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188 Chapter 5 Finite Word Length Effects

input signal. In most applications, it is necessary that the filter suppress such par-
asitic oscillations. This means that only filter structures that guarantee the sup-
pression of these oscillations are serious candidates for implementation. We
discussed earlier two such classes of structures—namely, nonrecursive FIR filters
and wave digital filters.
When implementing a filter in hardware, it is important to design the filter so
that short word lengths can be used for both the internal signals and the coeffi-
cients, since short word lengths generally result in less power consumption,
smaller chip area, and faster computation. We will later show that it is necessary
to use filter structures with low coefficient sensitivity in order to minimize the
internal data word length.
The major limitation when implementing a digital filter using standard digi-
tal signal processors with fixed-point arithmetic is their short data word length.
Typically, such processors have only 16-bit accuracy except for the Motorola 56000
family, which has 24-bit. A 24-bit word length is sufficient for most high-perfor-
mance applications while 16-bits is sufficient only for simple applications.
Modern standard signal processors support floating-point arithmetic in hard-
ware. The use of floating-point arithmetic will not alleviate the parasitic oscilla-
tion problems related to nonlinearities, but will provide a large number range that
can handle signals of widely varying magnitudes. The signal range requirements
are, however, usually modest in most well-designed digital signal processing algo-
rithms. Rather it is the accuracy of the calculations that is important. The accu-
racy is determined by the mantissa, which is 24 bits in IEEE standard 32-bit
floating-point arithmetic. This is long enough for most applications, but there is
not a large margin. Algorithms implemented with floating-point arithmetic must
therefore be carefully designed to maintain the accuracy. To summarize, floating-
point arithmetic provides a large dynamic range which is usually not required,
and the cost in terms of power consumption, execution time, and chip area is much
larger than that for fixed-point arithmetic. Hence, floating-point arithmetic is use-
ful in general-purpose signal processors, but it is not efficient for application-spe-
cific implementations.

5.2 PARASITIC OSCILLATIONS

The data word length increases when a signal value is multiplied by a coefficient
and would therefore become infinite in a recursive loop. The signal values must
therefore be quantized, i.e., rounded or truncated to the original data word length,
at least once in every loop. Quantization is therefore a nonavoidable nonlinear
operation in recursive algorithms. Another type of nonlinearity results from finite
number range.
Analysis of nonlinear systems is very difficult. Mainly first- and second-order
sections have therefore been studied for particular classes of input signals and
types of nonlinearities. Of particular interest are situations when the nonlinear
and the corresponding linear system behave markedly different, for example,
when the nonlinear system enters into a parasitic oscillation.
There are two kinds of parasitic oscillation, depending on the underlying
cause of the oscillation: overflow and granularity oscillations. Parasitic oscillations
are also called limit cycles. Granularity oscillations are caused by rounding or