Page 232 - DSP Integrated Circuits

5.9 Sensitivity and Noise 217

The round-off noise variance at the output of the filter is

2
We get, using Holders inequality , for p = 2

and Equation (5.18)

This is a lower bound of the noise variance for a filter scaled for wide-band
input signals. Using the fact that

and n = 2, we get another lower bound that is valid instead for Loo-norm
scaling:

This lower bound on the round-off noise corresponds to scaling for narrow-
band signals.
These two bounds are important, since they show that a structure with high
sensitivity will always have high round-off noise and requires a longer data word
length. It is sometimes argued that high-sensitivity structures can be imple-
mented using elaborate schemes for selecting the coefficients such that the fre-
quency specification is met. However, in such cases a penalty is paid in terms of
increased data word length.
The importance of low sensitivity to coefficient errors and low round-off noise
can be appreciated by considering the cost and execution time of multipliers,
which are essentially proportional to the sum of coefficients and data word
lengths. Using a low-sensitivity structure, short coefficients and a short data word
length can be employed. Low-sensitivity structures also tend to use the available
signal range efficiently. Therefore, we stress that low-sensitivity structures should
always be used for high-performance digital filters.