Digital Signal Processing Reference
In-Depth Information
1.7 FREQUENCY RESPONSE OF AN LTI SYSTEM
1.7.1 FROM IMPULSE RESPONSE
An interesting and useful result occurs by convolving a complex exponential
e
jωk
of radian frequency of
ω
=
ω
0
with an LTI system having an impulse response represented by
h
[
n
]
:
∞
(
h
e
jω
0
(k
−
n)
e
−
jω
0
n
)e
jω
0
k
y
[
k
]=
h
[
n
]
=
[
n
]
(1.8)
n
=−∞
The rightmost expression in Eq. (1.8) is the input signal
e
jω
0
k
scaled by the DTFT of
h
evaluated at
ω
0
. Since the DTFT evaluated at a single frequency is a complex number, it is sometimes convenient to
represent it as a magnitude and phase angle. Thus, for each frequency
ω
0
we would have the result
[
n
]
)(
H(e
jω
0
))e
jω
0
k
(1.9)
An interpretation of this result is that a sinusoidal excitation of frequency
ω
0
to an LTI system
produces an output that is a sinusoid of the same frequency, scaled by the magnitude of the DTFT of
h
H(e
jω
0
)e
jω
0
k
H(e
jω
0
)
y
[
k
]=
=
(
|
|
evaluated at
ω
0
. This is consistent with
the principle of Sinusoidal Fidelity discussed in Volume I of this series.
[
n
]
at
ω
0
, and phase shifted by the angle of the DTFT of
h
[
n
]
Example 1.8.
Consider the LTI system whose impulse response is
[
1
,
0
,
1
]
. Determine the magnitude
of the response to the complex exponential.
exp(j*2*pi*(0:1:31)*5/32)
We note that the normalized frequency is 5/16 = 0.3125 and obtain the magnitude of
H
at the
given frequency as 1.111 using the following code:
w = 5*pi/16; DTF T = 1+exp(-j*2*w);
magH = abs(DTF T)
To obtain the magnitude of response via time domain convolution, we make the following call:
tdMagResp = max(abs(conv([1,0,1],exp(j*2*pi*(0:1:31)*5/32))))
which produces the identical result, 1.111.
Equation (1.9) may be generalized and applied to real sinusoids. Thus, the
steady state
response
y
of an LTI system to a cosine (or sine) of magnitude
A
, frequency
ω
0
and phase angle
φ
, with the
DTFT magnitude and angle being
M
and
θ
is
[
n
]
y
[
n
]=
MA
cos
(ω
0
n
+
φ
+
θ)
(1.10)
Example 1.9.
Verify Eq. (1.10) using both cosine and sine waves having normalized frequencies of 0.53
for the LTI system whose impulse response is
[
]
1
,
0
,
1
.
We'll use a discrete cosine of the stated frequency with
φ
= 0 and
M
= 1, and then compute the
output according to Eq. (1.10) and then by convolution with the impulse response. The plot will show
that the results are identical except for the first two and last two samples (recall that Eq. (1.10) represents a
