# Optimal time for sampling symbol in digital receivers

I'm studying various publications about symbol timing recovery algorithms, Early-late, Mueller and Muller & Gardner. All documents I've read tells that, to minimize ISI :

The optimal time for sampling the signal is at the peak of the receiver matched filter

Looking at a real signal, it seems that a local peak is not always the right sampling time. For example, this figure shows the output of the receiver matched filter and the ideal sampling times : In this example, the ideal sampling times are not always at local peaks of the signal. Are there special requirements so that the "optimal sampling time is at local peaks" or could you point me in the right direction to understand this particular point ?

You are mixing up two different notions of sampling.

In a digital communications system, the received (analog, continuous-time) signal is passed through an A/D converter so that the needed further processing (e.g. matched filtering) can be implemented on a programmable DSP processor or as a MATLAB or C++ program or on a special purpose ASIC. The sampling that you show in your question is the one being carried out in the A/D converter, and as long as the sampling is above the Nyqvist rate, the fact that the samples are not at the peaks of the signal is irrelevant: the original signal is represented perfectly adequately by the samples regardless of where the peaks in the signal are with respect to the sampling instants.

The other notion of sampling has to do with the matched filter output which is a sequence of digital data. Which of these data should the decision-making device use to decide whether (in the simplest case) whether a 0 or a 1 was transmitted? The answer is what you quote: it is the datum when the signal output of the matched filter has a peak. That is, we are picking one value from the matched filter output sequence and making the decision based on that. Note that

• We are not going to average the matched filter output (or take a weighted average of the output) and make a decision on that quantity.

• We are not going to take the maximum value of the actual output sequence (which may, because of noise, be at a different location than the time when the signal output of the matched filter is supposed to peak) and make a decision based on that.

For more information on matched filters (including explanations of the above bulleted points), see this other answer of mine.

• In my example, I have over-sampled the signal on purpose, so that implicitly my question is not about A/D sampling, but symbol sampling (followed by decision logic). I'm still confused about sampling at peak, since it doesn't work in my example. 1) according to my understanding, this example is the output of a matched filter (a FIR with RRC taps, same as the transmitter filter, time reversed), is this correct ? If not, this is my problem. 2) If this is correct, what are the requirements, so that the peaks are at the optimal sampling time ? – omnit Mar 21 '16 at 20:06
• that answer doesn't help – omnit Apr 24 '19 at 4:45