Hardware and Software for Digital Signal Processors - Digital Signal Processing: Fundamentals and Applications - page 432

y f ðnÞ¼4y s ðnÞ

yðnÞ¼y f ðnÞ

We can develop these equations directly. First, we divide the original difference equation by a factor of 4 to scale
down all the coefficients to be less than 1, that is,

1
4

y f ðnÞ¼ 1

4 2x s ðnÞþ 1 4 0:5y f ðn 1Þ

and define a scaled output

y s ðnÞ¼ 1
4

y f ðnÞ

Finally, substituting y s ðnÞ on the left side of the scaled equation and rescaling up the filter output as
y f ðnÞ¼4y s ðnÞ, we have the same results as before.

The fixed-point implementation for direct-form II is more complicated. The developed direct-form
II implementation of the IIR filter is illustrated in Figure 9.17 .
As shown in the figure, two scale factors A and B are designated to scale denominator coefficients
and numerator coefficients to their Q-format representations, respectively. Here S is a special factor to
scale down the input sample so that the numerical overflow in the first sum in Figure 9.17 can be
prevented. The difference equations are given in Chapter 6 and listed here:

wðnÞ¼xðnÞa 1 wðn 1 Þa 2 wðn 2 Þ/ a M wðn MÞ

yðnÞ¼b 0 wðnÞþb 1 wðn 1 Þþ/þ b M wðn MÞ

The first equation is scaled down by the factor A to ensure that all the denominator coefficients are less
than 1, that is,

x () bB

0 / wn

()

s yn

()

BS

A y ()

w ()

+ +

1/ S 1/ A

− 1

− aA

1 /

bB

1 /

wn ( − 1 )

−1

bB

2 /

− aA

2 /

wn ( − 2 )

− aA

M /

−1

bB

M /

wn M ( − )

FIGURE 9.17
Direct-form II implementation of the IIR filter.