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Discrete Signal Representations In this chapter we discuss the fundamental concepts of discrete signal representations. Such representations are also known as discrete transforms, series expansions, or block transforms. Examplesof widely used discrete transforms are given in the next chapter. Moreover, optimal discrete representations will be discussed in Chapter 5. The term “discrete” refers to the fact that the signals are representedby discrete values,whereas the signals themselves may be continuous-time | Signal Analysis Wavelets Filter Banks Time-Frequency Transforms and Applications. Alfred Mertins Copyright 1999 John Wiley Sons Ltd Print ISBN 0-471-98626-7 Electronic ISBN 0-470-84183-4 Chapter 3 Discrete Signal Representations In this chapter we discuss the fundamental concepts of discrete signal representations. Such representations are also known as discrete transforms series expansions or block transforms. Examples of widely used discrete transforms are given in the next chapter. Moreover optimal discrete representations will be discussed in Chapter 5. The term discrete refers to the fact that the signals are represented by discrete values whereas the signals themselves may be continuous-time. If the signals that are to be transformed consist of a finite set of values one also speaks of block transforms. Discrete signal representations are of crucial importance in signal processing. They give a certain insight into the properties of signals and they allow easy handling of continuous and discrete-time signals on digital signal processors. 3.1 Introduction We consider a real or complex-valued continuous or discrete-time signal x assuming that x can be represented in the form n x aitpi. 3.1 2 1 47 48 Chapter 3. Discrete Signal Representations The signal a is an element of the signal space X spanned by y i . pn . The signal space itself is the set of all vectors which can be represented by linear combination of pn . For this the notation X span 3.2 will be used henceforth. The vectors tpi i 1 . n may be linearly dependent or linearly independent of each other. If they are linearly independent we call them a basis for X. The coefficients on i 1 . n can be arranged as a vector a ai . an T 3-3 which is referred to as the representation of x with respect to the basis One often is interested in finding the best approximation of a given signal a by a signal x which has the series expansion m x pi Pi with m n. 3.4 i l This problem will be discussed in Sections 3.2 and .
