4 Selection of $\gamma$ in the presence of noise

It has been proposed in [2] that the value of $\gamma$ for SE-CMA should be chosen based on the SNR, assuming the SNR can be measured or estimated. Specifically, the author proposes that $\gamma$ be chosen to solve

where a is a parameter and $\cal A$ is the set of all the alphabet symbols (i.e. for BPSK, ${\cal A}=\{-1,1\}$). For some of the common real-valued alphabets of unit source variance, (8) suggests the following choices for $\gamma$ at various SNR's:

 

$$\infty$$ SNR=10 SNR=15 SNR=25 SNR=50

BPSK 1.0012 1.0000 1.0000 1.0000 1

4-PAM 1.5124 1.6028 1.7356 1.7963 9/5

8-PAM 1.4817 1.4865 1.2960 1.1963 25/21

16-PAM 1.4764 1.4883 1.4715 1.4262 121/85

32-PAM 1.4753 1.4855 1.4997 1.5473 529/341

Notice that these values for $\gamma$ do not follow any pattern (i.e. they are not strictly increasing or decreasing with SNR). In fact, the relations between $\gamma$ and SNR are fairly peculiar, and are shown in Figure 10.

  

Figure 10: $\gamma$ as a function of SNR for common alphabets

Choosing $\gamma$ by (8) is better for some cases (i.e. those examples given in [2]), but does not always seem to be the best choice. Here are several specific examples where choosing $\gamma$ by (6) provides a better result.

Note that in [2], the notation is slightly different (e.g. $R_{SE}=\sqrt{\gamma}$). Furthermore, note that the examples in [2] use 8-PAM with a source variance of 21.