Signals and Systems - Electrical Engineering

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

An equivalent condition is that the poles ofH(z)are inside the unit circle. In this case,h[n] is

absolutely summable, indeed

∑∞

n= 0

0.5n=

1

1 −0.5

= 2

On the other hand,

H(z)=

1

1 −0.5z−^1

=

z

z−0.5

has a pole atz=0.5, inside the unit circle. Thus, the system is BIBO stable. As such, its transient

and steady-state responses exist. Asn→∞,y[n]=2 is the steady-state response, and−0.5nu[n] is

the transient solution. n

nExample 9.11

An FIR system has the input–output equation

y[n]=

1

3

[x[n]+x[n−1]+x[n−2]]

wherex[n] is the input andy[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

H(z)=

1

3

[1+z−^1 +z−^2 ]

=

z^2 +z+ 1

3 z^2

and the corresponding impulse response is

h[n]=

1

3

[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 thez-plane, very much inside the unit circle. n

Nonrecursive or FIR systems:The impulse responseh[n] of an FIR or nonrecursive system

y[n]=b 0 x[n]+b 1 x[n−1]+···+bMx[n−M]