Patent Publication Number: US-2010119017-A1

Title: Method and system for a 4x2 sfbc/stbc system with 2 spatial streams using angle feedback

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS/INCORPORATION BY REFERENCE 
     This application makes reference to, claims priority to, and claims the benefit of U.S. Provisional Application Ser. No. 61/113,145 filed Nov. 10, 2008. 
     This application also makes reference to U.S. patent application Ser. No. 11/864,611 filed Sep. 28, 2007. 
    
    
     Each of the above stated applications is hereby incorporated herein by reference in its entirety. 
     FIELD OF THE INVENTION 
     Certain embodiments of the invention relate to data communication. More specifically, certain embodiments of the invention relate to a method and system for a 4×2 SFBC/STBC system with 2 spatial streams using angle feedback. 
     BACKGROUND OF THE INVENTION 
     Quasi-orthogonal space time block coding (STBC) is a method for diversity transmission that is utilized in the field of wireless communication. The appeal of quasi-orthogonal STBC is that it seeks to enable wireless communication systems to utilize advantages of diversity transmission at a transmitting station, while allowing simplified decoding techniques at a receiving station. 
     Diversity transmission enables one or more streams of data to be transmitted via a plurality of transmitting antennas. Diversity transmission systems are described by the number of transmitting antennas and the number of receiving antennas. For example, a diversity transmission system, which utilizes four transmitting antennas to transmit signals and a single antenna to receive signals, may be referred to as a 4×1 diversity transmission system, while a diversity transmission system, which utilizes four transmitting antennas to transmit signals and two receiving antennas to receive signals, may be referred to as a 4×2 diversity transmission system. 
     Each data stream may comprise a sequence of data symbols. Each data symbol comprises at least a portion of the data from the data stream. In a diversity transmission system, which utilizes orthogonal frequency division multiplexing (OFDM), each data symbol is referred to as an OFDM symbol. Each OFDM symbol may utilize a plurality of frequency carrier signals, wherein the frequencies of the carrier signals span the bandwidth of an RF channel. RF channel bandwidths may be determined, for example, based on applicable communication standards utilized in various communication systems. Exemplary RF channel bandwidths for IEEE 802.11 wireless local area network (WLAN) systems are 20 MHz and 40 MHz. One or more of the frequency carrier signals within an RF channel bandwidth may be utilized to transmit at least a portion of the data contained in the OFDM symbol. The size of each portion of the data, as measured in bits for example, may be determined based on one or more constellation maps. The constellation map(s) may, in turn, be determined by one or more modulation types that are utilized to transport the data contained in the OFDM symbol via the RF channel. 
     In general, each of the data streams, which in turn comprise one or more OFDM symbols, may be referred to as a spatial stream. A diversity transmission system, which utilizes N TX  transmitting antennas to transmit signals and N RX  receiving antennas to receive signals, may be referred to as an N TX ×N RX  diversity transmission system. 
     In a diversity transmission system, each of the plurality of N TX  transmitting antennas may transmit data symbols from a corresponding plurality of N TX  space time streams. The N TX  space time streams may be generated from a number (N SS ) of spatial streams. Each of the data symbols in each space time stream may be referred to as a codeword. In a diversity transmission system, which utilizes quasi-orthogonal STBC, at any given time instant each of the plurality of N TX  transmitting antennas may transmit a codeword, which comprises one of the OFDM symbols, or a permutated version of the OFDM symbol, from a selected one of the N SS  spatial streams. 
     A variation of STBC is space frequency block coding (SFBC). In a diversity transmission system, which utilizes SFBC, each codeword may comprise a subset of the frequency carriers, or tones, and corresponding data portions, in an OFDM symbol. These subsets of frequency carriers may be referred to as tone groups. 
     In an STBC communication diversity system, each of the codewords may be generated based on an OFDM symbol, wherein each OFDM symbol is generated based on data from a selected spatial stream at a given time instant. In various embodiments of the invention, one or more of the concurrently codewords transmitted from a transmitting station may comprise a rotated and/or complex conjugate version of a corresponding OFDM symbol. A group of concurrently transmitted codewords may be transmitted during consecutive transmission opportunities (TXOPs) may comprise a codeword set. 
     In an SFBC communication diversity system, each of the codewords may be generated based on a portion of an OFDM symbol. Each portion may comprise one of a plurality of tone groups, where each tone group comprises a corresponding plurality of tones and where each tone represents a distinct frequency carrier, or frequency, within an OFDM symbol bandwidth. The collective set of tone groups comprise the set of frequency carriers within the OFDM symbol bandwidth. Each tone may be represented by f j (i), where i represents a tone group and j represents an index for each of the frequencies within the i th  tone group. Each of the codewords generated from an OFDM symbol may by transmitted concurrently via a single transmitting antenna. Each of the plurality of transmitting antennas in an SFBC communication diversity system may receive codewords via a corresponding transmit chain. Accordingly, the codewords may be communicated from a transmit chain to a corresponding transmitting antenna via transmit chain signals. Each of the transmitting antennas in the SFBC communication diversity system may transmit the chain signals concurrently with one or more of the remaining transmitting antennas. 
     In an SFBC communication diversity system, a codeword set comprises the set of codewords that are concurrently transmitted across the set of transmitting antennas. In other words, a codeword set comprises the collective plurality of codewords that are concurrently transmitted across the set of transmitting antennas. For each transmit chain, a plurality of codewords may be generated based on an OFDM symbol. 
     In the case of diversity transmission, with either STBC or SFBC, the transmitted signal may be modified as it travels across a communication medium to the receiving station. This signal-modifying property of the communication medium may be referred to as fading. Each of the signals transmitted by each of the plurality of transmitting antennas may experience differing amounts of fading as the signals travel through the communication medium. This variable fading characteristic may be represented by a transfer function matrix, H, which comprises a plurality of transfer function coefficients, h j , that represent the differing fading characteristics experienced by the transmitted signals. 
     The transmitted signals may be received by one or more receiving antennas located at a receiving station. The receiving station may process the received signals to determine estimated values for the codewords carried by the transmitted signals. However, the task of computing estimated values for the codewords may be computationally complex even when quasi-orthogonal STBC or SFBC are utilized. 
     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 system for a 4×2 SFBC/STBC system with 2 spatial streams using angle feedback, 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. 1  is an exemplary wireless communication system, which may be utilized in connection with an embodiment of the invention. 
         FIG. 2  is an exemplary transceiver comprising a plurality of transmitting antennas and a plurality of receiving antennas, which may be utilized in connection with an embodiment of the invention. 
         FIG. 3  is an exemplary diagram illustrating channel feedback, in accordance with an embodiment of the invention. 
         FIG. 4  is an exemplary block diagram of STBC with diversity reception, in accordance with an embodiment of the invention. 
         FIG. 5  is an exemplary block diagram of SFBC with diversity reception, in accordance with an embodiment of the invention. 
         FIG. 6  is a flowchart illustrating exemplary steps for a 4×2 STBC diversity transmission system for 2 spatial streams at a receiving station, in accordance with an embodiment of the invention. 
         FIG. 7  is a flowchart illustrating exemplary steps for a 4×2 STBC diversity transmission system for 2 spatial streams at a receiving station, 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 a 4×2 SFBC/STBC communication diversity system with 2 spatial streams using angle feedback. Various embodiments of the invention may comprise a method and system by which signals from 2 distinct spatial streams (N SS =2) are utilized to generate a plurality of 4 distinct transmit chain signals (N TX =4). Each of the transmit chain signals may be utilized by a transmitting station to transmit codewords. In various embodiments of the inventions, the plurality of transmit chain signals may be transmitted concurrently. The set of concurrently transmitted transmit chain signals may be received at a receiving station. 
     Based on the received signals, the receiving station may be operable to compute a channel estimate matrix. A channel estimate square matrix may be computed as a product of the computed channel estimate matrix and a Hermitian transformed version of the computed channel estimate matrix. The receiving station may be operable to determine a value for each distinct off-diagonal element in the computed channel estimate square matrix. In various embodiments of the invention, the receiving station may be operable to compute a rotation angle for which the sum of squares of the distinct off-diagonal elements is minimized. A rotation factor may be computed based on the computed rotation angle. In various embodiments of the invention, the rotation angle may be quantized based on a selected number of quantization bits. For example, in an exemplary embodiment of the invention, which utilizes n quantization bits, the rotation angle may be selected from among 2 n  candidate rotation angle values. 
     The receiving station may be operable to communicate the rotation angle and/or rotation factor to the transmitting station via feedback information that is transmitted from the receiving station to the transmitting station. The transmitting station may subsequently utilize a received rotation factor to compute a rotation factor. The transmitting station may utilize a received rotation factor or computed rotation factor to generate a subsequent plurality of transmit chain signals based on the 2 distinct spatial streams. The subsequent plurality of transmit chain signals may comprise a plurality of codewords, a plurality of rotated codewords and/or a plurality of complex conjugate codewords. A given complex conjugate codeword among the plurality of complex conjugate codewords may comprise a complex conjugate version of a corresponding codeword among the plurality of codewords. At least a portion of the plurality of rotated codewords may be generated based on the rotation factor and one of more of the plurality of codewords and/or one or more of the plurality of complex conjugate codewords. The subsequent plurality of transmit chain signals may be concurrently transmitted via the plurality of transmitting antennas from the transmitting station and subsequently received via a plurality of receiving antennas at the receiving station. 
     The receiving station may compute channel estimates based on the subsequently received signals, previously determined rotation angle(s) and/or previously determined rotation factor(s). Based on the computed channel estimates, the receiving station may process the subsequently received signals to generate a substantially orthogonal plurality of received signals. 
       FIG. 1  is an exemplary wireless communication system, which may be utilized in connection with an embodiment of the invention. Referring to  FIG. 1 , there is shown an access point (AP)  102 , a wireless local area network (WLAN) station (STA)  104 , and a network  108 . The WLAN STA  104  may comprise a decoder subsystem  104 A. 
     The AP  102  may comprise suitable logic, circuitry, interfaces or code that may be operable to communicate wirelessly via one or more radio frequency (RF) channels  106 . The STA  104  may comprise suitable logic, circuitry, interfaces or code that may be operable to communicate wirelessly via one or more radio frequency (RF) channels  106 . The AP  102  and STA  104  may each comprise a plurality of transmitting antennas and/or receiving antennas. The decoder subsystem  104 A may comprise suitable logic, circuitry, interfaces or code that may be operable to enable the STA  104  to concurrently receive a plurality of signals via the plurality of receiving antennas and generate a set of substantially orthogonal signals. The AP  102  may be communicatively coupled to the network  108 . The network  108  may comprise suitable devices, interfaces or code that may be operable to enable the AP  102  to communicate with other devices, either within the network  108  and/or communicatively coupled to the network  108 . The AP  102 , STA  104  and network  108  may enable communication based on one or more IEEE 802 standards, for example IEEE 802.11. 
     The STA  104  may utilize the RF channel  106  to communicate with the AP  102  by transmitting signals via an uplink channel. The transmitted uplink channel signals may comprise one of more frequencies associated with a channel as determined by a relevant standard, such as IEEE 802.11. The STA  104  may utilize the RF channel  106  to receive signals from the AP  102  via a downlink channel. Similarly, the received downlink channel signals may comprise one of more frequencies associated with a channel as determined by a relevant standard, such as IEEE 802.11. 
     The STA  104  and AP  102  may communicate via time division duplex (TDD) communications and/or via frequency division duplex communications. With TDD communications, the STA  104  may utilize the RF channel  106  to communicate with the AP  102  at a current time instant while the AP  102  may communicate with the STA  104  via the RF channel  106  at a different time instant. With TDD communications, the set of frequencies utilized in the downlink channel may be substantially similar to the set of frequencies utilized in the uplink channel. With FDD communications, the STA  104  may utilize the RF channel  106  to communicate with the AP  102  at the same time instant at which the AP  102  utilizes the RF channel  106  to communicate with the STA  104 . With FDD communications, the set of frequencies utilized in the downlink channel may be different from the set of frequencies utilized in the uplink channel. 
     In an exemplary embodiment of the invention, the STA  104  may concurrently receive a plurality of signals transmitted by the AP  102 , which utilizes a plurality of transmitting antennas, via the downlink portion of the RF channel  106 . The STA  104  may utilize a plurality of receiving antennas to receive the concurrently transmitted signals from the AP  102 . The STA  104  may compute channel feedback information based on the concurrently received plurality of signals. The computed feedback information may be represented as a feedback angle and/or as a complex-valued feedback factor. The STA  104  may transmit the computed feedback information to the AP  102  via the uplink portion of the RF channel  106 . In an exemplary embodiment of the invention, the feedback information may be represented as a single-bit binary value. In other exemplary embodiments of the invention, the feedback information may be represented as a two-bit, or more, binary value. In instances where the feedback angle is quantized as an m-bit binary value, the feedback information may comprise an m-bit feedback angle value that is selected by the STA  104  from one of 2 m  candidate feedback angle values. In various exemplary embodiments of the invention, m=1 or m=2. 
     The AP  102  may generate codewords and/or complex conjugate codewords based on OFDM symbols received via a plurality of two spatial streams. The AP  102  may utilize the feedback information, received from STA  104 , to generate rotated codewords and/or rotated complex conjugate codewords. The AP  102  may generate a subsequent plurality of transmit chain signals based on the generated codewords, rotated codewords, complex conjugate codewords and/or rotated complex conjugate codewords. The AP  102  may transmit the subsequent plurality of transmit chain signals to the STA  104  via a plurality of transmitting antennas. 
     The STA  104  may receive the transmitted subsequent plurality of transmit chain signals via a plurality of receiving antennas. The STA  104  may compute a channel estimate matrix based on the received signals. The STA  104  may generate a plurality of substantially orthogonal signals by processing the received signals based on a Hermitian transformed version of the computed channel estimate matrix. When represented as a matrix, the plurality of substantially orthogonal signal may comprise a plurality of off-diagonal matrix elements. In various embodiments of the invention, each of the plurality of off-diagonal matrix elements may comprise minimum values based on the previously computed rotation angle. 
     In various embodiments of the invention, the STA  104  may be operable to receive subsequent signals. One or more protocol data units (PDUs) may be communicated to the STA  104  via the subsequently received signals. In an exemplary embodiment of the invention, the STA  104  may be operable to compute one or more subsequent rotation angles and/or one or more subsequent rotation factors for each received PDU. The STA  104  may be operable to transmit the computed one or more subsequent rotation angles and/or the computed one or more subsequent rotation factors pursuant to the receipt of each PDU. In another exemplary embodiment of the invention, the STA  104  may be operable to compute one or more subsequent rotation angles and/or one or more subsequent rotation factors for each plurality of k (where k&gt;1) received PDUs. The STA  104  may be operable to transmit the computed one or more subsequent rotation angles and/or one or more subsequent rotation factors upon commencement of a subsequent TXOP. In another exemplary embodiment of the invention, the STA  104  may be operable to compute one or more subsequent rotation angles and/or one or more subsequent rotation factors for each duration of t time units. The STA  104  may be operable to transmit the computed one or more subsequent rotation angles and/or one or more subsequent rotation factors upon commencement of a subsequent TXOP. 
     In an exemplary embodiment of the invention, the decoder subsystem  104 A is operable to compute a rotation angle and/or rotation factor. The plurality of receiving antennas at the STA  104  may be coupled to the decoder subsystem. The rotation angle and/or rotation factor computed by the decoder subsystem may be transmitted, as feedback information, to the AP  102 . The decoder subsystem may compute a rotation factor c as represented in the following equation: 
       c=e j·θ     jb      [1] 
     where θ fb  represents the rotation angle computed by the decoder subsystem at the STA  104 . In various embodiments of the invention, the rotation angle may be represented as an m-bit binary value, which is selected at the STA  104  from 2 m  candidate rotation angle values. The feedback information transmitted by the STA  104  to the AP  102  may comprise the selected rotation angle θ fb  and or the corresponding computed rotation factor c. 
       FIG. 2  is an exemplary transceiver comprising a plurality of transmitting antennas and a plurality of receiving antennas, which may be utilized in connection with an embodiment of the invention. Referring to  FIG. 2 , there is shown a transceiver system  200 , a plurality of receiving antennas  222   a  . . .  222   n  and a plurality of transmitting antennas  232   a  . . .  232   n . The transceiver system  200  may comprise a receiver  202 , a transmitter  204 , a processor  206 , and a memory  208 . Although a transceiver is shown in  FIG. 2 , transmit and receive functions may be separately implemented. 
     In accordance with an embodiment of the invention, the processor  206  may enable digital receiver and/or transmitter functions in accordance with applicable communications standards. The processor  206  may also perform various processing tasks on received data. The processing tasks may comprise computing channel estimates, which may characterize the wireless communication medium, delineating PDU boundaries in received data, and computing PDU statistics, for example packet error rate statistics, which may be indicative of the presence or absence of detected bit errors in received PDUs. 
     The receiver  202  may perform receiver functions that may comprise, but are not limited to, the amplification of received RF signals, generation of frequency carrier signals corresponding to selected RF channels, for example uplink channels, the down-conversion of the amplified RF signals by the generated frequency carrier signals, demodulation of data contained in data symbols based on application of a selected demodulation type, and detection of data contained in the demodulated signals. The RF signals may be received via one or more receiving antennas  222   a  . . .  222   n . The data may be communicated to the processor  206 . 
     The transmitter  204  may perform transmitter functions comprising modulation of received data to generated data symbols based on application of a selected modulation type, generation of frequency carrier signals corresponding to selected RF channels, for example downlink channels, the up-conversion of the data symbols by the generated frequency carrier signals, and the generation and amplification of RF signals. The data may be received from the processor  206 . The RF signals may be transmitted via one or more transmitting antennas  232   a  . . .  232   n.    
     The memory  208  may comprise suitable logic, circuitry, interfaces and/or code that may enable storage and/or retrieval of data and/or code. The memory  208  may utilize any of a plurality of storage medium technologies, such as volatile memory, for example random access memory (RAM), and/or non-volatile memory, for example electrically erasable programmable read only memory (EEPROM). In the context of the present application, the memory  208  may enable storage of code for the computation and storage of rotation angles based on channel feedback information, the computation and storage of channel estimates based on the channel feedback information and/or the storage of channel feedback information, for example. 
     In operation, the processor  206  may enable the computation of rotation angles and/or rotation factors based on signals received at the receiver  202  via the plurality of receiving antennas  222   a  . . .  222   n . The received signals may enable the computation of channel estimates, which characterize the wireless communication medium through which the received signals were transmitted. The computed channel estimates may, in turn, enable the computation of the rotation angles and/or rotation factors. The processor  206  may enable the computed rotation angles and/or rotation factors to be transmitted by the transmitter  204  via the plurality of transmitting antennas  232   a  . . .  232   n . The computed rotation angles and/or rotation factors may enable generation of subsequent transmitted signals, in accordance with various embodiments of the invention. 
       FIG. 3  is an exemplary diagram illustrating channel feedback, in accordance with an embodiment of the invention. Referring to  FIG. 3 , there is shown a transmitting station  402 , a receiving station  422 , and a communications medium  444 . The communications medium  444  may represent a wireless communications medium. The transmitting station  402  may represent an AP  102  and the receiving station may represent an STA  104 , for example. The transmitting station  402  may transmit a signal vector X to the receiving station  422  via the communications medium  444 . The communications direction from the transmitting station  402  to the receiving station  422  may be referred to as a downlink direction. The signal vector X may comprise a plurality of signals, which are concurrently transmitted via one or more transmitting antennas that are located at the transmitting station  402 . The transmitted signals, which are represented in the signal vector X, may travel through the communications medium  444 . The transmitted signals may be altered while traveling through the communications medium  444 . The transmission characteristics associated with the communications medium  444  may be characterized by the transfer function matrix, H. The transmitted signals, which are represented by the signal vector S, may be altered based on the transfer function matrix H. In the downlink direction, the transfer function matrix H may be referred to as H down . The signals received at the receiving station  422  may be represented by the signal vector, Y. The signal vector Y may be generated based on the signal vector X and the transfer function matrix H as shown in the following equation: 
         Y=H   down   ×X    [2] 
     The coefficients, which are the matrix elements within the transfer function matrix H, may comprise channel estimate values, h m . The channel estimate values may be computed based on at least a portion of the received signals represented by the signal vector Y. In an exemplary embodiment of the invention, the channel estimate values may be computed based on the portion(s) of the signals, transmitted by the transmitting station  402 , which carry preamble data. 
     The receiving station  422  may compute a rotation angle θ fb  based on channel estimate values from the transfer function matrix H down . Based on the rotation angle, the receiving station  422  may compute at least one rotation factor, c. In various embodiments of the invention, the rotation factor may be computed as shown in equation [1]. The receiving station  422  may communicate the computed transfer function matrix H down  and/or rotation angle θ fb  to the transmitting station  402  as channel feedback information, as represented by the tuple (H down ,θ fb ), for example. In an exemplary embodiment of the invention, the rotation angle θ fb , in the tuple (H down ,θ fb ), may be represented as a 2-bit binary value. The receiving station  422  may communicate the channel feedback information (H down ,θ fb ) via one or more signals, which are represented by the transmitted signal vector X fb . The signals represented by the transmitted signal vector X fb  may be transmitted to the transmitting station  402  via the communications medium  444 . The signals represented by the signal vector X fb  may be altered while traveling through the communications medium  444 . The communications direction from the receiving station  422  to the transmitting station  402  may be referred to as an uplink direction. In the uplink direction the transfer function matrix may be referred to as H up . The signals received at the transmitting station 402 may be represented by the signal vector, Y fb . The signal vector Y fb  may be generated based on the signal vector X fb  and the transfer function matrix H up  as shown in the following equation: 
         Y   fb   =H   up   ×X   fb    [3] 
     The transmitting station  402  may utilize the rotation angle, θ fb , received in the channel feedback information to compute a corresponding rotation factor c. The transmitting station  402  may utilize the rotation factor, c, to generate subsequent transmitted signals. 
     In another exemplary embodiment of the invention, the receiving station  422  may communicate the channel feedback information, which comprises the computed transfer function matrix, H, and/or the computed rotation factor, c. The transmitting station  402  may utilize the received rotation factor, c, to generate subsequent transmitted signals. 
     In another exemplary embodiment of the invention, the signal vector X fb  may comprise a quantized version of the rotation angle, θ fb , a quantized version of the computed transfer function matrix, H, and/or a quantized version of the computed rotation factor, c. 
       FIG. 4  is an exemplary block diagram of STBC with diversity reception, in accordance with an embodiment of the invention. Referring to  FIG. 4 , there is shown a transmitting station  402  and a receiving station  422 . The transmitting station  402  may comprise an STBC encoder  502 . The transmitting station  402  may utilize diversity transmission by concurrently transmitting a plurality of RF output signals via at least a portion of the transmitting antennas  512   a ,  512   b ,  512   c  and  512   d . The concurrently transmitted plurality of RF output signals may form a signal group. Referring to  FIG. 4 , there is shown signal groups  532  and  534 . Each signal group may comprise a plurality of concurrently transmitted codewords. Signal group  532  may be transmitted at a given time instant while signal group  534  may be transmitted at a subsequent time instant. 
     In an exemplary embodiment of the invention, the signal groups  532  and  534  may be generated based on OFDM symbols received at the STBC encoder  502  via a plurality of 2 spatial streams. Spatial stream  1  comprises an OFDM symbol received by the STBC encoder  502  at a time instant t 0 , x[ 1 ](t[ 0 ]), and an OFDM symbol received at a time instant t 1 , x[ 1 ](t[ 1 ]). Spatial stream  2  comprises an OFDM symbol received by the STBC encoder  502  at a time instant t 0 , x[ 2 ](t[ 0 ]), and an OFDM symbol received at a time instant t 1 , x[ 2 ](t[ 1 ]). Based on the received OFDM symbols, the STBC encoder  502  may generate a plurality of transmit chain signals, each of which may comprise a plurality of codewords. For the transmit chain signal associated with transmitting antenna  512   a , the STBC encoder  502  may generate a codeword x[ 1 ](t[ 0 ]) at a time instant t 0 ∝ and a codeword x[ 1 ]* (t[ 1 ]) at a time instant t 1 ′, where x[ 1 ]*(t[ 1 ]) represents a complex conjugate version of x[ 1 ](t[ 1 ]). For the transmit chain signal associated with transmitting antenna  512   b , the STBC encoder  502  may generate a codeword c·x[ 1 ](t[ 1 ]) at a time instant t 0 ′ and a codeword −c·x[ 1 ]*(t[ 0 ]) at a time instant t 1 ′, where c·x[ 1 ](t[ 1 ]) represents a rotated version of x[ 1 ](t[ 1 ]) based on the rotation factor, c. Similarly, −c·x[ 1 ]*(t[ 0 ]) represents a rotated version of the complex conjugate version of x[ 1 ](t[ 0 ]). For the transmit chain signal associated with transmitting antenna  512   c , the STBC encoder  502  may generate a codeword x[ 2 ](t[ 0 ]) at a time instant t 0 ′ and a codeword x[ 2 ]*(t[ 1 ]) at a time instant t 1 ′, where x[ 2 ]*(t[ 1 ]) represents a complex conjugate version of x[ 2 ](t[ 1 ]). For the transmit chain signal associated with transmitting antenna  512   d , the STBC encoder  502  may generate a codeword x[ 2 ](t[ 1 ]) at a time instant t 0 ′ and a codeword −x[ 2 ]*(t[ 0 ]) at a time instant t 1 ′, where −x[ 2 ]*(t[ 0 ]) represents a rotated of the complex conjugate version of x[ 2 ](t[ 0 ]). The signal group  532  comprises code words x[ 1 ](t[ 0 ]), c·x[ 1 ](t[ 1 ]), x[ 2 ](t[ 0 ]) and x[ 2 ](t[ 1 ]). The signal group  534  comprises code words x[ 1 ]*(t[ 1 ]), −c·x[ 1 ]*(t[ 0 ]), x[ 2 ]*(t[ 1 ]) and −x[ 2 ]*(t[ 0 ]). Collectively, the OFDM symbols x[ 1 ](t[ 0 ]), x[ 1 ](t[ 1 ]), x[ 2 ](t[ 0 ]) and x[ 2 ](t[ 1 ]) may be represented as an original codeword vector, X as shown in the following equation: 
     
       
         
           
             
               
                 
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     The receiving station  422  may comprise an STBC decoder  504 . The receiving station  422  may receive signals via the receiving antennas  522   a  and  522   b.    
     Signals transmitted from the transmitting antennas  512   a ,  512   b ,  512   c  and  512   d  travel through a wireless communication medium and may be received at the receiving antennas  522   a  and  522   b . Signals traveling from the transmitting antenna  512   a  and received at the receiving antenna  522   a  may be modified based on the channel estimate value h[ 1 , 1 ]; signals traveling from the transmitting antenna  512   b  and received at the receiving antenna  522   a  may be modified based on the channel estimate value h[ 2 , 1 ]; signals traveling from the transmitting antenna  512   c  and received at the receiving antenna  522   a  may be modified based on the channel estimate value g[ 1 , 1 ]; and signals traveling from the transmitting antenna  512   d  and received at the receiving antenna  522   a  may be modified based on the channel estimate value g[ 2 , 1 ]. The aggregate of signals received at the receiving antenna  522   a  may be referred to as y[ 1 ]. Signals received at the receiving antenna  522   a  at a time instant t 0 ″ may be referred to by y[ 1 ](t[ 0 ]). Signals received at the receiving antenna  522   a  at a time instant t 1 ″ may be referred to by y[ 1 ](t[ 1 ]). 
     Signals traveling from the transmitting antenna  512   a  and received at the receiving antenna  522   b  may be modified based on the channel estimate value h[ 1 , 2 ]; signals traveling from the transmitting antenna  512   b  and received at the receiving antenna  522   b  may be modified based on the channel estimate value h[ 2 , 2 ]; signals traveling from the transmitting antenna  512   c  and received at the receiving antenna  522   b  may be modified based on the channel estimate value g[ 1 , 2 ]; and signals traveling from the transmitting antenna  512   d  and received at the receiving antenna  522   b  may be modified based on the channel estimate value g[ 2 , 2 ]. The aggregate of signals received at the receiving antenna  522   b  may be referred to as y[ 2 ]. Signals received at the receiving antenna  522   b  at a time instant t 0 ″ may be referred to by y[ 2 ](t[ 0 ]). Signals received at the receiving antenna  522   b  at a time instant t 1 ″ may be referred to by y[ 2 ](t[ 1 ]). In various embodiments of the invention, each of the channel estimate values h[ 1 , 1 ], h[ 1 , 2 ], h[ 2 , 1 ], h[ 2 , 2 ], g[ 1 , 1 ], g[ 1 , 2 ], g[ 2 , 1 ] and g[ 2 , 2 ] may comprise a plurality of distinct values, for example, a distinct value corresponding to each distinct carrier frequency within an RF channel bandwidth. Each distinct value may comprise a complex numerical value, a real numerical value and/or an imaginary numerical value. 
     The signals received at the STBC decoder  504 , Y, may be represented as shown in the following equation: 
     
       
         
           
             
               
                 
                   
                     [ 
                     
                         
                     
                      
                     
                       
                         
                           
                             
                               y 
                                
                               
                                 [ 
                                 1 
                                 ] 
                               
                             
                              
                             
                               ( 
                               
                                 t 
                                  
                                 
                                   [ 
                                   0 
                                   ] 
                                 
                               
                               ) 
                             
                           
                         
                       
                       
                         
                           
                             
                               
                                 y 
                                 * 
                               
                                
                               
                                 [ 
                                 1 
                                 ] 
                               
                             
                              
                             
                               ( 
                               
                                 t 
                                  
                                 
                                   [ 
                                   1 
                                   ] 
                                 
                               
                               ) 
                             
                           
                         
                       
                       
                         
                           
                             
                               y 
                                
                               
                                 [ 
                                 2 
                                 ] 
                               
                             
                              
                             
                               ( 
                               
                                 t 
                                  
                                 
                                   [ 
                                   0 
                                   ] 
                                 
                               
                               ) 
                             
                           
                         
                       
                       
                         
                           
                             
                               
                                 y 
                                 * 
                               
                                
                               
                                 [ 
                                 2 
                                 ] 
                               
                             
                              
                             
                               ( 
                               
                                 t 
                                  
                                 
                                   [ 
                                   1 
                                   ] 
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                     ] 
                   
                   = 
                   
                     
                       
                         [ 
                         
                             
                         
                          
                         
                           
                             
                               
                                 h 
                                  
                                 
                                   [ 
                                   
                                     1 
                                     , 
                                     1 
                                   
                                   ] 
                                 
                               
                             
                             
                               
                                 c 
                                 · 
                                 
                                   h 
                                    
                                   
                                     [ 
                                     
                                       2 
                                       , 
                                       1 
                                     
                                     ] 
                                   
                                 
                               
                             
                             
                               
                                 g 
                                  
                                 
                                   [ 
                                   
                                     1 
                                     , 
                                     1 
                                   
                                   ] 
                                 
                               
                             
                             
                               
                                 g 
                                  
                                 
                                   [ 
                                   
                                     2 
                                     , 
                                     1 
                                   
                                   ] 
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   - 
                                   
                                     c 
                                     * 
                                   
                                 
                                 · 
                                 
                                   
                                     h 
                                     * 
                                   
                                    
                                   
                                     [ 
                                     
                                       2 
                                       , 
                                       1 
                                     
                                     ] 
                                   
                                 
                               
                             
                             
                               
                                 
                                   h 
                                   * 
                                 
                                  
                                 
                                   [ 
                                   
                                     1 
                                     , 
                                     1 
                                   
                                   ] 
                                 
                               
                             
                             
                               
                                 - 
                                 
                                   
                                     g 
                                     * 
                                   
                                    
                                   
                                     [ 
                                     
                                       2 
                                       , 
                                       1 
                                     
                                     ] 
                                   
                                 
                               
                             
                             
                               
                                 
                                   g 
                                   * 
                                 
                                  
                                 
                                   [ 
                                   
                                     1 
                                     , 
                                     1 
                                   
                                   ] 
                                 
                               
                             
                           
                           
                             
                               
                                 h 
                                  
                                 
                                   [ 
                                   
                                     1 
                                     , 
                                     2 
                                   
                                   ] 
                                 
                               
                             
                             
                               
                                 c 
                                 · 
                                 
                                   h 
                                    
                                   
                                     [ 
                                     
                                       2 
                                       , 
                                       2 
                                     
                                     ] 
                                   
                                 
                               
                             
                             
                               
                                 g 
                                  
                                 
                                   [ 
                                   
                                     1 
                                     , 
                                     2 
                                   
                                   ] 
                                 
                               
                             
                             
                               
                                 g 
                                  
                                 
                                   [ 
                                   
                                     2 
                                     , 
                                     2 
                                   
                                   ] 
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   - 
                                   
                                     c 
                                     * 
                                   
                                 
                                 · 
                                 
                                   
                                     h 
                                     * 
                                   
                                    
                                   
                                     [ 
                                     
                                       2 
                                       , 
                                       2 
                                     
                                     ] 
                                   
                                 
                               
                             
                             
                               
                                 
                                   h 
                                   * 
                                 
                                  
                                 
                                   [ 
                                   
                                     1 
                                     , 
                                     2 
                                   
                                   ] 
                                 
                               
                             
                             
                               
                                 - 
                                 
                                   
                                     g 
                                     * 
                                   
                                    
                                   
                                     [ 
                                     
                                       2 
                                       , 
                                       2 
                                     
                                     ] 
                                   
                                 
                               
                             
                             
                               
                                 
                                   g 
                                   * 
                                 
                                  
                                 
                                   [ 
                                   
                                     1 
                                     , 
                                     2 
                                   
                                   ] 
                                 
                               
                             
                           
                         
                         ] 
                       
                       [ 
                       
                           
                       
                        
                       
                         
                           
                             
                               
                                 x 
                                  
                                 
                                   [ 
                                   1 
                                   ] 
                                 
                               
                                
                               
                                 ( 
                                 
                                   t 
                                    
                                   
                                     [ 
                                     0 
                                     ] 
                                   
                                 
                                 ) 
                               
                             
                           
                         
                         
                           
                             
                               
                                 x 
                                  
                                 
                                   [ 
                                   1 
                                   ] 
                                 
                               
                                
                               
                                 ( 
                                 
                                   t 
                                    
                                   
                                     [ 
                                     1 
                                     ] 
                                   
                                 
                                 ) 
                               
                             
                           
                         
                         
                           
                             
                               
                                 x 
                                  
                                 
                                   [ 
                                   2 
                                   ] 
                                 
                               
                                
                               
                                 ( 
                                 
                                   t 
                                    
                                   
                                     [ 
                                     0 
                                     ] 
                                   
                                 
                                 ) 
                               
                             
                           
                         
                         
                           
                             
                               
                                 x 
                                  
                                 
                                   [ 
                                   2 
                                   ] 
                                 
                               
                                
                               
                                 ( 
                                 
                                   t 
                                    
                                   
                                     [ 
                                     1 
                                     ] 
                                   
                                 
                                 ) 
                               
                             
                           
                         
                       
                       ] 
                     
                     + 
                     
                       [ 
                       
                           
                       
                        
                       
                         
                           
                             
                               n 
                               0 
                             
                           
                         
                         
                           
                             
                               n 
                               1 
                             
                           
                         
                         
                           
                             
                               n 
                               2 
                             
                           
                         
                         
                           
                             
                               n 
                               3 
                             
                           
                         
                       
                        
                       
                           
                       
                       ] 
                     
                   
                 
               
               
                 
                   [ 
                   5 
                   ] 
                 
               
             
           
         
       
     
     where Y is represented by the vector on the left hand side of equation [4] and n 0 , n 1 , n 2  and n 3  represent signal noise. Equation [5] may be represented as follows: 
     
       
         
           
             
               
                 
                   
                     [ 
                     
                         
                     
                      
                     
                       
                         
                           
                             
                               y 
                                
                               
                                 [ 
                                 1 
                                 ] 
                               
                             
                              
                             
                               ( 
                               
                                 t 
                                  
                                 
                                   [ 
                                   0 
                                   ] 
                                 
                               
                               ) 
                             
                           
                         
                       
                       
                         
                           
                             
                               
                                 y 
                                 * 
                               
                                
                               
                                 [ 
                                 1 
                                 ] 
                               
                             
                              
                             
                               ( 
                               
                                 t 
                                  
                                 
                                   [ 
                                   1 
                                   ] 
                                 
                               
                               ) 
                             
                           
                         
                       
                       
                         
                           
                             
                               y 
                                
                               
                                 [ 
                                 2 
                                 ] 
                               
                             
                              
                             
                               ( 
                               
                                 t 
                                  
                                 
                                   [ 
                                   0 
                                   ] 
                                 
                               
                               ) 
                             
                           
                         
                       
                       
                         
                           
                             
                               
                                 y 
                                 * 
                               
                                
                               
                                 [ 
                                 2 
                                 ] 
                               
                             
                              
                             
                               ( 
                               
                                 t 
                                  
                                 
                                   [ 
                                   1 
                                   ] 
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                     ] 
                   
                   = 
                   
                     
                       H 
                       × 
                       
                         [ 
                         
                           
                             
                               
                                 
                                   x 
                                    
                                   
                                     [ 
                                     1 
                                     ] 
                                   
                                 
                                  
                                 
                                   ( 
                                   
                                     t 
                                      
                                     
                                       [ 
                                       0 
                                       ] 
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   x 
                                    
                                   
                                     [ 
                                     1 
                                     ] 
                                   
                                 
                                  
                                 
                                   ( 
                                   
                                     t 
                                      
                                     
                                       [ 
                                       1 
                                       ] 
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   x 
                                    
                                   
                                     [ 
                                     2 
                                     ] 
                                   
                                 
                                  
                                 
                                   ( 
                                   
                                     t 
                                      
                                     
                                       [ 
                                       0 
                                       ] 
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   x 
                                    
                                   
                                     [ 
                                     2 
                                     ] 
                                   
                                 
                                  
                                 
                                   ( 
                                   
                                     t 
                                      
                                     
                                       [ 
                                       1 
                                       ] 
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                         
                         ] 
                       
                     
                     + 
                     
                       [ 
                       
                           
                       
                        
                       
                         
                           
                             
                               n 
                               0 
                             
                           
                         
                         
                           
                             
                               n 
                               1 
                             
                           
                         
                         
                           
                             
                               n 
                               2 
                             
                           
                         
                         
                           
                             
                               n 
                               3 
                             
                           
                         
                       
                        
                       
                           
                       
                       ] 
                     
                   
                 
               
               
                 
                   [ 
                   6 
                   ] 
                 
               
             
           
         
       
     
     In various embodiments of the invention, the receiving station  422  may compute the channel estimate matrix, H, as shown in equations [5] and [6]. The receiving station  422  may process the received signal vector Y by pre-multiplying the received signal vector Y by a Hermitian, or complex conjugate transpose, version of the matrix H. The Hermitian of matrix H may be represented as H H . The premultiplication of the signal vector Y by the Hermitian matrix H H , may result in a multiplication of the matrices H H  and H. This matrix product may be referred to as a square matrix, H sq , as shown in the following equation: 
     
       
         
           
             
               
                 
                   
                     
                       H 
                       sq 
                     
                     = 
                     
                       
                         
                           H 
                           H 
                         
                         × 
                         H 
                       
                        
                       
                         
 
                       
                        
                       
                           
                       
                       = 
                       
                         [ 
                         
                             
                         
                          
                         
                           
                             
                               
                                 
                                   ∑ 
                                   j 
                                 
                                  
                                 
                                   
                                     ∑ 
                                     i 
                                   
                                    
                                   
                                     ( 
                                     
                                       
                                          
                                         
                                           h 
                                            
                                           
                                             [ 
                                             
                                               i 
                                               , 
                                               j 
                                             
                                             ] 
                                           
                                         
                                          
                                       
                                       2 
                                     
                                     ) 
                                   
                                 
                               
                             
                             
                               0 
                             
                             
                               
                                 δ 
                                 1 
                               
                             
                             
                               
                                 δ 
                                 2 
                               
                             
                           
                           
                             
                               0 
                             
                             
                               
                                 
                                   ∑ 
                                   j 
                                 
                                  
                                 
                                   
                                     ∑ 
                                     i 
                                   
                                    
                                   
                                     ( 
                                     
                                       
                                          
                                         
                                           h 
                                            
                                           
                                             [ 
                                             
                                               i 
                                               , 
                                               j 
                                             
                                             ] 
                                           
                                         
                                          
                                       
                                       2 
                                     
                                     ) 
                                   
                                 
                               
                             
                             
                               
                                 - 
                                 
                                   δ 
                                   2 
                                 
                               
                             
                             
                               
                                 δ 
                                 1 
                                 * 
                               
                             
                           
                           
                             
                               
                                 δ 
                                 1 
                                 * 
                               
                             
                             
                               
                                 - 
                                 
                                   δ 
                                   2 
                                 
                               
                             
                             
                               
                                 
                                   ∑ 
                                   j 
                                 
                                  
                                 
                                   
                                     ∑ 
                                     i 
                                   
                                    
                                   
                                     ( 
                                     
                                       
                                          
                                         
                                           h 
                                            
                                           
                                             [ 
                                             
                                               i 
                                               , 
                                               j 
                                             
                                             ] 
                                           
                                         
                                          
                                       
                                       2 
                                     
                                     ) 
                                   
                                 
                               
                             
                             
                               0 
                             
                           
                           
                             
                               
                                 δ 
                                 2 
                                 * 
                               
                             
                             
                               
                                 δ 
                                 1 
                               
                             
                             
                               0 
                             
                             
                               
                                 
                                   ∑ 
                                   j 
                                 
                                  
                                 
                                   
                                     ∑ 
                                     i 
                                   
                                    
                                   
                                     ( 
                                     
                                       
                                          
                                         
                                           h 
                                            
                                           
                                             [ 
                                             
                                               i 
                                               , 
                                               j 
                                             
                                             ] 
                                           
                                         
                                          
                                       
                                       2 
                                     
                                     ) 
                                   
                                 
                               
                             
                           
                         
                          
                         
                             
                         
                         ] 
                       
                     
                   
                    
                   
                       
                   
                    
                   
                     
 
                   
                    
                   
                     where 
                     : 
                   
                 
               
               
                 
                   [ 
                   7 
                   ] 
                 
               
             
             
               
                 
                   
                     δ 
                     1 
                   
                   = 
                   
                     
                       
                         
                           h 
                           * 
                         
                          
                         
                           [ 
                           
                             1 
                             , 
                             1 
                           
                           ] 
                         
                       
                       · 
                       
                         g 
                          
                         
                           [ 
                           
                             1 
                             , 
                             1 
                           
                           ] 
                         
                       
                     
                     + 
                     
                       c 
                       · 
                       
                         h 
                          
                         
                           [ 
                           
                             2 
                             , 
                             1 
                           
                           ] 
                         
                       
                       · 
                       
                         
                           g 
                           * 
                         
                          
                         
                           [ 
                           
                             2 
                             , 
                             1 
                           
                           ] 
                         
                       
                     
                     + 
                     
                       
                         
                           h 
                           * 
                         
                          
                         
                           [ 
                           
                             1 
                             , 
                             2 
                           
                           ] 
                         
                       
                       · 
                       
                         g 
                          
                         
                           [ 
                           
                             1 
                             , 
                             2 
                           
                           ] 
                         
                       
                     
                     + 
                     
                       c 
                       · 
                       
                         h 
                          
                         
                           [ 
                           
                             2 
                             , 
                             2 
                           
                           ] 
                         
                       
                       · 
                       
                         
                           g 
                           * 
                         
                          
                         
                           [ 
                           
                             2 
                             , 
                             2 
                           
                           ] 
                         
                       
                     
                   
                 
               
               
                 
                   [ 
                   
                     8 
                      
                     a 
                   
                   ] 
                 
               
             
             
               
                 
                   
                     δ 
                     1 
                   
                   = 
                   
                       
                   
                    
                   
                     
                       c 
                       · 
                       
                           
                       
                        
                       
                         ( 
                         
                           
                             
                               
                                 
                                   
                                     h 
                                      
                                     
                                       [ 
                                       
                                         2 
                                         , 
                                         1 
                                       
                                       ] 
                                     
                                   
                                   · 
                                   
                                     
                                       g 
                                       * 
                                     
                                      
                                     
                                       [ 
                                       
                                         2 
                                         , 
                                         1 
                                       
                                       ] 
                                     
                                   
                                 
                                 + 
                               
                             
                           
                           
                             
                               
                                 
                                   h 
                                    
                                   
                                     [ 
                                     
                                       2 
                                       , 
                                       2 
                                     
                                     ] 
                                   
                                 
                                 · 
                                 
                                   
                                     g 
                                     * 
                                   
                                    
                                   
                                     [ 
                                     
                                       2 
                                       , 
                                       2 
                                     
                                     ] 
                                   
                                 
                               
                             
                           
                         
                         ) 
                       
                     
                     + 
                     
                         
                     
                      
                     
                       ( 
                       
                         
                           
                             
                               
                                 
                                   
                                     h 
                                     * 
                                   
                                    
                                   
                                     [ 
                                     
                                       1 
                                       , 
                                       1 
                                     
                                     ] 
                                   
                                 
                                 · 
                                 
                                   g 
                                    
                                   
                                     [ 
                                     
                                       1 
                                       , 
                                       1 
                                     
                                     ] 
                                   
                                 
                               
                               + 
                             
                           
                         
                         
                           
                             
                               
                                 
                                   h 
                                   * 
                                 
                                  
                                 
                                   [ 
                                   
                                     1 
                                     , 
                                     2 
                                   
                                   ] 
                                 
                               
                               · 
                               
                                 g 
                                  
                                 
                                   [ 
                                   
                                     1 
                                     , 
                                     2 
                                   
                                   ] 
                                 
                               
                             
                           
                         
                       
                        
                       
                           
                       
                       ) 
                     
                   
                 
               
               
                 
                   [ 
                   
                     8 
                      
                     b 
                   
                   ] 
                 
               
             
           
         
       
     
     where δ 1  may be represented using polar notation as a function of polar magnitudes r 1  and r 2  and polar angles θ 1  and θ 2  as shown in the following equation: 
       δ 1   =cr   1   e   jθ     1     +r   2   e   jθ     2      [9] 
     Similarly, δ 2  may be represented as shown in the following equations: 
       δ 1   =h*[ 1,1 ]·g [2,1 ]−c·h [2,1 ]·g*[ 1,1 ]+h*[ 1,2 ]·g [2,2 ]−c·h [2,2 ]·g*[ 1,2   ([10a] 
       δ 1 =( h*[ 1,1 ]·g[ 2,1 ]+h*[ 1,2 ]·g[ 2,2])− c ·( h[ 2,1 ]·g*[ 1,1 ]+h[ 2,2 ]·g*[ 1,2])   [10b] 
     where δ 2  may be represented using polar notation as a function of polar magnitudes r 3  and r 4  and polar angles θ 3  and θ 4  as shown in the following equation: 
       δ 2   =r   3   e   jθ     3     −cr   4   e   jθ     4      [11] 
     A feedback angle, θ fb , may be computed as shown in the following equations: 
     
       
         
           
             
               
                 
                   
                     θ 
                     fb 
                   
                   = 
                   
                     
                       argmin 
                       θ 
                     
                      
                     
                       ⌊ 
                       
                         ( 
                         
                           
                             
                                
                               
                                 δ 
                                 1 
                               
                                
                             
                             2 
                           
                           + 
                           
                             
                                
                               
                                 δ 
                                 2 
                               
                                
                             
                             2 
                           
                         
                         ⌋ 
                       
                     
                   
                 
               
               
                 
                   [ 
                   
                     12 
                      
                     a 
                   
                   ] 
                 
               
             
             
               
                 
                   
                     θ 
                     fb 
                   
                   = 
                   
                     
                       argmin 
                       θ 
                     
                      
                     
                       ⌊ 
                       
                         ( 
                         
                           
                             
                                
                               
                                 
                                   
                                     r 
                                     1 
                                   
                                    
                                   
                                      
                                     
                                       j 
                                        
                                       
                                         ( 
                                         
                                           
                                             θ 
                                             1 
                                           
                                           + 
                                           θ 
                                         
                                         ) 
                                       
                                     
                                   
                                 
                                 + 
                                 
                                   
                                     r 
                                     2 
                                   
                                    
                                   
                                      
                                     
                                       jθ 
                                       2 
                                     
                                   
                                 
                               
                                
                             
                             2 
                           
                           + 
                           
                             
                                
                               
                                 
                                   
                                     r 
                                     3 
                                   
                                    
                                   
                                      
                                     
                                       jθ 
                                       3 
                                     
                                   
                                 
                                 - 
                                 
                                   
                                     r 
                                     4 
                                   
                                    
                                   
                                      
                                     
                                       j 
                                        
                                       
                                         ( 
                                         
                                           
                                             θ 
                                             4 
                                           
                                           + 
                                           θ 
                                         
                                         ) 
                                       
                                     
                                   
                                 
                               
                                
                             
                             2 
                           
                         
                         ) 
                       
                       ⌋ 
                     
                   
                 
               
               
                 
                   [ 
                   
                     12 
                      
                     b 
                   
                   ] 
                 
               
             
           
         
       
     
     where θ represents an angle rotation value and θ fb  may represent the value for θ at which the lower bound value for the magnitude square values (|δ 1 | 2 +|δ 2 | 2 ) is minimized. 
     The angle rotation offset value θ may be represented by an m-bit value. In various exemplary embodiments of the invention, values m=1 and/or m=2 may be utilized. Consequently, the number of candidate values for θ may be 2 m . In an exemplary embodiment of the invention, the value θ fb  may be determined by computing 2 m  (|δ 1 | 2 +|δ 2 | 2 ) values in equation [12b], wherein the value θ fb  may be determined based on the value θ that corresponds to the minimum lower bound value for the expression (|r 1 e j(θ     1     +θ) +r 2 e jθ     2   | 2 +|r 3 e jθ     3   −r 4 e j(θ     r     +θ) | 2 ) among the 2 m  computed values. 
     The computed angle feedback value θ fb  may be computed in a receiving station  422  and communicated as an m-bit binary value to a transmitting station  402 . Based on the received angle feedback value, the transmitting station  402  may concurrently transmit a subsequent plurality of transmit chain signals, comprising a subsequent sequence of codewords, based on the corresponding angle rotation factor c as shown in  FIG. 4 . The receiving station  404  may receive the concurrently transmitted subsequent plurality of transmit chain signals and generate a substantially orthogonal plurality of received signals, {circumflex over (X)}, where {circumflex over (X)} is a vector representation of estimated codeword values from the original codeword vector X as shown in equation [4]. 
     Various embodiments of the invention may be practiced with SFBC communication diversity systems in a manner substantially as disclosed herein. 
       FIG. 5  is an exemplary block diagram of SFBC with diversity reception, in accordance with an embodiment of the invention. Comparing  FIG. 5  to  FIG. 4 , in  FIG. 5 , the transmitting station  402  comprises an SFBC encoder  602  and the receiving station  422  comprises an SFBC decoder  604 . Referring to  FIG. 5 , the SFBC encoder  602  generates transmit chain signals  632 ,  634 ,  636  and  638  based on OFDM symbols received via a plurality of  2  spatial streams received at a given time instant t 0 . Spatial stream  1  comprises an OFDM symbol x[ 1 ](t[ 0 ]) and spatial stream  2  comprises an OFDM symbol x[ 2 ](t[ 0 ]). Based on the received OFDM symbol, x[ 1 ](t[ 0 ]), the SFBC encoder may generate a transmit chain signal  632  that comprises a plurality of codewords, x[ 1 ](f[ 0 ]) and x[ 1 ]*(f[ 1 ]), where x[ 1 ]*(f[ 1 ]) represents a complex conjugate version of the codeword x[ 1 ](f[ 1 ]) and where f[ 0 ] and f[ 1 ] represent distinct tone groups within an RF channel bandwidth. Tone group f[ 0 ] may comprise a subset of the carrier frequencies within the RF channel bandwidth while tone group f[ 1 ] may comprise a distinct subset of the carrier frequencies within the RF channel bandwidth. Collectively, the tone groups f[ 0 ] and f[ 1 ] may comprise the set of carrier frequencies within the RF channel bandwidth. 
     Based on the received OFDM symbol, x[ 1 ](t[ 0 ]), the SFBC encoder may generate a transmit chain signal  634  that comprises a plurality of codewords, c·x[ 1 ](f[ 1 ]) and c·x[ 1 ]*(f[ 0 ]), where x[ 1 ]*(f[ 0 ]) represents a complex conjugate version of the codeword x[ 1 ](f[ 0 ]), c·x[ 1 ](f[ 1 ]) represents a rotated version of the codeword x[ 1 ](f[ 1 ]) and c·x[ 1 ]*(f[ 0 ]) represents a rotated version of the codeword x[ 1 ]*(f[ 0 ]). Based on the received OFDM symbol, x[ 2 ](t[ 0 ]), the SFBC encoder may generate a transmit chain signal  636  that comprises a plurality of codewords,x[ 2 ](f[ 0 ]) and x[ 2 ]*(f[ 1 ]), where x[ 2 ]*(f[ 1 ]) represents a complex conjugate version of the codeword x[ 2 ](f[ 1 ]). Based on the received OFDM symbol, x[ 2 ](t[ 0 ]), the SFBC encoder may generate a transmit chain signal  638  that comprises a plurality of codewords,x[ 2 ](f[ 1 ]) and −x[ 2 ]*(f[ 0 ]), where −x[ 2 ]*(f[ 0 ]) represents a rotated version of the complex conjugate version of the codeword x[ 2 ](f[ 0 ]). 
     Collectively, the OFDM symbols x[ 1 ](f[ 0 ]), x[ 1 ](f[ 1 ]), x[ 2 ](f[ 0 ]) and x[ 2 ](f[ 1 ]) may be represented as an original codeword vector, X as shown in the following equation: 
     
       
         
           
             
               
                 
                   X 
                   = 
                   
                     [ 
                     
                       
                         
                           
                             
                               x 
                                
                               
                                 [ 
                                 1 
                                 ] 
                               
                             
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                               ( 
                               
                                 f 
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                                   [ 
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                               ) 
                             
                           
                         
                       
                       
                         
                           
                             
                               x 
                                
                               
                                 [ 
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                   [ 
                   13 
                   ] 
                 
               
             
           
         
       
     
     The transmitting station  402  may concurrently transmit the transmit chain signals  632 ,  634 ,  636  and  638 , which may be received as a received signal vector, Y, at the receiving station  422 , where the received signal vector Y may be represented as shown in the following equation: 
     
       
         
           
             
               
                 
                   Y 
                   = 
                   
                     [ 
                     
                         
                     
                      
                     
                       
                         
                           
                             
                               y 
                                
                               
                                 [ 
                                 1 
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     where y[i][j] represents signals that comprise frequencies from a j th  tone group that may be concurrently received via an i th  receiving antenna and y*[i][j] represents a complex conjugate version of y[i][j]. 
     The receiving station may process the received signal vector Y substantially as disclosed above and as set forth above in equation [5] though equation [12]. 
       FIG. 6  is a flowchart illustrating exemplary steps for a 4×2 STBC diversity transmission system for  2  spatial streams at a receiving station, in accordance with an embodiment of the invention. Referring to  FIG. 6 , at step  702 , a receiving station  422  may concurrently receive a plurality of signals via a plurality of receiving antennas. At step  704 , the receiving station  422  may be operable to decode preamble data based on the received signals. In step  706 , the receiving station  422  may be operable to compute channel estimate matrices H and H H . At step  708 , the receiving station  422  may be operable to compute a square matrix H sq . At step  710 , the receiving station  422  may be operable to determine off-diagonal values in the square matrix. The off-diagonal values may correspond to crosstalk components among the received signals. At step  712 , the receiving station  422  may be operable to compute one or more rotation angle values, θ. One of the computed rotation angle values may be selected as a feedback rotation angle, θ fb . The feedback rotation angle value may correspond to a minimum value for the sum or squares for the distinct off-diagonal values in the square matrix. A corresponding rotation factor value, c=e jθ     jb   , may also be computed based on the selected feedback rotation angle value. At step  714 , the receiving station  422  may be operable to transmit the feedback rotation angle value, θ fb , and/or the corresponding rotation factor value, c, to the transmitting station  402 . 
       FIG. 7  is a flowchart illustrating exemplary steps for a 4×2 STBC diversity transmission system for 2 spatial streams at a receiving station, in accordance with an embodiment of the invention. Referring to  FIG. 7 , at step  802 , the transmitting station  402  may receive feedback information. At step  804 , the transmitting station  402  may be operable to determine a rotation angle and/or rotation factor based on the received feedback information. At step  805 , the transmitting station  402  may determine whether to utilize a received rotation factor when generating subsequent transmit chain signals. In instances where the transmitting station determines that a received rotation factor may not be utilized, and/or in instances where a rotation factor value is not determined from the feedback information, at step  806 , the transmitting station  402  may be operable to compute a rotation factor based on the received rotation angle. In instances where the transmitting station determines that a received rotation factor may be utilized and/or in instances where a rotation factor value is determined from the feedback information and/or in instances where a rotation angle value is not determined from the feedback information, at step  807 , the transmitting station  402  may utilize the received rotation factor to generate subsequent transmit chain signals. At step  808 , the transmitting station  402  may be operable to generate an original codeword sequence. The original codeword sequence may be generated based on OFDM symbols received via a plurality of 2 spatial streams. 
     At step  810 , the transmitting station  402  may be operable to generate a plurality of transmit chain signals based on the original codeword sequence and the rotation factor. The transmitting station  402  may generate a plurality of 4 transmit chain signals. At step  812 , the transmitting station  402  may be operable to concurrently transmit the generated transmit chain signals. 
     Aspects of a computer readable medium having stored thereon, a computer program having at least one code section for processing signals in a communication system, the at least one code section being executable by a computer for causing the machine to perform steps a 4×2 SFBC/STBC system with 2 spatial streams using angle feedback. 
     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.