Patent Publication Number: US-7724835-B2

Title: Space-time block coding in orthogonal frequency division communication systems

Description:
CROSS REFERENCE TO RELATED APPLICATIONS 
   This application claims priority from Provisional Application No. 60/572,160, filed May 17, 2004, entitled “Space-Time Block Coding for OFDM via Time Domain Processing,” which is assigned to the assignee of the present application and fully incorporated herein by reference in its entirety. 

   BACKGROUND OF THE DISCLOSURE 
   The present disclosure relates to wireless communication systems, and more particularly to transmission diversity in orthogonal frequency division multiplexing systems. 
   Demand for wireless digital communication and data processing systems is on the rise. Inherent in most digital communication channels are errors introduced when transferring frames, packets or cells containing data over a channel that has some characteristics. Such errors are often caused by interference or thermal noise. The bit error rates of wireless transmission systems pose certain difficulties in designing encoding and decoding schemes for data to be transmitted via such systems. Partly because of its mathematical tractability and partly because of its application to a broad class of physical communication channels, the additive white Gaussian noise (AWGN) model is often used to characterize the noise in most communication channels. 
   One type of wireless communication system is an Orthogonal Frequency Division Multiplexed (OFDM) system. OFDM is a multi-carrier modulation technique that partitions the overall system bandwidth into multiple (N) orthogonal frequency subcarriers. These subcarriers may also be called tones, bins, and frequency channels. Each subcarrier may be modulated with data. Up to N modulation symbols may be sent on the N total subcarriers in each OFDM symbol period. These modulation symbols are converted to the time-domain with an N-point inverse fast Fourier transform (IFFT) to generate a transformed symbol that contains N time-domain chips or samples. 
   To improve transmission diversity, space-time block coding in each of the two transmission paths has been developed, as described in Alamouti, “Space-Time Block Coding, A Simple Transmit Diversity Technique for Wireless Communications”, IEEE Journal on Selected Areas in Communications, Volume 16, pp. 1451-1458, October 1998, the content of which is incorporated herein by reference in its entirety. The channel is assumed to be time/frequency invariant (flat) and is further assumed to remain constant over at least two consecutive symbols. 
   In accordance with the transmission scheme described in Alamouti, the original symbol sequence x(n) is divided into blocks of two consecutive symbols x k (n) and x k+1 (n). In Alamouti every pair of symbols is subsequently mapped according to the following: 
                     [           x   k               x     k   +   1             ]     ⇒     [           x   k           -     x     k   +   1                   x     k   +   1     *           x   k   *           ]       =   ℵ           (   1.1   )               
where for simplicity, time-index n is not included in expression (1.1)
 
   Symbols x k  and x k+1 * are transmitted at time k respectively from the first and second transmit antennas. Symbols −x k+1  and x k * are transmitted at time k+1 respectively from the first and second transmit antennas. The corresponding received signal r k , r k+1  at times k and k+1 are defined by the following expressions:
 
 r   k   =x   k   h   1   +x   k+1   *h   2   +n   k  
 
 r   k+1   =−x   k+1   h   1   +x   k   *h   2   +n   k+1   (1.2)
 
where h 1  and h 2  respectively represent the channels associated with the first and second transmission paths, and are further assumed to be constant over two symbol periods. The received signals r k , r k+1  may be written as follows:
 
   
     
       
         
           
             
               
                 
                   
                     
                       
                         
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   It is understood that the channel matrix H is orthogonal and that an optimum receiver for this transmit diversity scheme multiplies r k  by H*, which is the matched filter receiver, to get two decision statistics for x k  and x k+1 , i.e., to recover the transmitted symbols. Using this method, a diversity order of two is achieved at a receiver with a single receive antenna. 
   The method described above may be adapted for use in OFDM systems by replacing the time-domain computations with frequency-domain computations. Assume X n  and X n+1  are two OFDM symbols to be transmitted on sub-carriers n and n+1 in an OFDM system. In addition, for each transmit antenna m assume the channel remains constant over two consecutive sub-carriers. That is
 
H m,n ≈H m,n+1 =H m   (1.4)
 
   By replacing the time-domain computations with frequency-domain computations, the received signal vector corresponding to sub-carriers n and n+1 may be written as: 
                     R   k     ⁢     •   ⁡     [           R   k               R     k   +   1     *           ]         =         [           H   1           -     H   2                 H   2   *           H   1   *           ]     ⁡     [           X   k               X     k   +   1     *           ]       +     [           V   k               V     k   +   1     *           ]               (   1.5   )               
thus achieving a diversity of 2.
 
     FIG. 1  is a block diagram of a portion of an OFDM transmitter  10  described above. Each OFDM symbol of size N is divided into N/2 groups of symbol pairs [X n  X n+1 ]. Each such pair of symbols is then encoded by the space-frequency encoder  12  to generate two different pairs of symbols [X n −X n+1 ] and [X n+1 * X n *]. Symbol pairs [X n −X n+1 ] are grouped into an N—symbol vector that is supplied to an inverse fast Fourier transform (IFFT) 18 block, which in response, generates an associated time-domain vector x 1  that is transmitted from antenna  14 . Similarly, symbol pairs [X n+1 *, X n *] are grouped into another N—symbol vector that is supplied to IFFT  20  block, which in response, generates an associated time-domain vector x 2  that is transmitted from antenna  16 . 
   As is seen from  FIG. 1  and described above, the space-frequency encoding is performed on the input symbols, i.e., in the frequency domain. Accordingly, space-encoder  12  is required to generate two different streams and hence two separate IFFT blocks  18 ,  20 , each associated with a different transmit antenna, are required for every transmitted OFDM symbol. 
   BRIEF SUMMARY OF THE DISCLOSURE 
   In an embodiment, a transmitter comprises at least two antennas and a processor. The processor causes a reversed complex conjugate of a second block to be transmitted from a first antenna during a first time slot and a first block to be transmitted from the first antenna during a second time slot after the first time slot, and causes the reversed complex conjugate of the first block to be transmitted from a second antenna during the first time slot and the second block to be transmitted from the second antenna during the second time slot. 
   In another embodiment, a method comprises generating a first block comprises a first sequence, generating a second block comprising a second sequence, forming a reversed complex conjugate of the first block, forming a reversed complex conjugate of the second block, providing the reversed complex conjugate of the second block followed by the first block for transmission from a first antenna, and providing the reversed complex conjugate of the first block followed by the second block for transmission from a second antenna. 
   In a further embodiment, a method of generating blocks for transmission comprises generating a first block, generating a second block, forming a complex conjugate of the second block, and providing the complex conjugate of the second block in an inverse of the first order followed by the first block for transmission from a first antenna. 

   
     BRIEF DESCRIPTION OF THE DRAWINGS 
       FIG. 1  is a simplified high-level block diagram of some blocks of an OFDM transmitter, as known in the prior art. 
       FIG. 2  is a simplified high-level block diagram of a transmitter system and a receiver system in a MIMO system in accordance with one embodiment. 
       FIG. 3  is a simplified high-level block diagram of a transmitter in accordance with one embodiment. 
       FIG. 4  shows symbols with respective cyclic prefixes for transmission in accordance with one embodiment. 
       FIG. 5  is a simplified high-level block diagram of some blocks of an OFDM receiver, in accordance with one embodiment. 
   

   DETAILED DESCRIPTION OF THE DISCLOSURE 
   Referring to  FIG. 2 , a block diagram of an embodiment of a transmitter system  110  and a receiver system  150  in a MIMO system  100  is illustrated. At transmitter system  110 , traffic data for a number of data streams is provided from a data source  112  to a transmit (TX) data processor  114 . In an embodiment, each data stream is transmitted over a respective transmit antenna. TX data processor  114  formats, codes, and interleaves the traffic data for each data stream based on a particular coding scheme selected for that data stream to provide coded data. 
   The coded data for each data stream may be multiplexed with pilot data using, for example, time division multiplexing (TDM) or code division multiplexing (CDM). The pilot data is typically a known data pattern that is processed in a known manner (if at all), and may be used at the receiver system to estimate the channel response. The multiplexed pilot and coded data for each data stream is then modulated (i.e., symbol mapped) based on a particular modulation scheme (e.g., BPSK, QSPK, M-PSK, or M-QAM) selected for that data stream to provide modulation symbols. The data rate, coding, and modulation for each data stream may be determined by controls provided by a processor  130 . 
   The modulation symbols for all data streams are then provided to a TX MIMO processor  120 , which may further process the modulation symbols (e.g., for OFDM). TX MIMO processor  120  then provides N T  modulation symbol streams to N T  transmitters (TMTR)  122   a  through  122   t . In an embodiment, TX MIMO processor  120  may provide the modulation symbols so that transmission symbols are arragned to be transmitted in pairs, where each pair is transmitted from at least two antennas and with each symbol being a sequentially reversed complex conjugate version of a symbol that is transmitted from another antenna as part of a same pair. 
   Each transmitter  122  receives and processes symbol pairs in the form of symbol streams and provides one or more analog signals, and further conditions (e.g., amplifies, filters, and upconverts) the analog signals to provide a modulated signal suitable for transmission over the MIMO channel. N T  modulated signals from transmitters  122   a  through  122   t  are then transmitted from N T  antennas  124   a  through  124   t , respectively. 
   At receiver system  150 , the transmitted modulated signals are received by N R  antennas  152   a  through  152   r , and the received signal from each antenna  152  is provided to a respective receiver (RCVR)  154 . Each receiver  154  conditions (e.g., filters, amplifies, and downconverts) a respective received signal, digitizes the conditioned signal to provide samples, and further processes the samples to provide a corresponding “received” symbol stream. 
   An RX MIMO/data processor  160  then receives and processes the N R  received symbol streams from N R  receivers  154  based on a particular receiver processing technique to provide N T  “detected” symbol streams. The processing by RX MIMO/data processor  160  is described in further detail below. Each detected symbol stream includes symbols that are estimates of the modulation symbols transmitted for the corresponding data stream. RX MIMO/data processor  160  then demodulates, deinterleaves, and decodes each detected symbol stream to recover the traffic data for the data stream. The processing by RX MIMO/data processor  160  is complementary to that performed by TX MIMO processor  120  and TX data processor  114  at transmitter system  110 . 
   RX MIMO processor  160  may derive an estimate of the channel response between the N T  transmit and N R  receive antennas, e.g., based on the pilot multiplexed with the traffic data. The channel response estimate may be used to perform space or space/time processing at the receiver. RX MIMO processor  160  may further estimate the signal-to-noise-and-interference ratios (SNRs) of the detected symbol streams, and possibly other channel characteristics, and provides these quantities to a processor  170 . RX MIMO/data processor  160  or processor  170  may further derive an estimate of the “operating” SNR for the system, which is indicative of the conditions of the communication link. Processor  170  then provides channel state information (CSI), which may comprise various types of information regarding the communication link and/or the received data stream. For example, the CSI may comprise only the operating SNR. The CSI is then processed by a TX data processor  178 , modulated by a modulator  180 , conditioned by transmitters  154   a  through  154   r , and transmitted back to transmitter system  110 . 
   At transmitter system  110 , the modulated signals from receiver system  150  are received by antennas  124 , conditioned by receivers  122 , demodulated by a demodulator  140 , and processed by a RX data processor  142  to recover the CSI reported by the receiver system. The reported CSI is then provided to processor  130  and used to (1) determine the data rates and coding and modulation schemes to be used for the data streams and (2) generate various controls for TX data processor  114  and TX MIMO processor  120 . 
   Processors  130  and  170  direct the operation at the transmitter and receiver systems that they are coupled with including the appropriate transmit and receive data processors. Memories  132  and  172  provide storage for program codes and data used by processors  130  and  170 , respectively. 
   Referring to  FIG. 3 , a functional block diagram of a transmitter system including multiple transmit antennae according to one embodiment is illustrated. In one embodiment, a separate data rate and coding and modulation scheme may be used for each of the N T  data streams to be transmitted on the N T  transmit antennae (i.e., separate coding and modulation on a per-antenna basis). The specific data rate and coding and modulation schemes to be used for each transmit antenna may be determined based on controls provided by processor  130  ( FIG. 2 ), and the data rates may be determined as described above. 
   Transmitter unit  100  includes, in one embodiment, a transmit data processor  202  that receives, codes, and modulates each data stream in accordance with a separate coding and modulation scheme to provide modulation symbols and transmit MIMO Transmit data processor  202  and transmit processor  204  are one embodiment of transmit data processor  114  and transmit processor  120 , respectively, of  FIG. 2 . 
   In one embodiment, as shown in  FIG. 3 , transmit data processor  202  includes demultiplexer  210 , N T  encoders  212   a  through  212   t , and N T  channel interleavers  214   a  through  214   t  (i.e., one set of demultiplexers, encoders, and channel interleavers for each transmit antenna). Demultiplexer  210  demultiplexes data (i.e., the information bits) into N T  data streams for the N T  transmit antennae to be used for data transmission. The N T  data streams may be associated with different data rates, as determined by rate control functionality, which in one embodiment may be provided by processor  130  or  170  ( FIG. 2 ). Each data stream is provided to a respective encoder  212   a  through  212   t.    
   Each encoder  212   a  through  212   t  receives and codes a respective data stream based on the specific coding scheme selected for that data stream to provide coded bits. In one embodiment, the coding may be used to increase the reliability of data transmission. The coding scheme may include in one embodiment any combination of cyclic redundancy check (CRC) coding, convolutional coding, Turbo coding, block coding, or the like. The coded bits from each encoder  212   a  through  212   t  are then provided to a respective channel interleaver  214   a  through  214   t , which interleaves the coded bits based on a particular interleaving scheme. The interleaving provides time diversity for the coded bits, permits the data to be transmitted based on an average SNR for the transmission channels used for the data stream, combats fading, and further removes correlation between coded bits used to form each modulation symbol. 
   The coded and interleaved bits from each channel interleaver  214   a  through  214   t  are provided to a respective symbol mapping block  222   a  through  222   t , of transmit processor  204 , which maps these bits to form modulation symbols. 
   The particular modulation scheme to be implemented by each symbol mapping block  222   a  through  222   t  is determined by the modulation control provided by processor  130  ( FIG. 1 ). Each symbol mapping block  222   a  through  222   t  groups sets of q j  coded and interleaved bits to form non-binary symbols, and further maps each non-binary symbol to a specific point in a signal constellation corresponding to the selected modulation scheme (e.g., QPSK, M-PSK, M-QAM, or some other modulation scheme). Each mapped signal point corresponds to an M j -ary modulation symbol, where M j  corresponds to the specific modulation scheme selected for the j-th transmit antenna and M j =2 q     j   . Symbol mapping blocks  422   a  through  222   t  then provide N T  streams of modulation symbols. 
   In the specific embodiment illustrated in  FIG. 3 , transmit processor  304  also includes a modulator  224  and inverse Fast Fourier transform (IFFT) block  226   a  through  226   t , along with symbol mapping blocks  222   a  through  222   t . Modulator  224  modulates the samples to form the modulation symbols for the N T  streams on the proper subbands and transmit antennas. In addition modulator  224  provides each of the N T  symbol streams at a proscribed power level. In one embodiment, modulator  224  may modulate symbols according to a FH sequence controlled by a processor, e.g. processor  130  or  170 . In such an embodiment, the frequencies with which the N T  symbol streams are modulated may vary for each group or block of symbols, frame, or portion of a frame of a transmission cycle. 
   Each IFFT block  226   a  through  226   t  receives a respective modulation symbol stream from modulator  224 . Each IFFT block  226   a  through  226   t  groups sets of NF modulation symbols to form corresponding modulation symbol vectors, and converts each modulation symbol vector into its time-domain representation (which is referred to as an OFDM symbol) using the inverse fast Fourier transform. IFFT blocks  226   a  through  226   t  may be designed to perform the inverse transform on any number of frequency subchannels (e.g., 8, 16, 32, . . . , N F ,). Each time-domain representation of the modulation symbol vector generated by IFFT blocks  226   a  through  226   t  is provided to encoder  228 . 
   In the embodiment of  FIG. 2 , modulated data includes symbols which may provided in a symbol stream, e.g. symbols X i , X i+1 , . . . X n . IFFT blocks  226   a  through  226   t  receive the symbol stream, symbols X i , X i+1 , . . . X n  and provide time domain sequences of each symbol that correspond to the samples of each symbol, e.g. sequence x i  for symbol X i , sequence x i+1  for symbol X i+1 , and sequence x n  for symbol X n . Encoder  228 , using the received sequences x i , x i+1 , . . . x n  generates sequences {tilde over (x)} i , −{tilde over (x)} i+1 , . . . −{tilde over (x)} N  Where sequence {tilde over (x)} i  is a reversed complex conjugate sequence of sequence x i , sequence {tilde over (x)} i+1  is a reversed complex conjugate sequence associated with sequence x i+1 , etc. Encoder  228  provides symbol pairs to transmitters  230   a  through  232   t , so that any symbol pair that is transmitted from two or more antennas is transmitted in the form of −{tilde over (x)} i+1 , x i  from a first antenna, e.g. antenna  232   a , in first and second time slots and is transmitted in the form of {tilde over (x)} i , x i+1  from a second antenna, e.g. antenna  232   b , in the first and second time slots. In other words, during time slot i, sequence −{tilde over (x)} i+1  is transmitted from transmit antenna  232   a  and sequence {tilde over (x)} i  is transmitted from transmit antenna  232   b . At time slot i+1, sequence {tilde over (x)} i  is transmitted from transmit antenna  232   a  and sequence x i+1  is transmitted from transmit antenna  232   a.    
   For a symbol stream or group of symbols X i (n)=X i (n), n=0, 1, . . . , N−1, is the n-th information symbol in the i-th OFDM symbol. The sequence for the i-th OFDM symbol may be defined, in vector format, as
 
 X   i   =[X   i (0) X   i (1) . . .  X   i ( N− 1)] T   (2.1)
 
   Let x i (k), k=0, 1, . . . , N−1 represent the corresponding IFFT output (i.e. the time domain samples of the symbol X i ), and let the symbol energy E s =E{X i (n)X i *(n)} be 1, i.e. the maximum energy allotted for transmission of the symbol. Further, let sequences x i  and x i+1  represent corresponding IFFT of consecutive OFDM symbols X i  and X i+1 . Using {tilde over (x)} i  and x i+1 , sequences {tilde over (x)} i  and −{tilde over (x)} i+1  are defined as below:
 
 {tilde over (x)}   i ( k )=   x     i ( N−K ) 0 ≦k≦N− 1
 
 {tilde over (x)}   i+1 ( k )=   x     i+1 ( N−K ) 0 ≦k≦N− 1  (2.2)
 
where  (•)  denotes a complex conjugate operation for scalars and element by element complex conjugate for vectors and matrices. Accordingly, {tilde over (x)} i  and −{tilde over (x)} i+1  are ordinally reversed and element by element complex conjugated sequences of x i  and x i+1 , respectively.
 
   The output of encoder  228  is coupled to cyclic prefix generators  230   a  through  230   t . The cyclic prefix generators  230   a  through  230   t  pre-pending a prefix of a fixed number of samples, which are generally a number of samples from the end of the OFDM symbol, to the N S  samples that constitute an OFDM symbol to form a corresponding transmission symbol. The prefix is designed to improve performance against deleterious path effects such as channel dispersion caused by frequency selective fading. 
   The symbols output by cyclic prefix generators  230   a  through  230   t  are provided to an associated transmitter  232   a  through  232   t  which causes the symbols to be transmitted by antennas  234   a  through  234   t.    
   It should be noted that while the above discussion refers to X i  and X i+1  as symbols and x i  and x i+1  as time domain sequences of symbols X i  and X i+1 , that the same approach may be applied to blocks of symbols or sequences. For example, X i  and X i+1  may each represent N symbols, where N may greater than or less than 1. In such a case, x i  and x i+1  would represent time-domain sequences of N symbols and {tilde over (x)} i  and {tilde over (x)} i+1  are reversed complex conjugates of N symbols. 
   While the above discussion relates to an embodiment utilizing two symbols transmitted over two time-slots, a greater number of symbols over a larger number of time slots may also be utilized in accordance with the embodiments described herein. In such embodiments, the matrix, which is defined by the number of transmission symbols and the number of antennas, is a unitary matrix. This allows for different rates to be utilized for transmission, i.e. n transmit symbols per m transmit antennas where n&gt;m. For example, a three antenna system consisting of antennas a 1 , a 2 , and a 3  may transmit symbols x 1 , x 2 , x 3 , and x 4  may utilize the following transmission scheme which is defined by an x by a matrix M t   
             (           x   1           x   2           x   3               -     x   2             x   1           -     x   4                 -     x   3             x   4           x   1               -     x   4             -     x   3             x   2                 x   ~     1             x   ~     2             x   ~     3               -       x   ~     2               x   ~     1           -       x   ~     4                 -       x   ~     3               x   ~     4             x   ~     1               -       x   ~     4             -       x   ~     3               x   ~     2           )     ⁢           ⁢     M   t           
where {tilde over (x)} 1 , {tilde over (x)} 2 , {tilde over (x)} 3 , and {tilde over (x)} 4  are time reversed complex conjugates of symbols x 1 , x 2 , x 3 , and x 4 , respectively, −x 2 , −x 3 , and −X 4  are inverted symbols x 2 , x 3 , and x 4 , respectively, and −{tilde over (x)} 2 , −{tilde over (x)} 3 , and −{tilde over (x)} 4  are inverted complex conjugates of symbols x 2 , x 3 , and x 4 , respectively.
 
The order of the symbols may be provided by encoder  228  in the order specified in M t  or any other scheme based upon a unitary matrix.
 
In some embodiments, encoder  228  may comprise a memory, e.g. one or more buffers, that stores the time domain symbols, their complex conjugates, their inverses, and inverted complex conjugates, and then may output them based upon a scheme based upon a unitary matrix to a plurality of transmit antennas.
 
   Referring to  FIG. 4 , symbols with respective cyclic prefixes for transmission in accordance with one embodiment are illustrated. At time slot i, time-domain sequence x i  is appended with its cyclic prefix and transmitted from a first transmit antenna, and time-domain sequence −{tilde over (x)} i+1  is appended with its cyclic prefix and transmitted from a second transmit antenna. At time slot i+1, time-domain sequence x i+1  is appended with its cyclic prefix and transmitted from the fist transmit antenna, and time-domain sequence {tilde over (x)}x i  is appended with its cyclic prefix and transmitted from the second transmit antenna. 
   Referring to  FIG. 5 , a simplified high-level block diagram of some blocks of an OFDM receiver, in accordance with one embodiment is illustrated. Receiver 400 is adapted to receive sequences y i  and y i+1  via receive antenna  402  and to demodulate and decode the sequences. As seen from  FIG. 5 , receiver  400  is shown as including, in part, a discrete Fourier transform block  404 , processing blocks  406  and  408 , each of which provides a complex conjugate function of the function the block receives, decoder/equalizer block  410 , and block  412  which performs time reverse operation. 
   In transmission of the symbols or blocks, h m (k) represents the symbol spaced channel impulse response for two transmit antennas m, m=1,2, where the first transmit antenna is represented by m=1 and the second transmit antenna is represented by m=2. In this case, h m (k) may be defined as: 
   
     
       
         
           
             
               
                 
                   
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   At the receiver of the blocks or symbols, sequences y i  and y i+1  represent the received time-domain sequences corresponding to time slots i and i+1, respectively, that are transmitted sequences x i  and x i+1  with their respective cyclic prefixes removed. 
   Sequences y i  and y i+1  received by receive antenna  402  are shown below:
 
 y   i   =[y   i (0) y   i (1) . . .  y   i ( N− 1)] T  
 
 y   i+1   =[y   i+1 (0) y   i+1 (1) . . .  y   i+1 ( N− 1)] T   (2.4)
 
and may be expressed as shown below:
 
 y   i   =H   1   ·x   i   −H   2   ·{tilde over (x)}   i+1   +v   i  
 
 y   i+1   =H   1   ·x   i+1   +H   2   ·{tilde over (x)}   i   +v   i+1   (2.5)
 
where both sequences v i  and v i+1  are white independent identically distributed (i.i.d.) Gaussian random noise vectors with covariance σ 2 ×I. Accordingly, the signal to noise ratio SNR is:
 
                 SNR   =     ρ   =     1     σ   2                 (   2.6   )               
where H m ,m=1,2 is the channel matrix corresponding to transmit antenna m and is given by:
 
   
     
       
         
           
             
               
                 
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                             0 
                           
                         
                       
                       
                         
                           h 
                           
                             m 
                             , 
                             1 
                           
                         
                       
                       
                         … 
                       
                       
                         
                           h 
                           
                             m 
                             , 
                             L 
                           
                         
                       
                     
                     
                       
                         
                           h 
                           L 
                         
                       
                       
                         0 
                       
                       
                         … 
                       
                       
                         … 
                       
                       
                         
                           h 
                           o 
                         
                       
                       
                         … 
                       
                       
                         
                           h 
                           
                             L 
                             - 
                             1 
                           
                         
                       
                     
                     
                       
                         ⋮ 
                       
                       
                         ⋰ 
                       
                       
                         ⋰ 
                       
                       
                         ⋰ 
                       
                       
                         ⋰ 
                       
                       
                         ⋰ 
                       
                       
                         ⋮ 
                       
                     
                     
                       
                         
                           h 
                           
                             m 
                             , 
                             1 
                           
                         
                       
                       
                         … 
                       
                       
                         
                           h 
                           
                             m 
                             , 
                             L 
                           
                         
                       
                       
                         0 
                       
                       
                         … 
                       
                       
                         0 
                       
                       
                         
                           h 
                           
                             m 
                             , 
                             0 
                           
                         
                       
                     
                   
                   ] 
                 
               
             
             
               
                 ( 
                 2.7 
                 ) 
               
             
           
         
       
     
   
   The matrix H m  is circulant and has the following eigenvalue decomposition:
 
 H   m   =Q′Λ   m   Q   (2.8)
 
where Q is the N×N discrete Fourier transform matrix (DFT) as shown below:
 
                   Q   ⁡     (     k   ,   n     )       =       1     N       ·     ⅇ       -   j     ⁢           ⁢   2   ⁢   π   ⁢           ⁢     kn   /   N                   (   2.9   )               
and Λ m  is the diagonal eigenvalue matrix whose diagonal is the N point DFT of h m,0 ,h m,1 , . . . ,h m,L .
 
   Using the DFT property that
 
 DFT ( {tilde over (x)}   i )= DFT (   x     i   [−n]   N )=  X   i  
 
where by definition:
 
 X   i   =DFT ( x   i )= Q·x   i  
 
 V   i   =DFT ( v   i )= Q·v   i  
 
the following expression (2.7) is attained. FFT block  402  receives symbol (signal vector) y i  and, in response, generates signal vector Y i . FFT block  402  also receives signal vector y i+1  and, in response, generates signal vector Y i+1 . Signal vectors Y i  and Y i+1  are expressed as shown below:
 
   
     
       
         
           
             
               
                 
                   
                     Y 
                     i 
                   
                   ⁢ 
                   • 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     Q 
                     · 
                     
                       y 
                       i 
                     
                   
                 
                 = 
                 
                   
                     
                       Q 
                       · 
                       
                         Q 
                         * 
                       
                     
                     ⁢ 
                     
                       Λ 
                       1 
                     
                     ⁢ 
                     
                       Q 
                       · 
                       
                         x 
                         i 
                       
                     
                   
                   - 
                   
                     
                       Q 
                       · 
                       
                         Q 
                         * 
                       
                     
                     ⁢ 
                     
                       Λ 
                       2 
                     
                     ⁢ 
                     
                       Q 
                       · 
                       
                         
                           x 
                           ~ 
                         
                         
                           i 
                           + 
                           1 
                         
                       
                     
                   
                   + 
                   
                     Q 
                     · 
                     
                       v 
                       i 
                     
                   
                 
               
             
             
               
                 ( 
                 2.10 
                 ) 
               
             
           
           
             
               
                 
                     
                 
                 ⁢ 
                 
                   = 
                   
                     
                       
                         Λ 
                         1 
                       
                       ⁢ 
                       
                         X 
                         i 
                       
                     
                     - 
                     
                       
                         Λ 
                         2 
                       
                       ⁢ 
                       
                         
                           X 
                           _ 
                         
                         
                           i 
                           + 
                           1 
                         
                       
                     
                     + 
                     
                       V 
                       i 
                     
                   
                 
               
             
             
               
                   
               
             
           
           
             
               
                 
                   
                     Y 
                     
                       i 
                       + 
                       1 
                     
                   
                   ⁢ 
                   • 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     Q 
                     · 
                     
                       y 
                       
                         i 
                         + 
                         1 
                       
                     
                   
                 
                 = 
                 
                   
                     
                       Q 
                       · 
                       
                         Q 
                         * 
                       
                     
                     ⁢ 
                     
                       Λ 
                       1 
                     
                     ⁢ 
                     
                       Q 
                       · 
                       
                         x 
                         
                           i 
                           + 
                           1 
                         
                       
                     
                   
                   + 
                   
                     
                       Q 
                       · 
                       
                         Q 
                         * 
                       
                     
                     ⁢ 
                     
                       Λ 
                       2 
                     
                     ⁢ 
                     
                       Q 
                       · 
                       
                         
                           x 
                           ~ 
                         
                         i 
                       
                     
                   
                   + 
                   
                     Q 
                     · 
                     
                       v 
                       
                         i 
                         + 
                         1 
                       
                     
                   
                 
               
             
             
               
                   
               
             
           
           
             
               
                 
                     
                 
                 ⁢ 
                 
                   = 
                   
                     
                       
                         Λ 
                         1 
                       
                       ⁢ 
                       
                         X 
                         
                           i 
                           + 
                           1 
                         
                       
                     
                     - 
                     
                       
                         Λ 
                         2 
                       
                       ⁢ 
                       
                         
                           X 
                           _ 
                         
                         i 
                       
                     
                     + 
                     
                       V 
                       
                         i 
                         + 
                         1 
                       
                     
                   
                 
               
             
             
               
                   
               
             
           
         
       
     
   
   Signal vector Y i  is delivered to decoder/equalizer block  410 . Signal Y i+1  is delivered to processing block  104 , which in response, generates and delivers to decoder/equalizer block  410 , complex conjugate vector signal  Y   i+1 . 
   Expression (2.10) may be written as: 
                         Y   i     =     [           Y   i                 Y   _       i   +   1             ]                 =         [           Λ   1           -     Λ   2                 Λ   2   *           Λ   1   *           ]     ⁡     [           X   i                 X   _       i   +   1             ]       +     [           V   i                 V   _       i   +   1             ]                   =       H   ·     X   i       +     V   i                     (   2.11   )               
where Y i  is a 2N×1 vector. Since the DFT matrix Q is an orthogonal matrix, the noise vector V i  is also white. Hence decoder/equalizer block  410 , which is adapted to perform a minimum mean-squared error (MMSE) as well as decoding/equalizing filter operation, is characterized by the following matrix filter W:
 
   
     
       
         
           
             
               
                 W 
                 = 
                 
                   
                     
                       [ 
                       
                         
                           H 
                           · 
                           
                             H 
                             * 
                           
                         
                         + 
                         
                           
                             1 
                             ρ 
                           
                           · 
                           I 
                         
                       
                       ] 
                     
                     
                       - 
                       1 
                     
                   
                   ⁢ 
                   H 
                 
               
             
             
               
                 ( 
                 2.12 
                 ) 
               
             
           
         
       
     
   
   Assume that the channel impulse response associated with the first and second transmission channels is respectively represented by Λ 1  and Λ 2 . Matrix D is defined as follows:
 
 D=Λ   I Λ 1 *+Λ 2 Λ 2 *
 
Matrix D is an N×N diagonal matrix whose (n,n) element d nn  is shown below:
 
|Λ 1 (i,i)| 2 +|Λ 2 (i,i)| 2  
 
Matrix {tilde over (D)} is defined as:
 
             D   ~     =     D   +       1   ρ     ⁢   I             
where ρ is the SNR. Accordingly:
   {tilde over (D)}   −1 Λ m =Λ m   {tilde over (D)}   −1  and  {tilde over (D)}   −1 Λ m *=Λ m   *{tilde over (D)}   −1 . 
Therefore, matrix W may be defined as shown below:
 
   
     
       
         
           
             
               
                 
                   
                     
                       W 
                       = 
                       
                         
                           [ 
                           
                             
                               
                                 
                                   Λ 
                                   1 
                                 
                               
                               
                                 
                                   - 
                                   
                                     Λ 
                                     2 
                                   
                                 
                               
                             
                             
                               
                                 
                                   Λ 
                                   2 
                                   * 
                                 
                               
                               
                                 
                                   Λ 
                                   1 
                                   * 
                                 
                               
                             
                           
                           ] 
                         
                         ⁡ 
                         
                           [ 
                           
                             
                               
                                 
                                   
                                     D 
                                     ~ 
                                   
                                   
                                     - 
                                     1 
                                   
                                 
                               
                               
                                 0 
                               
                             
                             
                               
                                 0 
                               
                               
                                 
                                   
                                     D 
                                     ~ 
                                   
                                   
                                     - 
                                     1 
                                   
                                 
                               
                             
                           
                           ] 
                         
                       
                     
                   
                 
                 
                   
                     
                       = 
                       
                         
                           W 
                           d 
                         
                         · 
                         
                           W 
                           e 
                         
                       
                     
                   
                 
               
             
             
               
                 ( 
                 2.13 
                 ) 
               
             
           
         
       
     
   
   As seen from expression (2.13), the matrix filter W includes two parts. The first part, W d , represents the decoding operation of the space-time block code. The second part, W e , represents the MMSE frequency domain equalizer part. Applying matrix filter W to the received signal vector Y i  provides the following: 
   
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   Z 
                                   i 
                                 
                               
                             
                             
                               
                                 
                                   Z 
                                   
                                     i 
                                     + 
                                     1 
                                   
                                 
                               
                             
                           
                           ] 
                         
                         ⁢ 
                         • 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         W 
                         ⁢ 
                         
                           
                             y 
                             * 
                           
                           · 
                           
                             Y 
                             i 
                           
                         
                       
                       = 
                       
                         
                           
                             D 
                             ~ 
                           
                           
                             - 
                             1 
                           
                         
                         · 
                         
                           [ 
                           
                             
                               
                                 
                                   
                                     
                                       Λ 
                                       1 
                                       * 
                                     
                                     ⁢ 
                                     
                                       Y 
                                       i 
                                     
                                   
                                   + 
                                   
                                     
                                       Λ 
                                       2 
                                     
                                     ⁢ 
                                     
                                       
                                         Y 
                                         _ 
                                       
                                       
                                         i 
                                         + 
                                         1 
                                       
                                     
                                   
                                 
                               
                             
                             
                               
                                 
                                   
                                     
                                       - 
                                       
                                         Λ 
                                         2 
                                         * 
                                       
                                     
                                     ⁢ 
                                     
                                       Y 
                                       i 
                                     
                                   
                                   + 
                                   
                                     
                                       Λ 
                                       1 
                                     
                                     ⁢ 
                                     
                                       
                                         Y 
                                         _ 
                                       
                                       
                                         i 
                                         + 
                                         1 
                                       
                                     
                                   
                                 
                               
                             
                           
                           ] 
                         
                       
                     
                   
                 
                 
                   
                     
                       = 
                       
                         
                           
                             
                               D 
                               ~ 
                             
                             
                               - 
                               1 
                             
                           
                           ⁢ 
                           
                             D 
                             · 
                             
                               [ 
                               
                                 
                                   
                                     
                                       X 
                                       i 
                                     
                                   
                                 
                                 
                                   
                                     
                                       
                                         X 
                                         _ 
                                       
                                       
                                         i 
                                         + 
                                         1 
                                       
                                     
                                   
                                 
                               
                               ] 
                             
                           
                         
                         + 
                         
                           [ 
                           
                             
                               
                                 
                                   
                                     V 
                                     ~ 
                                   
                                   i 
                                 
                               
                             
                             
                               
                                 
                                   
                                     V 
                                     ~ 
                                   
                                   
                                     i 
                                     + 
                                     1 
                                   
                                 
                               
                             
                           
                           ] 
                         
                       
                     
                   
                 
               
             
             
               
                 ( 
                 2.14 
                 ) 
               
             
           
         
       
     
   
   Vectors Z i  and Z i+1  are generated by decoder/equalizer block  410 . Expression (2.14) may be rewritten as shown below: 
                   [           Z   i                 Z   _       i   +   1             ]     =           D   ~       -   1       ⁢     D   ·     [           X   i               X     i   +   1             ]         +     [             V   ~     i                 V     ~   _         i   +   1             ]               (   2.15   )               
It is thus seen that matrix {tilde over (D)} −1 D is a diagonal matrix whose (n,n) element g nn  is shown below:
 
                   g   nn     =                  Λ   1     ⁡     (     n   ,   n     )            2     +              Λ   2     ⁡     (     n   ,   n     )            2                    Λ   1     ⁡     (     n   ,   n     )            2     +              Λ   2     ⁡     (     n   ,   n     )            2     +     1   /   ρ                 (   2.16   )               
It is also seen that the following expression applies:
   E{{tilde over (V)}   i   {tilde over (V)}   i   *}=E{{tilde over (  V )}   i+1   {tilde over (  V )}   i+1   *}=R   v    
where R v  is an (n,n) diagonal matrix (n,n), whose element ξ nn  is provided by the following expression:
 
                   ζ   nn     =       1   ρ     ·                  Λ   1     ⁡     (     n   ,   n     )            2     +              Λ   2     ⁡     (     n   ,   n     )            2           (                Λ   1     ⁡     (     n   ,   n     )            2     +              Λ   2     ⁡     (     n   ,   n     )            2     +     1   /   ρ       )     2                 (   2.17   )               
where both {tilde over (V)} i  and {tilde over (  V )} i+1  are independent identically distributed (i.i.d.) Gaussian random vectors.
 
   Using expressions (2.15), (2.16), and (2.17), the decision statistic {circumflex over (X)} i (n) for symbol X i (n), which is the n-th information symbol transmitted in the i-th OFDM block, may be expressed as shown below:
 
 s   i ( n )= g   nn   ·X   i ( n )+ v   i ( n )  (2.18)
 
and the corresponding signal-to-noise ratio (SNR) SNR i (n) may be expressed as shown below:
 
   
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           SNR 
                           i 
                         
                         ⁡ 
                         
                           ( 
                           n 
                           ) 
                         
                       
                       = 
                       
                         
                           g 
                           nn 
                           2 
                         
                         
                           ζ 
                           nn 
                         
                       
                     
                   
                 
                 
                   
                     
                       = 
                       
                         ρ 
                         · 
                         
                           ( 
                           
                             
                               
                                  
                                 
                                   
                                     Λ 
                                     1 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       n 
                                       , 
                                       n 
                                     
                                     ) 
                                   
                                 
                                  
                               
                               2 
                             
                             + 
                             
                               
                                  
                                 
                                   
                                     Λ 
                                     2 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       n 
                                       , 
                                       n 
                                     
                                     ) 
                                   
                                 
                                  
                               
                               2 
                             
                           
                           ) 
                         
                       
                     
                   
                 
                 
                   
                     
                       = 
                       
                         SNR 
                         · 
                         
                           ( 
                           
                             
                               
                                  
                                 
                                   
                                     Λ 
                                     1 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       n 
                                       , 
                                       n 
                                     
                                     ) 
                                   
                                 
                                  
                               
                               2 
                             
                             + 
                             
                               
                                  
                                 
                                   
                                     Λ 
                                     2 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       n 
                                       , 
                                       n 
                                     
                                     ) 
                                   
                                 
                                  
                               
                               2 
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
             
             
               
                 ( 
                 2.19 
                 ) 
               
             
           
         
       
     
   
   Similarly, the decision statistic {circumflex over (X)} i+1 (n) for symbol X i+1 (n), which is the n-th information symbol transmitted in the i+1 OFDM block, may be expressed as shown below:
 
 s   i+1 ( n )= g   nn   ·X   i+1 ( n )+ v   i+1 ( n )  (2.20)
 
and the corresponding signal to noise ration SNR i+1 (n) may be expressed as shown below:
 
                           SNR     i   +   1       ⁡     (   n   )       =       g   nn   2       ζ   nn                   =     ρ   ·     (                Λ   1     ⁡     (     n   ,   n     )            2     +              Λ   2     ⁡     (     n   ,   n     )            2       )                   =     SNR   ·     (                Λ   1     ⁡     (     n   ,   n     )            2     +              Λ   2     ⁡     (     n   ,   n     )            2       )                     (   2.21   )               
Thus a diversity gain of order 2 is achieved.
 
   In those cases, where more than two transmit antennas are utilized and more than two transmit symbols are grouped together, receiver includes additional outputs from decoder/equalizer block  410  which each provide the appropriate inversion and complex conjugation functions based upon the number of transmit antennas at the transmitter. 
   The functionality described with respect to  FIG. 5  may be implemented in receive processor  142  and processor  130  and receive processor  160  and processor  170 . In such a case, the functionality described with respect to elements  404 ,  406 ,  408 ,  410 , and  412  may provided in the processors. 
   Those skilled in the art will appreciate that the various illustrative logical blocks, modules, circuits, and algorithms described in connection with the embodiments disclosed herein may be implemented as electronic hardware, computer software, or combinations of both. To clearly illustrate this interchangeability of hardware and software, various illustrative components, blocks, modules, circuits, and algorithms have been described above generally in terms of their functionality. Whether such functionality is implemented as hardware or software depends upon the particular application and design constraints imposed on the overall system. Skilled artisans may implement the described functionality in varying ways for each particular application, but such implementation decisions should not be interpreted as causing a departure from the scope of the present invention. 
   The various illustrative logical blocks, processors, modules, and circuits described in connection with the embodiments disclosed herein may be implemented or performed with a general purpose processor, a digital signal processor (DSP), circuits, an application specific integrated circuit (ASIC), a field programmable gate array (FPGA) or other programmable logic device, discrete gate or transistor logic, discrete hardware components, or any combination thereof designed to perform the functions described herein. A general purpose processor may be a microprocessor, but in the alternative, the processor may be any conventional processor, processor, microprocessor, or state machine. A processor may also be implemented as a combination of devices, e.g., a combination of a DSP and a microprocessor, a plurality of microprocessors, one or more microprocessors in conjunction with a DSP core, multiple logic elements, multiple circuits, or any other such configuration. 
   The methods or algorithms described in connection with the embodiments disclosed herein may be embodied directly in hardware, in a software module executed by a processor, or in a combination of the two. A software module may reside in RAM memory, flash memory, ROM memory, EPROM memory, EEPROM memory, registers, hard disk, a removable disk, a CD-ROM, or any other form of storage medium known in the art. An exemplary storage medium is coupled to the processor such the processor can read information from, and write information to, the storage medium. In the alternative, the storage medium may be integral to the processor. The processor and the storage medium may reside in an ASIC. The ASIC may reside in a user terminal. In the alternative, the processor and the storage medium may reside as discrete components in a user terminal. 
   The previous description of the disclosed embodiments is provided to enable any person skilled in the art to make or use the present invention. Various modifications to these embodiments will be readily apparent to those skilled in the art, and the generic principles defined herein may be applied to other embodiments without departing from the spirit or scope of the invention. Thus, the present invention is not intended to be limited to the embodiments shown herein but is to be accorded the widest scope consistent with the principles and novel features disclosed herein.