36 SAMPLED-DATA CONTROL SYSTEMS 
the zero-order data hold. As will be seen later, it can be approximated 
by rational polynomials in s, but this procedure is not necessary in the 
theory of sampled-data systems, and the exact form will be used. 
It is of interest to compare the frequency response of a zero-order data 
hold with that of the cardinal hold as shown in Fig. 3.2. The frequency 
response is obtained by substituting jw for s in (3.14), resulting in G,(jw) 
as follows: 
He 
Gaaye en (3.15) 
qw 
By rearranging the terms of (3.15), G,(jw) can be expressed as 
Seven ge Sine / 2 
The amplitude and phase of G,(jw) are plotted in Fig. 3.7. 
The frequency response of the zero-order data hold is low pass with 
full cutoff occurring at frequencies of n/7' cps, where n is an integer. 
4m br 
T 
Frequency, radians per sec. 

Ang G, (ju) 
Fia. 3.7. Amplitude and phase frequency response of zero-order data hold. 
By contrast, the cardinal data hold has a frequency response which is 
unity in the passband with perfect cutoff at all frequencies beyond 
1/2T cps. In the frequency domain, the imperfections of the zero-order 
hold are seen to be the result of gradual cutoff up to 1/2T cps, as well as 
response, though at attenuated levels, beyond this frequency. The trans- 
mission of the higher-frequency components of the sampled signal 
accounts for the ripple in the time domain, as seen in Fig. 3.4. 
The phase shift introduced by G,(jw) is of interest to the designer of a 
control system incorporating a zero-order data hold. Since most control 
systems are low-pass in frequency response and lagging phase angles 
contribute to the instability of the system, the phase lag of Gr(jw) 
complicates the problem of stabilization of feedback control systems. 
As will be shown later, systems which are stable in the continuous 
