Time-varying biorthogonal filter banks: a state-space approach

Using state-space representations of biorthogonal time-varying filter banks, it is possible to come up with a theory for the transitional behavior between two stationary filter banks. The transition interval depends on the size of the common subspace spanned by the controllability operators of the initial and final filters on the decomposition sides, and the common subspace spanned by the observability operators of the filters on the reconstruction sides. When the respective operators span the same spaces, we can derive conditions under which the transition between the filter banks can be so controlled that both the decomposition and the reconstruction functions gently embrace. For such filters, the transition interval can be made arbitrarily short. If it is zero, then the special case of instantaneous transition is reached.