# Whitened Matched Filter

I am seeking for an advice on whitened matched filtering technique. I have looked into the literature and I do understand its the purpose and how to select the filter in order to achieve the desired response.

However, what I don't understand is that if the signal is matched to the channel filtered signal plus noise using the matched filter, and then if we apply the whitening filter, we get the inverse again which should actually be the channel filtered signal and noise. So in essence we end up where we started from.

I am sure I am missing something but any comments will be highly appreciated.

Thanks Milos

## 1 Answer

In the ideal AWGN channel we have the received signal is $r(t)=s(t)+n(t)$, where $s(t)$ is the transmitted signal and $n(t)$ is white Gaussian noise. In this case, the transmitted symbols can be estimated using a matched filter whose output is sampled at the symbol rate. Note that in general the noise at the output of the matched filter is correlated, and no longer white; however, at the sampling times the noise is uncorrelated.

In the ISI channel we have $r(t)=c(t) \ast s(t) + n(t)$, where $c(t)$ is the channel response. We can think of this system as an AWGN channel where the transmitted signal is $g(t)=c(t) \ast s(t)$, and then we can use $g^*(-t)$ as a matched filter. However, in this case we no longer have uncorrelated noise samples. Correlated noise is more harmful, and thus this situation is undesirable.

A whitening filter after the matched filter decorrelates the noise samples and improves the system's error rate performance.

Note that the whitening filter does not revert what the matched filter did. The reason is that the purpose of the filter is not to turn the noise back into white noise; its purpose is to decorrelate the noise at the sampling instants. If the transmitted symbols are $a_k$ for integer $k$, and the (discrete) whitening filter has taps $f_n,\,n=0,1,\ldots,L$, then the output of the whitening filter is $$v_k=\sum_{n=0}^L f_n a_{k-n} + w_k,$$ where the noise samples $w_k$ are uncorrelated. The symbols $a_k$ can then be optimally obtained from $v_k$ by the Viterbi algorithm, or (perhaps sub-optimally, but easily) from another type equalizer (ZF, LS, etc.).

• Thank you very much for your comments. Those are very valid points. Sorry but I was not very clear in my question. I was referring to the case when the signal received is corrupted only with the white noise. Then the match filtering is applied and the whitening to whiten isi is done post sampling (e.g. Proakis suggested that in one of his texts). I mean the match filter maximises the SNR (once again, the signal is corrupted only with white noise) and then whitening filter post sampling does the invert of the match filter. so we end up with the noisy signal again? – Milos Milosavljevic May 23 '17 at 22:34
• @MilosMilosavljevic Indeed I misunderstood your question. I have edited my answer; hopefully it's more useful now. The key point is that the whitening filter does not turn the noise back into white noise; it only decorrelates the noise samples at times $kT$ where $T$ is the symbol rate. – MBaz May 24 '17 at 1:59