# CHAPTER 2

# Time-Domain Analysis and *z* Transform

## 2.1 A LINEAR, TIME-INVARIANT SYSTEM

The purpose of analysis of a discrete-time system is to find the output in either the time or frequency domain of the system due to a discrete-time input signal. In Chapter 1, we defined the *discrete-time signal* as a function of the integer variable *n*, which represents discrete time, space, or some other physical variable. Given any integer value in −∞ < *n* < ∞, we can find the value of the signal according to some well-defined relationship. This can be described as a mapping of the set of integers to a set of values of the discrete-time signal. Description of this relationship varied according to the different ways of modeling the signal. In this chapter, we define the *discrete-time system* as a mapping of the set of discretetime signals considered as the input to the system, to another set of discrete-time signals identified as the output of the system. This mapping can also be defined by an analytic expression, formula, algorithm, or rule, in the sense that if we are given an input to the system, we can find the output signal. The mapping can therefore be described by several models for the system. The mapping or the input–output relationship may be linear, nonlinear, time-invariant, or time-varying. The system defined by this relationship is said to be linear if it satisfies the following conditions.

Assume that the output is *y*(*n*) due to an input *x*(*n*) according to this relationship. If an input ...

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