Applied Digital Signal Processing Theory and Practice

In this chapter we discuss the basic concepts and the mathematical tools that form the basis for the representation and analysis of discrete-time signals and systems. We start by showing how to generate, manipulate, plot, and analyze basic signals and systems using Matlab. Then we discuss the key properties of causality, stability, linearity, and time-invariance, which are possessed by the majority of systems considered in this book. We continue with the mathematical representation, properties, and implementation of linear time-invariant systems. The principal goal is to understand the interaction between signals and systems to the extent that we can adequately predict the effect of a system upon the input signal. This is extremely difficult, if not impossible, for arbitrary systems. Thus, we focus on linear time-invariant systems because they are amenable to a tractable mathematical analysis and have important signal processing applications.

Study objectives

After studying this chapter you should be able to:

• Describe discrete-time signals mathematically and generate, manipulate, and plot discrete-time signals using Matlab.

• Check whether a discrete-time system is linear, time-invariant, causal, and stable; show that the input-output relationship of any linear time-invariant system can be expressed in terms of the convolution sum formula.

• Determine analytically the convolution for sequences defined by simple formulas, write computer programs for the numerical computation of convolution, and understand the differences between stream and block processing.

• Determine numerically the response of discrete-time systems described by linear constant-coefficient difference equations.