Recursive Algorithms for Modulated and Extended Lapped Transforms

Modulated and extended lapped transforms (MLT, ELT) are shift-orthogonal transforms generated from parametric window functions. In this paper, new recursive algorithms are presented for the MLT and ELT. These invertible sliding lapped transforms are non-decimated and thus oversampled subband systems. The analysis bank input signal can be complex-valued and the optional filtering modes are even/odd-stacked subchannelization and real/complex-valued subsignals. The developed filter structure is similar to sliding discrete cosine and Fourier transform (DFT) filters but provides better frequency selectivity. In the presented running-sum implementation, the computations can be carried out with a bank of simple recursive filters as the windowing is embedded into the structure. The second objective is to propose a Goertzel-type of recursive algorithm for critically-sampled exponentially-modulated filter bank (EMFB). Since the basic Goertzel algorithm is a recursive filter implementation for the DFT it requires the following modifications: modulation phase shifting to meet the perfect reconstruction requirement and frequency shifting to accommodate odd-stacked configuration. The Goertzel-EMFB suits well for problems where only a few subbands are processed from a much larger set of available subchannels.