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Robust relative stability of time-invariant and time-varying lattice filters
Dasgupta, S.; Minyue Fu; Schwarz, C.;
Signal Processing, IEEE Transactions on [see also Acoustics, Speech, and Signal Processing, IEEE Transactions on]
Volume 46,
Issue 8,
Aug. 1998
Page(s):2088
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2100
Abstract:
We consider the relative stability of time-invariant and time-varying unnormalized lattice filters. First, we consider a set of lattice filters whose reflection parameters αi obey |αi|⩽δi and provide necessary and sufficient conditions on the δi that guarantee that each time-invariant lattice in the set has poles inside a circle of prescribed radius 1/ρ<1, i.e., is relatively stable with degree of stability ln ρ. We also show that the relative stability of the whole family is equivalent to the relative stability of a single filter obtained by fixing each αi to δi and can be checked with only the real poles of this filter. Counterexamples are given to show that a number of properties that hold for stability of LTI Lattices do not apply to relative stability verification. Second, we give a diagonal Lyapunov matrix that is useful in checking the above pole condition. Finally, we consider the time-varying problem where the reflection coefficients vary in a region where the frozen transfer functions have poles with magnitude less than 1/ρ and provide bounds on their rate of variations that ensure that the zero input state solution of the time-varying lattice decays exponentially at a rate faster than 1/ρ1>1/ρ
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