Final answer:
The MAP decision rule for the given signaling system with equiprobable symbols is achieved by using the ML decision, due to the equal prior probabilities. It involves comparing received signal samples against thresholds and choosing the symbol corresponding to the minimum Euclidean distance to the received signal.
Step-by-step explanation:
To find the MAP decision rule for single-shot detection of a signaling waveform with three equiprobable symbols, one has to compare the likelihood of each possible signal given the observed data, multiplied by the prior probability of that signal. Since the symbols are equiprobable, the prior probabilities are equal, and thus the MAP decision simply turns into a Maximum Likelihood (ML) decision.
The received samples are given as three possibilities, based on which symbol is sent:
- s_0: x[k] = -A + n[k]
- s_1: x[k] = n[k]
- s_2: x[k] = A + n[k]
The noise n[k] is White Gaussian Noise with zero mean and variance σ². The decision rule is to choose the symbol s_i that maximizes the probability of receiving x[k] given that symbol s_i was sent, which is equivalent to minimizing the Euclidean distance between the received signal x[k] and the signal representing each symbol.
The decision regions can be delineated by comparing the received sample x[k] to thresholds that are midway between the possible signal levels. The decision rule can be formulated as:
- If x[k] < -A/2, decide s_0
- If -A/2 ≤ x[k] < A/2, decide s_1
- If x[k] ≥ A/2, decide s_2