On the Stability of Time-Varying Recursive Filters
The problem of the bounded-input/bounded-output stability of time-varying recursive filters is discussed. While simple, well-known criteria exist for the stability of time-invariant filters, guaranteeing stability when the filter coefficients are allowed to vary is much more difficult. Better insight into the causes for instability can be gained by using the state-space representation of the filter and examining the singular values of the state transition matrix. Two simple criteria based on the state transition matrix can be derived that guarantee the stability of the time-varying filter. Moreover, in the second-order case the singular values of this matrix provide a useful estimate of the maximum and average signal gains that result from the modification of the filter coefficients. These estimates can be used in practice to keep a time-varying filter from blowing up. It is also shown that some filter topologies are better suited to time-varying filtering than others, and a few techniques are presented that can be used to stabilize an otherwise unstable time-varying filter.
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