So I am trying to show convergence of a filter, and in order for it to converge, I need the following condition to hold:
$ E\{ \mathbf{s} x sgn(\mathbf{h}^{T} \mathbf{s} x) \} \; \alpha \; E \{ \mathbf{s} x \}$
Where $\alpha$ is meant as 'proportional to up to a constant' and $\mathbf{s}$ and $\mathbf{h}$ are vectors of dimension Nx1 and x is a scalar. Now, we assume that:
$x \sim N(\mu, \sigma_{x}) \quad \mathbf{s} \sim N(\mathbf{\mu}, \mathbf{\sigma_{s}})$
Where $x$ and $\mathbf{s}$ are independent and $\mathbf{h}$ is assumed to be deterministic. Now I understand that:
$E \{ sgn(x) x \} = c \mu$
For some constant c. (I think $\frac{1}{2\pi}$ but its just a constant so I don't care too much really). So then what would be:
$E \{ sgn(\mathbf{s} x) \mathbf{s} \} = \; ??$
And:
$E \{ sgn(\mathbf{h}^{T} \mathbf{s}) \mathbf{s} \} = \; ??$
Also, perhaps more generally with regards to my actual question, what assumptions to I have to make in order for that first condition to hold? Remember, constants can be multiplied out in front.
Thanks!