Abstract:
A radio receiver  102  is provided. The radio receiver  102  comprises one or more data Fast Fourier Transformers, each data Fast Fourier Transformer operable to perform a Fast Fourier Transform on an input data block, one or more impulse response Fast Fourier Transformers, each impulse response Fast Fourier Transformer operable to perform a Fast Fourier Transform on a channel impulse response, one or more multiplier components operable to multiply a term of the output of one of the data Fast Fourier Transformers by a term of the output of one of the impulse response Fast Fourier Transformers, and one or more Inverse Fast Fourier Transformers, each Inverse Fast Fourier Transformer operable to perform an Inverse Fast Fourier Transform based on an output of one or more of the multipliers.

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
   This application claims priority to U.S. Provisional Application No. 60/600,014 filed Aug. 9, 2004, and entitled “Frequency Domain Equalization (FDE) for HSDPA,” by Anand G. Dabak et al, incorporated herein by reference for all purposes. 

   FIELD OF THE INVENTION 
   This invention relates in general to wireless communications and more specifically, but not by way of limitation, to frequency domain equalization. 
   BACKGROUND OF THE INVENTION 
   The downlink of high speed downlink packet access (HSDPA) for third generation partnership project (3GPP) compliant systems tends to suffer from multi-path and requires the use of a chip level equalizer instead of a conventional rake receiver architecture used in communication systems such as code division multiple access (CDMA) communication systems. Chip level equalization for HSDPA is typically performed using a time domain filter computation based upon various techniques such as normalized least mean square (NLMS), 2-stage NLMS, Griffith&#39;s, pre-filter rake, channel estimation based LMS (CHEST), Fast Fourier Transform (FFT) based minimum mean squared error (MMSE) filter calculation or channel response based minimum output energy (CR-MOE). 
   HSDPA delivers superior speed, capacity and efficiency improvements. Some advantages to HSDPA include improved spectrum efficiency, improved network capacity, higher peak data rates, and improved allocation of resources. 
   SUMMARY OF THE INVENTION 
   A radio receiver is provided. The radio receiver comprises one or more data Fast Fourier Transformers, each data Fast Fourier Transformer operable to perform a Fast Fourier Transform on an input data block, one or more impulse response Fast Fourier Transformers, each impulse response Fast Fourier Transformer operable to perform a Fast Fourier Transform on a channel impulse response, one or more multiplier components operable to multiply a term of the output of one of the data Fast Fourier Transformers by a term of the output of one of the impulse response Fast Fourier Transformers, and one or more Inverse Fast Fourier Transformers, each Inverse Fast Fourier Transformer operable to perform an Inverse Fast Fourier Transform based on an output of one or more of the multipliers. 
   A method for performing frequency domain equalization on data is also provided. The method comprises (a) receiving data that needs to be demodulated, the data including a first vector Y 1  and a first channel impulse response, H 1 , (b) performing Fast Fourier Transforms (FFT) on the first vector Y 1  and on the first channel impulse response H 1 , (c) multiplying the data by a term of the FFTs of the first vector Y 1  and the first channel impulse response H 1 , and (d) performing an Inverse FFT (IFFT) on the result of (c). 
   A method for performing frequency domain equalization on data is also provided. The method comprises (a) receiving a first vector Y 1 , a first channel impulse response H 11 , and a second channel impulse response H 21 , (b) receiving a second vector Y 2 , a third channel impulse response H 12 , and a fourth channel impulse response H 22 , (c) performing Fast Fourier Transforms (FFT) on the first vector Y 1 , the second vector Y 2 , the first channel impulse response H 11 , the second channel impulse response H 21 , the third channel impulse response H 12 , and the fourth impulse response H 22 , (d) determining a first product by multiplying the FFT of the first vector Y 1  by the FFT of the first channel impulse response H 11 , a second product by multiplying the FFT of the first vector Y 1  by the FFT of the second channel impulse response H 21 , a third product by multiplying the FFT of the second vector Y 2  by the FFT of the third channel impulse response H 12 , and a fourth product by multiplying the FFT of the second vector Y 2  by the FFT of the fourth impulse response H 22 , (e) determining a first sum based on summing the first, second, third, and fourth products, (f) determining a first output based on performing an Inverse FFT (IFFT) of the first sum, (g) determining a second sum based on summing the first, second, third, and fourth products, and (h) determining a second output based on performing an IFFT of the second sum. 

   
     BRIEF DESCRIPTION OF THE DRAWINGS 
     The features of the present invention, which are believed to be novel, are set forth with particularity in the appended claims. The invention may best be understood by reference to the following description, taken in conjunction with the accompanying drawings, in the several figures of which like reference numerals identify like elements, and in which: 
       FIG. 1  is a diagram of a multiple transmit and receive antenna system for HSDPA in accordance with an embodiment of the invention. 
       FIG. 2  shows a diagram highlighting the relationship between the block of interest and a block used in FDE in accordance with an embodiment of the invention. 
       FIG. 3  shows the new blocks used in demodulation by overlapping with the previous blocks shown in  FIG. 2  in accordance with an embodiment of the invention. 
       FIG. 4  shows a flowchart for performing the frequency domain equalization in accordance with an embodiment of the invention. 
       FIG. 5  shows a diagram of a radio in accordance with an embodiment of the invention. 
       FIG. 6  shows a block diagram for a one transmit and one receive antenna system in accordance with an embodiment of the invention. 
       FIG. 7  shows a diagram of the data used at the despreader input in accordance with an embodiment of the invention. 
       FIG. 8  shows a diagram of a frequency domain equalizer for a one transmit/two receive antenna system in accordance with an embodiment of the invention. 
       FIG. 9  shows a diagram for a frequency domain equalizer for a two transmit/one receive antenna system in accordance with an embodiment of the invention. 
       FIG. 10  shows a diagram for a frequency domain equalizer for a two transmit/two receive antenna system in accordance with an embodiment of the invention. 
       FIG. 11  shows a block diagram of a frequency domain equalizer architecture in accordance with an embodiment of the invention. 
   

   DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS 
   In one embodiment of the invention, a Fast Fourier Transform (FFT) based chip level equalization is described. The equalization does not require time domain filter computation of the chip level equalizer (CLE) as prior approaches. The equalization is performed in the frequency domain using in one illustrative example, a well-known digital signal processing (DSP) technique such as the windowing, overlap and save technique. The estimated complexity for the resulting chip level equalizer is expected to be about the same as previous FFT based chip level equalization computing time domain filters. 
   Referring now to  FIG. 1 , there is shown a multiple transmit and receive antenna system for HSDPA in accordance with an embodiment of the invention. A base station  104  having two antennas is shown communicating with a mobile receiver  102  having two antennas via a multi-path channel. An FFT-based frequency domain equalization (FDE) for various antenna-setups, such as 1×1, 1×2, 2×1, 2×2 are also described in order to better understand the invention. Extension to systems with more antennas at the transmitter and/or receiver than those described is also supported by the present invention by making simple adjustments. 
   One Transmit and One Receive Antenna System Case 
   If there is only a single transmit and a single receive antenna in the HSDPA system, the channel impulse response is assumed to be given by h T ={h 1 , h 2 , . . . , h N }. In the case of pedestrian A, pedestrian B, vehicular A channels found in reality, the various intermediate taps may be zero, but the receiver is assumed to estimate the channel for the full span for the channel response. The span of the channel response may be indicated by the 1999 release of the HSDPA modem standard and this is a long term non-real time phenomenon. The received data  202  and a block of interest for FDE  204  are shown in  FIG. 2 . 
   As shown in  FIG. 2 , the block to be used in FDE is given by Y T ={y 1 , y 2 , y 3 , . . . , y M } and and the block of interest for actual demodulation  204  is a subset of the above block Y and is given by {tilde over (Y)} T ={y k , y k+1 , . . . , y M−k+1 } where k&gt;0. The received data vector Y can now be written as: 
   
     
       
         
           
             
               
                 Y 
                 = 
                 
                   
                     
                       
                         [ 
                         
                           
                             
                               
                                 h 
                                 0 
                               
                             
                             
                               
                                   
                               
                             
                             
                               
                                   
                               
                             
                             
                               
                                   
                               
                             
                             
                               
                                   
                               
                             
                             
                               
                                   
                               
                             
                             
                               
                                   
                               
                             
                           
                           
                             
                               
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                                 h 
                                 1 
                               
                             
                             
                               
                                 h 
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                                 h 
                                 1 
                               
                             
                             
                               
                                   
                               
                             
                             
                               
                                   
                               
                             
                             
                               
                                   
                               
                             
                             
                               
                                   
                               
                             
                           
                           
                             
                               
                                 h 
                                 N 
                               
                             
                             
                               
                                   
                               
                             
                             
                               
                                   
                               
                             
                             
                               
                                   
                               
                             
                             
                               
                                   
                               
                             
                             
                               
                                   
                               
                             
                             
                               
                                   
                               
                             
                           
                           
                             
                               
                                   
                               
                             
                             
                               
                                 h 
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                                 h 
                                 N 
                               
                             
                             
                               
                                   
                               
                             
                             
                               
                                 h 
                                 1 
                               
                             
                             
                               
                                 h 
                                 0 
                               
                             
                             
                               
                                   
                               
                             
                           
                           
                             
                               
                                   
                               
                             
                             
                               
                                   
                               
                             
                             
                               
                                   
                               
                             
                             
                               
                                   
                               
                             
                             
                               
                                 h 
                                 2 
                               
                             
                             
                               
                                 h 
                                 1 
                               
                             
                             
                               
                                 h 
                                 0 
                               
                             
                           
                         
                         ] 
                       
                       
                         ︸ 
                         H 
                       
                     
                     ⁡ 
                     
                       [ 
                       
                         
                           
                             
                               x 
                               0 
                             
                           
                         
                         
                           
                             
                               x 
                               1 
                             
                           
                         
                         
                           
                             
                               x 
                               2 
                             
                           
                         
                         
                           
                             
                                 
                             
                           
                         
                         
                           
                             
                                 
                             
                           
                         
                         
                           
                             
                                 
                             
                           
                         
                         
                           
                             
                                 
                             
                           
                         
                         
                           
                             
                               x 
                               M 
                             
                           
                         
                       
                       ] 
                     
                   
                   + 
                   
                     N 
                     MX1 
                   
                 
               
             
             
               
                 Equation 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   ( 
                   1 
                   ) 
                 
               
             
           
         
       
     
   
   The channel matrix H is a Toeplitz matrix of dimension M×M. We now propose to make the matrix H circulant by copying certain terms of the channel response in the North East (top right) corner of the matrix H as given in the equation (2) below and call the resulting receive vector as Y circ  as given in equation (3) below: 
   
     
       
         
           
             
               
                 
                   
                     H 
                     circ 
                   
                   = 
                   
                     [ 
                     
                       
                         
                           
                             h 
                             0 
                           
                         
                         
                           
                               
                           
                         
                         
                           
                               
                           
                         
                         
                           
                               
                           
                         
                         
                           
                               
                           
                         
                         
                           
                             h 
                             2 
                           
                         
                         
                           
                             h 
                             1 
                           
                         
                       
                       
                         
                           
                             h 
                             1 
                           
                         
                         
                           
                             h 
                             0 
                           
                         
                         
                           
                               
                           
                         
                         
                           
                               
                           
                         
                         
                           
                             h 
                             N 
                           
                         
                         
                           
                               
                           
                         
                         
                           
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                             2 
                           
                         
                       
                       
                         
                           
                               
                           
                         
                         
                           
                             h 
                             1 
                           
                         
                         
                           
                             h 
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                             1 
                           
                         
                         
                           
                               
                           
                         
                         
                           
                               
                           
                         
                         
                           
                               
                           
                         
                         
                           
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                             N 
                           
                         
                       
                       
                         
                           
                             h 
                             N 
                           
                         
                         
                           
                               
                           
                         
                         
                           
                               
                           
                         
                         
                           
                               
                           
                         
                         
                           
                               
                           
                         
                         
                           
                               
                           
                         
                         
                           
                               
                           
                         
                       
                       
                         
                           
                               
                           
                         
                         
                           
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                             1 
                           
                         
                         
                           
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                             h 
                             2 
                           
                         
                         
                           
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                             1 
                           
                         
                         
                           
                             h 
                             0 
                           
                         
                       
                     
                     ] 
                   
                 
                 , 
                 and 
               
             
             
               
                 Equation 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   ( 
                   2 
                   ) 
                 
               
             
           
           
             
               
                 
                   Y 
                   circ 
                 
                 = 
                 
                   
                     
                       H 
                       circ 
                     
                     ⁢ 
                     X 
                   
                   + 
                   
                     N 
                     . 
                   
                 
               
             
             
               
                 Equation 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   ( 
                   3 
                   ) 
                 
               
             
           
         
       
     
   
   Since H circ  is a circulant matrix, it can be represented as H circ =F H  ΛF where Λ matrix and F is an FFT matrix. Using this in equation (3) the result is;
 
 Y   circ   =F   H   ΛFX+N=&gt;FY   circ   =ΛFX+FN.  
 
Since Λ is a diagonal matrix, the MMSE solution for X can now be seen to be
 
 {circumflex over (X)}   MMSE   =F   H ([Λ H Λ+σ 2   I ]) −1 Λ H   FY   circ .  Equation (4)
 
   Since A is a diagonal matrix, the matrix inverse in the above equation is a scalar inversion of the individual elements along the diagonal. The {circumflex over (X)} MMSE  is the estimate of the transmitted signal over the whole block (A) shown in  FIG. 2 . Because of the approximation in equation (2) of making the matrix H being circulant, the estimates of the transmitted signal towards the edge of the block A will not be reliable. Hence, only the estimates of X for a block of data (B) shown in  FIG. 2  and the remaining data from the block (A) is discarded. Next, in order to receive the remaining data in block (A)  202  the procedure is repeated where in another overlapping block A  304  is formed as shown in  FIG. 3 . Block  302  is the new block used in the frequency domain equalization, with  304  (B′) is the new block of interest for demodulation. 
   Referring now to  FIG. 4 , there is shown a flowchart highlighting the frequency domain equalization for HDSPA in accordance with one embodiment of the invention. In  402 , block for demodulation (B)  204  from the received data is selected. In  404 , a larger block (A)  202  is picked that overlaps (B)  204 , received data Y. In  406 , the FFT of vector Y and channel impulse response H are taken. In  408 , the term is multiplied by the term of the two FFT&#39;s and normalized using equation (4). In  410 , an inverse fast fourier transform (IFFT) of the resultant signal is computed. The edges of the resultant data are truncated in  412  in order to get the chip level estimates for the transmitted data for the block (B)  204 . Finally, in  414 , the data is despread in order to produce symbol level estimates. With judicious choice of equalizer parameters (e.g., the amount of overlapping, block size), it is expected that the FFT-based FDE performs at least as well as the time-domain counterparts (e.g. NLMS, Griffith, FFT-based TD-MMSE). 
   In  FIG. 5 , there is shown a diagram of a radio  502  that includes a receiver that implements the FDE of the present invention. In one illustrative example, radio  502  supports HSDPA. 
   Complexity Estimates 
   Complexity estimates for the FDE for HSDPA for typical values of the lengths for blocks A, B are now described. For a length, L, the total number of complex multiplies are FFT&#39;d if M/2 log 2(M) and M/2 log 2(M) are complex additions. The total of operations per chip is given in table 1 below, for the FDE for different chosen lengths for blocks A  202 , B  204  from  FIG. 2 . The length of block A  202  is denoted as M, and that for block B is denoted as, P. 
   For comparison purposes to the present invention, consider a basic NLMS time domain equalizer with length 16 taps filter. Using the NLMS equalizer, the number of operations needed per chip are; 16 complex multiplies per chip for the filtering, plus 16 complex multiplies per chip, for the NLMS update, implying a total of at least 32 complex multiplies per chip. Similarly, the number of complex additions per chip are 15 complex adds for filtering, plus 15 complex adds for the LMS update. This implies a total of 30 complex additions per chip. 
   Thus, the complexity of the frequency domain based equalizer of the present invention is less than a single stage of the NLMS time domain equalizer, while achieving practically the performance of an MMSE equalizer. It should be noted that the complexity estimate for the FDE does not take the channel estimation complexity into account. 
                                                             TABLE 1                       M = 256, P = 128   M = 128, P = 64   M = 64, P = 16   M = 32, P = 16                                    FFT(Y)   1024 X and +   448 X and +   192 X and +   80 X and +       FFT(h) with zero   1024 X and +   448 X and +   192 X and +   80 X and +       padding       Z = FFT(Y)*FFT(h)   256 X   128 X   64 X   32 X       z1 = z/([Λ H Λ + σ 2 I]) −1     512 X, 256 +   256 X, 128 +   128 X, 64 +   64 X, 32 +       IFFT(Z)   1024 X and +   448 X and +   192 X and +   80 X and +       Total   3840 complex X,   1728 complex X,   768 complex X,   304 complex X,           3584 complex +   1600 complex +   704 complex +   272 complex +       Operations/chip = Total/P   30 complex X,   27 complex X,   48 complex X,   19 complex X,           28 complex +   25 complex +   44 complex +   17 complex +                    
FDE as a Rake Receiver
 
   It is very important that the equalizer be able to do simple matched filtering operations and behave like a rake receiver so that the performance of the equalizer may not degrade compared to a rake receiver. In equation (4) this can be done simply by not taking the term Λ H Λ into account during the equalization phase. Thus, whenever the receiver is to be forced to behave as a rake receiver the equation becomes:
 
{circumflex over (X)} rake =F H Λ H FY circ .  Equation (5)
 
One Transmit, Two Receive Antenna
 
   Similar to  FIG. 2 , we again consider a block of interest for demodulation (B) from the two receive antennas and a larger sized block (A) for the block used in FDE. Skipping some of the obvious details, let us start by writing the equivalent equation (3) for the 1×2 antenna case; 
                     [           Y   circ   1               Y   circ   2           ]       2   ⁢   MXM       =           [           H   circ   1               H   circ   2           ]       2   ⁢   MXM       ⁢     X   MXM       +         [           N   1               N   2           ]       2   ⁢   MXM       .               Equation   ⁢           ⁢     (   6   )                 
Where the superscripts indicate the receive antenna. In this case, once exploiting the property of circulant matrices, we get;
 
   
     
       
         
           
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             Y 
                             circ 
                             1 
                           
                         
                       
                       
                         
                           
                             Y 
                             circ 
                             2 
                           
                         
                       
                     
                     ] 
                   
                   
                     2 
                     ⁢ 
                     MXM 
                   
                 
                 = 
                 
                   
                     
                       
                         
                           
                             
                               
                                 [ 
                                 
                                   
                                     
                                       
                                         F 
                                         H 
                                       
                                     
                                     
                                       0 
                                     
                                   
                                   
                                     
                                       0 
                                     
                                     
                                       
                                         F 
                                         H 
                                       
                                     
                                   
                                 
                                 ] 
                               
                               
                                 2 
                                 ⁢ 
                                 MX2M 
                               
                             
                             ⁡ 
                             
                               [ 
                               
                                 
                                   
                                     
                                       Λ 
                                       1 
                                     
                                   
                                 
                                 
                                   
                                     
                                       Λ 
                                       2 
                                     
                                   
                                 
                               
                               ] 
                             
                           
                           
                             2 
                             ⁢ 
                             MXM 
                           
                         
                         ⁢ 
                         
                           F 
                           MXM 
                         
                         ⁢ 
                         
                           X 
                           MXM 
                         
                       
                       + 
                       
                         
                           [ 
                           
                             
                               
                                 
                                   N 
                                   1 
                                 
                               
                             
                             
                               
                                 
                                   N 
                                   2 
                                 
                               
                             
                           
                           ] 
                         
                         
                           2 
                           ⁢ 
                           MXM 
                         
                       
                     
                     ⇒ 
                     
                       
                         
                           [ 
                           
                             
                               
                                 F 
                               
                               
                                 0 
                               
                             
                             
                               
                                 0 
                               
                               
                                 F 
                               
                             
                           
                           ] 
                         
                         ⁡ 
                         
                           [ 
                           
                             
                               
                                 
                                   Y 
                                   circ 
                                   1 
                                 
                               
                             
                             
                               
                                 
                                   Y 
                                   circ 
                                   2 
                                 
                               
                             
                           
                           ] 
                         
                       
                       
                         2 
                         ⁢ 
                         MXM 
                       
                     
                   
                   = 
                   
                     
                       
                         
                           
                             
                               [ 
                               
                                 
                                   
                                     
                                       Λ 
                                       1 
                                     
                                   
                                 
                                 
                                   
                                     
                                       Λ 
                                       2 
                                     
                                   
                                 
                               
                               ] 
                             
                             
                               2 
                               ⁢ 
                               MXM 
                             
                           
                           ⁢ 
                           
                             F 
                             MXM 
                           
                           ⁢ 
                           
                             X 
                             MXM 
                           
                         
                         + 
                         
                           
                             [ 
                             
                               
                                 
                                   
                                     N 
                                     1 
                                   
                                 
                               
                               
                                 
                                   
                                     N 
                                     2 
                                   
                                 
                               
                             
                             ] 
                           
                           
                             2 
                             ⁢ 
                             MXM 
                           
                         
                       
                       ⇒ 
                       
                         
                           X 
                           ^ 
                         
                         MMSE 
                       
                     
                     = 
                     
                       
                         
                           
                             F 
                             MXM 
                           
                           [ 
                           
                             
                               
                                 Λ 
                                 
                                   1 
                                   H 
                                 
                               
                               ⁢ 
                               
                                 Λ 
                                 1 
                               
                             
                             + 
                             
                               
                                 Λ 
                                 
                                   2 
                                   H 
                                 
                               
                               ⁢ 
                               
                                 Λ 
                                 2 
                               
                             
                             + 
                             
                               
                                 σ 
                                 2 
                               
                               ⁢ 
                               I 
                             
                           
                           ] 
                         
                         MXM 
                         
                           - 
                           1 
                         
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         ( 
                         
                           
                             
                               Λ 
                               
                                 1 
                                 H 
                               
                             
                             ⁢ 
                             F 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               Y 
                               circ 
                               1 
                             
                           
                           + 
                           
                             
                               Λ 
                               
                                 2 
                                 H 
                               
                             
                             ⁢ 
                             F 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               Y 
                               circ 
                               2 
                             
                           
                         
                         ) 
                       
                     
                   
                 
               
             
             
               
                 Equation 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   ( 
                   7 
                   ) 
                 
               
             
           
         
       
     
   
   The receiver performs a similar process to that shown in  FIG. 4 . In terms of complexity, looking at equation (7) one can note that the complexity for the 1×2 FDE case is less than twice the complexity of the single FDE since there is only one final IFFT. Hence, the expected complexity of the 1×2 FDE case is less than or comparable to a one stage NLMS, not including the channel estimation complexity for the FDE. 
   Two Transmit, One Receive Antenna Case 
   Similar to  FIG. 2 , a block of interest for demodulation (B) from the two receive antennas and a larger sized block (A) for the block used in FDE is considered. Starting by writing the equivalent equation (3) for the 2×1 antenna case: 
                   Y   circ     =             [           H   circ   1           H   circ   2           ]     MX2M     ⁢           [           X   1               X   2           ]       2   ⁢   MXM       +     N   MXM               Equation   ⁢           ⁢     (   8   )                 
Where the superscripts indicate the transmit antenna. In this case, exploiting the property of circulant matrices we get:
 
                         Y   circ     =               F   H     ⁡     [           Λ   circ   1           Λ   circ   2           ]       ⁡     [         F       0           0       F         ]       ⁡     [           X   1               X   2           ]       +   N     ⇒                   F   ⁢           ⁢     Y   circ       =             [           Λ   circ   1           Λ   circ   2           ]     ⁡     [         F       0           0       F         ]       ⁡     [           X   1               X   2           ]       +   FN     ⇒                   Equation   ⁢           ⁢     (   9   )                 
For a given Y=HX+N, the MMSE equalizer is given two equivalent formulations namely; (H H H+σ 2 I) −1 H H =H H (HH H +σ 2 I) −1 . Now using the second formulation in equation (9) we now get:
 
   
     
       
         
           
             
               
                 
                   
                     X 
                     ^ 
                   
                   MMSE 
                 
                 = 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   F 
                                   H 
                                 
                               
                               
                                 0 
                               
                             
                             
                               
                                 0 
                               
                               
                                 
                                   F 
                                   H 
                                 
                               
                             
                           
                           ] 
                         
                         
                           2 
                           ⁢ 
                           MX2M 
                         
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             
                               
                                 Λ 
                                 
                                   1 
                                   H 
                                 
                               
                             
                           
                           
                             
                               
                                 Λ 
                                 
                                   2 
                                   H 
                                 
                               
                             
                           
                         
                         ] 
                       
                     
                     
                       2 
                       ⁢ 
                       MXM 
                     
                   
                   ⁢ 
                   
                     
                       ( 
                       
                         
                           
                             Λ 
                             circ 
                             1 
                           
                           ⁢ 
                           
                             Λ 
                             circ 
                             
                               1 
                               H 
                             
                           
                         
                         + 
                         
                           
                             Λ 
                             circ 
                             2 
                           
                           ⁢ 
                           
                             Λ 
                             circ 
                             
                               2 
                               H 
                             
                           
                         
                         + 
                         
                           
                             σ 
                             2 
                           
                           ⁢ 
                           I 
                         
                       
                       ) 
                     
                     
                       - 
                       1 
                     
                   
                   ⁢ 
                   F 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     Y 
                     circ 
                   
                 
               
             
             
               
                 Equation 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   ( 
                   10 
                   ) 
                 
               
             
           
         
       
     
   
   Since the matrix inverse in the above equation is indeed a scalar inverse for each of the frequency terms, it can be interchanged with the rest of the operations. Comparing equation (10) to equation (7), it can be seen that structurally the two equations are similar, hence similar hardware architecture implementation for 1×2 and 2×1 can be used in practice. The receiver performs a similar process as shown in  FIG. 4 . In terms of complexity, looking at equation (10) it can be seen that the complexity for the 2×1 FDE case is less than twice the complexity of a single FDE since there is only 1 FFT. Hence, the expected complexity of the 2×1 FDE case is less than or comparable to the 1 stage NLMS case, not including the channel estimation complexity for the FDE. 
   Two Transmit, Two Receive Antenna Case 
   Similar to  FIG. 2 , we consider a block of interest for demodulation (B) from the two receive antennas and a larger sized block (A) for the block used in FDE. Skipping some of the basic steps, we start by writing the equivalent equation (3) for the 2×2 case: 
                   [           Y   circ   1               Y   circ   2           ]     =             [           H   circ   11           H   circ   21               H   circ   12           H   circ   22           ]       2   ⁢   MX2M       ⁡     [           X   1               X   2           ]         2   ⁢   MXM       +       [           N   1               N   2           ]       2   ⁢   MXM                 Equation   ⁢           ⁢     (   11   )                 
where superscript ij for the channel matrix implies the channel from transmit antenna i to receive antenna j. Similarly, superscripts on Y indicate the receive antenna and the superscript on X indicates the data on the transmit antenna. Exploiting the circulant property the equation becomes:
 
                   [           Y   circ   1               Y   circ   2           ]     =                   [           F   H         0           0         F   H           ]     ⁡     [           Λ   11           Λ   21               Λ   12           Λ   22           ]       ⁡     [         F       0           0       F         ]       ⁡     [           X   1               X   2           ]         2   ⁢   MXM       +     [           N   1               N   2           ]       ⇒     
     ⁢     [           F   ⁢           ⁢     Y   circ   1                 F   ⁢           ⁢     Y   circ   2             ]       =             [           Λ   11           Λ   21               Λ   12           Λ   22           ]     ⁡     [         F       0           0       F         ]       ⁡     [           X   1               X   2           ]         2   ⁢   MXM       +     [           N   1               N   2           ]                 Equation   ⁢           ⁢     (   12   )                 
The above equation now looks similar to that for a 2×2 Multi-Input Multi-Output (MIMO) system. In order to better understand the above equalizer, the individual equations for the different frequency terms are written down. Let y circ   1 (f) be the value of FY circ   1  at frequency f. Similarly, for other terms; we then get at frequency f:
 
   
     
       
         
           
             
               
                 
                   [ 
                   
                     
                       
                         
                           
                             y 
                             circ 
                             1 
                           
                           ⁡ 
                           
                             ( 
                             f 
                             ) 
                           
                         
                       
                     
                     
                       
                         
                           
                             y 
                             circ 
                             1 
                           
                           ⁡ 
                           
                             ( 
                             f 
                             ) 
                           
                         
                       
                     
                   
                   ] 
                 
                 = 
                 
                   
                     
                       [ 
                       
                         
                           
                             
                               
                                 Λ 
                                 11 
                               
                               ⁡ 
                               
                                 ( 
                                 f 
                                 ) 
                               
                             
                           
                           
                             
                               
                                 Λ 
                                 21 
                               
                               ⁡ 
                               
                                 ( 
                                 f 
                                 ) 
                               
                             
                           
                         
                         
                           
                             
                               
                                 Λ 
                                 12 
                               
                               ⁡ 
                               
                                 ( 
                                 f 
                                 ) 
                               
                             
                           
                           
                             
                               
                                 Λ 
                                 22 
                               
                               ⁡ 
                               
                                 ( 
                                 f 
                                 ) 
                               
                             
                           
                         
                       
                       ] 
                     
                     ⁡ 
                     
                       [ 
                       
                         
                           
                             
                               
                                 x 
                                 1 
                               
                               ⁡ 
                               
                                 ( 
                                 f 
                                 ) 
                               
                             
                           
                         
                         
                           
                             
                               
                                 x 
                                 2 
                               
                               ⁡ 
                               
                                 ( 
                                 f 
                                 ) 
                               
                             
                           
                         
                       
                       ] 
                     
                   
                   + 
                   
                     
                       [ 
                       
                         
                           
                             
                               
                                 n 
                                 1 
                               
                               ⁡ 
                               
                                 ( 
                                 f 
                                 ) 
                               
                             
                           
                         
                         
                           
                             
                               
                                 n 
                                 2 
                               
                               ⁡ 
                               
                                 ( 
                                 f 
                                 ) 
                               
                             
                           
                         
                       
                       ] 
                     
                     . 
                   
                 
               
             
             
               
                   
               
             
           
           
             
               
                 
                   Now 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   letting 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   Δ 
                 
                 = 
                 
                   
                     [ 
                     
                       
                         
                           
                             
                               Λ 
                               11 
                             
                             ⁡ 
                             
                               ( 
                               f 
                               ) 
                             
                           
                         
                         
                           
                             
                               Λ 
                               21 
                             
                             ⁡ 
                             
                               ( 
                               f 
                               ) 
                             
                           
                         
                       
                       
                         
                           
                             
                               Λ 
                               12 
                             
                             ⁡ 
                             
                               ( 
                               f 
                               ) 
                             
                           
                         
                         
                           
                             
                               Λ 
                               22 
                             
                             ⁡ 
                             
                               ( 
                               f 
                               ) 
                             
                           
                         
                       
                     
                     ] 
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   we 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   get 
                   ⁢ 
                   
                     : 
                   
                 
               
             
             
               
                   
               
             
           
           
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             
                               
                                 x 
                                 ^ 
                               
                               mmse 
                               1 
                             
                             ⁡ 
                             
                               ( 
                               f 
                               ) 
                             
                           
                         
                       
                       
                         
                           
                             
                               
                                 x 
                                 ^ 
                               
                               mmse 
                               2 
                             
                             ⁡ 
                             
                               ( 
                               f 
                               ) 
                             
                           
                         
                       
                     
                     ] 
                   
                   
                     2 
                     ⁢ 
                     X1 
                   
                 
                 = 
                 
                   
                     
                       ( 
                       
                         
                           
                             Δ 
                             H 
                           
                           ⁢ 
                           Δ 
                         
                         + 
                         
                           
                             σ 
                             2 
                           
                           ⁢ 
                           
                             I 
                             
                               2 
                               ⁢ 
                               X2 
                             
                           
                         
                       
                       ) 
                     
                     
                       2 
                       ⁢ 
                       X2 
                     
                     
                       - 
                       1 
                     
                   
                   ⁢ 
                   
                     
                       
                         Δ 
                         H 
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             
                               
                                 
                                   y 
                                   circ 
                                   1 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   f 
                                   ) 
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   y 
                                   circ 
                                   2 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   f 
                                   ) 
                                 
                               
                             
                           
                         
                         ] 
                       
                     
                     
                       2 
                       ⁢ 
                       X1 
                     
                   
                 
               
             
             
               
                 Equation 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   ( 
                   13 
                   ) 
                 
               
             
           
         
       
     
   
   Once the estimates of the transmitted signal are done, at all the frequencies we obtain the 
                 [             X   ^     mmse   1                 X   ^     mmse   2           ]           
by taking the inverse Fourier transform of all the stacked signals from antennas  1 ,  2  at all the frequencies from equation (13). The receiver block diagram is now similar to that shown in  FIG. 4 . It may be of concern that for the 2×2 case a real 2×2 matrix inversion is needed in equation (13). However, note that for the 2×2 case, there are only a total of 8 FFT&#39;s/IFFT&#39;s needed, 2 FFT&#39;s of received data, 4 FFT&#39;s of channel and 2 IFFT&#39;s. As compared to the 1×1 antenna case where 3 FFT/IFFT&#39;s are needed. Since the 1×1 FDE complexity is less than or comparable to NLMS, it is expected that despite the 2×2 matrix inversion needed in equation (13), the complexity for 2×2 FDE will be less than or comparable to a 2×2 NLMS equalizer. The complexity of NLMS is linear in the number of transmit and receive antennas. The complexity estimates for the FDE for the 2×2 antenna case is given in Table 2 below, which provides a comparison of complexity per chip for the present invention FDE approach and an NLMS based equalization approach
 
                                             TABLE 2                           Complexity           M = 32, P = 16   for NLMS                                    FFT(Y)   2*80 X and +           FFT(h) with zero padding   4*80 X and +       Z = FFT(Y)*FFT(h)   4*32 X       z1 = ([Δ H Δ + σ 2 I]) −1 Z   4*64 + 4*32 +           4*32 X, 4*64 +       IFFT(Z)   2*80 X and +       Total   1120 complex X,           736 complex +       Operations/chip = Total/P   70 complex X,   128 complex X, 120       (2X2 case)   46 complex +   complex adds                    
As can be seen from Table 2, the complexity of the proposed 2×2 FDE approach is less than the 2×2 NLMS equalizer.
 
   In  FIG. 6  there is shown a block diagram of a FDE for a one transmit/one receive antenna system (1×1). Thirty-two samples of received chips y={y 1 , y 2 , . . . , y 32 )  602  are sent to a length 32 FFT  604  that provides a 32 sample output. A channel response at chip level, h={h 1 , h 2 , . . . , h 32 }  608  is provided to a length 32 FFT whose 32 sample output is element wise multiplied with the output of FFT  604 . Block  610  is disabled if the FDE is to behave like a rake receiver, if not, a multipler  609  multiplies the output of block  607  with the 32 samples from multiplier  606 . An inverse FFT (IFFT) is performed by block  612 , and a length 16 despreading is performed by block  614 . The 16 sample output of the despreader  614  goes to a slicer or other similar circuitry. 
   Referring now to  FIG. 7 , there is shown the data  706  used at the input to despreader  614 , the data along the ends  702  and  704  is discarded at the output of the IFFT  612  and is not sent to the despeader  614 . 
   In  FIG. 8  there is shown a block diagram of a FDE for a one transmit/two receive antenna system (1×2). The receiver depicted in  FIG. 8  is substantially similar to that depicted in  FIG. 6 , and the signal received by each of the two antennas is processed similarly to that described with reference to  FIG. 6 , with the provision that the output of a first multiplier  816  and a second multiplier  818  is summed by a summing component  811 . 
   In  FIG. 9  there is shown a block diagram of a FDE for a two transmit/one receive antenna system (2×1). The receiver depicted in  FIG. 9  is substantially similar to that depicted in  FIG. 6 , and the signal received by the one antenna is processed similarly to that described with reference to  FIG. 6 , with the provision that the input y is processed by similar receiver stages each provisioned with an appropriate channel response. 
   In  FIG. 10  there is shown a block diagram of a FDE for a two transmit/two receive antenna system (2×2). The receiver depicted in  FIG. 8  is substantially similar to that depicted in  FIG. 6  and combines one or more aspects of the processing depicted in  FIG. 8  and FIG. 
   In  FIG. 11  there is shown a block diagram of a frequency domain equalizer architecture accordance with another embodiment. 
   While the preferred embodiments of the invention have been illustrated and described, it will be clear that the invention is not so limited. Numerous modifications, changes, variations, substitutions and equivalents will occur to those skilled in the art without departing from the spirit and scope of the present invention as defined by the appended claims.