Multirate Digital Signal Processing, Oversampling of Analog-to-Digital Conversion, and Undersampling of Bandpass Signals - Digital Signal Processing: Fundamentals and Applications - page 579

Table 12.1 Results of the Direct Interpolation Process in Figure 12.17 (8 multiplications and 6
additions for processing each input sample x(n))
n xðnÞ m wðmÞ yðmÞ

n ¼ 0 xð 0 Þ m ¼ 0 wð 0 Þ¼xð 0 Þ yð 0 Þ¼hð 0 Þxð 0 Þ

m ¼ 1 wð 1 Þ¼ 0 yð 1 Þ¼hð 1 Þxð 0 Þ

n ¼ 1 xð 1 Þ m ¼ 2 wð 2 Þ¼xð 1 Þ yð 2 Þ¼hð 0 Þxð 1 Þþhð 2 Þxð 0 Þ

m ¼ 3 wð 3 Þ¼ 0 yð 3 Þ¼hð 1 Þxð 1 Þþhð 3 Þxð 0 Þ

n ¼ 2 xð 2 Þ m ¼ 4 wð 4 Þ¼xð 2 Þ yð 4 Þ¼hð 0 Þxð 2 Þþhð 2 Þxð 1 Þ

m ¼ 5 wð 5 Þ¼ 0 yð 5 Þ¼hð 1 Þxð 2 Þþhð 3 Þxð 1 Þ

. . . . .

We assume that the FIR interpolation filter has four taps, shown as
H z ¼ h 0 þ h 1 z 1

þ h 2 z 2 þ h 3 z 3

and the filter output is

yðmÞ¼hð 0 ÞwðmÞþhð 1 Þwðm 1 Þþhð 2 Þwðm 2 Þþhð 3 Þwðm 3 Þ

For the purpose of comparison, the direct interpolation process shown in Figure 12.17 is summarized
in Table 12.1 , where wðmÞ is the upsampled signal and yðmÞ the interpolated output. Processing each
input sample xðnÞ requires applying the difference equation twice to obtain yð 0 Þ and yð 1 Þ . Hence, for
this example, we need eight multiplications and six additions.
The output results in Table 12.1 can be easily obtained by using the polyphase filters shown in
Figure 12.18 .
In general, there are L polyphase filters. With a designed interpolation filter HðzÞ of N taps, we can
determine each bank of filter coefficients as follows:

n ¼ h k þ nL for

n ¼ 0 ; 1 ; / ; N r k k ¼ 0 ; 1 ; / ; L 1 and

L 1

(12.12)

wn

0 () xn () ym ()

ym

0 ()

2

f s Lf s

wn

1 () ym

1 ()

1

2

FIGURE 12.18

Polyphase filter implementation for the interpolation in Figure 12.17 (4 multiplications and 3 additions for
processing each input sample x(n)).