Abstract:
A device includes circuitry configured to receive orthogonal frequency-division multiplexing (OFDM) symbols of a training sequence, and circuitry configured to correlate training samples of the OFDM symbols to determine a frequency offset and configured to provide the determined frequency offset to perform frequency offset compensation for at least one multiple-in-multiple-out (MIMO)-OFDM frame to correct samples of training symbols in the at least one MIMO-OFDM frame. The correlated training samples correspond to a number of receive antennas through which the training sequence was received.

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
     This application is a continuation application of and claims the benefit of priority to U.S. patent application Ser. No. 12/876,034, filed Sep. 3, 2010, now U.S. Pat. No. 8,306,094, which is a continuation of and claims the benefit of priority to U.S. patent application Ser. No. 12/179,830, filed Jul. 25, 2008, now U.S. Pat. No. 7,796,681, which is a continuation of and claims the benefit of priority to U.S. patent application Ser. No. 10/912,829, filed Aug. 5, 2004, now U.S. Pat. No. 7,408,976, which claims the benefit of priority to U.S. Provisional Application Ser. No. 60/572,934, filed on May 19, 2004, the disclosure of each of which is incorporated herein by reference in its entirety. 
    
    
     BACKGROUND 
     Wireless phones, laptops, PDAs, base stations and other systems may wirelessly transmit and receive data. A single-in-single-out (SISO) system may have two single-antenna transceivers in which one predominantly transmits and the other predominantly receives. The transceivers may use multiple data rates depending on channel quality. 
     An M R ×M T  multiple-in-multiple-out (MIMO) wireless system uses multiple transmit antennas (M T ) and multiple receive antennas (M R ) to improve data rates and link quality. The MIMO system may achieve high data rates by using a transmission signaling scheme called “spatial multiplexing,” where a data bit stream is demultiplexed into parallel independent data streams. The independent data streams are sent on different transmit antennas to obtain an increase in data rate according to the number of transmit antennas used. Alternatively, the MIMO system may improve link quality by using a transmission signaling scheme called “transmit diversity,” where the same data stream (i.e., same signal) is sent on multiple transmit antennas after appropriate coding. The receiver receives multiple copies of the coded signal and processes the copies to obtain an estimate of the received data. 
     The number of independent data streams transmitted is referred to as the “multiplexing order” or spatial multiplexing rate (r s ). A spatial multiplexing rate of r s =1 indicates pure diversity and a spatial multiplexing rate of r s =min(M R ,M T ) (minimum number of receive or transmit antennas) indicates pure multiplexing. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  is a block diagram of a wireless MIMO-OFDM communication system according to an embodiment. 
         FIG. 2  is a block diagram of a receive section in a transceiver in the MIMO-OFDM communication system. 
         FIG. 3  illustrates an IEEE 802.11a frame format. 
         FIG. 4  illustrates a frame format for the MIMO-OFDM communication system. 
         FIGS. 5A and 5B  show a flowchart describing a MIMO-OFDM receiver processing operation according to an embodiment. 
         FIG. 6  is a block diagram of a portion of a time/frequency synchronization module in the receive section of the transceiver. 
         FIG. 7  is a flowchart describing a symbol timing estimation operation according to an embodiment. 
     
    
    
     DETAILED DESCRIPTION 
       FIG. 1  illustrates a wireless multiple-in-multiple-out (MIMO) communication system  100 , which includes a first transceiver  102  with M T  transmit (T X ) antennas  104  and a second transceiver  106  with M R  receive (Rx) antennas  108 , forming an M R ×M T  MIMO system. For the description below, the first transceiver  102  is designated as a “transmitter” because the transceiver  102  predominantly transmits signals to the transceiver  106 , which predominantly receives signals and is designated as a “receiver”. Despite the designations, both “transmitter”  102  and “receiver”  106  may include a transmit section  110  and a receive section  112  and may transmit and receive data. 
     The transmitter  100  and receiver  102  may be implemented in a wireless local Area Network (WLAN) that complies with the IEEE 802.11 standards (including IEEE 802.11, 802.11a, 802.11b, 802.11g, and 802.11n). The IEEE 802.11 standards describe orthogonal frequency-division multiplexing (OFDM) systems and the protocols used by such systems. In an OFDM system, a data stream is split into multiple substreams, each of which is sent over a different subcarrier frequency (also referred to as a “tone”). For example, in IEEE 802.11a systems, OFDM symbols include 64 tones (with 48 active data tones) indexed as {−32, −31, . . . , −1, 0, 1, . . . 30, 31}, where 0 is the DC tone index. The DC tone is not used to transmit information. 
     The antennas in the transmitter  102  and receiver  106  communicate over channels in a wireless medium. In  FIG. 1 , H represents the reflections and multi-paths in the wireless medium, which may affect the quality of the channels. The system may perform channel estimation using known training sequences which are transmitted periodically (e.g., at the start of each frame). A training sequence may include one or more pilot symbols, i.e., OFDM symbols including only pilot information (which is known a priori at the receiver) on the tones. The pilot symbol(s) are inserted in front of each transmitted frame. The receiver  106  uses the known values to estimate the medium characteristics on each of the frequency tones used for data transmission. For example, on the receiver side, the signal Y k , for tone k in an SISO system can be written as,
 
 Y   k   =H   k   X   k   +N   k ,
 
     where H k  is the channel gain for the k-th tone, X k  is the symbol transmitted on the k-th tone, and N k  is the additive noise. An estimate of the channel may be determined at the receiver by dividing Y k  by X k . 
     The number of independent data streams transmitted by the transmit antennas  104  is called the “multiplexing order” or “spatial multiplexing rate” (r S ). A spatial multiplexing rate of r S =1 indicates pure diversity, and a spatial multiplexing rate of r s =min(M R ,M T ) (minimum number of receive or transmit antennas) indicates pure multiplexing. 
     In an embodiment, the MIMO system  100  may use combinations of diversity and spatial multiplexing, e.g., 1≦r s ≦min(M R ,M T ). For example, in a 4×4 MIMO system, the system may select one of four available multiplexing rates (r s ε[1, 2, 3, 4]) depending on the channel conditions. The system may change the spatial multiplexing rate as channel conditions change. 
       FIG. 2  shows a block diagram of the receive section  112 . The receive section  112  includes stages similar to those in the receive section of an IEEE 802.11a receiver, but with some modifications to account for the multiple receive antennas. 
     Signals received on the multiple receive antennas are input to corresponding processing chains  200 . Each processing chain includes a radio-frequency (RF) module  201  for RF-to-baseband and analog-to-digital (A/D) conversion. The receiver may have a common automatic gain control (AGC) for all antennas to provide minimal gain across all the receive antennas. A time/frequency synchronization module  202  performs synchronization operations and extracts information from the multiple substreams (for r S &gt;1) for channel estimation  203 . Each processing chain  200  includes a cyclic prefix removal module  204 , serial-to-parallel (S/P) converter  206 , fast Fourier transform (FFT) module  208 , a common phase error (CPE) correction module  210 , a space-frequency detection module  212 , and a parallel-to-serial (P/S) converter  214 . The multiple substreams are input to a space-frequency deinterleaver and decoding module  216  which de-interleaves the substreams into a single data stream  217  and performs soft Viterbi decoding. The single stream is then input to a descrambler  218 . 
     The MIMO-OFDM system may be compatible with IEEE 802.11a systems, and consequently may have many similarities to an IEEE 802.11a system. For example, like IEEE 802.11a systems, the MIMO-OFDM system may use 52 tones (48 data tones and 4 pilot tones), 312.5 kHz subcarrier spacing, an FFT/inverse FFT (IFFT) period of 3.2 μs, a cyclic prefix with a duration of 0.8 μs, and an OFDM symbol duration of 4.0 μs. The MIMO-OFDM system may also use a frame format  300  similar to that specified by IEEE 802.11a, which is shown in  FIG. 3 . In addition, variations of the MIMO-OFDM systems are also possible, including using different numbers of tones, different guard intervals, different forward error correction codes, and different constellations. 
     An IEEE 802.11a frame  300  includes a short preamble  301 , a long preamble  302 , a header  304 , and a DATA field  306 . The short preamble  302  includes of a short training symbol  308  with a duration of 0.8 μs repeated ten times. The short preamble may be used for signal detection, AGC, coarse frequency offset estimation, and symbol timing estimation. 
     The long preamble  302  includes two long training symbols  310 , each of duration 3.2 μs, which are separated from the short training symbols  508  by a long guard interval (1.6 μs)  312 . The long preamble is used for fine frequency offset estimation and channel estimation. 
     The header  304  includes a SIGNAL symbol  314 , which is encoded at 6 Mbps. The SIGNAL symbol  314  is 12 bits in length and includes 4 bits for the data rate, 1 reserved bit, 1 parity bit, and 6 tail bits (set to “0” to return the convolutional decoder to State 0). 
     The DATA field  306  includes OFDM symbols including the data bits to be transmitted. The data bits are prepended by a 16-bit SERVICE field and are appended by 6 tail bits. The resulting bits are appended by a number of pad bits needed to yield an integer number of OFDM symbols. 
     The MIMO-OFDM system  100  may use a similar frame format  400 , as shown in  FIG. 4 . The illustrated frame format  400  is for systems with three transmit antennas (M T =3), but can be modified for other M T . Each transmit antenna transmits a different MIMO-OFDM frame  400 . Like the IEEE 802.11a frame  300 , the MIMO-OFDM frames  400  include a short preamble  402  with a series of short training symbols  404 , a long preamble  405  with a set of two long training symbols  406 , a header  408  including a SIGNAL symbol  410 , and a data field  412 . In addition, the header  408  may include a second SIGNAL symbol (SIGNAL 2 )  414 , which may be used to transmit MIMO-OFDM-specific information, such as the number of transmit antennas and the spatial multiplexing rate. The frame may also include a supplemental long preamble  416  including M T −1 additional long training symbols to train the other antennas. 
     As in IEEE 802.11a, a short OFDM training symbol consists of 12 tones, which are modulated by the elements of the following frequency-domain sequence: 
     
       
         
           
             
               S 
               
                 
                   - 
                   26 
                 
                 , 
                 26 
               
             
             = 
             
               
                 
                   13 
                   6 
                 
               
               × 
               
                 { 
                 
                   0 
                   , 
                   0 
                   , 
                   
                     1 
                     + 
                     j 
                   
                   , 
                   0 
                   , 
                   0 
                   , 
                   0 
                   , 
                   
                     
                       - 
                       1 
                     
                     - 
                     j 
                   
                   , 
                   0 
                   , 
                   0 
                   , 
                   0 
                   , 
                   
                     1 
                     + 
                     j 
                   
                   , 
                   0 
                   , 
                   0 
                   , 
                   0 
                   , 
                   
                     
                       - 
                       1 
                     
                     - 
                     j 
                   
                   , 
                   0 
                   , 
                   0 
                   , 
                   0 
                   , 
                   
                     
                       - 
                       1 
                     
                     - 
                     j 
                   
                   , 
                   0 
                   , 
                   0 
                   , 
                   0 
                   , 
                   
                     1 
                     + 
                     j 
                   
                   , 
                   0 
                   , 
                   0 
                   , 
                   0 
                   , 
                   0 
                   , 
                   0 
                   , 
                   0 
                   , 
                   0 
                   , 
                   
                     
                       - 
                       1 
                     
                     - 
                     j 
                   
                   , 
                   0 
                   , 
                   0 
                   , 
                   0 
                   , 
                   
                     
                       - 
                       1 
                     
                     - 
                     j 
                   
                   , 
                   0 
                   , 
                   0 
                   , 
                   0 
                   , 
                   
                     1 
                     + 
                     j 
                   
                   , 
                   0 
                   , 
                   0 
                   , 
                   0 
                   , 
                   
                     1 
                     + 
                     j 
                   
                   , 
                   0 
                   , 
                   0 
                   , 
                   0 
                   , 
                   
                     1 
                     + 
                     j 
                   
                   , 
                   0 
                   , 
                   0 
                   , 
                   0 
                   , 
                   
                     1 
                     + 
                     j 
                   
                   , 
                   0 
                   , 
                   0 
                 
                 } 
               
             
           
         
       
     
     The multiplication by √{square root over (13/6)} is in order to normalize the average power of the resulting OFDM symbol. The short training symbol has a duration of 0.8 μs and is repeated 10 times. 
     As in IEEE 802.11a, a long training OFDM symbol includes 52 tones, which are modulated by the following frequency-domain BPSK training sequence:
 
 L   −26,26 ={1,1,−1,−1,1,1,−1,1,−1,1,1,1,1,1,1−1,−1,1,1,−1,1,−1,1,1,1,1,0,1,−1,−1,1,1,−1,1,−1,1,−1,−1,−1,−1,−1,1,1,−1,−1,1,−1,1,−1,1,1,1,1}
 
     The number of sets of long training symbols (or “long preambles”) may be M T  for all spatial multiplexing rates. The additional long training symbols may be used to estimate the full M R ×M T  channel matrix. This estimation may be used for link adaptation, in which modulation, coding rate, and/or other signal transmission parameters may be dynamically adapted to the changing channel conditions. 
       FIGS. 5A-5B  show a flowchart describing a MIMO-OFDM signal processing operation  500  performed by the receiver  106 .  FIG. 6  shows a portion of the time/frequency synchronization module  202 . The module  202  may perform adjacent channel rejection (ACR) and low pass filtering (LPF) operations (using ACR LPF module  602 ) and frequency offset correction on the received signals (block  502 ). 
     The time/frequency synchronization module  202  may use the short training symbols to estimate symbol timing (block  504 ). The received signal for the i-th receive antenna and n-th sample (r i,n ) may be used by a computation module  604  to generate a quantity q i,n  using the following equation:
 
 q   i,n   =sgn[Re ( r   i,n )]+ jsgn[Im ( r   i,n )].
 
     The quantity q i,n  calculated for each of the M R  antennas may be used by a summing module  606  to generate a metric P n  for the n-th sample using the following equation: 
     
       
         
           
             
               
                 P 
                 n 
               
               = 
               
                 
                   ∑ 
                   
                     i 
                     = 
                     0 
                   
                   
                     
                       M 
                       R 
                     
                     - 
                     1 
                   
                 
                 ⁢ 
                 
                   
                     ∑ 
                     
                       m 
                       = 
                       1 
                     
                     L 
                   
                   ⁢ 
                   
                     
                       q 
                       
                         i 
                         , 
                         
                           n 
                           + 
                           m 
                           - 
                           2 
                         
                       
                       * 
                     
                     ⁢ 
                     
                       Lq 
                       
                         i 
                         , 
                         
                           n 
                           + 
                           m 
                           - 
                           L 
                         
                       
                     
                   
                 
               
             
             , 
           
         
       
     
     where L denotes the number of samples in one short training symbol. A value M n  for the n-th sample may then be computed using the following equation:
 
 M   n =(1−α s ) M   n-1 +α S (| Re ( P   n )|+| Im ( P   n )|),
 
     where the parameter α s  may have a value in the range of (0, . . . , 31/64), e.g., 3/32 for a 40 MHz analog-to-digital (A/D) conversion rate. 
     The value M n  may be used to estimate the symbol timing as in IEEE 802.11a, with the exception of using a more flexible threshold τ 3  to check for n r  (right endpoint of the plateau), as shown in  FIG. 7 . Typical parameter values for a 40 MHz analog-to-digital (A/D) conversion rate are given in Table 1. 
     
       
         
               
               
               
             
               
               
               
             
           
               
                 TABLE 1 
               
               
                   
               
               
                 Parameter 
                 Exemplary value 
                 Range 
               
               
                   
               
             
             
               
                   
               
             
          
           
               
                 L 
                 32 
                   
               
               
                 τ 1   
                 0.375 
                 (0, . . . , 255/256) 
               
               
                 A 
                 64 * M R   
                   
               
               
                 τ 2   
                 0.890625 
                 (0, . . . , 255/256) 
               
               
                 τ 3   
                 0.5 
                 (0, . . . , 255/256) 
               
               
                 B 
                 15 
                 (0, . . . , 63) 
               
               
                 n D   
                 25 
                 (0, . . . , 63) 
               
               
                   
               
             
          
         
       
     
     The time/frequency synchronization module  202  may estimate the fine frequency offset (block  506 ) by correlating the received M R ×1 vectors from the two long training symbols in a long preamble using the following equation: 
     
       
         
           
             
               C 
               = 
               
                 
                   ∑ 
                   
                     i 
                     = 
                     0 
                   
                   
                     
                       M 
                       R 
                     
                     - 
                     1 
                   
                 
                 ⁢ 
                 
                   
                     ∑ 
                     
                       n 
                       = 
                       0 
                     
                     
                       N 
                       - 
                       1 
                     
                   
                   ⁢ 
                   
                     
                       r 
                       
                         i 
                         , 
                         n 
                       
                       * 
                     
                     ⁢ 
                     
                       r 
                       
                         i 
                         , 
                         
                           n 
                           + 
                           N 
                         
                       
                     
                   
                 
               
             
             , 
           
         
       
     
     where N is the FFT size. The angle of the correlation result (Δ{circumflex over (f)}) may be used to estimate the fine frequency offset using the following equation: 
     
       
         
           
             
               
                 Δ 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   f 
                   ^ 
                 
               
               = 
               
                 
                   arg 
                   ⁡ 
                   
                     ( 
                     C 
                     ) 
                   
                 
                 
                   2 
                   ⁢ 
                   π 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     NT 
                     s 
                   
                 
               
             
             , 
           
         
       
     
     where T S  is the sampling period at the FFT  208  output. The time/frequency synchronization module  202  may correct samples of long training symbols using the estimated fine frequency offset (block  508 ). 
     The channel estimation module  203  may perform channel estimation by averaging the corrected samples corresponding to the two long training symbols in a long preamble for all receive antennas  108 . The channel estimation module  203  may compute the relative CPEs of the M T  long preambles (block  510 ) using the following equation: 
     
       
         
           
             
               
                 CPE 
                 p 
               
               = 
               
                 
                   
                     ∑ 
                     
                       k 
                       ∈ 
                       
                         K 
                         pilots 
                       
                     
                   
                   ⁢ 
                   
                     
                       ∑ 
                       
                         i 
                         = 
                         0 
                       
                       
                         
                           M 
                           R 
                         
                         - 
                         1 
                       
                     
                     ⁢ 
                     
                       
                         R 
                         
                           i 
                           , 
                           k 
                         
                         
                           
                             ( 
                             0 
                             ) 
                           
                           * 
                         
                       
                       ⁢ 
                       
                         R 
                         
                           i 
                           , 
                           k 
                         
                         
                           ( 
                           p 
                           ) 
                         
                       
                     
                   
                 
                 
                   
                     ∑ 
                     
                       k 
                       ∈ 
                       
                         K 
                         pilots 
                       
                     
                   
                   ⁢ 
                   
                     
                       ∑ 
                       
                         i 
                         = 
                         0 
                       
                       
                         
                           M 
                           R 
                         
                         - 
                         1 
                       
                     
                     ⁢ 
                     
                       
                          
                         
                           R 
                           
                             i 
                             , 
                             k 
                           
                           
                             ( 
                             0 
                             ) 
                           
                         
                          
                       
                       2 
                     
                   
                 
               
             
             , 
             
               p 
               = 
               1 
             
             , 
             … 
             ⁢ 
             
                 
             
             , 
             
               
                 M 
                 T 
               
               - 
               1 
             
             , 
           
         
       
     
     where CPE 0 =1, R i,k   (p)  is the k-th FFT output for the i-th receive antenna and p-th preamble, and K pilots  are the indices of the pilot tones. As in IEEE 802.11a, tones k=−21, −7, 7, and 21 are used for pilot tones in each data MIMO-OFDM symbol. 
     The channel estimation module  203  may generate channel estimates for the pilot tones and data tones using the frequency domain BPSK (Biphase Shifting Key) long training symbols (L k ) (block  512 ). For the data tones, the subcarrier channel estimates may be calculated using the following equation:
 
 ĥ   i,k   (p)   =R   i,k   (p) /( L   k   CPE   p √{square root over ( r   S )}),
 
     where ĥ i,k   (p)  is the channel estimate for the k-th tone, i-th receive antenna, and p-th preamble. For the pilot tones, which are always sent on the same tone, the subcarrier channel estimates may be calculated using the following equation:
 
 ĥ   i,k   (0)   =R   i,k   (0)   /L   k .
 
     An equalizer  220  may perform MIMO equalization (block  514 ) by forming an M R ×r S  effective channel matrix for data tone k: 
     
       
         
           
             
               
                 H 
                 ^ 
               
               k 
             
             = 
             
               
                 [ 
                 
                   
                     
                       
                         
                           h 
                           ^ 
                         
                         
                           0 
                           , 
                           k 
                         
                         
                           ( 
                           0 
                           ) 
                         
                       
                     
                     
                       … 
                     
                     
                       
                         
                           h 
                           ^ 
                         
                         
                           0 
                           , 
                           5 
                         
                         
                           ( 
                           
                             
                               r 
                               s 
                             
                             - 
                             1 
                           
                           ) 
                         
                       
                     
                   
                   
                     
                       ⋮ 
                     
                     
                       ⋱ 
                     
                     
                       ⋮ 
                     
                   
                   
                     
                       
                         
                           h 
                           ^ 
                         
                         
                           
                             
                               M 
                               R 
                             
                             - 
                             1 
                           
                           , 
                           k 
                         
                         
                           ( 
                           0 
                           ) 
                         
                       
                     
                     
                       … 
                     
                     
                       
                         
                           h 
                           ^ 
                         
                         
                           
                             
                               M 
                               R 
                             
                             - 
                             1 
                           
                           , 
                           k 
                         
                         
                           ( 
                           
                             
                               r 
                               s 
                             
                             - 
                             1 
                           
                           ) 
                         
                       
                     
                   
                 
                 ] 
               
               . 
             
           
         
       
     
     The equalizer  220  may use a zero forcing equalizer per tone:
 
 G   k =( Ĥ   k   *Ĥ   k ) −1   Ĥ   k *.
 
     The equalizer  220  may then compute a bit-metric weight for the l-th substream, which equals the normalized post-processing signal-to-noise ratio (SNR) of the l-th substream: 
     
       
         
           
             
               
                 W 
                 
                   l 
                   , 
                   k 
                 
               
               = 
               
                 1 
                 
                   
                     [ 
                     
                       
                         ( 
                         
                           
                             
                               H 
                               ^ 
                             
                             k 
                             * 
                           
                           ⁢ 
                           
                             
                               H 
                               ^ 
                             
                             k 
                           
                         
                         ) 
                       
                       
                         - 
                         1 
                       
                     
                     ] 
                   
                   
                     l 
                     , 
                     l 
                   
                 
               
             
             , 
           
         
       
     
     where l,l represents the diagonal element. 
     The CPE correction module  210  may generate a scalar CPE estimate for the d-th data symbol (block  518 ) using the following equation: 
     
       
         
           
             
               
                 CPE 
                 
                   ( 
                   d 
                   ) 
                 
               
               = 
               
                 
                   
                     ∑ 
                     
                       k 
                       ∈ 
                       
                         K 
                         pilots 
                       
                     
                   
                   ⁢ 
                   
                     
                       ∑ 
                       
                         i 
                         = 
                         0 
                       
                       
                         
                           M 
                           R 
                         
                         - 
                         1 
                       
                     
                     ⁢ 
                     
                       
                         P 
                         k 
                       
                       ⁢ 
                       
                         
                           h 
                           ^ 
                         
                         
                           i 
                           , 
                           k 
                         
                         
                           
                             ( 
                             0 
                             ) 
                           
                           * 
                         
                       
                       ⁢ 
                       
                         Y 
                         
                           i 
                           , 
                           k 
                         
                         
                           ( 
                           d 
                           ) 
                         
                       
                     
                   
                 
                 
                   
                     ∑ 
                     
                       k 
                       ∈ 
                       
                         K 
                         pilots 
                       
                     
                   
                   ⁢ 
                   
                     
                       ∑ 
                       
                         i 
                         = 
                         0 
                       
                       
                         
                           M 
                           R 
                         
                         - 
                         1 
                       
                     
                     ⁢ 
                     
                       
                          
                         
                           
                             h 
                             ^ 
                           
                           
                             i 
                             , 
                             k 
                           
                           
                             ( 
                             0 
                             ) 
                           
                         
                          
                       
                       2 
                     
                   
                 
               
             
             , 
           
         
       
     
     where Y i,k   (p)  is the k-th output for the i-th receive antenna and d-th data symbol, and P k ε{1, −1} is the BPSK pilot symbol for tone k. 
     Using this CPE value, the CPE correction module  210  can determine the residual frequency offset tracking for the d-th OFDM data symbol (Δ{circumflex over (f)} (d) ) (block  518 ) using the following equation:
 
Δ {circumflex over (f)}   (d)   =Δ{circumflex over (f)}   (d-1)   +βIm[CPE   (d) ].
 
     The space-frequency detection module  212  may generate the k-th output for the d-th data symbol by concatenating the corresponding outputs of the M R  receive antennas:
 
 Y   k   (d)   =[Y   0,k   (d)   Y   1,k   (d)    . . . Y   M     R     -1,k   (d) ] T .
 
     The space-frequency detection module  212  may then form an equalized signal for data tone k using the zero forcing equalizer for the tone (G k ):
 
 {tilde over (X)}   k   (d)   =G   k   Y   k   (d) .
 
     The space-frequency detection module  212  may then compensate for CPE (block  520 ) using the following equations:
 
 {circumflex over (X)}   k   (d)   ={circumflex over (X)}   k   (d)   /CPE   (d) ;
 
 Ŵ   l,k   =|CPE   (d) | 2   W   l,k .
 
     The CPE compensated weight value for the l-th substream (Ŵ l,k ) may be used to obtain log-likelihood ratios (LLRs) for soft Viterbi decoding. As in IEEE 802.11a, the space-frequency deinterleaving and decoding module  216  may concatenate LLRs for each substream into a single sequence, deinterleave the LLR sequence, and decode data bits using soft Viterbi decoding (block  522 ). The descrambler may then descramble data bits using scrambler state estimation obtained from the SERVICE field (block  524 ). 
     A number of embodiments have been described. Nevertheless, it will be understood that various modifications may be made without departing from the spirit and scope of the invention. For example, blocks in the flowcharts may be skipped or performed out of order and still produce desirable results. Accordingly, other embodiments are within the scope of the following claims.