Fixed Detector Examples

  
Figure 5.1: Fixed Detectors' Squared Output Error under ideal conditions

• Ideal Model: I consider the ideal model where there is no channel ISI, asynchronism, or non-orthogonal spreading codes and an SNR of 50 dB. We expect the fixed receivers to perform very closely to one another in this situation. All the other parameters used in this simulation are listed in table 5.1. Figure 5.1 shows the squared output error for the matched filter, MMSE, and zero-forcing detectors. They are almost indistinguishable from one another.
    
  Table 5.1: Parameters for ideal simulation of fixed receivers
  

  These results follow from the derivations of the detectors. Under the ideal conditions both the zero-forcing and MMSE detectors converge to the matched filter receiver. To see that they actually converge to the same detector, we can also compare the detector impulse responses.

• Bad Cross Correlation Properties of spreading codes: To see how the effects of codes with bad cross correlation properties (i.e. non-orthogonal) effect the detectors, I ran a simulation with random spreading codes of length 8. All other parameters were as in the ideal case(see Table 5.2). Figure 5.2 shows the squared output error results when using random codes. As we might expect, the matched filter does not decorrelate the codes and as a result performs much worse than the the zero-forcing or MMSE detectors. (Note: The zero-forcing and MMSE squared output error are indistinguishable)
    
  Figure 5.2: Fixed Receivers Squared Output Error using random spreading codes.
  
    
  Table 5.2: Parameters for Cross Correlation Effects on Fixed Detector Performance
  

• Non Existence of Zero-Forcing Solution: The true zero-forcing solution does not always necessarily exist. Since the solution depends on the invertability of , must be full column rank. It so happens with the parameters set as in table 5.3, has more rows than columns, implying that it is not full rank. A solution is generated, but it is a minimum norm pseudo-inverse solution, not a true zero-forcing solution. In this case, the MMSE detector is the best of the fixed detectors. Figure 5.3 shows the squared output error plots of the three detectors. Note that the asynchronism and non-orthogonal Gold codes are responsible for the poor performance of the matched filter detector.
    
  Figure 5.3: Fixed Detector performance when zero-forcing solution does not exist
  
    
  Table 5.3: Parameters for Non-Existence of Zero-Forcing Solution
  

• The Near-Far Effect This next simulation shows the effects of unequal user powers. In this particular example, the user to be detected is 5 dB lower than all the other 15 users in the system. The spreading codes are Gold codes and the system is synchronous. The non-orthogonality of the spreading codes in combination with the power disparity causes the matched filter to perform inadequately. The final BER for the matched filter is .14. The performance of the matched filter is improved by using orthogonal Hadamard codes in a synchronous system. However, if asynchronism is introduced, the near-far effect becomes even worse. Figure 5.4 shows the results of the near-far effect and Table 5.4 shows the system parameters.
    
  Figure 5.4: Fixed Detector performance under the Near-Far Effect
  
    
  Table 5.4: Parameters for Near-Far Effects on Fixed Detectors
  

• Fractionally Spaced vs. Baud Spaced MMSE equalizer: On average, it has been shown that fractionally spaced MMSE detectors perform better than baud spaced MMSE detectors [1]. This example demonstrates a case where under severe multipath and out of cell interference, the FS-MMSE equalizer gives a substantial gain of about 6 dB. The simulations were performed by setting the random seed buttons inactive, allowing reproducible channels for the baud and fractionally spaced simulations. Figure 5.5 shows the squared output error results and Table 5.5 shows the parameter settings used.
    
  Figure 5.5: Fractionally Spaced vs. Baud Spaced MMSE Detectors
  
    
  Table 5.5: Parameters for Fractionally Spaced vs. Baud Spaced Detection