Patent Publication Number: US-8116242-B2

Title: Receiver having multi-antenna log likelihood ratio generation with channel estimation error

Description:
FIELD OF THE DISCLOSURE 
     The present disclosure relates generally to communication systems, and more particularly to communication system receivers and improved methods and apparatus for providing channel decoder inputs. 
     BACKGROUND 
     Orthogonal Frequency Division Multiplexing (OFDM) provides good data throughput for a given bandwidth and is therefore widely employed for wireless systems. OFDM has been adopted by various wireless standards such as IEEE 802.11a, 802.16, ETSI HIPERLAN/2 as well as digital video broadcasting (DVB). 
     Channel estimation is of critical importance to OFDM because the channel varies across the frequency domain sub-carriers and also across OFDM symbols in time. 
     Channel decoders (e.g., turbo or Low-Density Parity Check (LDPC) decoders) need inputs to be able to properly decode the received coded waveform and one type of input is the Logarithm of the Likelihood Ratio (LLR). A standard means of computing LLRs given a channel estimate in an OFDM system is to generate the LLRs using the channel estimates based on a flat Rayleigh fading assumption (because an OFDM channel estimate on a particular subcarrier looks like it is flat Rayleigh faded). However, such techniques assume correct channel estimates and will not work well when there is significant channel estimation error. 
     Techniques do exist for accounting for channel estimation error when computing LLRs in OFDM systems. For example, in M. M. Wang, W. Xiao, &amp; T. Brown, “Soft Decision Metric Generation for QAM With Channel Estimation Error,” IEEE Transactions On Communications, Vol. 50, No. 7 (July 2002), a system is described which considers channel estimation Mean Square Error (MSE) when computing LLRs in OFDM. However, the disclosed techniques do not consider the multi-user aspect, and do not consider the frequency-domain channel estimation MSE, which is not uniform across frequency. Further, such techniques have not considered LLR generation for antenna combining algorithms such as Minimum Mean Square Error (MMSE), successive cancellation, and maximum likelihood detection (which is also known as joint detection). 
     Therefore, such systems have been limited to a single data source and a single receiving antenna. Other techniques, for example LLR generation for LDPC codes in Multiple-Input/Multiple-Output (MIMO) OFDM, also neglect channel estimation error, or otherwise neglect the fact that channel estimation error can vary greatly across frequency, and therefore such techniques will not work well for cases having significant channel estimation error. 
     Further, the performance of turbo-coded or LDPC-coded OFDM channels can be seriously degraded when the channel estimation error is not accounted for especially for higher order Quadrature Amplitude Modulations (QAM) such as 16-QAM and 64-QAM. 
     Therefore what is needed is an apparatus and method for computing LLRs for single or multiple data streams and multiple antenna combining techniques while accounting for channel estimation error. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  is a block diagram of a Spatial Division Multiple Access (SDMA) network wherein multiple transmitters communicate with a receiver having multiple receiving antennas. 
         FIG. 2  is a block diagram of a Multiple Input, Multiple Output (MIMO) network wherein a transmitter having multiple transmit antennas transmits multiple data streams to a receiver having multiple receive antennas. 
         FIG. 3  is a block diagram illustrating high level components of a receiver in accordance with the various embodiments. 
         FIG. 4  is a flow chart illustrating a high level operation of a receiver in accordance with the various embodiments. 
         FIG. 5  is a flow chart illustrating a high level operation of a receiver in accordance with an embodiment employing linear Minimum Mean Square Error combing. 
         FIG. 6  is a flow chart illustrating a high level operation of a receiver in accordance with an embodiment employing successive cancellation. 
         FIG. 7  is a flow chart illustrating a high level operation of a receiver in accordance with an embodiment employing joint detection reception. 
         FIGS. 8   a  and  8   b  are graphs of simulation results for a receiver in accordance with an embodiment, showing channel estimation error versus subcarrier for joint channel estimation of four signals using a single OFDM symbol of a pilot. 
         FIG. 9   a  is a graph of simulation results for prior receivers showing Frame Error Rate versus Signal to Noise Ratio per 16-QAM symbol. 
         FIG. 9   b  is a graph of simulation results for a receiver in accordance with an embodiment, showing Frame Error Rate versus Signal to Noise Ratio per 16-QAM symbol. 
         FIG. 10   a  is a graph of simulation results for a prior receiver, showing Frame Error Rate versus Signal to Noise Ratio per 64-QAM symbol. 
         FIG. 10   b  is a graph of simulation results for a receiver in accordance with an embodiment, showing Frame Error Rate versus Signal to Noise Ratio per 64-QAM symbol. 
     
    
    
     DETAILED DESCRIPTION 
     A method and apparatus which provides channel decoder input generation for various antenna combining techniques while accounting for channel estimation error is provided herein. 
     In some embodiments, channel decoder inputs will be Log-Likelihood Ratios (LLRs) which may be considered generally to be a codeword component, the codeword being an encoded message encoded on a transmitting side, and a noise component. 
     In the various embodiments herein disclosed, LLR calculation may be performed for arbitrary channel estimators with linear MMSE combining, successive cancellation combining, and joint detection. Further, in some embodiments, computational complexity may be greatly reduced by the use of channel estimation other than the full MMSE channel estimator. In additional embodiments, the use of linear MMSE or successive cancellation combining may be employed to greatly lower the computational complexity over joint detection. In yet other embodiments, the use of approximate maximum likelihood methods such as sphere decoding may be employed to greatly lower the computational complexity over joint detection. 
     It will be appreciated that LLR calculation, channel estimation and otherwise processing received signals may be performed in a dedicated device such as a receiver having a dedicated processor, a processor coupled to an analog processing circuit or receiver analog “front-end” with appropriate software for performing a receiver function, an application specific integrated circuit (ASIC), a digital signal processor (DSP), or the like, or various combinations thereof, as would be appreciated by one of ordinary skill. Memory devices may further be provisioned with routines and algorithms for operating on input data and providing output such as operating parameters to improve the performance of other processing blocks associated with, for example, reducing noise and interference, and otherwise appropriately handling the input data. 
     It will further be appreciated that wireless communications units may refer to subscriber devices such as cellular or mobile phones, two-way radios, messaging devices, personal digital assistants, personal assignment pads, personal computers equipped for wireless operation, a cellular handset or device, or the like, or equivalents thereof provided such units are arranged and constructed for operation in accordance with the various inventive concepts and principles embodied in exemplary receivers, and methods for generating or determining channel decoder inputs including, but not limited to LLRs, channel estimation, and accounting for channel estimation error as discussed and described herein. 
     The inventive functionality and inventive principles herein disclosed are best implemented with or in software or firmware programs or instructions and integrated circuits (ICs) such as digital signal processors (DSPs) or application specific ICs (ASICs) as is well known by those of ordinary skill in the art. Therefore, further discussion of such software, firmware and ICs, if any, will be limited to the essentials with respect to the principles and concepts used by the various embodiments. 
     Turning now to the drawings wherein like numerals represent like components,  FIG. 1  illustrates a spatial division multiple access (SDMA) or spatial division multiplexed (SDM) network  100 . In such SDMA networks, a receiver  101  includes a number of antennas  109  which may be referred to as a smart antenna system. The multiple antennas  109 , or smart antenna system  109 , may receive several data streams or input signals on the same frequency simultaneously, that is, on the same time-frequency resource. 
     For example, transmitter unit  103  and transmitter unit  110  may transmit data stream  106  and data stream  111  respectively, which may be received by receiver  101  simultaneously using multiple antennas  109 . The SDMA network approach increases the aggregate data throughput nearly proportionally to the number of antennas at the receiver. In the various embodiments, LLRs are determined for each transmitter, such as transmitter  103  and transmitter  110 , transmitting on the same time-frequency resource or channel. Thus, in the various embodiments receiver  101  will have receiver components  107 , comprising receiving components appropriate for, and corresponding to the multiple antennas  109 . Also in the various embodiments, receiver  101  will have components  108  which comprises a channel estimation component, an LLR calculation component, a channel decoding component and a storage component. 
       FIG. 2  illustrates a Multiple Input, Multiple Output (MIMO) network  200  in which data throughput between a receiver and transmitter is improved using multiple antennas or smart antennas. In  FIG. 2 , receiver  201  has multiple antennas  209  and transmitter  203  likewise has multiple antennas  210 . Transmitter  203  may transmit a set of data streams, such as data streams  206  through  210 , simultaneously. Similar to  FIG. 1 , in the various embodiments receiver  201  will have receiver components  207 , comprising receiving components appropriate for, and corresponding to the multiple antennas  209 ; and components  208  which comprises a channel estimation component, an LLR calculation component, a channel decoding component and a storage component. 
     Further network  100  and network  200 , may employ any of various modulation and coding schemes for the air interfaces between transmitters and receivers. For example, Quadrature Amplitude Modulation (QAM) may be employed including, but not limited to, 16-QAM, 64-QAM, etc. Additionally, various approaches to channelization of signals and/or subcarriers may be employed, such as but not limited to, Code Division Multiple Access (CDMA), Time Division Multiple Access (TDMA), etc. Further, such approaches may be used in combination with each other and/or other techniques such as Orthogonal Frequency Division Multiplexing (OFDM) such that various sub-carriers employ various channelization techniques. The air interfaces may also conform to various interfaces such as, but not limited to, 802.11, 802.16, etc. 
       FIG. 3  illustrates high level components in accordance with a receiver embodiment for example, components  107  and  108  in  FIG. 1  and components  207  and  208  in  FIG. 2 . A number of antennas such as antenna  301  and antenna  303  provide inputs to respective receiver circuitries  305  and  307 . Received inputs are, in general, demodulated such that a channel estimation vector, via channel estimation circuitry  309 , and set of LLRs, via LLR calculation circuitry  313 , is computed for each channel resulting in a channel decoder  315  channel decoder input  317 , that is, in general, a combination of a codeword and noise power from storage unit  311 . In the embodiment illustrated by  FIG. 3 , the LLR circuitry  313  employs channel estimation MSE in addition to modulation type and noise power to determine the channel decoder input  317 . The channel decoder  315  then determines bit estimates  319  for each transmitter. 
     The high level operation of the receiver of  FIG. 3  is illustrated in  FIG. 4 . Thus, in  401  a channel estimation MSE as a function of frequency, the transmitter modulation type, and a noise power are provided to the LLR circuitry  313 . A signal from at least one transmitter is received at one of the various antennas  301 ,  303  and respective receiver circuitry  305 ,  307  in block  403 . The channel estimation circuitry  309  computes a channel estimate for the signal from the transmitter, or computes multiple channel estimates for multiple transmitter sources, in block  405 . In block  407 , decoder inputs are determined as a function of the channel estimates, the received signal, noise power, and channel estimation MSE. 
     LLR generation with channel estimation error for embodiments employing linear MMSE combining is illustrated at a high level by  FIG. 5  and is described in detail as follows. 
     The received M×1 signal, or input signal, on subcarrier k and symbol b, Y(k,b), is modeled as: 
                     Y   ⁡     (     k   ,   b     )       =         ∑     u   =   1       N   s       ⁢         H   u     ⁡     (     k   ,   b     )       ⁢       x   u     ⁡     (     k   ,   b     )           +     N   ⁡     (     k   ,   b     )                 (   1   )               
where H u (k,b) is signal u&#39;s M×1 channel vector on subcarrier k and symbol b, x u (k,b) is signal u&#39;s symbol on subcarrier k and symbol b, and N(k,b) is an M×1 vector of additive noise with correlation matrix σ n   2 I.
 
     Because there is a channel estimate, the model of the received signal that will be used to calculate the LLRs is given as: 
                     Y   ⁡     (     k   ,   b     )       =         ∑     u   =   1       N   s       ⁢           H   ^     u     ⁡     (     k   ,   b     )       ⁢       x   u     ⁡     (     k   ,   b     )           +       ∑     u   =   1       N   s       ⁢       {         H   u     ⁡     (     k   ,   b     )       -         H   ^     u     ⁡     (     k   ,   b     )         }     ⁢       x   u     ⁡     (     k   ,   b     )           +     N   ⁡     (     k   ,   b     )                 (   2   )               
where Ĥ u (k,b) is the channel estimate for signal u, as shown in block  505 . Note that the combining weights assume the channel estimate is correct so that the received signal looks like the original one given in Equation (1) with an additional “noise” term that accounts for the channel estimation error.
 
     The linear MMSE combining weights, as determined in block  507 , are a function of the channel estimates from block  505  and are given for signal u as: 
                       w   u     ⁡     (     k   ,   b     )       =         (         ∑     v   =   1       N   s       ⁢           H   ^     v     ⁡     (     k   ,   b     )       ⁢         H   ^     v   H     ⁡     (     k   ,   b     )           +       σ   n   2     ⁢     I   M         )       -   1       ⁢         H   ^     u     ⁡     (     k   ,   b     )                 (   3   )               
Defining E u (k,b)=H u (k,b)−Ĥ u (k,b) and using Equation (2) the symbol estimate for signal u, as computed in block  509 , is given as:
 
     
       
         
           
               
             
               
                 
                   
                     
                       
                         
                           
                             
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     To compute the LLRs as in block  511 , E u (k,b) is assumed to be a zero-mean Gaussian random vector with correlation matrix, δ u   2 (k,b)I M  (i.e., δ u   2 (k,b) is the channel estimation MSE for signal u on subcarrier k and symbol b, the MSE being stored as shown in storage unit  311  of  FIG. 3 ), and E{|K x u (k,b)| 2 } is assumed to be one. Using these assumptions the frequency-domain symbol estimate for signal u, as illustrated in block  509  is modeled as:
 
 r   u ( k,b )= w   u   H ( k,b ) Ĥ   u ( k,b ) x   u ( k,b )+ n   u ( k,b )   (5)
 
where n u (k,b) is modeled as a zero-mean Gaussian random variable with variance given by:
 
                       σ     r   ,   u     2     ⁡     (     k   ,   b     )       =         w   u   H     ⁡     (     k   ,   b     )       ⁢     {           δ   u   2     ⁡     (     k   ,   b     )       ⁢              x   u     ⁡     (     k   ,   b     )            2       +         ∑     v   =   1       N   s         v   ≠   u       ⁢           H   ^     v     ⁡     (     k   ,   b     )       ⁢         H   ^     v   H     ⁡     (     k   ,   b     )           +         ∑     v   =   1       N   s         v   ≠   u       ⁢       δ   v   2     ⁡     (     k   ,   b     )         +     σ   n   2       }     ⁢       w   u     ⁡     (     k   ,   b     )                 (   6   )               
Note that this variance accounts for the residual cross talk after applying the weights as well as the channel estimation error from storage unit  311  and as shown in block  509 . Note also that the variance depends on the magnitude of signal u&#39;s symbol but not on the amplitude of the other signals&#39; symbols because the received symbol estimate for user u is conditioned on x u (k,b) and thus x v (k,b)&#39;s for v≠u are treated as random variables.
 
     The probability density function (pdf) of r u (k,b) given x u (k,b) is then given by: 
                     f   ⁡     (         r   u     ⁡     (     k   ,   b     )       |       x   u     ⁡     (     k   ,   b     )         )       =     C   ⁢           ⁢     exp   (     -                  r   u     ⁡     (     k   ,   b     )       -         w   u   H     ⁡     (     k   ,   b     )       ⁢         H   ^     u     ⁡     (     k   ,   b     )       ⁢       x   u     ⁡     (     k   ,   b     )                2                 w   u   H     ⁡     (     k   ,   b     )       ⁢     {           δ   u   2     ⁡     (     k   ,   b     )       ⁢              x   u     ⁡     (     k   ,   b     )            2       +                             ∑     v   =   1       N   s         v   ≠   u       ⁢         H   ^     v     ⁢     (     k   ,   b     )     ⁢         H   ^     v   H     ⁡     (     k   ,   b     )           +         ∑     v   =   1       N   s         v   ≠   u       ⁢       δ   v   2     ⁡     (     k   ,   b     )         +     σ   n   2       }     ⁢       w   u     ⁡     (     k   ,   b     )                   )               (   7   )               
where C is a constant that is not important for the LLR computation. Using Equation (7), the LLR for bit l for user u on subcarrier k and symbol b, LLR{b u,l (k,b)}, is found as shown in block  511  as (assuming equal-probable symbol values):
 
                     LLR   ⁢     {       b     u   ,   l       ⁡     (     k   ,   b     )       }       =     log   (       ∑         x   u     ⁡     (     k   ,   b     )       ∈     Ω   l   +         ⁢     f   (           r   u     ⁡     (     k   ,   b     )       ⁢            x   u     ⁡     (     k   ,   b     )       )       -     log   (       ∑         x   u     ⁡     (     k   ,   b     )       ∈     Ω   l   -         ⁢       f   (       r   u     ⁡     (     k   ,   b     )            ⁢       x   u     ⁡     (     k   ,   b     )           )                     (   8   )               
where Ω l   +  is the set of symbols where bit l of symbol x u (k,b) equals plus one and Ω l    −  is the set of symbols where bit l of symbol x u (k,b) equals minus one (or zero).
 
     If the link is cyclic-prefix single carrier then the received time-domain signal is the IFFT of the frequency-domain symbol estimates in Equation (5) (spread-OFDM will also have similar modifications with, for example, the IFFT operation being replaced by Walsh de-spreading). Thus the time-domain symbol estimates are modeled as (assuming that the combining weights are unbiased):
 
 ŝ   u ( n,b )= s   u ( n,b )+ v   u ( n,b )  (9)
 
where s u (0,b) through s u (N f −1,b) are the N f -point IFFT of the frequency-domain symbols (0≦k≦N f 31 1), x u (k,b), and v u (n,b) is the IFFT of n u (k,b) for 0≦k&lt;N f 31 1.
 
     The IFFT operation on n u (k,b) will equalize the channel estimation error across all time-domain symbols (i.e., all symbol estimates will have similar quality unlike the frequency-domain symbols where the symbol estimates at edge sub-carriers can be significantly worse than symbol estimates at non-edge sub-carriers). This means that v u (n,b) is modeled as a zero-mean Gaussian random variable with variance given by (using Parseval&#39;s theorem): 
     
       
         
           
             
               
                 
                   
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     LLR generation with channel estimation error, for embodiments employing successive cancellation, is illustrated at a high level by  FIG. 6  and is described in detail as follows. 
     In the various embodiments employing successive cancellation, N s  received signals are decoded in some order and a signal is decoded, re-encoded, and mapped back to symbol values before being cancelled using the channel estimate for that signal. The decoding order may in some embodiments be chosen to pick the stream with the best average post-detection SINR (Signal to Interference plus Noise Ratio) at each iteration or can be decoded in order in embodiments employing techniques such as weighted-BLAST or MCR-selection BLAST. Thus, the decoding order is determined as illustrated in block  607 . 
     Assuming that the decoding order determined in block  607  is u 1  through u N     s    and setting the decoding index “n” to one as in block  609 , the successive cancellation combining weights for signal u n , for corresponding transmitter u n , is determined as shown in block  611  and by: 
                       w     u   n       ⁡     (     k   ,   b     )       =         (         ∑     v   =   n       N   s       ⁢           H   ^       u   v       ⁡     (     k   ,   b     )       ⁢         H   ^       u   v     H     ⁡     (     k   ,   b     )           +       σ   n   2     ⁢     I   M         )       -   1       ⁢         H   ^       u   n       ⁡     (     k   ,   b     )                 (   11   )               
Wherein it is assumed that each signal is decoded without errors so that the symbol estimate for signal u n  which will be determined as shown in block  613 , may be expressed as:
 
                                   r     u   n       ⁡     (     k   ,   b     )       =       ⁢         w     u   n     H     ⁡     (     k   ,   b     )       ⁢     {       Y   ⁡     (     k   ,   b     )       -       ∑     v   =   1       n   -   1       ⁢           H   ^       u   v       ⁡     (     k   ,   b     )       ⁢       x     u   v       ⁡     (     k   ,   b     )             }                   =       ⁢       w     u   n     H     ⁡     (     k   ,   b     )                       ⁢     (         ∑     v   =   n       N   s       ⁢           H   ^       u   v       ⁡     (     k   ,   b     )       ⁢       x     u   v       ⁡     (     k   ,   b     )           +       ∑     u   =   1       N   s       ⁢         E   u     ⁡     (     k   ,   b     )       ⁢       x   u     ⁡     (     k   ,   b     )           +     N   ⁡     (     k   ,   b     )         )                     (   12   )               
It is important to note that in the various embodiments, only the portion of Y(k,b) in Equation (2) corresponding to the channel estimate times the symbol value is cancelled, not the channel estimation error term.
 
     However, unlike the linear MMSE embodiments as illustrated in  FIG. 5  and described in detail above, the amplitude of the decoded signal&#39;s symbols are known. As discussed above with respect to the computation of LLRs in block  511  of  FIG. 5 , that is, in the embodiments employing linear MMSE, it is assumed that E{|x u (k,b)| 2 }=1, since the other stream&#39;s symbols are unknown. Of course, in the embodiments employing successive cancellation, for the signals that have yet to be decoded, it will still be assumed that E{|x u (k,b)| 2 }=1. 
     Therefore, to calculate the LLRs as shown in block  615  of  FIG. 6 , the symbol estimate for signal u n  is computed in block  613  and is further modeled as:
 
 r   u     n   ( k,b )= w   u     n     H ( k,b ) Ĥ   u     n   ( k,b ) x   u     n   ( k,b )+ n   u     n   ( k,b )  (13)
 
where n u     n   (k,b) is assumed to be a zero-mean Gaussian random variable with variance given by:
 
     
       
         
           
             
               
                 
                   
                     
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                   ( 
                   14 
                   ) 
                 
               
             
           
         
       
     
     The Probability Density Function (pdf) of r n     n   (k,b) given x u     n   (k,b) is then given by: 
                   f   (           r   u     ⁡     (     k   ,   b     )       ⁢            x   u     ⁡     (     k   ,   b     )       )       =     C   ⁢           ⁢   exp   ⁢           (     -                  r     u   n       ⁡     (     k   ,   b     )       -         w     u   n     H     ⁡     (     k   ,   b     )       ⁢         H   ^       u   n       ⁡     (     k   ,   b     )       ⁢       x     u   n       ⁡     (     k   ,   b     )                2                 w     u   n     H     ⁡     (     k   ,   b     )       ⁢     {         ∑     v   =     n   +   1         N   s       ⁢     {             H   ^       u   v       ⁡     (     k   ,   b     )       ⁢         H   ^       u   v     H     ⁡     (     k   ,   b     )         +       δ     u   v     2     ⁡     (     k   ,   b     )         }       +                           ∑     v   =   1     n     ⁢         δ     u   v     2     ⁡     (     k   ,   b     )       ⁢              x     u   v       ⁡     (     k   ,   b     )            2         +     σ   n   2       }     ⁢       w     u   n       ⁡     (     k   ,   b     )                   )                   (   15   )               
where C is constant that is not important for the LLR computation. Using Equation (15), the LLR for bit l for user u n  on subcarrier k and symbol b, LLR{b u     n     ,l (k,b)}, is found as shown in block  615  as (and assuming equal-probable symbol values):
 
                     LLR   ⁢     {       b       u   n     ,   l       ⁡     (     k   ,   b     )       }       =     log   (         ∑     x         u   n     ⁡     (     k   ,   b     )       ∈     Ω   l   +           ⁢     f   ⁡     (         r     u   n       ⁡     (     k   ,   b     )       |       x     u   n       ⁡     (     k   ,   b     )         )         -     log   (       ∑         x     u   n       ⁡     (     k   ,   b     )       ∈     Ω   l         ⁢     f   ⁡     (         r     u   n       ⁡     (     k   ,   b     )       |       x     u   n       ⁡     (     k   ,   b     )         )                       (   16   )               
As discussed above with respect to the embodiments employing linear MMSE, if the link is cyclic-prefix single carrier, the model of the symbol estimates will have to change similar to Equation (9).
 
     LLR generation with channel estimation error, for embodiments employing joint detection, is illustrated at a high level by  FIG. 7  and is described in detail as follows. 
     In the various embodiments employing joint detection, similar to Equation (2) the received signal, or input signal at antennas  301  and  303 , is modeled as: 
                     Y   ⁡     (     k   ,   b     )       =         ∑     u   =   1       N   s       ⁢           H   ^     u     ⁡     (     k   ,   b     )       ⁢       x   u     ⁡     (     k   ,   b     )           +       ∑     u   =   1       N   s       ⁢         E   u     ⁡     (     k   ,   b     )       ⁢       x   u     ⁡     (     k   ,   b     )           +     N   ⁡     (     k   ,   b     )                 (   17   )               
where E u (k,b) models the channel estimation error for signal u, as shown in block  701 , and is assumed to be a zero-mean Gaussian random vector with correlation matrix δ u   2 (k,b)I M . The pdf of Y(k,b) given x 1 (k,b) through X N     s   (k,b) is given as:
 
                     f   ⁡     (         Y   ⁡     (     k   ,   b     )       |       x   1     ⁡     (     k   ,   b     )         ,   …   ⁢           ,       x     N   s       ⁡     (     k   ,   b     )         )       =     C   ⁢           ⁢   exp   ⁢     {     -                Y   ⁡     (     k   ,   b     )       -       ∑     u   =   1       N   s       ⁢           H   ^     u     ⁡     (     k   ,   b     )       ⁢       x   u     ⁡     (     k   ,   b     )                  2         σ   n   2     +       ∑     u   =   1       N   s       ⁢         δ   u   2     ⁡     (     k   ,   b     )       ⁢              x   u     ⁡     (     k   ,   b     )            2               }               (   18   )               
where C is a constant that will not be important in the LLR calculation.
 
     As shown in block  709  and using Equation (18), the LLR for bit l of user u at subcarrier k and symbol b is given as: 
                     LLR   ⁢     {       b     u   ,   l       ⁡     (     k   ,   b     )       }       =       log   (       ∑         x   u     ⁡     (     k   ,   b     )       ∈     Ω   l   +         ⁢         ∑     v   =   1       N   s         v   ≠   u       ⁢       ∑         x   v     ⁡     (     k   ,   b     )       ∈   Ω       ⁢     exp   ⁢     {     -                Y   ⁡     (     k   ,   b     )       -       ∑     w   =   1       N   s       ⁢           H   ^     w     ⁡     (     k   ,   b     )       ⁢       x   w     ⁡     (     k   ,   b     )                  2         σ   n   2     +       ∑     w   =   1       N   s       ⁢                x   w     ⁡     (     k   ,   b     )            2     ⁢       δ   w   2     ⁡     (     k   ,   b     )                 }             )     -     log   ⁢           (       ∑         x   u     ⁡     (     k   ,   b     )       ∈     Ω   l   -         ⁢         ∑     v   =   1       N   s         v   ≠   u       ⁢       ∑         x   v     ⁡     (     k   ,   b     )       ∈   Ω       ⁢     exp   ⁢     {     -                Y   ⁡     (     k   ,   b     )       -       ∑     w   =   1       N   s       ⁢           H   ^     w     ⁡     (     k   ,   b     )       ⁢       x   w     ⁡     (     k   ,   b     )                  2           σ   n   2     +       ∑     w   =   1       N   s       ⁢                x   w     ⁡     (     k   ,   b     )            2     ⁢       δ   w   2     ⁡     (     k   ,   b     )             )         }                             (   19   )               
where Ω is the set of possible symbols values, which may be for example in some embodiments, all sixteen 16-QAM constellation points, Ω l   +  is the set of symbols where bit l on symbol x u (k,b) equals plus one, and Ω l  is the set of symbols where bit l on symbol x u (k,b) equals minus one (or zero). Since in equation (19) the summation over all possible symbols values (i.e., all symbol values in Ω, Ω l   + , or Ω l   − ) may have excessively high computational complexity, methods with lower complexity but nearly the same performance may be employed such as sphere decoding.
 
     In sphere decoding, certain symbol values are removed from the set of all symbol values because it is determined that it was very improbable that they were sent from the transmitters. Thus the LLR calculation for sphere decoding would use a similar formula to equation (19) except that the summations would be done only over the likely symbol values (i.e., not over the symbol values that were determined to be sent with very low probability). In other words, instead of using the set of all possible data symbol combinations to determine the LLRs, a set of possible data symbol combinations is determined (e.g., using sphere decoding ideas) for each transmitter and the set of possible data symbol combinations is used to determine the LLRs in place of the set of all possible data symbol combinations. 
     In order to calculate the LLRs given in Equations (8), (16) and (19), that is, for any of the various embodiments, for example as shown in blocks  407 ,  511 ,  615  and  709 , the channel estimation error for each signal at each subcarrier and symbol time is needed. For linear frequency domain channel estimators such as an MMSE channel estimator, including an MMSE Finite Impulse Response (FIR) channel estimator, a time-tap Least Squares (LS) channel estimator, and Discrete Fourier Transformation (DFT) type channel estimators, the channel estimation error is readily found in the various embodiments. Further in the various embodiments, by using the expected channel conditions, the channel estimation error may be anticipated and thus may be pre-computed and stored in memory at the receiver, for example storage unit  311  of  FIG. 3 . 
     Therefore, in the various embodiments, the channel estimator is assumed to have the following form:
 
 Ĥ   u,m ( k,b )= q   u   H ( k,b ) Y   p,m   (20)
 
where q u (k,b) is a P×1 vector of channel estimation coefficients (P is the number of pilot symbols) for signal u at subcarrier k and symbol time b and Y p,m  is the following P×1 vector of the received pilot data on antenna m for example, antenna  301 :
 
                     Y     p   ,   m       =       [             Y   m     ⁡     (       k   1     ,     b   1       )               ⋮               Y   m     ⁡     (       k   P     ,     b   P       )             ]     =         ∑     u   =   1       N   s       ⁢       [             H     u   ,   m       ⁡     (       k   1     ,     b   1       )               ⋮               H     u   ,   m       ⁡     (       k   P     ,     b   P       )             ]     ⁢     X     p   ,   u           +     [             N   m     ⁡     (       k   1     ,     b   1       )               ⋮               N   m     ⁡     (       k   P     ,     b   P       )             ]                 (   21   )               
where {k 1 ,b 1 } through {k P ,b P } are the pilot locations and
     X p,u =diag(x u (k 1 ,b 1 ), . . . , x u (k P ,b P )).
 
The channel estimation error for user u is determined as:
 
δ u   2 ( k,b )= E{|H   u,m ( k,b )− Ĥ   u,m ( k,b )| 2 }  (22)
   

     Using Equations (20), (21) and (22), the channel estimation error becomes: 
                       δ   u   2     ⁡     (     k   ,   b     )       =     E   ⁢                 {              H     u   ,   m       ⁡     (     k   ,   b     )       -         q   u   H     ⁡     (     k   ,   b     )       ⁢     {         ∑     u   =   1       N   s       ⁢       [             H     u   ,   m       ⁡     (       k   1     ,     b   1       )               ⋮               H     u   ,   m       ⁡     (       k   P     ,     b   P       )             ]     ⁢     X     p   ,   u           +     [             N   m     ⁡     (       k   1     ,     b   1       )               ⋮               N   m     ⁡     (       k   P     ,     b   P       )             ]       }     ⁢                2         }                     (   23   )               
Simplifying Equation (23), the channel estimation error becomes:
 
                       δ   u   2     ⁡     (     k   ,   b     )       =       r   ⁡     (     0   ,   0     )       +         q   u   H     ⁡     (     k   ,   b     )       ⁢       Rq   u     ⁡     (     k   ,   b     )         -     2   ⁢   Re   ⁢     {           q   u   H     ⁡     (     k   ,   b     )       ⁡     [           r   ⁡     (         k   1     -   k     ,       b   1     -   b       )               ⋮             r   ⁡     (         k   P     -   k     ,       b   P     -   b       )             ]       ⁢     X     p   ,   u         }                 (   24   )               
where Re{a} means the real part of a, r(k−f,b−t)=E{H m (k,b)H m *(f,t)}, and P×P R is given by:
 
     
       
         
           
               
             
               
                 
                   
                     
                       
                         
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     To further facilitate the understanding of the various embodiments disclosed herein, examples of the expected channel estimation error are provided in  FIGS. 8   a  and  8   b , for an OFDM system with 768 subcarriers that span 19.2 MHz with an FFT size of 1024. There are four signals (i.e., a receive SDMA of four) with pilot symbols spanning an entire OFDM symbol. Signal one&#39;s pilots are random 16-PSK symbols and the other signal&#39;s pilot symbols are encoded using Steiner&#39;s method as follows:
 
 x   u ( k,b )= x   1 ( k,b ) e   −j2π(u−1)k/4   (26)
 
     All channel estimates are designed assuming each user&#39;s delay profile was a 10 μsec flat profile. For the simulation results, each SDMA user&#39;s channel uses a COST-259-style spatial channel model consisting of a single scattering zone having 50 discrete multi-path rays and a 2 μsec RMS delay spread. The SNR was 10 dB for both the theoretical results and simulations. The channel estimation strategies compared are the MMSE FIR channel estimation with a 17 tap filter, the MMSE channel estimator as described in Vook and Thomas,  MMSE Multi - User Channel Estimation for Broadband Wireless Communications , IEEE Globecom-2001 in San Antonio Tex. (November 2001), which is incorporated by reference herein, a Least Squares (LS) time-tap estimator with an FFT size of 800, and a DFT-based channel estimator (labeled “FFT” estimator). 
       FIGS. 8   a  and  8   b  compare the theoretical channel estimation error using Equation (24) to simulation results. Thus  FIG. 8   a  illustrates channel estimation error versus subcarrier number for joint channel estimation of four signals using a single OFDM symbol of Steiner&#39;s encoded pilots, and shows theoretical results for a 10 μs flat delay profile.  FIG. 8   b  illustrates simulated results using an exponential decaying 2 μs RMS delay spread channel. The SNR for  FIGS. 8   a  and  8   b  was 10 dB. It is to be noted that the theoretical results closely match the simulation results despite the simulations using an exponentially decaying power-delay profile and the channel estimators assuming a flat power delay profile. As expected, the performance of all channel estimators severely degrades at the edge subcarriers indicating that the LLRs are less reliable at these subcarriers. Note that the MMSE FIR filter example has difficulties obtaining channel estimates because of the small number of taps used. 
       FIGS. 9   a  and  10   a  provide simulation results without the various embodiments and for comparison,  FIGS. 9   b  and  10   b  provide simulation results with the various embodiments wherein channel estimation error is tracked in an OFDM uplink. The simulations use a COST-259 style spatial channel model consisting of a single scattering zone having  100  discrete multi-path rays, a 2 μs RMS delay spread, and a 15° multi-path angular spread with respect to the base antenna array. 
     In examples illustrated by  FIGS. 9   a ,  9   b ,  10   a  and  10   b , the base has a uniform linear array of four antennas with a five-wavelength spacing between the antenna elements. The OFDM system uses a 1024-point FFT with a 25 kHz subcarrier spacing at a 3.7 GHz carrier frequency. The cyclic prefix length is 256 (10 μs) and the total OFDM symbol duration is 50 μs. 
     For the simulations, the 3GPP turbo code with max-log-map decoding was used. There are four SDMA users and the pilot format consists of a single OFDM symbol with the pilot structure as per Equation (26). One-thousand (1000) channel realizations were run for each SNR point and there are ten data frames (separately coded) following the pilot sequence. The FER curves are averaged over all SDMA users. 
       FIGS. 9   a ,  9   b ,  10   a  and  10   b  show FER results for rate 1/2 turbo-coded 16-QAM and rate 1/2 turbo-coded 64-QAM respectively for three channel estimators; MMSE, Least Squares (LS) time-tap estimator, and a DFT-based estimator. All three estimators assume a flat delay profile with a maximum delay spread of 10 μs. The receiver simulated employed successive cancellation with the optimal stream decoding order. For the successive cancellation operation, before a stream is cancelled it is first decoded, then re-encoded, and mapped back to symbol values. A clear improvement is illustrated in  FIG. 9   b  and  FIG. 10   b , wherein the embodiments herein disclosed are employed, over the previous methods illustrated in  FIG. 9   a  and  FIG. 10   a.    
     It is to be understood that the various embodiments and inventive principles and concepts discussed and described herein may be particularly applicable to receivers and associated communication units, devices, and systems providing or facilitating voice communications services or data or messaging services over wide area networks (WANs), such as conventional two way systems and devices, various cellular phone systems including, but not limited to, analog and digital cellular, and any networks employing Spatial Division Multiple Access (SDMA), Spatial Division Multiplexing (SDM), Orthogonal Frequency Division Multiple Access (OFDMA), Orthogonal Frequency Division Multiplexing (OFDM) and any variants thereof. 
     Principles and concepts described herein may further be applied in devices or systems with short range communications capability normally referred to as W-LAN capabilities, such as IEEE 802.11, Hiper-LAN or 802.16, WiMAX, digital video broadcasting (DVB), and the like that may further utilize CDMA, frequency hopping, orthogonal frequency division multiplexing, or TDMA technologies and one or more of various networking protocols, such as TCP/IP (Transmission Control Protocol/Internet Protocol), IPX/SPX (Inter-Packet Exchange/Sequential Packet Exchange), Net BIOS (Network Basic Input Output System) or other protocol structures. 
     While the preferred embodiments of the invention have been illustrated and described, it is to be understood 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.