• Background

The purpose of the detector stage  is to frame up observation vectors composed of samples of the received signal and to make reliable estimates of the transmitted symbol based upon various decision rules.

The detector is actually composed of two subsections.

Firstly, the incoming signal must be sampled at discrete intervals and formed into observation vectors.

Detectors can be classified as coherent, non-coherent or differentially coherent as well.  Thus, the data in the observation vector will vary in accordance with how the incoming signal was sampled and accumulated.  For example, if a stable, synchronized carrier reference is available at the receiver, then coherent detection can be used.  Otherwise, non-coherent detection with differential encoding can be employed.    In the later case, it is the phase difference between successive symbols that is used rather than knowledge of the absolute phase.  Finally, non-coherent detection can be employed.  While non-coherent detection is less complex to implement, over most modulation schemes and channel types, coherent detection delivers superior P_error performance.

Then, the observation vectors are fed into decision logic where estimates are made as to what was the most likely transmitted symbol.  These estimates are based upon various decision rules.  The particular decision criterion chosen is dependent upon, among other things, the characteristics of the transmitted symbol alphabet, the class of receiver employed and the characteristics of the operating channel.

• Decision Rules

There are many different classes of decision rule, each with it's own optimality criterion.  For example, one might ask "what the optimal receiver" is.  In actuality, the optimal receiver is dependent upon the operating channel.  The optimal receiver for the AWGN channel is one which employs a matched filter or a correlating detector.  The optimal receiver for the ISI fading channel is one which employs a sequence detector.

The particular decision rule chosen is dependent upon what type of detector is required.  For example, if a matched filter or correlating detector is employed, then the decision rule would be to maximize the S/N ratio across all test statistics.  For binary signaling, a decision rule that minimizes the probability of error is called a Maximum Likelihood detector and is optimal in the ISI channel when a sequence detector is used.

The Maximum Likelihood Sequence Estimator (MLSE) is actually a subset of a larger class of decision rules called Bayesian decision rules.   The Bayes decision rule is one which seeks to minimize the Minimum Mean Square Estimate (MMSE).

There are also the Maximum a Posteriori (MAP) Estimates as well as the Neyman-Pearson and Minimax decision rules.  Each one with it's own applicability.

• Derivation of Link Performance

One typical measure of communication link performance is the familiar BER vs SNR curve (as a function of detection type, fading characteristics, etc.) .  These families of parametric curves are actually divined from the decision regions found above.  Once the decision regions are derived, the probability of error is computed.  The probability of error is simply the probability that a given decision variable U2 exceeds the decision variable U1.  This probability is related to the distance between the decision regions via the Q function.