
85-16 Signal Processing
where a
0
= 1. An equivalent relation between the input and output variables can be given through the
co
nvolution
s
um
i
n
t
erms
o
f
t
he
i
mpulse
r
esponse
s
equence
h(k
T):
y kT h kT x kT T
N
( ) ( ) ( )= −
∑
µ
µ
=0
(8
5
.44
)
e
c
orresponding
t
ransfer
f
unction
i
s
g
iven
b
y
H z
Y z
X z
b z
a z
H z b
z z
z z
N
v
N
( )
( )
( )
( )= =
+
⇔ =
−
−
−
−
∑
µ
µ
µ
µ
µ
µ
∞µ
=0
=1
0
1
0
∏
µ
=1
N
(8
5
.45
)
where
H(z) is the z-transform of the impulse response h(kT)
X(z) and Y(z) are the z-transform of the input signal x(kT) and the output or the ltered signal y(kT)
As
ca
n
be se
en
fr
om
Eq
uation
85.44
,
if fo
r
at le
ast
on
e
μ, a
μ
≠ 0, the corresponding system is recursive; its
im
pulse
re ...