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A limit cycle, sometimes referred to as a multiplier roundoff limit cycle, is a low-level oscillation that can exist in an otherwise stable filter as a result of the nonlinearity associated with rounding (or truncating) internal filter calculations.

**Limit Cycle Oscillations:**

A limit
cycle, sometimes referred to as a multiplier roundoff limit cycle, is a
low-level oscillation that can exist in an otherwise stable filter as a result
of the nonlinearity associated with rounding (or truncating) internal filter
calculations. Limit cycles require recursion to exist and do not occur in
nonrecursive FIR filters. As an example of a limit cycle, consider the
second-order filter realized by

where *Q _{r}* {} represents quantization
by rounding. This is stable filter with poles at 0.4375 ±

Notice
that while the input is zero except for the first sample, the output oscillates
with amplitude 1/8 and period 6.

Limit
cycles are primarily of concern in fixed-point recursive filters. As long as
floating-point filters are realized as the parallel or cascade connection of
first - and second-order subfilters, limit cycles will generally not be a
problem since limit cycles are practically not observable in first- and
second-order systems implemented with 32- b floating-point arithmetic [12]. It
has been shown that such systems must have an extremely small margin of
stability for limit cycles to exist at anything other than underflow levels,
which are at an amplitude of less than 10^{— 3 8} . There are at
least three ways of dealing with limit cycles when fixed-point arithmetic is
used. One is to determine a bound on the maximum limit cycle amplitude,
expressed as an integral number of quantization steps [13]. It is then possible
to choose a word length that makes the limit cycle amplitude acceptably low.
Alternately, limit cycles can be prevented by randomly rounding calculations up
or down [14]. However, this approach is complicated to implement. The third
approach is to properly choose the filter realization structure and then
quantize the filter calculations using magnitude truncation [15,16]. This
approach has the disadvantage of producing more roundoff noise than truncation
or rounding [see (3.12)—(3.14)].

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