As shown in (3.5.11), the rounding of 2B-bit product to B bits introduces the roundoff
noise, which has zero-mean and its power is defined by (3.5.6). Roundoff errors can be
trapped into the feedback loops of IIR filters and can be amplified. In the cascade
realization, the output noise power due to the roundoff noise produced at the previ-
ous section may be evaluated using (6.6.13). Therefore the order in which individual
sections are cascaded also influences the output noise power due to roundoff.
Most modern DSP chips (such as the TMS320C55x) solve this problem by using
double-precision accumulator(s) with additional guard bits that can perform many
multiplication±accumulation operations without roundoff errors before the final result
in the accumulator is rounded.
As discussed in Section 3.6, when digital filters are implemented using finite word-
length, we try to optimize the ratio of signal power to the power of the quantization
noise. This involves a trade-off with the probability of arithmetic overflow. The most
effective technique in preventing overflow of intermediate results in filter computation is
by introducing appropriate scaling factors at various nodes within the filter stages. The
optimization is achieved by introducing scaling factors to keep the signal level as high as
possible without getting overflow. For IIR filters, since the previous output is fed back,
arithmetic overflow in computing an output value can be a serious problem. A detailed
analysis related to scaling is available in a reference text .
Example 6.17: Consider the first-order IIR filter with scaling factor a described by
1 À az
where stability requires that jaj < 1. The actual implementation of this filter is
illustrated in Figure 6.21. The goal of including the scaling factor a is to ensure
that the values of yn will not exceed 1 in magnitude. Suppose that x(n) is a
sinusoidal signal of frequency !
, then the amplitude of the output is a factor of
j. For such signals, the gain of H(z) is
1 À jaj
Thus if the signals being considered are sinusoidal, a suitable scaling factor is
a < 1 À jaj:
Figure 6.21 Implementation of a first-order IIR section with scaling factor
DESIGN AND IMPLEMENTATION OF IIR FILTERS