m23 - Frequency Response of Discrete-Time Systems
1.1
2006/08/01 13:36:51.812 GMT-5
2006/09/17 13:25:31.499 GMT-5
C.
Sidney
Burrus
csb@rice.edu
C.
Sidney
Burrus
csb@rice.edu
Doug
Kochelek
kochelek@rice.edu
discrete time system
frequency response
transfer function
The frequency response of a linear, time invariant is a powerful and complete description of the system. It can be measured or it can be calculated by evaluating the transfer function on the unit circle.
Frequency Response of Discrete-Time Systems
The frequency response of a Discrete-Time system is something experimentally
measurable and something that is a complete description of s linear,
time-invariant system in the same way that the impulse response is. The
frequency response of a linear, time-invariant system is defined as the
magnitude and phase of the sinusoidal output of the system with a sinusoidal
input. More precisely, if
x
(
n
)
=
cos
(
ω
n
)
and the output of the system is expressed as
y
(
n
)
=
M
(
ω
)
cos
(
ω
n
+
φ
(
ω
)
)
+
T
(
n
)
where
T
(
n
)
contains no components at
ω
,
then
M
(
ω
)
is called the magnitude frequency response and
φ
(
ω
)
is called the phase frequency response. If the system is causal, linear,
time-invariant, and stable,
T
(
n
)
will approach zero as
n
→
∞
and the only output will be the pure sinusoid at the same frequency as the
input. This is because a sinusoid is a special case of
z
n
and, therefore, an eigenvector.
If
z
is a complex variable of the special form
z
=
e
j
ω
then using Euler's relation of
e
j
x
=
cos
(
x
)
+
j
sin
(
x
)
,
one has
x
(
n
)
=
e
j
ω
n
=
cos
(
ω
n
)
+
j
sin
(
ω
n
)
and therefore, the sinusoidal input of (3.22) is simply the real part of
z
n
for a particular value of
z
,
and, therefore, the output being sinusoidal is no surprise.