Sampling and quantization - Continuous and Discrete Time Signals and Systems - page 398

9.1.1 Reconstruction of a band-limited signal from its samples

Figure 9.3(b) illustrates that the CTFT X s ( ω ) of the sampled signal x s ( t )isa
periodic extension of the CTFT of the original signal x ( t ). By eliminating the
replicas in X s ( ω ), we should be able to reconstruct x ( t ). This is accomplished
by applying the sampled signal x s ( t ) to the input of an ideal lowpass filter (LPF)
with the following transfer function:

T s ω≤ω s / 2

H ( ω ) = (9.7)

0 elsewhere.

The CTFT Y ( ω ) of the output y ( t ) of the LPF is given by Y ( ω ) = X s ( ω ) H ( ω ),
and therefore all shifted replicas at frequencies ω>ω s / 2 are eliminated. All
frequency components within the pass band ω ≤ ω s / 2 of the LPF are amplified
by a factor of T s to compensate for the attenuation of 1/ T s introduced during
sampling. The process of reconstructing x ( t ) from its samples in the frequency
domain is illustrated in Fig. 9.4. We now proceed to analyze the reconstruction
process in the time domain.
According to the convolution property, multiplication in the frequency
domain transforms to convolution in the time domain. The output y ( t )of
the lowpass filter is therefore the convolution of its impulse response h ( t )
with the sampled signal x s ( t ). Based on entry (17) of Table 5.2, the impulse
response of an ideal lowpass filter with the transfer function given in Eq. (9.7) is
given by

ω s t
2 π

h ( t ) = sinc . (9.8)

Convolving the impulse response h ( t ) with the sampled signal, x s ( t ) =

∞

x ( kT s ) δ ( t − kT s ) yields

k =−∞

∞

Fig. 9.4. Reconstruction of the
original baseband signal x ( t )by
ideal lowpass filtering.
(a) Spectrum of the sampled
signal x s ( t ); (b) transfer function
H (ω) of the lowpass filter;
(c) spectrum of the
reconstructed signal x ( t ).

ω s t
2 π

y ( t ) = sinc ∗ x ( kT s ) δ ( t − kT s ) , (9.9)

k =−∞

which reduces to

∞

ω s t
2 π

y ( t ) = x ( kT s ) sinc ∗ δ ( t − kT s ) (9.10)

k =−∞

X s ( w ) H ( w )

Y ( w )

1/ T s 1

T s

w w w

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