Finite-time-domain synthesis of recursive linear time-variant causal digital filters by separable sequences

A linear time-variant (LTV) digital filter having a separable sequence as its impulse response has been proved to be recursive. Such filters have the potential of saving computation time and storage. Two techniques are presented for synthesizing a desired LTV digital filter given in numerical form in a finite time domain by a separable sequence. The first is a realization technique which decomposes the impulse response of a desired filter into a separable sequence and theoretically leads to the solution of a separable sequence with the lowest order. For a numerical solution, the order of the separable sequence is dependent on an error tolerance that reflects the realization accuracy. If the desired filter is recursive, then the exact order can be solved. The solved order, as well as the decomposition error, is independent of the error tolerance in a wide range. The second is an approximation technique which finds a separable sequence of a given order by minimizing the normalized mean squared error between the impulse response of the desired filter and the separable sequence. The technique searches the separable sequence by alternately using two nonlinear restrictions that an optimally approximated filter must satisfy. The performances of the proposed techniques are illustrated and compared with other available techniques through numerical examples. The results of the comparisons show that our approximation technique results in smaller approximation errors than those of the others