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State space behavior in time-varying biorthogonal filter banks

Using state space representations of biorthogonal filter banks, it is possible to come up with a compact theory for the transition between two time-invariant filter banks. The transition interval depends on the sizes of the common subspaces spanned by the controllability operators of the decomposition filters and by the observability operators of the reconstruction filters. When the respective operators span the same space, the transition can be made arbitrarily short. If it is zero, then the special case of instantaneous transition is reached.