Signal Sampling and Quantization - Digital Signal Processing: Fundamentals and Applications - page 22

yt

s ()

y ()

Digital signal
y ()

Lowpass
reconstruction
filter

DAC

y ()

y ()
y (1

yt

s ()

y (0 y s (0

yT

s ()

y () 2

y T

s ()

2

n t t

T

a.Digital signal processed b.Sampled signal recovered c.Analog signal recovered.

Y ()

1.

f B
max

=

f

− B 0 B

d.Recovered signal spectrum

FIGURE 2.8

Signal notations at the reconstruction stage.

reconstruction filter is applied to the ideally recovered sampled signal y s ðtÞ to obtain the recovered
analog signal.
To study the signal reconstruction, we let yðnÞ¼xðnÞ for the case of no DSP, so that the recon-
structed sampled signal and the input sampled signal are ensured to be the same; that is, y s ðtÞ¼x s ðtÞ .
Hence, the spectrum of the sampled signal y s ðtÞ contains the same spectral content of the original
spectrum Xðf Þ , that is, YðfÞ¼Xðf Þ , with a bandwidth of f max ¼ B Hz (described in Figure 2.8 d )
and the images of the original spectrum (scaled and shifted versions). The following three cases are
discussed for recovery of the original signal spectrum Xðf Þ .
Case 1: f s ¼ 2 f max
As shown in Figure 2.9 , where the Nyquist frequency is equal to the maximum frequency of the
analog signal xðtÞ , an ideal lowpass reconstruction filter is required to recover the analog signal
spectrum. This is an impractical case.

Xf

s ()

Idea l l o wpass filter

1
T

f

− f B
s

− f s f s f B

s

+

− B 0 B

FIGURE 2.9
Spectrum of the sampled signal when f s ¼ 2f max .