2 The SE-CMA Cost Function, Gradient, and Hessian

In order to be able to plot accurate SE-CMA cost contours, I felt it necessary to derive an expression for the cost in terms of $\ensuremath{\mathbf{s}}, \ensuremath{\mathbf{C}}, \ensuremath{\mathbf{f}}, \gamma, M, N_s,$ and $\sigma_w$. From this expression, I was hoping to be able to learn something about the locations SE-CMA minima; however, the cost expression is fairly complex, and I was unable to use it for any rigorous analysis. In any case, the expressions for the cost, gradient, and hessian for SE-CMA are included below. The variable $\cal S$ is the set of all M^N_s source symbol possibilities. For example, for a system with a BPSK source, a channel length of 6, and an equalizer length of 2 ($M=2, N_c=6, N_f=2 \Longrightarrow N_s=3$), the set of possible symbol combinations would be:

$${\cal S}=\left\{ \left[\begin{array} {r}1\ 1\ 1\end{array... ...left[\begin{array} {r}-1\ -1\ -1\end{array}\right] \right\}$$

The following expressions for the SE-CMA cost, gradient, and hessian make use of the statistical functions: ${\rm\: erf}\left( x \right)=\frac{2}{\sqrt{\pi}}\int_{0}^{x}{\rm\: exp}\left( -t^2 \right)dt$, ${\rm\: Q}\left( x \right)=\frac{1}{\sqrt{2\pi}}\int_{x}^{\infty}{\rm\: exp}\left( \frac{-t^2}{2} \right)dt$.
where , etc

and $g_0=\left[ \begin{array} {cccc} 1 & 0 & \cdots & 0 \end{array} \right]\ensur... ...\cdots & 0 \end{array} \right]\ensuremath{\mathbf{C}}^T\ensuremath{\mathbf{s}}$, etc

Even for the seemingly simple case of BPSK with a two-tap equalizer, the cost expression has 6 error function terms (erf's) and 6 exponential terms (exp's) - still a very unmanageable expression. The correctness of the above formulae was verified through simulation by comparison with stochasticly generated cost surfaces. Similar cost functions for CMA 2-2 can be found in [4].

I tried to make some approximations of the cost function, but still could not simplify the expression enough to perform any analysis. One approximation that was tested was to replace the erf's with

$$erf(x)\approx tanh(1.2x)=\frac{{\rm\: exp}\left( 1.2x \right... ...\: exp}\left( 1.2x \right) +{\rm\: exp}\left( -1.2x \right)}$$

This did not help. It was also determined that some of the terms in the above cost function could be neglected for SNR's greater than 15 dB. However, this too provided little solace since the expressions were still too complicated to manipulate.