Signal Sampling and Quantization - Digital Signal Processing: Fundamentals and Applications - page 23

Xf
s

()

P ractical lowpass filter

T

f

−− f B

s

− f s −+ f B

s

− B f B

s

− f B

s

+

f s

0 B

FIGURE 2.10

Spectrum of the sampled signal when f s > 2f max .

Case 2: f s > 2 f max
In this case, as shown in Figure 2.10 , there is a separation between the highest-frequency edge of
the baseband spectrum and the lower edge of the first replica. Therefore, a practical lowpass recon-
struction (anti-image) filter can be designed to reject all the images and achieve the original signal
spectrum.
Case 3: f s < 2 f max
Case 3 violates the condition of the Shannon sampling theorem. As we can see, Figure 2.11 depicts
the spectral overlapping between the original baseband spectrum and the spectrum of the first replica
and so on. Even when we apply an ideal lowpass filter to remove these images, in the baseband there
are still some foldover frequency components from the adjacent replica. This is aliasing, where the
recovered baseband spectrum suffers spectral distortion, that is, it contains an aliasing noise spectrum;
in the time domain, the recovered analog signal may consist of the aliasing noise frequency or
frequencies. Hence, the recovered analog signal is incurably distorted.

Xf

s ()

Ideal lo wpass filter

1
T

f

f B
s

+

− f s − f B

s

f B
s

− f s

− f B
s

− B 0 B

FIGURE 2.11

Spectrum of the sampled signal when f s < 2f max .

Note that if an analog signal with a frequency f is undersampled, the aliasing frequency component

f alias in the baseband is simply given by the following expression:

f alias ¼ f s f
The following examples give a spectrum analysis of the signal recovery.

EXAMPLE 2.2
Assume that an analog signal is given by
xðtÞ¼5cosð2p$2; 000tÞþ3cosð2p$3; 000tÞ; for t 0