Lecture Notes 3: Error Probability for M Signals

For the special case of orthogonal signals the error probability can be expressed as a single integral. Because of the difficulty of evaluating the error probability for general signal sets, bounds are needed to determine the performance. Different bounds have different complexity of evaluation. This first bound we derive is known as the Gallager bound. We apply this bound to the case of orthogonal signals (for which the true answer is already known). The Gallager bound has the property that when the number of signals become large the bound becomes tight. However, the bound is fairly difficult to evaluate for many signal sets. A special case of the Gallager bound is the Union-Bhattacharayya bound. This is simpler than the Gallager bound to evaluate but also is looser than the Gallager bound. The last bound considered is the union bound. This bound is tighter than the Union-Bhattacharayya bound and the Gallager bound for sufficiently high signal-to-noise ratios.