Lecture 29 - Optimal Signal Detection

1 Optimal PAM Detection in AWGN

Assume that time delay/frame detection, frequency, and phase estimation and correction have been accomplished. Then the only non-ideality left in our model is additive noise.

Noise entering the system.

We would like to design the last stage of our receiver to minimize the average probability of a symbol or bit error \(P_s = \frac{1}{N} \sum_{n=1}^N \mathbb{P} [\hat{a} [n] \neq a[n]]\).

There are two key facts to keep in mind. First, if our symbols are in some constellation (\(a[n] \in A\)) and are uniformly distributed over that constellation \(A\), then minimum probability of error corresponds to maximum likelihood detection of \(a[n]\) from \(r(t)\).

Second, if \(r_p(t) = p(t) * p^*(t)\) satisfies the Nyquist condition for zero ISI, then \(r_p(nT_s) = c \delta [n-M]\) for some constants \(c, M\). This means that as long as we know what \(M\) and \(c\) are, we know what the effects of the simplified model are on the signal, and that each received symbol corresponds to a transmitted one. When the Nyquist condition for zero ISI holds, the receiver simplifies to

Matched filter receiver with delay.

This receiver is extremely simple because we have already accounted for all of the other non-idealities. The final step in the simplification of the receiver is to notice that, when we have Nyquist pulse shaping, the pulse shaping filter on the transmit and receive side will combine, and in effect, disappear. This leaves us with the following:

Simplified receiver model.

where \(\tilde{w}(t)\) is circularly symmetric complex Gaussian noise.