DTFT, DFT, ZT, and FFT 461
R.5.6 Time invariant linear systems are completely characterized by its impulse response.
If the impulse response is known, then the output sequence g(n) can easily be eval-
uated for any arbitrary input sequence f(n) (similar to the continuous case).
R.5.7 The notation h(n) is reserved to denote the impulse {f(n) = δ(n)} system response
{h(n) = g(n) = L[δ(n)]} (analogous to the continuous case h(t)).
R.5.8 When the discrete system is both linear and time invariant (LTI), a number of
techniques can be used to transform the linear difference equations that relate the
system input to the system output, using the time and frequency domains. A par-
ticular useful transform from the time into the frequency domain and vice versa
from the frequency into the time domain is the Z transform (ZT).
R.5.9 Before introducing more concepts, let us gain some insight and experience by
analyzing some discrete systems. For example, analyze if the following systems,
excited by a real input sequence f(n) and producing an output gi(n), for i = 1, 2, 3 are
LT I.
The systems are
a. g 1 (n) = f(n)
b. g 2 (n) = nf(n)
c. g 3 (n) = f^2 (n)
FIGURE 5.3
Discrete operators and system elements.
g(n) = kf(n)
g(n) = f(n − 1 )
g(n) = f(n + 1)
g(n) = f 1 (n) + f 2 (n)
f 2 (n)
g(n) = f 1 (n)f 2 (n)
f 2 (n)
Z^1
Z−^1
+
X
k
f 1 (n)
f 1 (n)
f(n)
f(n)
f(n)