Robust relative stability of time-invariant and time-varying lattice filters

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 i obey j ij i and provide necessary and sufficient conditions on the i that guarantee that each time-invariant lattice in the set has poles inside a circle of prescribed radius 1= < 1, i.e., is relatively stable with degree of stability ln : 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 i to i 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= 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= 1> 1= :