Limit cycle effect in recursive systems

Limit cycle: in a recursive (IIR) digital filter, finite word-length arithmetic (rounding or truncation of products, or overflow of sums) can keep the output non-zero, or make it oscillate with a constant amplitude, even after the input becomes zero (or constant). The output then does not decay to zero as it would in the ideal filter. Such a persistent oscillation is called a limit cycle.

Types

• Granular (zero-input) limit cycles: caused by rounding/truncation of the product in the feedback path. The amplitude stays within a small "dead band", of the order of the quantisation step.
• Overflow limit cycles: caused by adder overflow (two's-complement wrap-around); these are large-amplitude oscillations. They can be avoided by scaling or by saturation arithmetic.

Example: first-order filter y(n) = 0.9y(n−1) + x(n), rounding to integers

Let the input be x(n) = 7δ(n), then zero. The ideal output 7(0.9)ⁿ decays to zero. With the product 0.9y(n−1) rounded to the nearest integer (0.5 rounds up):

At n = 3, 0.9 × 5 = 4.5 is rounded back to 5, so y(n) = 5 forever even though the input is zero: a limit cycle of period 1 (a dc offset). The output can get stuck at any value inside the dead band

|y| ≤ (Δ/2)/(1 − |a|) = 0.5/(1 − 0.9) = 5

(Δ = 1 is the quantisation step). For a second-order section with complex poles, the limit cycle is an oscillation, e.g. a sustained ±1 sequence, with dead band (Δ/2)/(1 − |a₂|).

Remedies

Use more bits (smaller Δ), magnitude truncation (which drives the output to zero), scaling to avoid overflow, saturation arithmetic, or structures with lower sensitivity (cascade/parallel, lattice, wave digital filters).