Multirate Digital Signal Processing, Oversampling of Analog-to-Digital Conversion, and Undersampling of Bandpass Signals - Digital Signal Processing: Fundamentals and Applications - page 558

x () w () y ()

Anti-aliasing
filter H(z)

M

f
M

f s

f s

f

s

sM

X ()

fH ()

f s f s /2 f s

f s /2

0

H ()

f f M

s /

1

sM

fH ()

f sM
2

f sM
2

f s f s /2 f s

0 f s /2

W ()

fH ()

f sM
2

f sM
2

f s f s

f s /2 f s /2

0

Y ()

fH ()

0

f sM
2

f sM
2

2 f sM f sM f sM 2 f sM

FIGURE 12.2

Spectrum after downsampling.

To verify this principle, let us consider a signal xðnÞ generated by the following:
x n ¼ 5 sin 2 p 1 ; 000 n
8 ; 000

þ cos 2 p 2 ; 500
8 ; 000

þ cos 5 n 8 ¼ 5 sin n 4 (12.8)

With a sampling rate of f s ¼ 8 ; 000 Hz, the spectrum of xðnÞ is plotted in the first graph in
Figure 12.3A , where we observe that the signal has components at frequencies of 1,000 Hz and
2,500 Hz. Now we downsample xðnÞ by a factor of 2, that is, M ¼ 2. According to Equation (12.3) , we
know that the new folding frequency is 4,000/2 ¼ 2,000 Hz. Hence, without using the anti-aliasing
lowpass filter, the spectrum would contain an aliasing frequency of 4 kHz - 2.5 kHz ¼ 1.5 kHz
introduced by 2.5 kHz, plotted in the second graph in Figure 12.3A .
Now we apply a finite impulse response (FIR) lowpass filter designed with a filter length of N ¼
27 and a cutoff frequency of 1.5 kHz to remove the 2.5 kHz signal before downsampling to avoid