10.8 SUMMARY

In this chapter, we have discussed the problem of detection of signal waveforms and parameter estimation of signals in the presence of additive noise. We first covered binary and M-ary detection. The approach adopted was to decompose the signal waveform into a set of K independent random variables, and write the signal in Karhunen-Lo ve expansion. The coefficients of Karhunen-Lo ve expansion are in a sense samples of the received signal. Since the additive noise was white and Gaussian, the coefficients of the Karhunen-Lo ve expansion were uncorrelated and jointly Gaussian. Consequently, the problem was reduced to an equivalent decision problem, as developed in Chapter 5.

In Sections 10.4 and 10.5, we assumed that the received signals may contain some unknown parameters that needed to be estimated. Linear and nonlinear estimation were considered. When the parameter to be estimated was nonrandom, we used maximum likelihood estimation. The maximum a posteriori estimation was used for a random parameter. The "goodness" of the estimation techniques was studied as well.

The general binary detection with unknown parameters was presented in Section 10.6. Again using Karhunen-Lo ve coefficients, we obtained the aproximated K-term likelihood ratio, and then we let K ? ? to obtain the likelihood ratio. This approach of obtaining a K-term approximation of Karhunen-Lo ve coefficients and letting K ? ? was also used in solving for the parameter-estimates discussed in Sections 10.4 and 10.5. Specifically, we considered signals with random phase, and derived the incoherent matched filter. Then,...