Abstract:
A method and system for processing communication signals is provided and may comprise, minimizing a cost function in a MIMO pre-coding system by choosing a smallest scalar cost from a plurality of scalar costs, wherein each one of the scalar costs is generated from one or more sums and one or more products of elements of a product matrix, and wherein the product matrix is generated from one of the plurality of unitary matrices, a matrix comprising the channel estimate, and their respective Hermitian Transposes. The codebook is selected based on the minimized cost function. The channel estimate is quantized onto the selected codebook.

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS/INCORPORATION BY REFERENCE 
     This is a continuation of U.S. application Ser. No. 12/910,485 filed on Oct. 22, 2010, which in turn makes reference to, claims priority to and claims the benefit of U.S. application Ser. No. 11/767,158, filed on Jun. 22, 2007, which application makes reference to, claims priority to, and claims the benefit of U.S. Provisional Application Ser. No. 60/884,132, filed on Jan. 9, 2007. 
     This application also makes reference to:
     U.S. Application Ser. No. 60/884,118, filed on Jan. 9, 2007;   U.S. Application Ser. No. 60/884,133, filed on Jan. 9, 2007;   U.S. Application Ser. No. 60/884,113, filed on Jan. 9, 2007;   U.S. application Ser. No. 11/767,108, filed on Jun. 22, 2007;   U.S. application Ser. No. 11/767,123, filed on Jun. 22, 2007; and   U.S. application Ser. No. 11/767,071, filed on Jun. 22, 2007;   

     Each of the above referenced applications is hereby incorporated herein by reference in its entirety. 
    
    
     FIELD OF THE INVENTION 
     Certain embodiments of the invention relate to signal processing for communication systems. More specifically, certain embodiments of the invention relate to a method and system for codebook design of MIMO pre-coders with finite rate channel state information feedback. 
     BACKGROUND OF THE INVENTION 
     Mobile communications have changed the way people communicate and mobile phones have been transformed from a luxury item to an essential part of every day life. The use of mobile phones is today dictated by social situations, rather than hampered by location or technology. While voice connections fulfill the basic need to communicate, and mobile voice connections continue to filter even further into the fabric of every day life, the mobile Internet is the next step in the mobile communication revolution. The mobile Internet is poised to become a common source of everyday information, and easy, versatile mobile access to this data will be taken for granted. 
     Third generation (3G) cellular networks have been specifically designed to fulfill these future demands of the mobile Internet. As these services grow in popularity and usage, factors such as cost efficient optimization of network capacity and quality of service (QoS) will become even more essential to cellular operators than it is today. These factors may be achieved with careful network planning and operation, improvements in transmission methods, and advances in receiver techniques. To this end, carriers need technologies that will allow them to increase downlink throughput and, in turn, offer advanced QoS capabilities and speeds that rival those delivered by cable modem and/or DSL service providers. 
     In order to meet these demands, communication systems using multiple antennas at both the transmitter and the receiver have recently received increased attention due to their promise of providing significant capacity increase in a wireless fading environment. These multi-antenna configurations, also known as smart antenna techniques, may be utilized to mitigate the negative effects of multipath and/or signal interference on signal reception. It is anticipated that smart antenna techniques may be increasingly utilized both in connection with the deployment of base station infrastructure and mobile subscriber units in cellular systems to address the increasing capacity demands being placed on those systems. These demands arise, in part, from a shift underway from current voice-based services to next-generation wireless multimedia services that provide voice, video, and data communication. 
     The utilization of multiple transmit and/or receive antennas is designed to introduce a diversity gain and to raise the degrees of freedom to suppress interference generated within the signal reception process. Diversity gains improve system performance by increasing received signal-to-noise ratio and stabilizing the transmission link. On the other hand, more degrees of freedom allow multiple simultaneous transmissions by providing more robustness against signal interference, and/or by permitting greater frequency reuse for higher capacity. In communication systems that incorporate multi-antenna receivers, a set of M receive antennas may be utilized to null the effect of (M−1) interferers, for example. Accordingly, N signals may be simultaneously transmitted in the same bandwidth using N transmit antennas, with the transmitted signal then being separated into N respective signals by way of a set of N antennas deployed at the receiver. Systems that utilize multiple transmit and receive antennas may be referred to as multiple-input multiple-output (MIMO) systems. One attractive aspect of multi-antenna systems, in particular MIMO systems, is the significant increase in system capacity that may be achieved by utilizing these transmission configurations. For a fixed overall transmitted power and bandwidth, the capacity offered by a MIMO configuration may scale with the increased signal-to-noise ratio (SNR). For example, in the case of fading multipath channels, a MIMO configuration may increase system capacity by nearly M additional bits/cycle for each 3-dB increase in SNR. 
     The widespread deployment of multi-antenna systems in wireless communications has been limited by the increased cost that results from increased size, complexity, and power consumption. This poses problems for wireless system designs and applications. As a result, some work on multiple antenna systems may be focused on systems that support single user point-to-point links, other work may focus on multiuser scenarios. Communication systems that employ multiple antennas may greatly improve the system capacity. To obtain significant performance gains using MIMO technology, it may however be desirable to supply information on the channel to the transmitter. Such channel data is called channel state information (CSI). In many wireless systems, the uplink and the downlink operate in frequency division duplex (FDD) mode, that is, the uplink and the downlink use different frequencies. When this is the case, channel measurements of the uplink may not be applicable to the downlink and vice versa. In these instances, the channel may be measured only by a signal receiver and channel state information may be fed back to the transmitter. In the case of a large number of antennas, feeding back channel state information on the uplink channel may involve data transfers of large amounts of data. Since bandwidth on the uplink channel may be limited, it may be undesirable to transfer large amounts of channel information data on the uplink channel. Furthermore, CSI-related processing may impose a significant computational burden on a wireless communications device. 
     Further limitations and disadvantages of conventional and traditional approaches will become apparent to one of skill in the art, through comparison of such systems with some aspects of the present invention as set forth in the remainder of the present application with reference to the drawings. 
     BRIEF SUMMARY OF THE INVENTION 
     A method and/or system for an efficient channel quantization method for MIMO pre-coding systems, substantially as shown in and/or described in connection with at least one of the figures, as set forth more completely in the claims. 
     These and other advantages, aspects and novel features of the present invention, as well as details of an illustrated embodiment thereof, will be more fully understood from the following description and drawings. 
    
    
     
       BRIEF DESCRIPTION OF SEVERAL VIEWS OF THE DRAWINGS 
         FIG. 1A  is a diagram illustrating exemplary cellular multipath communication between a base station and a mobile computing terminal, in connection with an embodiment of the invention. 
         FIG. 1B  is a diagram illustrating an exemplary MIMO communication system, in accordance with an embodiment of the invention. 
         FIG. 2  is a block diagram illustrating an exemplary MIMO pre-coding transceiver chain model, in accordance with an embodiment of the invention. 
         FIG. 3  is a block diagram of an exemplary MIMO pre-coding system with finite rate channel state information feedback, in accordance with an embodiment of the invention. 
         FIG. 4  is a performance plot of an exemplary 2×2 MIMO system with an MMSE receiver and finite-rate feedback, in accordance with an embodiment of the invention. 
         FIG. 5  is a flow chart illustrating an exemplary codebook selection and channel quantization algorithm, in accordance with an embodiment of the invention. 
     
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
     Certain embodiments of the invention may be found in a method and system for an efficient channel quantization method for MIMO pre-coding systems. Aspects of a method and system for an efficient channel quantization method for MIMO pre-coding systems may include quantizing channel state information in a MIMO pre-coding system onto a codebook, which comprises one or more unitary matrices, using a cost function; selecting a quantization function that enables the quantizing based on a number of receive antennas and a number of transmit antennas in the MIMO pre-coding system; and selecting the cost function from a set of cost functions based on the number of receive antennas and the number of transmit antennas in the MIMO pre-coding system. The channel state information may be a matrix Ĥ. The set of cost functions may comprise function f(A) which may be defined by the following relationships: 
               f   ⁡     (   A   )       =         ∑     i   =   1     N     ⁢       ∑       j   =   1     ,     j   ≠   i       N     ⁢              a     i   ,   j            2     ⁢           ⁢   and   ⁢     /     ⁢   or   ⁢           ⁢     f   ⁡     (   A   )             =         ∑     i   =   1     N     ⁢       ∑       j   =   1     ,     j   ≠   i       N     ⁢            a     i   ,   j            2         -       ∑     i   =   1     N     ⁢     a     i   ,   i                   
where A is a matrix of size N by N and a ij  is element (i,j) of matrix A. The quantization function may be defined by the following relationship: V q =arg min {circumflex over (V)}     i     εC ƒ({circumflex over (V)} i   H ·Ĥ H ·Ĥ·{circumflex over (V)} i ), where V q , {circumflex over (V)} i εC, and Ĥ may be matrices of appropriate dimensions and C may be the codebook. An index of an element of the codebook onto which the channel state information is quantized may be transmitted from a receiver to a transmitter in the MIMO pre-coding system. The number of transmit antennas and the number of receive antennas may be at least one each, in the MIMO pre-coding system. A vector signal at the transmitter of the MIMO pre-coding system may be linearly transformed with one of the unitary matrices. The set of codebooks may be updated dynamically or chosen statically.
 
       FIG. 1A  is a diagram illustrating exemplary cellular multipath communication between a base station and a mobile computing terminal, in connection with an embodiment of the invention. Referring to  FIG. 1A , there is shown a house  120 , a mobile terminal  122 , a factory  124 , a base station  126 , a car  128 , and communication paths  130 ,  132  and  134 . 
     The base station  126  and the mobile terminal  122  may comprise suitable logic, circuitry and/or code that may be enabled to generate and process MIMO communication signals. 
     Wireless communications between the base station  126  and the mobile terminal  122  may take place over a wireless channel. The wireless channel may comprise a plurality of communication paths, for example, the communication paths  130 ,  132  and  134 . The wireless channel may change dynamically as the mobile terminal  122  and/or the car  128  moves. In some cases, the mobile terminal  122  may be in line-of-sight (LOS) of the base station  126 . In other instances, there may not be a direct line-of-sight between the mobile terminal  122  and the base station  126  and the radio signals may travel as reflected communication paths between the communicating entities, as illustrated by the exemplary communication paths  130 ,  132  and  134 . The radio signals may be reflected by man-made structures like the house  120 , the factory  124  or the car  128 , or by natural obstacles like hills. Such a system may be referred to as a non-line-of-sight (NLOS) communications system. 
     A communication system may comprise both LOS and NLOS signal components. If a LOS signal component is present, it may be much stronger than NLOS signal components. In some communication systems, the NLOS signal components may create interference and reduce the receiver performance. This may be referred to as multipath interference. The communication paths  130 ,  132  and  134 , for example, may arrive with different delays at the mobile terminal  122 . The communication paths  130 ,  132  and  134  may also be differently attenuated. In the downlink, for example, the received signal at the mobile terminal  122  may be the sum of differently attenuated communication paths  130 ,  132  and/or  134  that may not be synchronized and that may dynamically change. Such a channel may be referred to as a fading multipath channel. A fading multipath channel may introduce interference but it may also introduce diversity and degrees of freedom into the wireless channel. Communication systems with multiple antennas at the base station and/or at the mobile terminal, for example MIMO systems, may be particularly suited to exploit the characteristics of wireless channels and may extract large performance gains from a fading multipath channel that may result in significantly increased performance with respect to a communication system with a single antenna at the base station  126  and at the mobile terminal  122 , in particular for NLOS communication systems. 
       FIG. 1B  is a diagram illustrating an exemplary MIMO communication system, in accordance with an embodiment of the invention. Referring to  FIG. 1B , there is shown a MIMO transmitter  102  and a MIMO receiver  104 , and antennas  106 ,  108 ,  110 ,  112 ,  114  and  116 . There is also shown a wireless channel comprising communication paths h 11 , h 12 , h 22 , h 21 , h 2 NTX , h 1 NTX , h NRX 1 , h NRX 2 , h NRX NTX , where h mn  may represent a channel coefficient from transmit antenna n to receiver antenna m. There may be N TX  transmitter antennas and N RX  receiver antennas. There is also shown transmit symbols x 1 , x 2  and x NTX , and receive symbols y 1 , y 2  and y NRX . 
     The MIMO transmitter  102  may comprise suitable logic, circuitry and/or code that may be enabled to generate transmit symbols x i  i ε {1, 2, . . . N TX } that may be transmitted by the transmit antennas, of which the antennas  106 ,  108  and  110  may be depicted in  FIG. 1B . The MIMO receiver  104  may comprise suitable logic, circuitry and/or code that may be enabled to process the receive symbols y i  i ε {1, 2, . . . N RX } that may be received by the receive antennas, of which the antennas  112 ,  114  and  116  may be shown in  FIG. 1B . An input-output relationship between the transmitted and the received signal in a MIMO system may be written as:
 
 y=Hx+n  
 
where y=[y 1 , y 2 , . . . y NRX ] T  may be a column vector with N RX  elements, . T  may denote a vector transpose, H=[h ij ]: i ε {1, 2, . . . N RX }; j ε {1, 2, . . . N TX } may be a channel matrix of dimensions N RX  by N TX , x=[x 1 , x 2 , . . . x NTX ] T  is a column vector with N TX  elements and n is a column vector of noise samples with N RX  elements. The channel matrix H may be written, for example, as H=UΣV H  using the Singular Value Decomposition (SVD), where . H  denotes the Hermitian transpose, U is a N RX  by N TX  unitary matrix, Σ is a N TX  by N TX  diagonal matrix and V is N TX  by N TX  unitary matrix. Other matrix decompositions that may diagonalize or transform the matrix H may be used instead of the SVD. If the receiver algorithm implemented in MIMO receiver  104  is, for example, an Ordered Successive Interference Cancellation (OSIC), other matrix decompositions that convert the matrix H to lower/upper triangular may be appropriate. One such decomposition may comprise Geometric Mean Decomposition (GMD), where H=QRP H , where R may be upper triangular with the geometric mean of the singular values of H on the diagonal, and Q and P may be unitary.
 
       FIG. 2  is a block diagram illustrating an exemplary MIMO pre-coding transceiver chain model, in accordance with an embodiment of the invention. Referring to  FIG. 2 , there is shown a MIMO pre-coding system  200  comprising a MIMO transmitter  202 , a MIMO baseband equivalent channel  203 , a MIMO receiver  204 , and an adder  208 . The MIMO transmitter  202  may comprise a transmitter (TX) baseband processing block  210  and a transmit pre-coding block  214 . The MIMO baseband equivalent channel  203  may comprise a wireless channel  206 , a TX radio frequency (RF) processing block  212  and a receiver (RX) RF processing block  218 . The MIMO receiver  204  may comprise a pre-coding decoding block  216  and a RX baseband processing block  220 . There is also shown symbol vector s, pre-coded vector x, noise vector n, received vector y and channel-decoded vector y′. 
     The MIMO transmitter  202  may comprise a baseband processing block  210 , which may comprise suitable logic, circuitry and/or code that may be enabled to generate a MIMO baseband transmit signal. The MIMO baseband transmit signal may be communicated to a transmit pre-coding block  214 . A baseband signal may be suitably coded for transmission over a wireless channel  206  in the transmit pre-coding block  214  that may comprise suitable logic, circuitry and/or code that may enable it to perform these functions. The TX RF processing block  212  may comprise suitable logic, circuitry and/or code that may enable a signal communicated to the TX RF processing block  212  to be modulated to radio frequency (RF) for transmission over the wireless channel  206 . The RX RF processing block  218  may comprise suitable logic, circuitry and/or code that may be enabled to perform radio frequency front-end functionality to receive the signal transmitted over the wireless channel  206 . The RX RF processing block  218  may comprise suitable logic, circuitry and/or code that may enable the demodulation of its input signals to baseband. The adder  208  may depict the addition of noise to the received signal at the MIMO receiver  204 . The MIMO receiver  204  may comprise the pre-coding decoding block  216  that may linearly decode a received signal and communicate it to the RX baseband processing block  220 . The RX baseband processing block  220  may comprise suitable logic, circuitry and/or logic that may enable to apply further signal processing to baseband signal. 
     The MIMO transmitter  202  may comprise a baseband processing block  210 , which may comprise suitable logic, circuitry and/or code that may be enabled to generate a MIMO baseband transmit signal. The MIMO baseband transmit signal may be communicated to a transmit pre-coding block  214  and may be the symbol vector s. The symbol vector s may be of dimension N TX  by 1. 
     The transmit pre-coding block  214  may be enabled to apply a linear transformation to the symbol vector s, so that x=Ws, where W may be of dimension N TX  by length of s, and x=[x 1 , x 2 , . . . x NTX ] T . Each element of the pre-coded vector x may be transmitted on a different antenna among N TX  available antennas. 
     The transmitted pre-coded vector x may traverse the MIMO baseband equivalent channel  203 . From the N RX  receiver antennas, the received signal y may be the signal x transformed by the MIMO baseband equivalent channel  203  represented by a matrix H, plus a noise component given by the noise vector n. As depicted by the adder  208 , the received vector y may be given by y=Hx+n=HWs+n. The received vector y may be communicated to the pre-coding decoding block  216 , where a linear decoding operation B may be applied to the received vector y to obtain the decoded vector y′=B H y=B H HWs+B H n, where B may be a complex matrix of appropriate dimensions. The decoded vector y′ may then be communicated to the RX baseband processing block  220  where further signal processing may be applied to the output of the pre-coding decoding block  216 . 
     If the transfer function H of the MIMO baseband equivalent channel  203  that may be applied to the transmitted pre-coded vector x is known both at the MIMO transmitter  202  and the MIMO receiver  204 , the channel may be diagonalized by, for example, setting W=V and B=U, where H=UΣV H  may be the singular value decomposition. In these instances, the channel decoded vector y′ may be given by the following relationship:
 
 y′=U   H   UΣV   H   Vs+U   H   n=Σs+U   H   n  
 
Since Σ may be a diagonal matrix, there may be no interference between the elements of symbol vector s in y′ and hence the wireless communications system may appear like a system with up to N TX  parallel single antenna wireless communication systems, for each element of s, up to the rank of channel matrix H which may be less or equal to N TX .
 
       FIG. 3  is a block diagram of an exemplary MIMO pre-coding system with finite rate channel state information feedback, in accordance with an embodiment of the invention. Referring to  FIG. 3 , there is shown a MIMO pre-coding system  300 , comprising a partial MIMO transmitter  302 , a partial MIMO receiver  304 , a Wireless channel  306 , an adder  308 , and a feedback channel  320 . The partial MIMO transmitter  302  may comprise a transmit pre-coding block  314 . The partial MIMO receiver  304  may comprise a pre-coding decoding block  316 , a channel estimation block  322 , a channel quantization block  310 , and a codebook processing block  318 . There is also shown a symbol vector s, a pre-coded vector x, a noise vector n, a received vector y, and a decoded vector y′. 
     The transmit pre-coding block  314 , the wireless channel  306 , the adder  308  and the pre-coding decoding block  316  may be substantially similar to the transmit pre-coding block  214 , the MIMO baseband equivalent channel  203 , the adder  208  and the pre-coding decoding block  216 , illustrated in  FIG. 2 . The channel estimation block  322  may comprise suitable logic, circuitry and/or logic to estimate the transfer function of the wireless channel  306 . The channel estimate may be communicated to the codebook processing block  318  and the channel quantization block  310 . In this regard, the decomposed channel may be communicated to the channel quantization block  310 . The channel quantization block  310  may comprise suitable logic, circuitry and/or code, which may be enabled to partly quantize the channel onto a codebook. The codebook processing block  318  may comprise suitable logic, circuitry and/or logic, which may be enabled to generate a codebook. The feedback channel  320  may represent a channel that may be enabled to carry channel state information from the partial MIMO receiver  304  to the partial MIMO transmitter  302 . 
     In many wireless systems, the channel state information, that is, knowledge of the channel transfer matrix H, may not be available at the transmitter and the receiver. However, in order to utilize a pre-coding system as illustrated in  FIG. 2 , it may be desirable to have at least partial channel knowledge available at the transmitter. In the exemplary embodiment of the invention disclosed in  FIG. 2 , the MIMO transmitter  302  may require the unitary matrix V for pre-coding in the transmit pre-coding block  214  of MIMO transmitter  202 . 
     In frequency division duplex (FDD) systems, the frequency band for communications from the base station to the mobile terminal, downlink communications, may be different from the frequency band in the reverse direction, uplink communications. Because of a difference in frequency bands, a channel measurement in the uplink may not generally be useful for the downlink and vice versa. In these instances, the measurements may only be made at the receiver and channel state information (CSI) may be communicated back to the transmitter via feedback. For this reason, the CSI may be fed back to the transmit pre-coding block  314  of the partial MIMO transmitter  302  from the partial MIMO receiver  304  via the feedback channel  302 . The transmit pre-coding block  314 , the wireless channel  306 , and the adder  308  are substantially similar to the corresponding blocks  214 ,  203  and  208 , illustrated in  FIG. 2 . 
     At the partial MIMO receiver  304 , the received signal y may be used to estimate the channel transfer function H by H in the channel estimation block  322 . The estimate may further be decomposed into, for example, a diagonal or triangular form, depending on a particular receiver implementation, as explained for  FIG. 2 . However, explicitly decomposing the channel matrix may be computationally expensive and it may therefore be advantageous to avoid explicit computation of, for example, the SVD or the GMD. Furthermore, in the case of a plurality of antennas, the dimensions of the matrices U, Σ and V may grow quickly. In these instances, it may be desirable to quantize the matrix {circumflex over (V)} into a matrix V q  of dimensions N TX  by N TX , where V q  may be chosen from pre-defined finite set of unitary matrices C={V i }. The set of unitary matrices C may be referred to as the codebook. By finding a matrix V q  from the codebook that may be, in some sense, closest to the matrix {circumflex over (V)}, it may suffice to transmit the index q to the transmit pre-coding block  314  via the feedback channel  320  from the channel quantization block  310 , if the partial MIMO transmitter  302  may know the codebook C. The codebook C may be varying much slower than the channel transfer function H and it may suffice to periodically update the codebook C in the transmit pre-coding block  314  from the codebook processing block  318  via the feedback channel  320 . The codebook C may be chosen to be static or adaptive. Furthermore, the codebook C may also be chosen, adaptively or non-adaptively, from a set of codebooks, which may comprise adaptively and/or statically designed codebooks. The matrix {circumflex over (V)} may be quantized into V q  without explicitly decomposing the channel matrix H, as described by the following relationships: 
                     v   q     =     arg   ⁢           ⁢       min         V   ^     i     ∈   C       ⁢     f   ⁡     (         V   ^     i   H     ·       H   ^     H     ·     H   ^     ·       V   ^     i       )                   (   1   )                 f   ⁡     (   A   )       =     {             ∑     i   =   1       N   RX       ⁢       ∑       j   =   1     ,     j   ≠   i         N   RX       ⁢            a     i   ,   j            2                 if   ⁢           ⁢     N   TX       ≤     N   RX                     ∑     i   =   1       N   RX       ⁢       ∑       j   =   1     ,     j   ≠   i         N   RX       ⁢            a     i   ,   j            2         -       ∑     i   =   1       N   RX       ⁢     a     i   ,   i                   if   ⁢           ⁢     N   TX       &gt;     N   RX                       (   2   )               
where A=[a ij ]. Hence, the matrix V q  may be chosen as the matrix {circumflex over (V)} i  in the codebook C that may minimize the function ƒ({circumflex over (V)} i   H ·Ĥ H ·Ĥ·{circumflex over (V)} i ) as defined above. As illustrated in equation (1) above, if {circumflex over (V)} i ={circumflex over (V)}, the product {circumflex over (V)} i   H ·Ĥ H ·Ĥ·{circumflex over (V)} i  may become a diagonal matrix. By minimizing f(.), the matrix V q  is chosen so that the product {circumflex over (V)} i   H ·Ĥ H ·Ĥ·{circumflex over (V)} i  may be most like a diagonal matrix, in some sense. Hence, the channel H may be estimated in the channel estimation block  322  and the channel matrix H may be quantized in block  318  into a matrix V q  that may, in some sense, most resemble V. Using equation (1) above, this may be achieved without having to decompose the wireless channel matrix H first.
 
     In the channel quantization block  310 , a matrix, for example Ĥ may be quantized into a matrix V q  and the index q may be fed back to the partial MIMO transmitter  302  via the feedback channel  320 . Less frequently than the index q, the codebook C from the codebook processing block  318  may be transmitted to the partial MIMO transmitter  302  via the feedback channel  320 . The codebook C may be chosen to be static or adaptive. Furthermore, the codebook C may also be chosen, adaptively or non-adaptively, from a set of codebooks, which may comprise adaptively and/or statically designed codebooks. To feedback the index q, M bits may suffice when the cardinality |C| of the codebook C may be less or equal to |C|≦2 M . 
     The transmit pre-coding block  314  may perform, for example, the linear transformation x=V q s. The pre-coding decoding block  316  at the receiver may implement the linear transformation y′=Û H y. 
       FIG. 4  is a performance plot of an exemplary 2×2 MIMO system with an MMSE receiver and finite-rate feedback, in accordance with an embodiment of the invention. Referring to  FIG. 4 , there is shown an SNR axis and a spectral efficiency axis (Seff). There is further shown a line plot for a Singular Value Decomposition with Full feedback SVD (Flt-FB)  402 , a line plot for a SVD with 2 bits of feedback (SVD-2B)  404 , a line plot for SVD-3B  406 , a line plot for SVD-4B  408 , a line plot for SVD-5B  410 , and a line plot for SVD-6B  412 . 
     As explained with respect to  FIG. 3 , the number of bits available for feedback may determine how finely the unitary matrix V may be quantized. For M bits, the codebook C may comprise |C|≦2 M  elements {V i }. Therefore, a larger number of feedback bits may allow better quantization of the channel and may offer better performance. It may be seen in  FIG. 4  that even few bits of feedback using an exemplary embodiment of the invention may offer performance that may be close to full feedback, SVD (Flt-FB)  402 . The line plots SVD-2B  404 , SVD-3B  406 , SVD-4B  408 , SVD-5B  410  and SVD-6B  412  may be close to SVD (Flt-FB)  402 . A Minimum Mean Square Error (MMSE) receiver may be enabled in  FIG. 4 . 
       FIG. 5  is a flow chart illustrating an exemplary codebook selection and channel quantization algorithm, in accordance with an embodiment of the invention. An exemplary codebook selection and quantization algorithm may be started in step  502 . In step  504 , a channel estimate may be obtained in, for example, the channel estimation block  322  of the partial MIMO receiver  304  illustrated in  FIG. 3 . In step  506 , if there are a lesser or equal number of transmit antennas than receive antennas, that is N TX ≦N RX , then the cost function f(.) may be set as illustrated in step  508  and the quantization of the channel estimate Ĥ in step  510  may be enabled according to the relationship shown in step  510 . In step  506 , if there are a higher number of transmit antennas than receive antennas, that is N TX ≦N RX , then the cost function f(.) may be set as illustrated in step  512  and the quantization of the channel estimate Ĥ in step  510  may be enabled according to the relationship shown in step  510 . The index q of the quantized matrix V q  may be fed back in step  516  from, for example, the partial MIMO receiver  304  to the partial MIMO transmitter  302  via the feedback channel  320 , as illustrated in  FIG. 3 . The algorithm may loop back from step  516  to step  504  and recommence by obtaining a channel estimate. 
     In accordance with an embodiment of the invention, a method and system for an efficient channel quantization method for MIMO pre-coding systems may comprise quantizing channel state information in the channel quantization block  310  in a MIMO pre-coding system  300  onto a codebook, which comprises one or more unitary matrices, using a cost function, as illustrated for  FIG. 3 ; selecting a quantization function, as illustrated in  FIG. 5 , that enables the quantizing based on a number of receive antennas and a number of transmit antennas, as shown in steps  506  through  514 , in the MIMO pre-coding system; and selecting the cost function in step  508  and/or  512  from a set of cost functions based on the number of receive antennas and the number of transmit antennas in the MIMO pre-coding system  300 . The channel state information may be a matrix Ĥ, that may be obtained, for example, in channel estimation block  322  as shown in  FIG. 3 . The set of cost functions may comprise function f(A), which may be defined by the following relationships: 
               f   ⁡     (   A   )       =         ∑     i   =   1     N     ⁢       ∑       j   =   1     ,     j   ≠   i       N     ⁢              a     i   ,   j            2     ⁢           ⁢   and   ⁢     /     ⁢   or   ⁢           ⁢     f   ⁡     (   A   )             =         ∑     i   =   1     N     ⁢       ∑       j   =   1     ,     j   ≠   i       N     ⁢            a     i   ,   j            2         -       ∑     i   =   1     N     ⁢     a     i   ,   j                   
where A is a matrix of size N by N and a ij  is element (i,j) of matrix A. The quantization function may be defined by the following relationship: V q =arg min {circumflex over (V)}     i     εC ƒ({circumflex over (V)} i   H ·Ĥ H ·Ĥ·{circumflex over (V)} i ), where V q , {circumflex over (V)} i εC, and Ĥ may be matrices of appropriate dimensions and C may be the codebook. An index of an element of the codebook onto which the channel state information is quantized may be transmitted from a receiver  304  to a transmitter  302  in the MIMO pre-coding system  300 . The number of transmit antennas and the number of receive antennas may be at least one each, in the MIMO pre-coding system  300 . A vector signal s at the transmitter  302  of the MIMO pre-coding system  300  may be linearly transformed with one of the unitary matrices in the transmit pre-coding block  314 . In the codebook processing block  318 , for example, the set of codebooks may be updated dynamically or chosen statically.
 
     Another embodiment of the invention may provide a machine-readable storage, having stored thereon, a computer program having at least one code section executable by a machine, thereby causing the machine to perform the steps as described above for a method and system for an efficient channel quantization method for MIMO pre-coding systems. 
     Accordingly, the present invention may be realized in hardware, software, or a combination of hardware and software. The present invention may be realized in a centralized fashion in at least one computer system, or in a distributed fashion where different elements are spread across several interconnected computer systems. Any kind of computer system or other apparatus adapted for carrying out the methods described herein is suited. A typical combination of hardware and software may be a general-purpose computer system with a computer program that, when being loaded and executed, controls the computer system such that it carries out the methods described herein. 
     The present invention may also be embedded in a computer program product, which comprises all the features enabling the implementation of the methods described herein, and which when loaded in a computer system is able to carry out these methods. Computer program in the present context means any expression, in any language, code or notation, of a set of instructions intended to cause a system having an information processing capability to perform a particular function either directly or after either or both of the following: a) conversion to another language, code or notation; b) reproduction in a different material form. 
     While the present invention has been described with reference to certain embodiments, it will be understood by those skilled in the art that various changes may be made and equivalents may be substituted without departing from the scope of the present invention. In addition, many modifications may be made to adapt a particular situation or material to the teachings of the present invention without departing from its scope. Therefore, it is intended that the present invention not be limited to the particular embodiment disclosed, but that the present invention will include all embodiments falling within the scope of the appended claims.