
Invert z -transforming Equation 4.579,
(y
k
)
impulse response
¼ h
k
¼ z
1
{H(z)} (4:580)
Equation 4.580 tells us the impulse response of an LTI discrete-time system is equal to the inverse
z-transform of the z-domain transfer function of the system. Henceforth, the impulse response
sequence will be denoted h
k
, k ¼0, 1, 2,. . . . This most important property of discrete-time systems
is illustrated in Figure 4.63.
The z-domain transfer function of the first-order system in Example 4.29 is
H(z) ¼
Y(z)
U(z)
¼
bz
z a
(4:581)
0
0.2
0.4
0.6
0.8
h
k
0 5 10 15 20
0
0.5
1
y
k
α = 0.3 (fast)
α = 0.7 (moderate)
α = 0.9 (slow)
α = 0.3 (fast)
α = 0.7 (moderate)
α = 0.9 (slow)
Unit step response ...