Correlated noise in digital filters

Theoretical analysis of noise generation in finite precision digital filters (or more generally, in finite precision numerical algorithms) is usually carried out assuming that the effects of round-off or truncation associated with each numerical operation can be considered as equivalent to the corresponding injection of random, uniformly distributed, white noise signal sources. These sources are usually assumed to be uncorrelated in time and uncorrelated with each other. A detailed examination of the situation reveals that this is not the case even when the filter is driven by a random sequence which, in itself, may be correlated. When the filter is driven by a highly correlated input signal (such as a singularity function or a periodic function), or when the filter is in a limit-cycle oscillation, errors generated by finite precision arithmetic are highly correlated as is apparent from their periodic nature. Equations for absolute bounds on limit cycles have been formulated by several authors. However, these formulas are generally overly pessimistic for design purposes.