Assume $y=Hx+n$, how to design an optimal linear filter $W$ such that $$\mathbb{E}\left(\frac{\|x^{\dagger}Wy\|^2}{\|Wy\|^2}\right)$$ is maximized, where $\frac{Wy}{\|Wy\|}$ can be reviewed as a normalized estimator of $x$.
Conditions:
$H$ is a known matrix of size $M\times N$ ($M\leq N$);
$x$ (of size $N\times 1$) is an unknown but deterministic vector;
$y$ (of size $M\times 1$) is the observations;
and $n$ is AWGN with noise power $N_0$.
The expectation is taken over noise $n$ (one can also assume that $x$ is random and then the expectation is also taken over $x$).
Note 1: This problem is close to, but not exactly the same as an LMMSE estimate, while in the latter case we have $$W=H^{\dagger}/\left(HH^{\dagger}+N_0 I\right).$$
Note 2: When $N_0\!\to\!0$, the LMMSE becomes optimal since in this case $$W=H^{\dagger}/\left(HH^{\dagger}\right),$$ and $$Wy=x.$$ Hence, $$\frac{\|x^{\dagger}Wy\|^2}{\|Wy\|^2}=\frac{\|x\|^4}{\|x\|^2}=\|x\|^2,$$ which attains the maximum by noting that $$\frac{\|x^{\dagger}Wy\|^2}{\|Wy\|^2}\leq\frac{\|x\|^2\|Wy\|^2}{\|Wy\|^2}=\|x\|^2.$$