Whitening Matched Filters for Equalizer

Question

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

Answer 1

You are right that in scenarios where the channel is unknown and/or time-varying, we cannot implement a fixed matched filter. The first filter in the signal chain is usually an analog band pass filter eliminating out-of-band noise. Then after A/D conversion one would use an adaptive equalizer that performs part of the function of the matched filter, but that will usually also be optimized for further noise suppression. This optimality criterion for equalizers that combines minimization of intersymbol-interference and noise power is commonly referred to as minimum mean-square error (MSE) criterion.

Answer 2

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.

See my answer here for difference between the matched filter, zero forcing, and MMSE receivers: https://dsp.stackexchange.com/a/66139/31316.