Abstract:
A receiver includes a channel providing information received from wireless communications, including information from a selected user. A minimum mean square error combiner is coupled to the channel for receiving samples in a symbol duration and minimizes mean square error in such samples. The combiner has coefficients derived from a training sequence. Information of the selected user is extracted from a mixture consisting of information from unintended users, interference and noise.

Description:
BACKGROUND 
       [0001]    Ultra-Wideband (UWB) is a technology for transmitting information spread over a large bandwidth, such as &gt;500 MHz. UWB is an emerging technology inviting major advances in wireless communication, networking, radar and positioning systems. UWB technology has drawn the attention of industry for its attractive features such as low power density, rich multi-path diversity, low complexity base band processing, multi-access capability, timing precision etc. On the other hand the stringent timing requirement and frequency selective nature of the UWB channels pose challenge in the receiver processing. Moreover, as the UWB is emerging as a technology, it is finding its use in multiuser systems and hence efficient multi-user detection is also a challenging area to pursue for the researchers. 
         [0002]    The ability to resolve multipath is one of the most attractive features of UWB. A Rake receiver can be employed to exploit the multipath diversity. The high data nature of UWB coupled with frequency selective nature of the channel make the system suffer from severe intersymbol interference (ISI). To combat the effect of ISI, Rake may be followed by a equalizer. In some prior systems, combined RAKE and equalization methods were proposed for direct sequence UWB (DS-UWB) systems. 
         [0003]    Problems in wireless communication include multipath fading, intersymbol interference (ISI), MUI and other interference which leads to severe distortion in the transmitted waveform when it arrives at a receiver. The multipath fading and ISI effect are due to the hostile nature of the wireless medium. Moreover, modern wireless systems support multiple users and also have to be interoperable with other systems. They suffer severely by the problem of MUI and other interferences. Traditionally these problems have been considered separately and receiver architecture for alleviating them have been proposed in a one-by-one manner. However, this approach increases the complexity of the receiver owing to the inclusion of several functional blocks. 
         [0004]    Problems in RAKE and equalization methods are further aggravated in multiuser communications where the desired information is embedded with multiuser interference (MUI). The Rake receiver, using maximum ratio combining (MRC), is optimum only when the disturbance to the desired signal is sourced by additive white Gaussian noise (AWGN). In the presence of MUI the Rake combiner will exhibit an error floor depending on the signal-to-interference-plus-noise ratio (SINR). 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0005]      FIG. 1A  is a block diagram of a linear detector according to an example embodiment. 
           [0006]      FIG. 1B  is a block diagram of a nonlinear detector according to an example embodiment. 
           [0007]      FIG. 1C  is a block diagram of a combiner according to an example embodiment. 
           [0008]      FIG. 2  is  FIG. 2  is a graph illustrating bit error rates for a linear multi-user detector according to an example embodiment. 
           [0009]      FIG. 3  is a graph illustrating bit error rates for a non-linear multi-user detector according to an example embodiment. 
           [0010]      FIG. 4  is a graph comparing bit error rates of a multi-user detector based receiver with conventional receivers based on RAKE and an MMSE. equalizer according to an example embodiment. 
           [0011]      FIG. 5  is a graph illustrating bit error rates for a fixed SNR non-linear multiuser detector according to an example embodiment. 
           [0012]      FIG. 6  is a table comparing characteristics of various detectors according to an example embodiment. 
           [0013]      FIG. 7  a block diagram of an example computer system that may perform methods and algorithms according to an example embodiment. 
       
    
    
     DETAILED DESCRIPTION 
       [0014]    In the following description, reference is made to the accompanying drawings that form a part hereof, and in which is shown by way of illustration specific embodiments which may be practiced. These embodiments are described in sufficient detail to enable those skilled in the art to practice the invention, and it is to be understood that other embodiments may be utilized and that structural, logical and electrical changes may be made without departing from the scope of the present invention. The following description of example embodiments is, therefore, not to be taken in a limited sense, and the scope of the present invention is defined by the appended claims. 
         [0015]    The functions or algorithms described herein may be implemented in software or a combination of software and human implemented procedures in one embodiment. The software may consist of computer executable instructions stored on computer readable media such as memory or other type of storage devices. The term “computer readable media” is also used to represent any means by which the computer readable instructions may be received by the computer, such as by different forms of wired or wireless transmissions. Further, such functions correspond to modules, which are software, hardware, firmware or any combination thereof. Multiple functions may be performed in one or more modules as desired, and the embodiments described are merely examples. The software may be executed on a digital signal processor, ASIC, microprocessor, or other type of processor operating on a computer system, such as a personal computer, server or other computer system. 
         [0016]    A method for multiuser detection is based on minimum mean square error (MMSE) for a DS-UWB multiuser communication system. The method exploits the inherent multi path diversity and also mitigates the effects of both inter symbol interference (ISI) and multiuser interference (MUI). Simulation results show that the given algorithms perform better than the other known detectors in literature. A closed form expression for the bit error rate (BER) results of the above method is also provided. In one embodiment, a direct sequence-code division multiple access (DS-CDMA) based UWB multi-user communication system and simple linear and nonlinear receiver structures combine multipath diversity and reject ISI as well as multiuser interference. A system model is presented. A linear detection scheme is described and analyzed. Similarly, a non-linear detector is described and analyzed. 
         [0017]    System Model 
         [0018]    Consider a DS-CDMA based UWB multi-user system where there are N u  active users. At the transmitter u, BPSK symbols a k   u ε{−1,1} are spread and modulated with chip pulses g c (t). Defining the chip waveform 
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         [0000]    where {c i   u }, c i   u ε{−1,+1} denotes the spreading code of length N c  and chip duration T c . The transmit signal with the channel response 
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         [0019]    where T s  denotes the symbol duration, p u (t)=g u (t)*h u (t) and * denotes the convolution operator. 
         [0020]    Assuming perfect synchronism, the signal at the receiver input is given by 
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         [0021]    In one embodiment, a receiver is designed that extracts the information of the desired user from this mixture. Traditional receiver architectures, which employ RAKE based reception, entail knowledge about the channel and the spreading sequence of the user. The designed receiver does not impose any of these requirements. Two types of detectors are described as shown in  FIGS. 1A and 1B . 
         [0022]    Linear Multiuser Detector (Linear MUD) 
         [0023]    In one embodiment, the architecture of a linear receiver may be based on a simple MMSE linear combiner (Weiner Filter) which optimally combines the received samples in a symbol duration so as to minimize the mean square error. The receiver structure is as shown at  110  in  FIG. 1A , which depicts a block diagram of a linear detector. The receiver has a feed-forward filter (FFF)  115 . The input to the FFF  115  at time kT s  is the vector r k  (cf.(6)) that has been converted from serial to parallel at  120 . MMSE criterion is again applied to optimize the coefficients of the filter. 
         [0024]    The determination of the coefficients of the combiner may be carried through a training sequence method  150  as shown in  FIG. 1C .  FIG. 1C  illustrates signals sampled from a channel  155  and provided to an MMSE-MUD block  160  that has tap coefficients derived from training sequence  150 . Combiner tap coefficients may be determined based on MMSE criterion. 
         [0025]    Let the sampling rate T r  at the receiver be chosen such that such that 
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         [0000]    Then the discrete equivalent of the received waveform becomes 
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         [0026]    Assume u=1 is the user of interest at the receiver. By dropping the superscript for the user  1 , (3) may be rewritten as 
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         [0000]    where the second term represents the multiuser interference (MUI) part. Representing MUI as m l , 
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         [0027]    The received sequence may be fragmented into frames of N s  samples each frame representing the sampled version of the received symbol of the user  1 . That is, r k =[r (k+1).N     s    . . . r k.N     s=1   ] T , where r k  represents the samples of the received waveform corresponding to the a k  transmitted. The objective is to choose the linear filter taps ŵ=[w 1 , . . . w N     s   ] which minimizes the mean square error E∥a k −wr k ∥ 2 . The solution to this problem is given by the Weiner-Hopf equation [10], 
         [0000]      ŵ=γ ar Γ rr   −1 . 
         [0000]    where Γ rr =E[r k r k   T ] represents the received vector autocorrelation matrix and γ ar =E[a k r k   T ] is a vector representing the cross correlation between the desired symbol and the corresponding received samples. 
         [0028]    The symbol estimate at the receiver can be computed as 
         [0000]      â k =sign(ŵr k ) 
         [0029]    Expression for the Linear MUD 
         [0030]    Analysis of the linear receiver is provided along with consideration of some special cases. With the slight abuse of the notation, let us denote p k (j)=[p((k−j)N s +N s ) . . . p((k−j)N s +1)] T , m k =[m (k+1).N     s    . . . m k.N     s     +1 ] T  and n k =[n (k+1).N     s    . . . n k.N     s     +1 ] T . The discussion is restricted to the channels with a finite memory of L s  symbol durations. Then, the sampled vector at the receiver r k  can be represented by 
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         [0031]    Further, by defining a k =[a k-L     s     +1 ] T  and a matrix P=[p 0 (0), . . . , p 0 (L s −1)], the above equation may be rewritten as 
         [0000]        r   k   =[r   (k+1).N     s      . . . T   k.N     s     +1]   T   =Pa   k   +m   k   +n   k .  (6) 
         [0000]    Remark: Consider the waveform response p(t) of the channel. The length of the channel is of duration L s  symbol periods. The corresponding discrete form {p(n)} then, spans for a duration of N s ·L s  samples. P is essentially the matrix form of the sequence {p(n)}. The dimension of P is N s ×L s . Note that P does not depend on the frame index k. 
         [0032]    Evaluation of the correlation matrices is now provided. It is assumed that the signal received at the receivers due to different users are uncorrelated. 
         [0000]      Γ rr   =E[r   k   r   k   T   ]=PE[a   k   a   k   T   ]P   T   +R   m   +R   n , 
         [0000]    where R m  and R n  represent the correlation matrices of MUI and noise respectively. 
         [0033]    The training sequence {a k }, chosen is assumed to be pseudo random so that E[a i a j ]=σ a   2 δ(l−j). Therefore, 
         [0000]      σ rr =σ a   2   P.P   T   +R   m   αR   n . 
       Similarly, 
       [0034]      γ ar σ a   2   p   0   T (0). 
         [0035]    The Weiner filter 
         [0000]        w=σ   a   2   p   0   T (0)[σ a   2   P.P   T   +R   m   +R   n ] −1 .  (7) 
         [0036]    Next, some special cases are considered. 
         [0037]    Single User AWGN and fading with no ISI: 
         [0038]    Given that N u =1, R n =σ n   2 ·I N     s    and L s =1. The P=p 0 (0). From (7) we have, 
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         [0000]    which essentially represents an MRC combiner. 
         [0039]    Multi User AWGN and Fading with no ISI: 
         [0040]    Given N u  users, R n =σ n   2 ·I N     s    and L s =1. Then P u =p o   u (0), u=1, . . . N u . From (7) for the user m, 
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         [0000]    which essentially represents a blind multiuser detector. 
       BER Analysis 
       [0041]    The probability of error of the linear MUD is now derived. The Beaulieu Series method is adopted for the evaluation of the probability of error. The output of the combiner y k  at the k th  symbol period can be written as 
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         [0000]    where q kj   u =wp k   u (j) and N k =wn k ˜N(0,∥w∥ 2 .σ n   2 ). 
         [0042]    Given that a i   u ε{−1,+1} and are assumed to iid and uniformly distributed, effect of other terms in (8) on the first term is evaluated. Fixing the value of a k . Then, the characteristic function of x k  is 
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                               u 
                               ≠ 
                               1 
                             
                           
                            
                           
                             
                               ∏ 
                               
                                 l 
                                 = 
                                 
                                   k 
                                   - 
                                   
                                     L 
                                     s 
                                     u 
                                   
                                   + 
                                   1 
                                 
                               
                               k 
                             
                              
                             
                               
                                 cos 
                                  
                                 
                                   ( 
                                   
                                     ω 
                                      
                                     
                                         
                                     
                                      
                                     
                                       q 
                                       kl 
                                       u 
                                     
                                   
                                   ) 
                                 
                               
                               . 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   9 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Now, the probability of a bit error P b  can be given by [1] 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       P 
                       b 
                     
                     ≈ 
                     
                       
                         1 
                         2 
                       
                       - 
                       
                         
                           ∑ 
                           
                             n 
                             = 
                             
                               { 
                               
                                 1 
                                 , 
                                 3 
                                 , 
                                 5 
                                 , 
                                 
                                     
                                 
                                  
                                 … 
                               
                               } 
                             
                           
                           ∞ 
                         
                          
                         
                           
                             
                               2 
                                
                               
                                 sin 
                                  
                                 
                                   ( 
                                   
                                     n 
                                      
                                     
                                         
                                     
                                      
                                     
                                       ω 
                                       0 
                                     
                                      
                                     
                                       q 
                                       00 
                                     
                                   
                                   ) 
                                 
                               
                                
                               
                                 exp 
                                 ( 
                                 
                                   
                                     
                                       - 
                                       
                                         n 
                                         2 
                                       
                                     
                                      
                                     
                                       ω 
                                       0 
                                       2 
                                     
                                      
                                     
                                       
                                          
                                         w 
                                          
                                       
                                       2 
                                     
                                      
                                     
                                       σ 
                                       n 
                                       2 
                                     
                                   
                                   2 
                                 
                                 ) 
                               
                             
                             
                               n 
                                
                               
                                   
                               
                                
                               π 
                             
                           
                           × 
                           
                             
                               ∏ 
                               
                                 j 
                                 = 
                                 
                                   k 
                                   - 
                                   
                                     L 
                                     s 
                                   
                                   + 
                                   1 
                                 
                               
                               
                                 k 
                                 - 
                                 1 
                               
                             
                              
                             
                               
                                 cos 
                                  
                                 
                                   ( 
                                   
                                     n 
                                      
                                     
                                         
                                     
                                      
                                     
                                       ω 
                                       0 
                                     
                                      
                                     
                                       q 
                                       kj 
                                     
                                   
                                   ) 
                                 
                               
                                
                               
                                 
                                   ∏ 
                                   
                                     u 
                                     ≠ 
                                     1 
                                   
                                 
                                  
                                 
                                   
                                     ∏ 
                                     
                                       l 
                                       = 
                                       
                                         k 
                                         - 
                                         
                                           L 
                                           s 
                                           u 
                                         
                                         + 
                                         1 
                                       
                                     
                                     k 
                                   
                                    
                                   
                                     cos 
                                      
                                     
                                       ( 
                                       
                                         n 
                                          
                                         
                                             
                                         
                                          
                                         
                                           ω 
                                           0 
                                         
                                          
                                         
                                           q 
                                           kl 
                                           u 
                                         
                                       
                                       ) 
                                     
                                   
                                 
                               
                             
                           
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   10 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where 
         [0000]    
       
         
           
             
               
                 ω 
                 0 
               
               = 
               
                 
                   2 
                    
                   π 
                 
                 T 
               
             
             , 
           
         
       
     
         [0000]    parameter T governs the sampling rate in the frequency domain. Higher values of T ensure negligible approximation error.  FIG. 2  shows the comparison of the analytically obtained BER against the simulated one. It can be seen from the figure that the analytical curve is in close approximation with that of simulation. This is expected because the curves are evaluated without making Gaussian assumption over the interference part. 
       Non-Linear Multiuser Detector (Non-Linear MUD) 
       [0043]    In the previous section, the linear receiver which performs linear MMSE estimation of the symbol by processing the received samples of the corresponding symbol duration was considered. However, due to the memory of the channel, one could potentially use the estimates of the previous symbols while attempting to estimate the current symbol. This can be achieved by exploiting the familiar Decision Feedback principle.  FIG. 1B  depicts a block diagram of a nonlinear receiver  130 . The receiver has two filters one feed-forward filter (FFF)  135  and the other is a feedback filter (FBF)  140 . The input to the FFF  135  at time kT s  is the vector r k  (equation 6) which is provided by serial to parallel converter  120 . The FFF  135  is similar to the Weiner combiner studied in the previous section which generates a test statistic from the received samples. The FBF  140  has at its input the sequence of decisions on previously detected symbols. FBF  140  is intended to remove the part of the ISI from the current symbol caused by previously detected symbols. MMSE criterion is again applied to optimize the coefficients of the two filters. Note that the input samples to FFF  135  are spaced T r  seconds apart while the input samples of the FBF  140  are spaced T s  seconds apart. 
         [0044]    The equalizer output is represented as 
         [0000]    
       
      
       x 
       k 
       =w 
       ff 
       r 
       k 
       +w 
       fb 
       â 
       k-1  
      
     
         [0000]    where, the row vector w ff  represents N s  length FF filter and w fb  is a N b  length vector representing the FB filter. The set of past decisions is represented by â k-1 =[â k-1  . . . â k-N     b   ] T . The estimate â k  of a k  symbol can be obtained by passing x k  through the detector i.e., â k-1 =sign(x k ). The objective is to choose the filter coefficient set w=[w ff ,w fb ] which minimizes the MSE 
         [0000]        E[∥a   k   −x   k ∥ 2   ]=E[∥a   k   −wY   k ∥ 2 ], 
         [0000]    where Y k =[r k   T , â k-1   T ] T . Proceeding similar to previous sections, we have the optimal solution for w as 
         [0000]        ŵ=γ   ay Γ yy   −1 , 
         [0045]    where Γ yy =E[Y k Y k   T ] represents the autocorrelation matrix and γ ay =E[a k Y k   T ] represents the cross correlation matrix. Assuming that there is no error in the feedback, the matrices can be written as 
         [0000]    
       
         
           
             
               Γ 
               yy 
             
             = 
             
               [ 
               
                 
                   
                     
                       Γ 
                       rr 
                     
                   
                   
                     
                       E 
                        
                       
                         [ 
                         
                           
                             Y 
                             k 
                           
                            
                           
                             a 
                             
                               k 
                               - 
                               1 
                             
                             T 
                           
                         
                         ] 
                       
                     
                   
                 
                 
                   
                     
                       E 
                        
                       
                         [ 
                         
                           
                             a 
                             
                               k 
                               - 
                               1 
                             
                           
                            
                           
                             Y 
                             k 
                             T 
                           
                         
                         ] 
                       
                     
                   
                   
                     
                       
                         σ 
                         a 
                         2 
                       
                       · 
                       
                         I 
                         
                           N 
                           b 
                         
                       
                     
                   
                 
               
               ] 
             
           
         
       
     
         [0000]    and 
         [0000]    
       
         
           
             
               γ 
               ay 
             
             = 
             
               
                 [ 
                 
                   
                     
                       
                         γ 
                         ar 
                       
                     
                   
                   
                     
                       0 
                     
                   
                 
                 ] 
               
               . 
             
           
         
       
     
         [0000]    Proceeding on similar lines with the previous section, the bit error rate of the nonlinear MUD can be evaluated using Beaulieu series. Therefore, by denoting q kj   u =w ff p k   u (j) we have, 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           P 
                           b 
                         
                         ≈ 
                           
                          
                         
                           
                             1 
                             2 
                           
                           - 
                           
                             
                               ∑ 
                               
                                 n 
                                 = 
                                 
                                   { 
                                   
                                     1 
                                     , 
                                     3 
                                     , 
                                     5 
                                     , 
                                     
                                         
                                     
                                      
                                     … 
                                   
                                   } 
                                 
                               
                               ∞ 
                             
                              
                             
                               
                                 2 
                                  
                                 
                                   sin 
                                    
                                   
                                     ( 
                                     
                                       n 
                                        
                                       
                                           
                                       
                                        
                                       
                                         ω 
                                         0 
                                       
                                        
                                       
                                         q 
                                         00 
                                       
                                     
                                     ) 
                                   
                                 
                                  
                                 
                                   exp 
                                   ( 
                                   
                                     
                                       
                                         - 
                                         
                                           n 
                                           2 
                                         
                                       
                                        
                                       
                                         ω 
                                         0 
                                         2 
                                       
                                        
                                       
                                         
                                            
                                           
                                             w 
                                             ff 
                                           
                                            
                                         
                                         2 
                                       
                                        
                                       
                                         σ 
                                         n 
                                         2 
                                       
                                     
                                     2 
                                   
                                   ) 
                                 
                               
                               
                                 n 
                                  
                                 
                                     
                                 
                                  
                                 π 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                           
                          
                         
                           
                             ∏ 
                             
                               j 
                               = 
                               1 
                             
                             
                               N 
                               fb 
                             
                           
                            
                           
                             
                               cos 
                                
                               
                                 ( 
                                 
                                   n 
                                    
                                   
                                       
                                   
                                    
                                   
                                     ω 
                                     0 
                                   
                                    
                                   
                                     ω 
                                     
                                       fb 
                                       , 
                                       j 
                                     
                                   
                                 
                                 ) 
                               
                             
                              
                             
                               
                                 ∏ 
                                 
                                   j 
                                   = 
                                   
                                     k 
                                     - 
                                     
                                       L 
                                       s 
                                     
                                     + 
                                     1 
                                   
                                 
                                 
                                   k 
                                   - 
                                   1 
                                 
                               
                                
                               
                                 
                                   cos 
                                    
                                   
                                     ( 
                                     
                                       n 
                                        
                                       
                                           
                                       
                                        
                                       
                                         ω 
                                         0 
                                       
                                        
                                       
                                         q 
                                         kj 
                                       
                                     
                                     ) 
                                   
                                 
                                 × 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                           
                          
                         
                           
                             
                               ∏ 
                               
                                 u 
                                 ≠ 
                                 1 
                               
                             
                              
                             
                               
                                 ∏ 
                                 
                                   l 
                                   = 
                                   
                                     k 
                                     - 
                                     
                                       L 
                                       s 
                                       u 
                                     
                                     + 
                                     1 
                                   
                                 
                                 k 
                               
                                
                               
                                 cos 
                                  
                                 
                                   ( 
                                   
                                     n 
                                      
                                     
                                         
                                     
                                      
                                     
                                       ω 
                                       0 
                                     
                                      
                                     
                                       q 
                                       kl 
                                       u 
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                           , 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   11 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where, w fb,j  represents the j th  tap of the feedback filter. Note that the first product part of the summation represents the effect due to the feedback.  FIG. 3  shows the comparison of the analytically obtained BER against the simulated one. As before, the curves match very closely. 
         [0046]    Results and Discussion 
         [0047]    In this section, the BER performance of the proposed detectors is presented for a DS-UWB system. Comparison with the other detectors in literature is also provided. For the simulation, a multiuser DS-UWB system with antipodal modulation scheme was considered. The DS sequences had a processing gain of 8. The signal at the receiver was sampled at twice the chip rate, T r =T chip /2. The channel is simulated using the IEEE 802.15.3a channel model. For comparison, a) RAKE-MMSE-Linear equalizer, b) RAKE-MMSE-DFE, c) Linear MUD and d) Non-Linear MUD were considered.  FIG. 4  shows the BER vs Average SNR plot of the different detectors when N u =6. It is clear from the figure that the proposed schemes perform significantly better than the other detectors. The BER performance as a function of users for a fixed SNR is shown in  FIG. 5 .  FIG. 5  illustrates that Linear/Non-linear MUD yield a significant gain with the number of users. This is expected because as N u  increases, MUI increases. The RAKE receiver with MRC is optimal only in the single user case with the Gaussian interference whereas the proposed MUDs have the inherent ability to mitigate the effects of the multiuser interference. For N u =1 there is no significant difference in the performance of the receivers considered. Non-linear detectors outperform linear detectors due to the presence of feedback. 
         [0048]    Various parameters for comparison are presented in  FIG. 6 . The proposed detectors entail no explicit knowledge of the channel parameter and hence are less sensitive to estimation errors. They also do not need explicit knowledge of the spreading sequence and the transmitted pulse shape. 
       CONCLUSIONS 
       [0049]    Two multiuser detection methods are described based on minimum mean square error (MMSE) for a DS-UWB multiuser communication system. The methods exploit the inherent multipath diversity and also mitigates the effects of both intersymbol interference (ISI) and multiuser interference (MUI). BER performance for various detectors was described. Closed form expressions for the BER results of the above methods were also provided. Further, a comparison table was presented to underline the simplicity of the proposed algorithms over others. 
         [0050]    A block diagram of a computer system that executes programming for performing the above methods and algorithms is shown in  FIG. 7 . A general computing device in the form of a computer  710 , may include a processing unit  702 , memory  704 , removable storage  712 , and non-removable storage  714 . Memory  704  may include volatile memory  706  and non-volatile memory  708 . Computer  710  may include—or have access to a computing environment that includes—a variety of computer-readable media, such as volatile memory  706  and non-volatile memory  708 , removable storage  712  and non-removable storage  714 . Computer storage includes random access memory (RAM), read only memory (ROM), erasable programmable read-only memory (EPROM) &amp; electrically erasable programmable read-only memory (EEPROM), flash memory or other memory technologies, compact disc read-only memory (CD ROM), Digital Versatile Disks (DVD) or other optical disk storage, magnetic cassettes, magnetic tape, magnetic disk storage or other magnetic storage devices, or any other medium capable of storing computer-readable instructions. Computer  710  may include or have access to a computing environment that includes input  716 , output  718 , and a communication connection  720 . The computer may operate in a networked environment using a communication connection to connect to one or more remote computers. The remote computer may include a personal computer (PC), server, router, network PC, a peer device or other common network node, or the like. The communication connection may include a Local Area Network (LAN), a Wide Area Network (WAN) or other networks. 
         [0051]    Computer-readable instructions stored on a computer-readable medium are executable by the processing unit  702  of the computer  710 . A hard drive, CD-ROM, and RAM are some examples of articles including a computer-readable medium. 
         [0052]    The Abstract is provided to comply with 37 C.F.R. §1.72(b) to allow the reader to quickly ascertain the nature and gist of the technical disclosure. The Abstract is submitted with the understanding that it will not be used to interpret or limit the scope or meaning of the claims.