Digital Signal Processing Reference
In-Depth Information
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−10
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(a)
(b)
Fig. 12.6. Spectral estimation of
decaying exponential signal
h
(
t
)
= 2 exp(j18π
t
) + exp(−j8π
t
)
using the DFT in Example 12.7.
(a) Estimated magnitude
spectrum; (b) estimated
phase spectrum.
>> dw = 2*pi*f1/N; % CTFT frequency resolution
>> w = -pi*f1:dw:pi*f1-dw; % compute CTFT frequencies
>> stem(w,abs(H));
% plot CTFT magnitude spectrum
>> stem(w,angle(H));
% plot CTFT phase spectrum
The resulting plots are shown in Fig. 12.6, and they have a frequency resolution
of
ω =
2
π
. We know that the CTFT for
h
(
t
)isgivenby
CTFT
←−−→
2
δ
(
ω −
18
π
)
+ δ
(
ω +
8
π
)
.
2e
j18
π
t
−
j8
π
t
+
e
We observe that the two impulses at
ω =−
8
π
and 18
π
radians/s are accurately
estimated in the magnitude spectrum plotted in Fig. 12.6(a). Also, the relative
amplitude of the two impulses corresponds correctly to the area enclosed by
these impulses in the CTFT for
h
(
t
).
The phase spectrum plotted in Fig. 12.6(b) is unreliable except for the two
frequencies
ω =−
8
π
and 18
π
radians/s. At all other frequencies, the magni-
tude
H
(
ω
)
is zero, therefore the phase
<
H
(
ω
) carries no information. This
is because the phase is computed as the inverse tangent of the ratio between
the imaginary and real components of
H
(
ω
). When
H
(
ω
)
is close to zero,
the argument of the inverse tangent is given by
ε
1
/ε
2
, with
ε
1
and
ε
2
approach-
ing zero. In such cases, incorrect results are obtained for the phase. The phase
<
H
(
ω
) is therefore ignored when
H
(
ω
)
is close to zero.
Example 12.8
Using the DFT, estimate the frequency characteristics of the CT signal
x
(
t
)
=
2 exp( j19
π
t
).
Solution
The three steps involved in computing the CTFT are as follows.
Step 1: Impulse-train sampling
The CT signal
x
(
t
) constitutes a complex
exponential with fundamental frequency 9.5 Hz. The Nyquist sampling rate
f
1
is therefore given by
f
1
≥
2
9
.
5
=
19 samples
/
s
.
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