The sliding discrete Fourier transform (DFT) is a well-known algorithm for obtaining a few frequency components of the DFT spectrum with a low computational cost. However, the conventional sliding DFT cannot be applied to practical conditions, e.g., using the sine window and the zero-padding DFT, with preserving the computational efficiency. This paper discusses the extension of the sliding DFT to such cases. Expressing the window function by complex sinusoids, a recursive algorithm for computing a frequency component of the DFT spectrum using an arbitrary sinusoidal window function is derived. The algorithm can be easily extended to the zero-padding DFT. Computer simulations using very long signals show the validity of our algorithm.

If the signal is not Gaussian, then the power spectral density (PSD) approach is insufficient to analyze signals and we resort to estimate the higher order spectra of the signal. However, estimation of the higher order spectra is even more time consuming, for example, the complexity of trispectrum is O(N 4). This problem becomes even more serious when short time Fourier transform (STFT) is computed - computation of the trispectrum is required after every shift of the window. In this paper, a method to recursively compute trispectrum has been presented and it is shown that the computational complexity, for a window size of N, is reduced to be O(N 3) and is the same as the space complexity.