Digital Signal Processing Reference
In-Depth Information
zeros of ] must also lie in the left half-plane. Thus, both a system and its inverse
are stable, provided that the poles and zeros of the system transfer function are in
the left half-plane.
H(s)
Recall from (5.1) the definition of the Fourier transform:
q
f(t)e
-jvt
dt.
F(v) =
[f(t)] =
L
(7.67)
-
q
From Section 5.5, using the Fourier transform, we find that the transfer function for
a causal system with the impulse response
h(t)
is given by
q
h(t)e
-jvt
dt.
H
f
(v) =
[h(t)] =
L
(7.68)
0
Comparing this transfer function with that based on the Laplace transform, namely,
q
h(t)e
-st
dt,
H
l
(s) =
l
[h(t)] =
L
(7.69)
0
we see that the two transfer functions are related by
`
H
f
(v) = H
l
(s)
s = jv
= H
l
(jv).
(7.70)
Here we have subscripted the transfer functions for clarity, and we see a problem in
notation. If we do not subscript the transfer functions, (7.70) is expressed as
which is inconsistent, to say the least. However, in using the Fourier
transform, we commonly denote the frequency response as When using the
Laplace transform, we commonly denote the same frequency response as
The reader should note this inconsistency; it is not likely to be changed. We will use
the same confusing custom here.
For the system of (7.49), the frequency response is given by
H(v) = H(jv),
H(v).
H(jv).
Á
b
n
(jv)
n
+ b
n- 1
(jv)
n- 1
Y(jv)
X(jv)
=
+
+ b
1
(jv) + b
0
H(jv) =
.
(7.71)
Á
a
n
(jv)
n
+ a
n- 1
(jv)
n- 1
+
+ a
1
(jv) + a
0
Recall from Sections 3.7 and 5.5 that this frequency response can be measured
experimentally on a stable physical system. With the input
x(t) = cos
vt,
from
(3.73), the steady-state output is given by
y
ss
(t) = ƒH(jv) ƒ cos
(vt + arg H(jv)),
(7.72)
