Digital Signal Processing Reference
In-Depth Information
Figure 8.1
FIR filter frequency response using truncated coefficients.
8.1.3 Retaining Accuracy and Stability
There is no effective way to directly relate the accuracy of the coefficients to the
degree of degradation of the frequency response. At this time, trial and error
techniques are all that can be offered when determining the necessary accuracy of
coefficients. However, there are some helpful practices that can be observed when
dealing with the implementation of these coefficients to reduce the effects of
truncation. The following suggestions relate to IIR filters unless stated otherwise
since they have some unique problems due to their recursive nature. The fact that
IIR filters are implemented using feedback leads to special problems that the FIR
filter does not experience.
Probably the most important implementation rule when dealing with IIR
filters is that it is much better to implement the filter as a cascade of quadratic
factors (as we have done) than to combine the transfer function into a quotient of
high-order polynomials. (Even some experimentation has been done with this idea
for FIR filters.) This technique provides better control of the stability of the filter.
By quantizing the coefficients that represent the filter, we are actually specifying a
fixed number of positions within the unit circle where the poles can be located.
The fewer bits used for quantization, the fewer the positions for the poles. In
addition, these pole locations are not uniformly distributed within the unit circle.
However, by choosing different topologies for the implementation (such as a
coupled form of quadratic), more uniformly distributed pole locations can be
achieved. Also, it has been shown that those poles which either lie close to the unit
circle or to each other are the most critical to represent properly, and therefore
may need some special implementation method. Another technique that can be
used to reduce errors caused by computations using finite accumulators is to pair
poles and zeros located near each other in the same quadratic factor to reduce
large fluctuations. In addition, sections with poles closest to the unit circle can be
