Hardware and Software for Digital Signal Processors - Digital Signal Processing: Fundamentals and Applications - page 433

1
A wðnÞ¼

1
A xðnÞ

1
A a 1 wðn 1 Þ

1
A a 2 wðn 2 Þ/

1
A a M wðn MÞ

w s ðnÞ¼

wðnÞ¼A w s ðnÞ
Similarly, scaling the second equation yields

1
B yðnÞ¼

1
B b 0 wðnÞþ

1
B b 1 wðn 1 Þþ/þ

1
B b M wðn MÞ

y s ðnÞ¼

and

yðnÞ¼B y s ðnÞ
To avoid the first adder overflow (first equation), the scale factor S can be safely determined by
Equation (9.3) :

S ¼ I max ðjhð 0 Þj þ jhð 1 Þj þ jhð 2 Þj þ/Þ (9.3)

where hðkÞ is the impulse response due to the denominator polynomial of the IIR filter, where the poles
can cause a larger value to the first sum. Hence, h(k) is given by
hðnÞ¼Z 1

1

(9.4)

1 þ a 1 z 1 þ/þ az M

All the scale factors A , B , and S are usually chosen to be a power of 2, respectively, so that the shift
operations can be used in the coding process. Example 9.15 serves as illustration.

EXAMPLE 9.15
Given the IIR filter
yðnÞ¼0:75xðnÞþ1:49xðn 1Þþ0:75xðn 2Þ1:52yðn 1Þ0:64yðn 2Þ

with a passband gain of 1 and a full range of input, use the direct-form II implementation to develop the DSP
implementation equations in the Q-15 fixed-point system.
Solution:
The difference equations without scaling in the direct-form II implementation are given by
wðnÞ¼xðnÞ1:52wðn 1Þ0:64wðn 2Þ

yðnÞ¼0:75wðnÞþ1:49wðn 1Þþ0:75wðn 2Þ

To prevent overflow in the first adder, we obtain the reciprocal of the denominator polynomial as

1
1 þ 1:52z 1 þ 0:64z 2

AðzÞ¼

Using the MATLAB function impz() leads to:
>> h ¼ impz(1,[1 1.52 0.64]);
>> sf

¼ sum(abs(h))

sf ¼ 10.4093