# 7

# System Time Response Characteristics

In this chapter we investigate the time response of a sampled data system and compare it with the response of a similar continuous system. In addition, the mapping between the *s*-domain and the *z*-domain is examined, the important time response characteristics of continuous systems are revised and their equivalents in the discrete domain are discussed.

## 7.1 TIME RESPONSE COMPARISON

An example closed-loop discrete-time system with a zero-order hold is shown in Figure 7.1(a). The continuous-time equivalent of this system is also shown in Figure 7.1(b), where the sampler (A/D converter) and the zero-order hold (D/A converter) have been removed. We shall now derive equations for the step responses of both systems and then plot and compare them.

As described in Chapter 6, the transfer function of the above discrete-time system is given by

where

and the *z*-transform of the plant is given by

Expanding by means of partial fractions, we obtain

**Figure 7.1** (a) ...