Sampling and quantization - Continuous and Discrete Time Signals and Systems - page 396

Fig. 9.2. Time-domain
illustration of sampling as a
product of the band-limited
signal and an impulse train.
(a) Original signal x ( t );
(b) sampled signal x s ( t ) with
sampling interval T s = T ;
(c) sampled signal x s ( t ) with
sampling interval T s

x ( t ) x s ( t ) with T s = T

t t

2 T 4 T 6 T

0 −6 T −4 T −2 T 0

(a) (b)

x s ( t ) with T s = 2 T

= 2 T .

t

−6 T −4 T −2 T 0 2 T 4 T 6 T

(c)

Calculating the CTFT of Eq. (9.4), the CTFT X s ( ω ) of the sampled signal
x s ( t )isgivenby

k =−∞ δ ( t − kT s )

∞ ∞

1
2 π F x ( t ) ∗ℑ

X s ( ω ) =ℑ x ( t ) = δ ( t − kT s )

k =−∞

ω −

∞ ∞

1
2 π

2 π
T s

2 m π
T s

1
T s

2 m π
T s

= X ( ω ) ∗ δ ω − = X

m =−∞ m =−∞

(9.5)

where ∗ denotes the CT convolution operator. In deriving Eq. (9.5), we used
the following CTFT pair:

ω −

∞ ∞

2 π
T s

2 m π
T s

CTFT
←→

δ ( t − kT s ) δ

Fig. 9.3. Frequency-domain
illustration of the impulse-train
sampling. (a) Spectrum X (ω)of
the original signal x ( t );
(b) spectrum X s (ω)ofthe
sampled signal x s ( t ) with
sampling rate ω s ≥ 4πβ; (c)
spectrum X s (ω) of the sampled
signal x s ( t ) with sampling rate
ω s < 4πβ.

k =−∞ m =−∞

based on entry (19) of Table 5.2. Equation (9.5) implies that the spectrum X s ( ω )
of the sampled signal x s ( t ) is a periodic extension, consisting of the shifted
replicas of the spectrum X ( ω ) of the original baseband signal x ( t ). Figure 9.3
illustrates the frequency-domain interpretation of Eq. (9.5). The spectrum of the
original signal x ( t ) is assumed to be an arbitrary trapezoidal waveform and is
shown in Fig. 9.3(a). The spectrum X s ( ω ) of the sampled signal x s ( t ) is plotted

X s ( w ) with w s ≥4 pb X s ( w ) with w s <4 pb

X ( w )

1 1/ T s

1/ T s

w w

w

−2 pb 0 2 pb − w s −2 pb 0 2 pb w s −2 w s − w s 0 w s 2 w s

( w s −2 pb ) 2 pb

(a) (b) (c)