Signal Sampling and Quantization - Digital Signal Processing: Fundamentals and Applications - page 28

According to Figure 2.16 , we can derive the aliasing level percentage using the symmetry of the
Butterworth magnitude function and its first replica. It follows that

s

f a
f c

2 n

1 þ

Xðf Þj f ¼f a ¼ jHðf Þj f ¼f s f a

X a

aliasing level % ¼ ¼ s for 0 f f c (2.11)

f s f a
f c

2 n

jHðf Þj f ¼f a

1 þ

With Equation (2.11) , we can estimate the aliasing noise percentage, or choose a higher-order anti-
aliasing filter to satisfy the requirement for the aliasing level percentage.

EXAMPLE 2.4
Given the DSP system shown in Figures 2.16 to 2.18 , where a sampling rate of 8,000 Hz is used and the anti-
aliasing filter is a second-order Butterworth lowpass filter with a cutoff frequency of 3.4 kHz, determine
a. the percentage of aliasing level at the cutoff frequency;
b. the percentage of aliasing level at a frequency of 1,000 Hz.
Solution:

f s ¼ 8; 000; f c ¼ 3; 400; and n ¼ 2

a. Since f a ¼ f c ¼ 3; 400 Hz, we compute

s

1 þ

3:4
3:4

22

¼ 1:4142 aliasing level % ¼ s

1 þ

2:0858 ¼ 67:8%

8 3:4
3:4

22

b. With f a ¼ 1; 000 Hz, we have

s

1 þ

1
3:4

22

¼ 1:03007
4:3551

aliasing level % ¼ s

1 þ

¼ 23:05%

8 1
3:4

22

Let us examine another example with an increased sampling rate.

EXAMPLE 2.5
Given the DSP system shown in Figures 2.16 to 2.18 , where a sampling rate of 16,000 Hz is used and the anti-
aliasing filter is a second-order Butterworth lowpass filter with a cutoff frequency of 3.4 kHz, determine the
percentage of aliasing level at the cutoff frequency.