# Whitening Matched Filters for Equalizer

I was looking at the answer Whitened Matched Filter. It mentions that in case of ISI channel, we can use $$g^∗(−t)$$ as matched filter. For ISI channel, $$r(t)=c(t)∗s(t)+n(t)$$ where $$r(t)$$ is the received signal, $$c(t)$$ is the channel response, $$s(t)$$ transmit signal and $$n(t)$$ is the noise signal. The receiver structure for ISI channel consists of matched filter $$g^*(-t) = c^*(-t) * s^*(-t)$$ followed by a sampler (A/D).

My question is, that if this receiver structure practically implemented or is it used for analysis only? Since in a real receiver, the matched filtering is implemented at the front-end does not include the channel response but only signal response ($$g^*(-t) = s^*(-t)$$). This is because we don't know the channel response before the sampler and we are not adapting analog filters to match to the channel response. How does a practical receiver structure looks like with adaptive filters after the sampler on the digital side? What am I missing here?

Thanks strong text

We need the $$g_{TX}(t)*c(t)*g_{RX}(t)$$ to be a Nyquist pulse for the zero-ISI condition. The raised root cosine (RRC) pulse is a Nyquist pulse and is generally used over the $$\text{sinc}$$ pulse for its faster decay in time domain, making it less susceptible to timing errors.
Since the designer can control $$g_{TX}(t)$$ and $$g_{RX}(t)$$, those are commonly chosen to be square root raised cosine (SRRC) pulses. Choosing those to be SRRC means that $$g_{TX}(t)*g_{RX}(t)$$ will be RRC. This is why we have channel estimation and equalizers, with the goal to reversing the distortion that the channel (the channel can also be thought of as both the transmit filter and channel impulse response) imposed to try and get back to zero-ISI. The channel estimation and equalization, is done in the discrete time domain on the samples. You can look up fractionally spaced equalizers (FSE), this is using the oversampled signal to perform the equalization.