On the crossband filter representations of LTI systems: Algorithms and inversion

Since convolution operators characterize linear time-invariant (LTI) systems, convolution is one of the most fundamental operations in acoustic signal processing. Likewise, the analysis and processing of acoustic signals in the time-frequency domain is commonplace. In this context, LTI systems are often approximated with simple time-frequency domain operations, such as channel-wise multiplicative weights or channel-wise convolution. On the other hand, accurate computation of LTI systems in the time-frequency domain usually requires using crossband filters. However, the algorithms for connecting the LTI systems and the corresponding crossband filters have not been well-established, which has hindered their practical applications. In this paper, we introduce and compare several algorithms for computing the crossband filter representation of a given LTI system. Furthermore, we propose an inverse scheme for converting crossband filters into LTI systems, which is proven to recover the correct LTI system from its crossband filter representation. Efficient algorithms for this conversion are presented. We demonstrate numerically that the conversion of crossband filters into LTI systems is faithful: If the crossband filter represents an LTI system, then the conversion recovers the corresponding impulse response, confirming our theoretical result. Finally, we compare the computation time of forward and inverse conversion for all the presented algorithms.