One-Sided Z-Transform 535

9.4 One-Sided Z-Transform 535

It is important to remember the relations

Y(z)

Z [y[n]]

H(z) = Z[h[n]] =

X(z) = Z [x[n]]

where H(z) is the transfer function and h[n] is the impulse response of the system, with x[n] as the input and y[n] as the output.

■ Example 9.10

Consider a discrete-time IIR system represented by the difference equation

(9.25) with x[n] as the input and y[n] as the output. Determine the transfer function of the system and

y[n] = 0.5y[n − 1] + x[n]

from it find the impulse and the unit-step responses. Determine under what conditions the system is BIBO stable. If stable, determine the transient and steady-state responses of the system.

Solution

The system transfer function is given by

Y(z)

H(z) =

X(z) = 1 − 0.5z −1

and its impulse response is

h[n] = Z −1

[H(z)] = 0.5 n u[n]

The response of the system to any input can be easily obtained by the transfer function. If the input is x[n] = u[n], we have

1 Y(z) = H(z)X(z) =

( 1 − 0.5z −1 )( 1−z −1 )

so that the total solution is

y[n] = −0.5 n u[n] + 2u[n]

From the transfer function H(z) of the LTI system, we can test the stability of the system by finding the location of its poles—very much like in the analog case. An LTI system is BIBO stable if and only if the impulse response of the system is absolutely summable—that is,

X |h[n]| ≤ ∞

C H A P T E R 9: The Z-Transform

An equivalent condition is that the poles of H(z) are inside the unit circle. In this case, h[n] is absolutely summable, indeed

On the other hand,

H(z) =

1 − 0.5z −1 = z − 0.5

has a pole at z = 0.5, inside the unit circle. Thus, the system is BIBO stable. As such, its transient and steady-state responses exist. As n → ∞, y[n] = 2 is the steady-state response, and −0.5 n u[n] is

the transient solution. ■

■ Example 9.11

An FIR system has the input–output equation

1 y[n] = [x[n] + x[n − 1] + x[n − 2]]

where x[n] is the input and y[n] is the output. Determine the transfer function and the impulse response of the system, and from them indicate whether the system is BIBO stable or not.

Solution

The transfer function is

1 H(z) = [1 + z −1 +z −2 ]

3 z 2 +z+1

3z 2

and the corresponding impulse response is

1 h[n] = [δ[n] + δ[n − 1] + δ[n − 2]]

The impulse response of this system only has three nonzero values, h[0] = h[1] = h[2] = 1/3, and the rest of the values are zero. As such, h[n] is absolutely summable and the filter is BIBO stable. FIR filters are always BIBO stable given their impulse responses will be absolutely summable, due to their final support, and equivalently because the poles of the transfer function of these system are at the origin of the z-plane, very much inside the unit circle.

■ Nonrecursive or FIR systems: The impulse response h[n] of an FIR or nonrecursive system

y[n] = b 0 x[n] + b 1 x[n − 1] + · · · + b M x[n − M]