Signal Sampling and Quantization - Digital Signal Processing: Fundamentals and Applications - page 25

Xf

s ()

Aliasing noise

2./ T

f

kHz

5 6 8 10 11 13 14 16 18 19

−11 −10 −6 −5 −3 −2 2 3

FIGURE 2.14

Spectrum of the sampled signal in Example 2.3.

Y ()

Aliasing noise

f kHz

−3 −2 2 3

FIGURE 2.15
Spectrum of the recovered signal in Example 2.3.

Solution:
a. The spectrum for the sampled signal is sketched in Figure 2.14 .
b. Since the maximum frequency of the analog signal is larger than that of the Nyquist frequencydthat is, twice
the maximum frequency of the analog signal is larger than the sampling ratedthe sampling theorem condition is
violated. The recovered spectrum is shown in Figure 2.15 , where we see that aliasing noise occurs at 3 kHz.

2.2.1 Practical Considerations for Signal Sampling: Anti-Aliasing Filtering
In practice, the analog signal to be digitized may contain other frequency components in addition to the
folding frequency, such as high-frequency noise. To satisfy the sampling theorem condition, we apply
an anti-aliasing filter to limit the input analog signal, so that all the frequency components are less than
the folding frequency (half of the sampling rate). Considering the worst case, where the analog signal
to be sampled has a flat frequency spectrum, the band limited spectrum Xðf Þ and sampled spectrum
X s ðf Þ are depicted in Figure 2.16 , where the shape of each replica in the sampled signal spectrum is the
same as that of the anti-aliasing filter magnitude frequency response.
Due to nonzero attenuation of the magnitude frequency response of the anti-aliasing lowpass filter,
the aliasing noise from the adjacent replica still appears in the baseband. However, the amount of
aliasing noise is greatly reduced. We can also control the aliasing noise by either using a higher-order
lowpass filter or increasing the sampling rate. For illustrative purpose, we use a Butterworth filter. The
method can also be extended to other filter types such as the Chebyshev filter. The Butterworth
magnitude frequency response with an order of n is given by

1

s

jHðf Þj ¼ (2.7)

f
f c

2 n

1 þ