The Impact of Finite Wordlength Implementation - Digital Signal Processing: An Introduction with MATLAB and Applications - page 200

needs to devise strategies to deal with overflows properly if and when they do take
place. It is worth noting that floating-point implementation of digital filters
overcomes the problem of overflow. However, fixed-point implementation is still
the more popular implementation option due to its attractiveness in terms of cost
and speed.
By scaling in fixed-point arithmetic, one seeks to ensure that the signals do not
overflow the dynamic range permitted by the number system. This scaling can
effectively be implemented by adopting the fixed-point fractional representation.

4.5.2.1 Scaling of Direct Form IIR Filter

To illustrate the scaling technique, consider a first-order IIR filter as shown in
Fig. 4.10 . To be consistent with the fixed-point fractional format, let x(n) be in the
range [-1,1) and let its impulse response and transfer function be h(n) and H(z),
respectively. Then, the output signal w(n) at node d is given by the convolution

w ð n Þ¼ X

1

f ð n k Þ x ð k Þ;

ð 4 : 27 Þ

k ¼ 0

where f(n) is the impulse response from the input to the node d (which is equals
h(n) in this simple case). The signal w(n) will not in general be in the required
range. One may put a general bound for w(n) as follows

j w ð n Þj X

j x ð n k Þjj f ð k Þj x max X

1 1

j f ð k Þj; ð 4 : 28 Þ

k ¼ 0 k ¼ 0

where x max represents the upper bound of the input signal.
Now in the case at hand, x max \ 1, and so the bound reduces to
j w ð n Þj X

1

j f ð k Þj: ð 4 : 29 Þ

k ¼ 0

Now, for |w(n)| \ 1, the summation P k= ? |f(k)| must be less than unity, and
therefore the node d must be scaled down by a factor of c such that

1
P k ¼ 0 j f ð k Þj :

c ¼ ð 4 : 30 Þ

Fig. 4.10 First-order IIR

node

d

w ( n )

filter

x ( n )

+

y ( n )

z -

1

a