Page 219 - DSP Integrated Circuits

204 Chapter 5 Finite Word Length Effects

A wide-band input signal is characterized by the Loo-norm, || S x (U- See Prob-
lem 5.14. Hence, q = °o => p = 1 and the filter should therefore be scaled such that
II F v || 2 = c, where c < 1 for all critical nodes.

EXAMPLE 5.6

Scale the filter derived in Example 4.5 for wide-band input signals. Two's-complement
representation is used. Only those inputs to the multipliers with noninteger coeffi-
cients, and the output, need to be scaled.
The purpose of scaling is to assure that overflow does not occur too often at
multiplier inputs and at the output. Generally, we assume that the input signal
level is properly scaled. Hence, the inputs to the coefficients

are assumed to have a proper signal level. However, the output of the filter and the
input nodes to all other coefficients must be scaled. In order to scale the filter, the
Z/2-norms to the critical node have to be determined. This is easily accomplished by
the program shown in Box 5.1. The program iterates the system of difference
equations with an impulse sequence as input. The L2-norms are determined by
summing the squared impulse response in the critical nodes. A formal method of
deriving the difference equations will be discussed in Chapter 6.
We get the L2-norms of the outputs of the second-order sections by using the
program:

First, the coefficients CQI, an, and 021 are divided by || FS \\% = 1.892948 so
that the output of the first section and thereby the input to the second section
become properly scaled. The new coefficients are

This will affect the signal levels in the subsequent sections and, of course, the
measured Z/2-norms. The Z/2-norms to the outputs of the remaining sections are
obtained by dividing the original L2-norms by || ^3 \\^. We get