Recursive Wiener-Khintchine Theorem

Authors: Khalid M. Aamir, Mohammad A. Maud

Abstract:

Power Spectral Density (PSD) computed by taking the Fourier transform of auto-correlation functions (Wiener-Khintchine Theorem) gives better result, in case of noisy data, as compared to the Periodogram approach. However, the computational complexity of Wiener-Khintchine approach is more than that of the Periodogram approach. For the computation of short time Fourier transform (STFT), this problem becomes even more prominent where computation of PSD is required after every shift in the window under analysis. In this paper, recursive version of the Wiener-Khintchine theorem has been derived by using the sliding DFT approach meant for computation of STFT. The computational complexity of the proposed recursive Wiener-Khintchine algorithm, for a window size of N, is O(N).

Keywords: Power Spectral Density (PSD), Wiener-KhintchineTheorem, Periodogram, Short Time Fourier Transform (STFT), TheSliding DFT.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1061912

References:

[1] W. T. Cochran, J. W. Cooley et al., "What is the Fast Fourier Transform?" IEEE Transactions on Audio and Electroacoustics, Vol. AU-15, No. 2, June 1967.
[2] M. T. Heideman, D. H. Johnson and C. S. Burrus, "Gauss and the History of the Fast Fourier Transform," IEEE ASSP Magazine, October 1984.
[3] J. C. Goswami, and A. K Chan, "Fundamentals of Wavelets: Theory, Algorithms and Applications," John Wiley & Sons, Inc., USA, 1999.
[4] H. V. Sorensen and C. S. Burrus, "Efficient Computation of Short Time Fast Fourier Transform," Proc. ICASSP-88, April 1988, pp. 1894-1897.
[5] E. Jacobsen, and R. Lyons, "The sliding DFT," Signal Processing Magazine, IEEE, Vol. 20 , No. 2 , Mar 2003, pp. 74 - 80.
[6] E. A. Robinson, "A Historical Perspective of Spectrum Estimation," Proceedings of the IEEE, vol. 70, No. 9, Sep.1982, pp. 885-907.
[7] A. Papoulis, "Levinson-s Algorithm, Wold-s Decomposition and Spectral Estimation," SIAM Review, vol. 27, No. 3, Sep. 1985, pp. 405-441.