Hardware and Software for Digital Signal Processors - Digital Signal Processing: Fundamentals and Applications - page 431

Next, the direct-form I implementation of the IIR filter is illustrated in Figure 9.16 . As shown in the figure, the
purpose of the scale factor C is to scale down the original filter coefficients to the Q-format. The factor C is usually
chosen to be a power of 2 for using a simple shift operation in DSP.

x () C

1/ S b C

0 / s yn ()

()

f yn

S

s xn

()

yn ()

+

bC

1 /

−1

− aC

1 /

−1

+

−1

− aC

2 /

−1

−1

b C

M /

− aC

N /

−1

FIGURE 9.16

Direct-form I implementation of the IIR filter.

EXAMPLE 9.14
The IIR filter

yðnÞ¼2xðnÞþ0:5yðn 1Þ

uses the direct-form I realization, and for a particular application, the maximum input is

I max ¼ 0:010.:0 2 ¼ 0:25. Develop the DSP implementation equations in the Q-15 fixed-point system.
Solution:
This is an IIR filter whose transfer function is

2
1 0:5z 1 ¼

2z
z 0:5

HðzÞ¼

Applying the inverse z-transform, we obtain the impulse response
hðnÞ¼2 ð0:5Þ

n uðnÞ

To prevent overflow in the adder, we can compute the S factor with the help of the Maclaurin series or approximate
Equation (9.2) numerically. We get

2ð0:5Þ 0 þ 2ð0:5Þ 1 þ 2ð0:5Þ 2 þ/

¼ 0:25 2 1
1 0:5

S ¼ 0:25

¼ 1

The MATLAB function impz() can also be applied to find the impulse response and the S factor:
>> h ¼ impz(2,[1 0.5]); % Find the impulse response
>> sf ¼ 0.25*sum(abs(h)) % Determine the sum of absolute values of h(k)
sf ¼ 1
Hence, we do not need to perform input scaling. However, we need scale down all the coefficients to use the
Q-15 format. A factor of C ¼ 4 is selected. From Figure 9.16 , we get the difference equations as
x s ðnÞ¼xðnÞ

y s ðnÞ¼0:5 s xðnÞþ0:125y f ðn 1Þ