claims.