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I'm working on a multiuser wireless system simulation and have run into something regarding sampling rates. From everything I've known for a matched filter receiver, the correlation step is performed followed by symbol rate sampling. Say the symbol rate is $T_s$. My question is: do we ever sample at higher than the symbol rate? What if I was able to sample at $\frac{T_s}{5}$? It seems to me then I'd have five samples of the same symbol, which is in the same spirit of repetition coding? Is there a difference between repetition coding and just sampling faster?

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You are mixing up notions of matched filtering (e.g. using an LTI system) with a correlation receiver implementation of the same. The correlation receiver is an "integrate and dump" device -- a time-varying circuit -- whose output resets to zero (that's the dumping part) immediately after the output is sampled at the correct time (once each symbol interval) so that the correlator starts with initial condition $0$ for each symbol correlation. Note that the correlation receiver output does not equal the matched filter output except at the sampling instants. A simple illustration of this point is available towards the end of my answer to Understanding the Matched Filter

So, if you sample the correlation receiver output multiple times (five times in your case) during one symbol interval, the question arises as to what happens to the correlator after each sampling? Is it dumped after each sample, or does it continue to accumulate the correlation until the final sample is taken and the correlator is dumped right afterwards? In the former case, the samples are independent random variables while in the latter case, they are dependent. Also, how do you propose to process these samples? In the former case, the optimum thing to do would be to add up the five samples (which has the same result as running the correlator over the entire symbol interval and sampling just once at the end of the interval) while in the latter case, the optimum thing to do would be be discard the first four samples and just keep the last one. Here, optimum is with respect to the signals being received in white Gaussian noise; if your noise is something else, then you need to specify what it is, and to what extent is the concept of matched filtering or correlation even justified as the right thing to do.

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Two things to keep in mind:

  1. In bandlimited scenarios (such as wireless communications), one uses long pulses that overlap in time. This means that all samples out of the matched filter, except those taken at $T_s$, have intersymbol interference.

  2. All the noise samples out of the matched filter are correlated, except the samples taken at the symbol rate. Correlated noise is much worse than uncorrelated noise.

These two things make it very hard (impossible?) to extract any useful information from the matched filter that is not already present in the samples taken at $T_s$.

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  • $\begingroup$ The OP is not doing a matched filter operation as in using an LTI filter where there is intersymbol interference at times other than the sampling instants even when the pulses are nit overlapping in time at all. Instead, the OP is using a correlator to serve as the matched filter, and the correlator output is quite different from the LTI matched filter output at instants other than the sampling instants at $T$ second intervals. $\endgroup$ – Dilip Sarwate Dec 28 '18 at 21:46
  • $\begingroup$ @DilipSarwate Your mind-reading powers are much more developed than mine. $\endgroup$ – MBaz Dec 28 '18 at 22:59

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