Adaptive Detector Examples

• LMS convergence to the MMSE detector: In single user applications, the LMS property of converging to the MMSE solution is what makes the algorithm desirable. This example shows that the LMS algorithm still converges to the MMSE detector in the multiuser case. A step size of and 100% training bandwidth were used to generate figure 5.6. The algorithm does not converge exactly to the MMSE detector because of the excess MSE caused by the large step size and the fact that the noise power is 30 dB. Table 5.6 displays the other parameter settings.
    
  Figure 5.6: (a) Square Output Error trajectory (b) MMSE Detector Impulse responses
  
    
  Table 5.6: Parameters for LMS Convergence to the MMSE Detector
  

• Effects of Power Distribution on Convergence Rates: Figure 5.7 shows the squared output error of three different users. Each of the three users has a different amount of power, which is reflected in the corresponding achievable minimum MSE. If we look at the initial slope of each of the three users' squared output error trajectories, we can see a noticeable variance. It seems that the highest power user has the largest initial slope, converging close to the MMSE quickly. As the user power decreases, there is a noticeable decline in the initial convergence rate. This would lead me to believe that higher powered users have a larger initial convergence rate. However, there is not enough information to make an educated decision about overall convergence rate of the different users. The overall convergence rate depends on the initialization as well as other system dynamics. Monte Carlo simulations may produce more conclusive results about the overall convergence rate of the individual users. Table 5.7 shows the other selected parameters for the simulation.
    
  Figure 5.7: (a) Square Output Error trajectories (b) Power Distribution of in cell users. The filled circles correspond with the trajectories in (a).
  
    
  Table 5.7: Parameters for Power Distribution Effects on Convergence of LMS
  

• Effects of Training Bandwidth on Convergence Rate: When the training bandwidth is varied, we can see the different effects it has on powerful users and weak users. The plots in figure 5.8 show the differences for two users separated by approximately 6dB. For the lower powered user, a 1% training bandwidth results in very slow convergence in this particular example. After 5000 iterations the SE is not even -5dB, which on average corresponds to an undesirable bit error rate. It appears that if a user has a high SNR, then a reduction in bandwidth will not affect the bit error rate as much as the lower powered user, however the initial convergence rate is adversely affected. The overall effect of training bandwidth will again depend on the system dynamics. For average performance, Monte Carlo simulations are necessary. Table 5.8 shows the other system parameters for this simulation.
    
  Figure 5.8: (a) Square Output Error trajectories for User 1 @ 6dB (b) Square Output Error trajectories for User 3 @ 0dB
  
    
  Table 5.8: Parameters for Power Distribution Effects on Convergence of LMS under different training bandwidths
  

• Near-Far Resistant Minimum Entropy Initialization of CMA: The initialization technique for CMA presented in [9] is based on a pre-whitening of the received data to eliminate near-far effects. Near-far effects cause CMA minima to become very small or disappear. This is an example showing what happens when the initialization technique is used for pre-whitened and non-pre-whitened CMA. I chose a user that was -3 dB from the mean power of in cell users. Figure 5.9 shows the squared error tap trajectories for both the whitened and non-whitened implementations. Both were initialized according to the minimum entropy initialization scheme, but CMA was run on the whitened and non-whitened received data respectively. Also shown is the kurtosis measure for both the non-whitened and pre-whitened initialization. From this figure we can see that by pre-whitening, we have initialized in the correct location. (Note: The solid stem is the desired choice of delay in the Kurtosis plots. Notice that it corresponds to the largest value in the pre-whitened case, but not in the non-whitened case.) We can see by figure 5.10 that the eye is opened in the pre-whitened case and not in the non-whitened case. A Decision Directed algorithm could be implemented along with the pre-whitened CMA to achieve MMSE like performance.
    
  Figure 5.9: (a) Square Output Error trajectories for whitened and non-whitened CMA (b) Kurtosis Measure and Power Distribution
  
    
  Figure 5.10: (a) Constellation and Eye Diagram for Pre-whitened CMA (b) Constellation and Eye Diagram for Non-whitened CMA
  
    
  Table 5.9: Parameters for Whitened vs. Non-Whitened CMA