Multirate Digital Signal Processing, Oversampling of Analog-to-Digital Conversion, and Undersampling of Bandpass Signals - Digital Signal Processing: Fundamentals and Applications - page 557

In this way, before downsampling, we can guarantee the maximum frequency of the filtered signal
satisfies

f max < f s
2 M

(12.5)

such that no aliasing noise is introduced after downsampling. A general block diagram of decimation is
given in Figure 12.1 , where the filtered output in terms of the z-transform can be written as

WðzÞ¼HðzÞXðzÞ (12.6)

where XðzÞ is the z-transform of the sequence to be decimated, xðnÞ , and HðzÞ is the lowpass filter
transfer function. After anti-aliasing filtering, the downsampled signal yðmÞ takes its value from the
filter output as

yðmÞ¼wðmMÞ (12.7)

The process of reducing the sampling rate by a factor of 3 is shown in Figure 12.1 . The corresponding
spectral plots for xðnÞ , wðnÞ , and yðmÞ in general are shown in Figure 12.2 .

y ()

x () w ()

Anti-aliasing
filter H(z)

M

f
M

f s f s

s

x ()

n

0 2 4 6 8 10 12 14 16

w ()

n

0 2 4 6 8 10 12 14 16

y ()

m

1 2 3 4

0 5

FIGURE 12.1
Block diagram of the downsampling process with M ¼ 3.