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5.8 Coefficient Sensitivity 215

where AW is the loss due the round-off noise. The required data word lengths for
the two scaling strategies are 22 (21.93) and 20 (19.73) bits, respectively.

5.8 COEFFICIENT SENSITIVITY

In analog systems, sensitivity to element errors caused by unavoidable inaccura-
cies resulting from manufacturing tolerances, temperature variations, and aging
is of major concern. These particular aspects are essentially irrelevant in digital
signal processing. In fact, independence of element variation is the single most
important reason for the success of digital signal processing. Sensitivity to coeffi-
cient variations nevertheless plays an important role in determining the cost of
digital filters.
The frequency response is directly influenced by the accuracy of the coeffi-
cients. Although coefficient word length can in principle be chosen arbitrarily
large, it is important to use a minimum word length since it has a major impact on
the amount of hardware resources, maximal speed, and power consumption
required.
The effect of coefficient errors on fre-
quency response can be treated separately
from the influence of round-off noise. To ana-
lyze the influence of finite coefficient word
length effects on frequency response, many
different measures have been used. The deriv-
ative of the transfer function H(z) with
respect to a coefficient in a filter structure has Figure 5.22 LSI network
both practical and theoretical significance. We
will therefore derive an important expression
for the derivative.
Consider the LSI network shown in Fig. 5.22 which is described by

From these equations we get the transfer function

The transfer function from the input to the multiplier is

and the transfer function from the output of the multiplier to the output of the
network is: