Exhaustive-Search and Decorrelative Detection

Exhaustive-Search and Decorrelative Detection

Consider the following complex-valued discrete-time synchronous CDMA signal model. At any time instant, the received signal is the superposition of K users' signals, plus the ambient noise, given by

Equation 4.96

Equation 4.97

where, as before, , c_i,k {+1, -1}, is the normalized signature sequence of the kth user; N is the processing gain; b_k {+1, -1} and A k are, respectively, the data symbol and the complex amplitude of the kth user; ; and is a complex vector of i.i.d. ambient noise samples with independent real and imaginary components. Denote

where u is a real noise vector consisting of 2N i.i.d. samples. Then (4.97) can be written as

Equation 4.98

It is assumed that each element u_j of u follows a two-term Gaussian mixture distribution:

Equation 4.99

with 0 < 1 and k > 1. Note that the overall variance of the noise sample u_j is

Equation 4.100

We have Cov (u) = (s²/2) I_2N and Cov (n) = s²I_N.

Recall from the preceding sections that there are primarily two categories of multiuser detectors for estimating b from y in (4.98), all based on minimizing the sum of a certain function r(·) of the chip residuals

Equation 4.101

where denotes the jth row of the matrix Y. These are as follows.

• Exhaustive-search detectors:

  Equation 4.102

  

  
  

• Decorrelative detectors:

  Equation 4.103

  

  
  

  Equation 4.104

  

  
  

Note that exhaustive-search detection is based on the discrete minimization of the cost function C (b;y), over 2^K candidate points, whereas decorrelative detection is based on the continuous minimization of the same cost function. As before, let y = r' be the derivative of the penalty function r. In general, the optimization problem (4.103) can be solved iteratively according to the following steps [553]:

Equation 4.105

Equation 4.106

Recall further from Section 4.2 the following three choices of the penalty function r(·) in (4.101), corresponding to different forms of detectors:

• Log-likelihood penalty function:

  Equation 4.107

  

  
  

  Equation 4.108

  

  
  

  where f(·) denotes the pdf of the noise sample u_j. In this case, the exhaustive-search detector (4.102) corresponds to the ML detector, and the decorrelative detector (4.104) corresponds to the ML decorrelator.

• Least-squares penalty function:

  Equation 4.109

  

  
  

  Equation 4.110

  

  
  

  In this case, the exhaustive-search detector (4.102) corresponds to the ML detector based on a Gaussian noise assumption, and the decorrelative detector (4.104) corresponds to the linear decorrelator.

• Huber penalty function:

  Equation 4.111

  

  
  

  Equation 4.112

  

  
  

  where s²/2 is the noise variance given by (4.100) and

  Equation 4.113

  

  
  

  is a constant. In this case, the exhaustive-search detector (4.102) corresponds to the discrete minimizer of the Huber cost function, and the decorrelative detector (4.104) corresponds to the robust decorrelator.