Page 231 - DSP Integrated Circuits

216 Chapter 5 Finite Word Length Effects

Taking the derivative of the transfer function, H, with respect to the coeffi-
cient, a, leads to the main result:

where F is the transfer function from the input of the filter to the input of the mul-
tiplier and G is the transfer function from the output of the multiplier to the out-
put of the filter.
Note that the Fourier transform corresponding to F is the function used in
scaling the filter. Thus, the sensitivity depends on the scaling criteria used. In fact,
great caution should be taken when evaluating and comparing different struc-
tures. Unfortunately, it is not uncommon that erroneous sensitivity comparisons
and unjustified low-sensitivity claims are made in the literature.

5.8.1 Coefficient Word Length
It is usually advantageous to minimize the word length required to represent all
the coefficients if the arithmetic operations are executed on a set of flexible pro-
cessing elements. The coefficient with the longest word length therefore deter-
mines the size of the multiplier in a standard or ASIC signal processor. For a
standard signal processor the coefficient word length is fixed, and it is therefore
sufficient to find a set of binary coefficients that can be represented with this word
length and that satisfies the frequency response requirements.
If instead each multiplication is implemented using a dedicated multiplier, it
may be advantageous to minimize the number of nonzero bits in the coefficients
and thereby achieve a significant reduction in cost for the multiplier. The cost is
proportional to the number of nonzero bits and increases slowly with increasing
coefficient word length.
In practice, the binary coefficient values are found by searching in the neigh-
borhood of the theoretical coefficient values. Experience indicates that optimal
binary values may be found far away from the theoretical values. Hence, the
search for binary values may be very time consuming for filters with many coeffi-
cients. However, the time may be reduced if the search starts with a good estimate
of the word length. The required coefficient word length may be estimated using
statistical methods [1].

5.9 SENSITIVITY AND NOISE

Fettweis has shown that coefficient sensitivity and round-off noise are closely
related [7]. Jackson [17] has derived the following lower bounds on round-off noise
in terms of the sensitivity.
Let FI be the scaled frequency response from the input of the filter to the input
of the multiplier ai, and G; the frequency response from the output of the multi-
plier to the output of the filter. For a scaled filter we have