474 Practical MATLAB® Applications for Engineers
R.5.48 The power of ZT is that any difference equation can be easily transformed into
an algebraic expression, which can then be easily solved by algebraic manipula-
tions. This transformation can be accomplished based on two important proper-
ties, which can be easily learned and applied. They are
a. Linearity
b. Time shift
It is obvious that ZT is linear, since DTFT is also linear. Then, based on the linear-
ity principles
Zaf n af n[^11 ( ) 2 2( )] Zaf n[^11 ( )] Zaf n[2 2( )] aFz^11 ( ) aF z^22 ( )
and the time-shift property yields
Zfn n z Fz n
[()] n ()
0 0 for any integer (^0)
R.5. 49 Let us consider the case where the input to a discrete system is f(n) = r(n) = rn (a ramp),
and its system impulse response is h(n). Then the output g(n) can be evaluated by
g(n) = f(n) ⊗ h(n) (convolution of f(n) with h(n) in time), analyzed as follows:
gn rhn kk
k
()( )
∑or
gn r hknk
k() ()
∑and
gn rnkhkr
k() ()
∑where clearly
rhkk Hz
k
∑ () ()Then g(n) = rnH(z). Its block diagram is illustrated in Figure 5.11.
FIGURE 5.11
System block diagram of R.5.49.
f(n) = rn h(n) ↔ H(z) g(n) = rnH(z