Page 221 - DSP Integrated Circuits

206 Chapter 5 Finite Word Length Effects

The Z/2-norms to the outputs of the remaining sections are obtained by divid-
ing the previous Z/2-norms by || F§ \\2/\\ FS Ifo. We get

The third set of coefficients OQS, 013, and a^ are scaled by dividing the Z/2-norms
obtained in the previous step by || Fj \\2/\\ F$ \\2- We get the scaled coefficients

Finally, the last set of coefficients is divided by || F y \\2/\\ Fj \\%. The scaled coef-
ficients are

Finally, running the program again, but now with the new set of coefficients,
shows that all the Z/2-norms are unity. The resulting coefficients are slightly differ-
ent from the ones given in Example 4.5 due to rounding errors. Note that rounding
shall be done such that the zeros remain on the unit circle, i.e., OQ = °2- The
denominator coefficients are not affected by the scaling.

5.5.5 Scaling of Narrow Band Signals
The Z/i-norm, || S x \\i characterizes, according to Equation (5.7), a sinusoidal or nar-
row-band signal. From Equation (5.11) we find, with q = 1 andp = °°, that the upper
bound for the variance of the signal in the critical node is determined by || F v IU.
c
Thus, the filter should be scaled such that the maximum value of Wv\max = II FV IU = >
where c < 1 for all critical nodes. The probability of overflow can be adjusted by
choosing an appropriate value for c.

EXAMPLE 5.7
Scale the signal levels for the filter used in Example 5.6, but assume now that a
narrow-band input signal is used.
The appropriate scaling criterion is the Loo-norm of the frequency responses to
the critical nodes. It is difficult to determine the Loo-norm without using a com-
puter. In practice, the Loo-norm is therefore determined using a program based on,
for example, the DFT of the impulse responses.
We first measure the Loo-norm from the input to the first critical node, i.e., the
output of the first second-order section. We get the first Loo-norm, || F% IU = 6.951637.
The coefficients OQI, an, and a^i are then divided by || ^3 (U so that the output of
the first section and the input to the second section become properly scaled. The
new coefficients are