Signal-Space Analysis

Abstract

In this chapter we move slightly away from the conventional time and frequency-domain analysis of signals and systems by adopting a new domain of analysis which we can call the vector-domain. We shall be able to synthesize M waveforms by adequately combining a (possible) smaller number N of base waveforms. Nevertheless, instead of working predominately with functions of time or frequency, we shall work with vectors. As a consequence, we shall have a group of tools that will make the communication system analysis easier than usual. This chapter may be viewed as a smooth transition or interface between the study of baseband transmission, covered in Chap. 6, and the study of passband transmission to be discussed in Chap. 6, where we shall make an extensive use the concepts covered here. Chapter 5 starts with a brief introduction and then moves to the development of tools for signal representation and analysis in the vector-domain. The Gram-Schmidt procedure for generating base-functions will be considered in the sequel. The vector AWGN channel model will be revisited in light of the new concepts, and the generalized maximum likelihood receiver structure for an arbitrary M-ary signaling over the AWGN channel will be developed. Its error probability analysis will be made by applying the union bound concept. The rotation and translation invariance properties will be studied in the sequel and, finally, the relation between the symbol error probability and the bit error probability will be analyzed.