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Next: CONCLUSION Up: Adaptive Equalization of Non-linear Previous: SIMULATION RESULTS

   
SIMULATION RESULTS

The complex radial basis function (CRBF) was tested on 64-QAM signal with ISI and non-linear distortion and it was compared with the ordinary least mean square (LMS) algorithm, to check its performance in non-linear environment. To check how RBF performs in this environment and how it compares to the performance of LMS equalizer, we take three examples :

Example 1: Here the linear channel was taken with only ISI and no non-linear terms. The channel transfer function was given by :

 
H(z)= (0.0485+j0.0194)+(0.0573+j0.0253)z-1+(0.0786+j0.0282)z-2+    
  (0.0874+j0.0447)z-3+(0.9222+j0.3031)z-4+(0.1427+j0.0349)z-5+    
  (0.0835+j0.0157)z-6+(0.0621+j0.0078)z-7+(0.0359+j0.0049)z-8+    
  $\textstyle \mbox{} (0.0214+j0.0019)z^{-9}.$    

The 64-QAM signal received after corrupting with ISI is as shown in Fig. 3. The total no. of symbols considered in each case is 20000. The SER which was obtained by using a complex sigmoid function was 0.848. Then LMS was applied to it whose order was set to M=10 and the learning rate mu=0.0001. The result using LMS for equalization is shown in Fig. 4. The SER obtained with LMS is 0. Then RBF equalizer was used on the same set of signals, and the result obtained is as shown in Fig. 5. The SER in this case was also 0. The NCRBF used had no. of centers as 350 and order N=6. The width was chosen as $\sigma=4.25$.


  
Figure 3: 64-QAM signal corrupted by ISI given in (10)
\begin{figure}\hspace{-5mm} \mbox{\centerline{\psfig{figure=qam64_lin1.ps,height=8cm}} } \end{figure}


  
Figure 4: The constellation of the 64-QAM after the complex LMS equalizer
\begin{figure}\hspace{-5mm} \mbox{\centerline{\psfig{figure=lms64_lin1.ps,height=8cm}} } \end{figure}


  
Figure 5: The constellation of the 64-QAM after CRBF equalizer
\begin{figure}\hspace{-5mm} \mbox{\centerline{\psfig{figure=ncrbf64_lin1.ps,height=8cm}} } \end{figure}

Example 2: In this case we retain all the ISI terms and make the channel slightly non-linear, the ISI terms remain as in (10) but along with that here non-linear terms are added which are

 
yn=on+0.02on2+0.005on3 (11)

To see the real improvements in performance with complex RBF, this nonlinear distortion was given. The SER with complex sigmoid function is 0.8275. The distortion in the channel is visible from Fig. 6. Fig. 7 shows the result after LMS equalizer is applied to the same channel. The SER in this case was 0.1111, which is significantly high. This is not surprising because the complex LMS equalizer can reduce only the linear channel distortion. When RBF is applied for the above channel, then the result is visible from Fig. 8. Here is we compare the results with that of LMS equalizer we see a vast improvement. In this case the SER is reduced to 0.033. The parameters of CRBF is the same as that for example 1, but now the width is $\sigma=3.4$.


  
Figure 6: 64-QAM signal corrupted by ISI (10) and nonlinearity (11)
\begin{figure}\hspace{-5mm} \mbox{\centerline{\psfig{figure=qam64_nonlin1.ps,height=8cm}} } \end{figure}


  
Figure 7: The constellation of the 64-QAM after the complex LMS equalizer
\begin{figure}\hspace{-5mm} \mbox{\centerline{\psfig{figure=lms64_nonlin1.ps,height=8cm}} } \end{figure}


  
Figure 8: The constellation of the 64-QAM after the CRBF Equalizer
\begin{figure}\hspace{-5mm} \mbox{\centerline{\psfig{figure=ncrbf64_nonlin1.ps,height=8cm}} } \end{figure}

Example 3: In this case more emphasis is places on non-linearity as compared to that of ISI, so the transfer function (TF) as well as the nonlinearity has been modified. The TF is :
 
H(z)= (0.0874+j0.0447)+(0.9222+j0.3031)z-1    
  (0.1427+j0.0349)z-2+(0.0835+j0.0157)z-3    

and

 
yn=on+0.05on2+0.01on3 (12)

The SER due to complex sigmoid is 0.8, while that due to LMS with the same parameters as in previous examples, is 0.45. The resultant due to LMS equalization is shown in Fig. 10. Finally CRBF was applied for this channel and the result can be seen from Fig. 11.

  
Figure 9: 64-QAM signal corrupted by ISI (12) and non-linearity (13)
\begin{figure}\hspace{-5mm} \mbox{\centerline{\psfig{figure=qam64_nonlin2.ps,height=8cm}} } \end{figure}


  
Figure 10: The constellation of the 64-QAM after the complex LMS equalizer
\begin{figure}\hspace{-5mm} \mbox{\centerline{\psfig{figure=lms64_nonlin2.ps,height=8cm}} } \end{figure}


  
Figure 11: The constellation of the 64-QAM after CRBF equalizer
\begin{figure}\hspace{-5mm} \mbox{\centerline{\psfig{figure=ncrbf64_nonlin2.ps,height=8cm}} } \end{figure}

The parameters of the CRBF are the same as in previous examples, but the width was kept 3.6. The SER obtained using this is 0.1077. From the fig. itself we can see that CRBF performs quite better in non-linear environments, as compared to linear algorithms like LMS.
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Next: CONCLUSION Up: Adaptive Equalization of Non-linear Previous: SIMULATION RESULTS
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