Signals and Systems - Electrical Engineering

724 CHAPTER 12: Applications of Discrete-Time Signals and Systems

where x(nTs)=x(t)|t=nTsare the sample values of the continuous-time signal x(t),

Xs(s)=L[xs(t)]=

∑

n

x(nTs)e−snTs

2. The Laplace transform Xs(s)is related to the Z-transform of the discrete-time signal x(nTs)by letting
   z=esTs.
   n Recall that the ideal sampler is a time-varying system and that the quantizer is a nonlinear system;
   thus sampled-data and digital control systems are time varying and time-varying nonlinear, respectively.
   Therefore, the complexity of their analyses.

12.3.1 Open-Loop Sampled-Data System

Consider the system shown in Figure 12.6. Assume the discrete-time signalx(nTs)coming from a

computer is used to drive an analog plant with a transfer functionG(s). To change the state of the

plant,x(nTs)is converted into a continuous signal that holds the value of the sample for the duration

of the sample periodTs. This can be implemented using a DAC with azero-order hold(ZOH), which

holds the value ofx(nTs)until the next sample arrives at(n+ 1 )Ts. Furthermore, to allow the output

signal to be possibly processed by a computer, assume the output of the planty(t)is also sampled

to gety(nTs). We are interested in the transfer function that relates the discrete inputx(nTs)to the

discrete outputy(nTs)whereTsis the sampling period chosen according to the maximum frequency

present in the analog inputx(t).

As we saw in Chapter 7, the transfer function of a zero-order hold is

Hzoh(s)=

1 −e−sTs

s

(12.11)

which corresponds to an impulse response

hzoh(t)=u(t)−u(t−Ts) (12.12)

or a pulse of durationTsand unit amplitude. If the sampled signal is written as

xs(t)=

∑

n

x(nTs)δ(t−nTs) (12.13)

FIGURE 12.6

Open-loop sampled-data system for an

analog plantG(s). The output of the DAC

with a ZOH is illustrated.

DAC

ZOH

Plant

G(s) ADC

x(nTs) y(t) y(nTs)

x(t)

t

ZOH signal

0 Ts 2 Ts 3 Ts 4 Ts 5 Ts 6 Ts