Digital Signal Processing Reference
In-Depth Information
Substituting
the
value
of
the
CTFT
G
(
ω
)
=
5
π
[
δ
(
ω −
8000
π
)
+ δ
(
ω +
8000
π
)] in the above equation, we obtain
∞
G
s
(
ω
)
=
12 000
5
π
[
δ
(
ω −
8000
π −
24 000
m
π
)
m
=−∞
+ δ
(
ω +
8000
π −
24 000
m
π
)]
=
12 000(5
π
)
+δ
(
ω +
40 000
π
)
+
δ
(
ω +
56 000
π
)
m
=−
2
+ δ
(
ω +
16 000
π
)
+
δ
(
ω +
32 000
π
)
m
=−
1
+ δ
(
ω −
8000
π
)
+
δ
(
ω +
8000
π
)
+ δ
(
ω −
32 000
π
)
+
δ
(
ω −
16 000
π
)
m
=
0
m
=
1
+ δ
(
ω −
56 000
π
)
+
δ
(
ω −
40 000
π
)
+
.
m
=
2
To reconstruct the original sinusoidal signal, the sampled signal is passed
through an ideal LPF
H
2
(
ω
). The frequency components outside the pass-band
range
ω≤
12 000
π
radians/s are eliminated from the ouput. The CTFT
Y
(
ω
)
of the output
y
(
t
) of the LPF is therefore given by
Y
(
ω
)
=
5
π
[
δ
(
ω +
8000
π
)
+ δ
(
ω −
8000
π
)]
,
which results in the reconstructed signal
y
(
t
)
=
5 cos(8000
π
t
)
.
The graphical interpretation of the aforementioned sampling and reconstruction
process is illustrated in Fig. 9.7.
As the signal
g
(t) is a sinusoidal signal with frequency 4 kHz, the Nyquist
sampling rate is 8 kHz. In part (i), the sampling rate (6 kHz) is lower than the
Nyquist rate, and consequently the reconstructed signal is different from the
original signal due to the aliasing effect. In part (ii), the sampling rate is higher
than the Nyquist rate, and as a result the original sinusoidal signal is accurately
reconstructed.
9.1.2 Aliasing in sampled sinusoidal signals
As demonstrated in Example 9.1, undersampling of a baseband signal at a
sampling rate less than the Nyquist rate leads to aliasing. Under such conditions,
perfect reconstruction of the baseband signal is not possible from its samples.
In this section, we consider undersampling of a sinusoidal signal
x
(
t
)
=
cos(2
π
f
0
t
)
with a fundamental frequency of
f
0
Hz. The sampling rate
f
s
, in samples/s,
is assumed to be less than the Nyquist rate of 2
f
0
, i.e.
f
s
<
2
f
0
. We show
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