Signals and Systems - Electrical Engineering

9.4 One-Sided Z-Transform 537

has finite length and is given by

h[n]=b 0 δ[n]+b 1 δ[n−1]+···+bMδ[n−M]

Its transfer function is

H(z)=

Y(z)

X(z)

=b 0 +b 1 z−^1 +···+bMz−M

=

b 0 zM+b 1 zM−^1 +···+bM

zM

with all its poles at the originz=0 (multiplicityM), and as such the system is BIBO stable.

Recursive or IIR systems:The impulse responseh[n] of an IIR or recursive system

y[n]=−

∑N

k= 1

aky[n−k]+

∑M

m= 0

bmx[n−m]

has (possible) infinite length and is given by

h[n]=Z−^1 [H(z)]

=Z−^1

[ ∑M

m= 0 bmz

−m

1 +

∑N

k= 1 akz−k

]

=Z−^1

[

B(z)

A(z)

]

=

∑∞

`= 0

h[`]δ[n−`]

whereH(z)is the transfer function of the system. If the poles ofH(z)are inside the unit circle, or

A(z)6=0 for|z|≥1, the system is BIBO stable.

9.4.4 Interconnection of Discrete-Time Systems

Just like with analog systems, two discrete-time LTI systems with transfer functionsH 1 (z)andH 2 (z)

(or with impulse responsesh 1 [n] andh 2 [n]) can be connected in cascade, parallel, or feedback. The

first two forms result from properties of the convolution sum.

The transfer function of the cascading of the two LTI systems is

H(z)=H 1 (z)H 2 (z)=H 2 (z)H 1 (z) (9.26)