4.2 8-PAM with undermodeled channel

For this example, the SNR is 16 dB, the source is unit variance 8-PAM, a 2-tap FSE is used, and the (undermodeled) channel is given by $\ensuremath{\mathbf{c}}=\left[\begin{array} {cccccc}0.1&0.3&1&-0.1&0.5&0.2\end{array}\right]^T$. With these system parameters, $\gamma$ is chosen to be $\gamma_1=1.1905$ via (6) or $\gamma_2=1.4763$ via (8). Using (7), I again plotted the cost contour and found that the location of the SE-CMA minima are closer to the MMSE minima when $\gamma$ is chosen via (6).

A numeric simulation was again performed. For 400,000 iterations with $\mu=10^{-4}$, a portion of the squared-error history is shown in Figure 12.

  

Figure 12: Squared error history for undermodeled example

It is obvious that by using $\gamma_1$ a lower error is obtained.

Therefore, I have shown that in a particular undermodeled case, using (8) to choose $\gamma$ does not always yield a superior result over the noiseless selection of $\gamma$ via (6).