The system consists of multiple low-pass filter and noise, $h_1$, $h_2$, $h_3$, ..., $n_1$, $n_2$, $n_3$, ...

At the transmitter, the power of signal is fixed; and at the receiver, MMSE equalizer is used to minimize the MSE.

My question is that if we can pre-emphasis the signal in the transmitter to improve the overall performance. If the distribution of signal and noise is known, what is the optimal pre-emphasis system?



As MMSE estimators are unbiased, try reducing Cramér–Rao bound and maybe your MMSE equalizer can achieve a better performance. Note that in general the MMSE estimator is known to be asymptotically efficient.

Note also that you must specify criteria before say something is "optimal". Even so, it is usually not easy to prove the optimality.


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