Sampling and quantization - Continuous and Discrete Time Signals and Systems - page 401

Fig. 9.6. Sampling and
reconstruction of a sinusoidal
signal g ( t ) = 5 cos(8000π t )at
a sampling rate of
6000 samples/s. CTFTs of:
(a) the sinusoidal signal g ( t );
(b) the impulse train s ( t ); (c) the
sampled signal g s ( t );
and (d) the signal reconstructed
with an ideal LPF H 1 (ω) with a
cut-off frequency of
6000π radians/s.

S ( w )
2 p (6000)

G ( w )
5 p

…

…

w w

(1000 p ) (1000 p )

−24 −16 −8 0 8 16 24 −24 −16 −8 0 8 6 4

(a) (b)

G s ( w )

Y ( w )
5 p

5 p (6000)

H 1 ( w )

…

…

w w

−24 −16 −8 0 8 16 24 (1000 p ) −24 −16 −8 0 8 16 24 (1000 p )

(c) (d)

The graphical representation of the sampling and reconstruction of the sinu-
soidal signal in the frequency domain is illustrated in Fig. 9.6. The CTFTs of
the sinusoidal signal g ( t ) and the impulse train s ( t ) are plotted, respectively,
in Fig. 9.6(a) and Fig. 9.6(b). Since the CTFT S ( ω )of s ( t ) consists of several
impulses, the CTFT G s ( ω ) of the sampled signal g s ( t ) is obtained by convolving
the CTFT G ( ω ) of the sinusoidal signal g ( t ) separately with each impulse in
G s ( ω ) and then applying the principle of superposition. To emphasize the results
of individual convolutions, a different pattern is used in Fig. 9.6(b) for each
impulse in S ( ω ). For example, the impulse δ ( ω ) located at origin in S ( ω )is
shown in Fig. 9.6(b) by a solid line. Convolving G ( ω ) with δ ( ω ) results in two
impulses located at ω = 8000 π , which are shown in Fig. 9.6(c) by solid lines.
Similarly for the other impulses in S ( ω ).
The output y ( t ) is obtained by applying G s ( ω ) to the input of an ideal LPF
with a cut-off frequency of 6000 π radians/s. Clearly, only the two impulses at
ω = 4000 π , corresponding to the sinusoidal signal cos(4000 π t ), lie within
the pass band of the lowpass filter. The remaining impulses are eliminated from
the output. This results in an output, y ( t ) = cos(4000 π t ), which is different

from the original signal.
(ii)

The CTFT G s ( ω ) of the sampled signal with ω s = 2 π (12 000) radians/s

( T s = 1 / 12 000 s) is given by

∞

G s ( ω ) = 12 000 G ( ω − 2 π m (12 000))

m =−∞

∞

= 12 000

G ( ω − 24 000 m π ) .

m =−∞