Multirate Digital Signal Processing, Oversampling of Analog-to-Digital Conversion, and Undersampling of Bandpass Signals - Digital Signal Processing: Fundamentals and Applications - page 582

Table 12.2 Results of the Direct Decimation Process in Figure 12.20 (6 multiplications and 4
additions for obtaining each output yðmÞ)

n xðnÞ wðnÞ m yðmÞ

n ¼ 0 xð 0 Þ wð 0 Þ¼hð 0 Þxð 0 Þ m ¼ 0 yð 0 Þ¼hð 0 Þxð 0 Þ

n ¼ 1 xð 1 Þ wð 1 Þ¼hð 0 Þxð 1 Þþhð 1 Þxð 0 Þ discard

yð 1 Þ¼hð 0 Þxð 2 Þþ

n ¼ 2 xð 2 Þ wð 2 Þ¼hð 0 Þxð 2 Þþhð 1 Þxð 1 Þþhð 2 Þxð 0 Þ m ¼ 1

hð 1 Þxð 1 Þþhð 2 Þxð 0 Þ

n ¼ 3 xð 3 Þ wð 3 Þ¼hð 0 Þxð 3 Þþhð 1 Þxð 2 Þþhð 2 Þxð 1 Þ discard

yð 2 Þ¼hð 0 Þxð 4 Þþ

n ¼ 4 xð 5 Þ wð 4 Þ¼hð 0 Þxð 4 Þþhð 1 Þxð 3 Þþhð 2 Þxð 2 Þ m ¼ 2

hð 1 Þxð 3 Þþhð 2 Þxð 2 Þ

n ¼ 5 xð 6 Þ wð 5 Þ¼hð 0 Þxð 5 Þþhð 1 Þxð 4 Þþhð 2 Þxð 3 Þ discard

... ..

wm

0 ()

xn () ym ()

ym

0 ()

2

f
M

f s

wm

1 ()

s

ym

1 ()

2

FIGURE 12.21
Polyphase filter implementation for the decimation in Figure 12.20 . (3 multiplications and 1 addition for obtaining
each output y(m)).

The efficient way to implement a polyphase filter is given in Figure 12.21 .
Similarly, there are M polyphase filters. With the designed decimation filter HðzÞ of N taps, we can
obtain filter bank coefficients by

n ¼ h k þ nM for

N
M 1

r k k ¼ 0 ; 1 ; / ; M 1 and n ¼ 0 ; 1 ; / ; (12.13)

For our example, we see that M 1 ¼ 1and N=M 1 ¼ 1 ð rounded up Þ . Thus, we have two
filter banks. Since k ¼ 0and n ¼ 1, k þ nM ¼ 0 þ 1 2 ¼ 2. The time index upper limit
required for hðk þ nMÞ is 2 for the first filter bank r 0 ðzÞ . Hence r 0 ðzÞ has filter coefficients hð 0 Þ
and hð 2 Þ .
However, when k ¼ 1 and n ¼ 1, k þ nM ¼ 1 þ 1 2 ¼ 3, the time index upper limit required
for hðk þ nMÞ is 3 for the second filter bank, and the corresponding filter coefficients are required to be
hð 1 Þ and hð 3 Þ . Since our direct interpolation filter hðnÞ does not contain the coefficient hð 3 Þ , we set
hð 3 Þ¼ 0 to get the second filter bank with one tap only, as shown in Figure 12.21 . Also as shown in
that figure, achieving each yðmÞ requires three multiplications and one addition. In general, the number
of multiplications can be reduced by a factor of M.
The commutative model for the polyphase decimator is shown in Figure 12.22 .