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Round-off Effects in IIR Digital Filters 551
We will restrict ourselves to the rounding operation of the quantizer
due to its superior statistical qualities (no bias or average value).
From (6.59), we know that, for the rounding operation, the quantizer
error, e R , has the same characteristics across all three number representation
formats. Hence for MATLAB simulation purposes, we will consider
the sign-magnitude format because it is easy to program and simulate
for arithmetic operation. However, in practice, two’s-complement format
number representation has advantages over the others in terms of hardware
implementation.
Digital filter implementation requires arithmetic operations of multiplication
and addition. If two B-bit fractional numbers are multiplied,
the result is a 2B-bit fractional number that must be quantized to B bits.
Similarly, if two B-bit fractional numbers are added, the sum could be
more than one, which results in an overflow, which in itself is a nonlinear
characteristic; or the sum must be corrected using a saturation strategy,
which is also nonlinear. Thus, a finite word-length implementation of the
filter is a highly nonlinear filter and must be analyzed carefully for any
meaningful results.
In this section, we will consider two approaches to deal with errors due
to finite word-length representation. The first type of error can occur when
error samples become correlated with each other due to the nonlinearity
of the quantizer. This is called limit-cycle behavior, and it can exist only
in IIR filters. We will analyze this problem using the nonlinear quantizer
model rather than the statistical model of the quantizer. In the second
type of error, we assume that more nonlinear effects in the quantizer have
been suppressed. Then, using the statistical model of the quantizer, we
develop a quantization noise model for IIR filters that is more useful in
predicting the finite word-length effects.
10.2.1 LIMIT CYCLES
Digital filters are linear systems, but when quantizers are incorporated
in their implementation, they become nonlinear systems. For nonlinear
systems it is possible to have an output sequence even when there is no
input. Limit cycles is one such behavior that creates an oscillatory periodic
output that is highly undesirable.
DEFINITION 1
Limit cycle
A zero-input limit cycle is a nonzero periodic output sequence produced
by nonlinear elements or quantizers in the feedback loop of a
digital filter.
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