Meeting the Welch Bound with Equality

A signal set whose root-mean-square inner product magnitude equals the Welch lower bound is called a WBE signal set. WBE signal sets are of interest in synchronous CDMA communication systems. This chapter surveys the known results on WBE signal sets and extends them in several ways. In particular, WBE signal sets over signal alphabets whose size is a prime power (the most important case is the quaternary alphabet), and arbitrary-size WBE signal sets over small alphabets are constructed. Constructions are also described for signal sets whose maximum inner product magnitudes equal (or are only very slightly larger than) the Welch bound. Similarly, signal sets whose root-mean-square correlation magnitudes equal the Welch lower bound on correlations are of interest in asynchronous CDMA communication systems. It is shown that the root-mean-square correlation magnitude of all WBE signal sets (and many more) equals the Welch bound. Finally, the signal-to-noise ratio for CDMA communication systems using WBE signal sets is studied.