Multi-Resolution Fourier Analysis Part I: Fundamentals

DOI: 10.4236/ijcns.2011.46042   PDF   HTML     4,576 Downloads   8,417 Views   Citations


In the first paper of this series, we propose a multi-resolution theory of Fourier spectral estimates of finite duration signals. It is shown that multi-resolution capability, achieved without further observation, is obtained by constructing multi-resolution signals from the only observed finite duration signal. Achieved resolutions meet bounds of the uncertainty principle (Heisenberg inequality). In the forthcoming parts of this series, multi-resolution Fourier performances are observed, applied to short signals and extended to time-frequency analysis.

Share and Cite:

N. Bey, "Multi-Resolution Fourier Analysis Part I: Fundamentals," International Journal of Communications, Network and System Sciences, Vol. 4 No. 6, 2011, pp. 364-371. doi: 10.4236/ijcns.2011.46042.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] M. Shahram and P. Milanfar, “On the Resolvability of Sinuoids with Nearby Frequencies in the Presence of Noise,” IEEE Transactions on Signal Processing, Vol. 53, No. 7, 2005, pp. 2579-2588. doi:10.1109/TSP.2005.845492
[2] D. V. Anderson, “Speech Analysis and Coding Using a Multi-Resolution Sinusoidal Transform,” Acoustics, Speech, and Signal Processing, ICASSP96, Conference Proceedings, Vol. 2, 1996, pp. 1037-1040. doi:10.1161/01.CIR.82.4.118
[3] R. Haberl, H. F. Schels, P. Steinbigler, G. Jilge and G. Steinbeck, “Top-Resolution Frequency Analysis of Electrocardiogram with Adaptive Frequency Determination. Identification of Late Potentials in Patients with Coronary Artery Disease,” Circulation, Vol. 82, 1990, pp. 1183-1192. doi: 10.1109/ICASSP.1996.543301
[4] L. Marple, “Resolution of Conventional Fourier, Autoregressive, and Special ARMA Methods of Spectrum Analysis,” Acoustics, Speech, and Signal Processing, IEEE International Conference on ICASSP77, May 1977, pp. 74-77. doi:10.1109/ICASSP.1977.1170219
[5] D. L. Donoho and P. H. Stark, “Uncertainty Principles and Signal Recovery,” SIAM Journal on Applied Mathematics, Vol. 49, 1989, pp. 906-931. doi:10.1137/0149053
[6] S. M. Kay and S. L. Marple, “Spectrum Analysis—A Modern Perspective,” Proceedings of the IEEE, Vol. 69, No. 11, 1981, pp. 1380-1419. doi:10.1109/PROC.1981.12184
[7] A. Bruckstein, T. J. Shan and T. Kailath, “The Resolution of Overlapping Echos,” IEEE Transactions on Acoustic, Speech and Signal Processing, Vol. 33, No. 6, 1985, pp. 1357-1367.
[8] Y. Meyer, “Wavelets: Algorithms and Applications,” Society for Industrial and Applied Mathematics, Philadelphia, 1993.
[9] O. Rioul and M. Vitterli, “Wavelets and Signal Processing,” IEEE Signal Processing Magazine, Vol. 8, No. 4, 1991, pp. 14-38. doi:10.1109/79.91217
[10] I. Daubechies, “Ten Lectures on Wavelets,” Siam, Philadaphia, 1992.
[11] M. Kunt, “Traitement Numérique des Signaux,” Dunod, Paris, 1981.
[12] F. J. Harris, “On the Use of Windows for Harmonic Analysis with the Discrete Fourier Transform,” Proceeding of the IEEE, Vol. 66, No. 1, 1978, pp. 51-83. doi:10.1109/PROC.1978.10837
[13] M. Unser, “Sampling—50 Years after Shanon,” Proceedings of the IEEE, Vol. 88, No. 4, 2000, pp. 569-587. doi:10.1109/5.843002
[14] S. M. Kay, “The Effects of Noise on the Autoregressive Spectral Estimator,” IEEE Transactions on Acoustics, Speech and Signal Processing, Vol. 27, No. 5, 1979, pp. 478-485.
[15] S. M. Kay, “Noise Compensation for Autoregressive Spectral Estimates”, IEEE Transactions on Acoustics, Speech and Signal Processing, Vol. 28, No. 3, 1980, pp. 292-303.
[16] Q. T. Zhang, “Reduction of Asymptotic Bias of AR and Spectral Estimates by Tapering,” Journal of Time Series Analysis, Vol. 13, No. 5, 1992, pp. 451-69. doi:10.1111/j.1467-9892.1992.tb00120.x
[17] k. Kaluzynski, “Order Selection in Doppler Blood Flow Signal Spectral Analysis Using Autoregressive Modeling”, Medical and Biological Engineering and Computing, Vol. 27, No. 1, 1989, pp. 89-82. doi:10.1007/BF02442176
[18] N. Y. Bey, “Recovering of Autoregressive Spectral Estimates of Signals Buried in Noise,” Signal, Image and Video Processing, Vol. 1, No. 4, 2007, pp. 321-331. doi:10.1007/s11760-007-0026-3
[19] V. F. Pisarenko, “On the Estimation of Spectra by Means of Non-Linear Functions of the Covariance Matrix,” Geophysical J. Royal Astronomical Soc., Vol. 28, 1972, pp. 511-531.
[20] V.F. Pisarenko, “The Retrieval of Harmonics from a Covariance Function,” Geophysical Journal of the Royal Astronomical Society, Vol. 33, No. 5, 1973, pp. 347-366.
[21] M. L. van Blabicum and R. Mittra, “Problems and Solutions Associated with Prony’s Method for Processing Transient Data”, IEEE Transactions on Electromagnetic Compatibility, EMC-20, No. 1, 1978, pp. 174-182. doi:10.1109/TEMC.1978.303708
[22] M. Kaveh and A. Barabell, “The Statistical Performance of the MUSIC and the Minimum Norm Algorithms in Resolving Plane Waves in Noise,” IEEE Transactions on Acoustic, Speech and Signal Processing, Vol. 34, No. 2, 1986, pp. 331-341. doi:10.1109/TASSP.1986.1164815
[23] A. Barabell, “Improving the Resolution Performance of Eigenstructure-Based Direction-Finding Algorithms,” IEEE International Conference on ICASSP’83: Acoustics, Speech, and Signal Processing, Vol. 8, 1983, pp. 336-339.
[24] N. Y. Bey, “Extraction of Signals Buried in Noise, Part I: Fundamentals,” Signal Processing, Vol. 86, No. 9, 2006, pp. 2464-2478. doi:10.1016/j.sigpro.2005.11.014
[25] N. Y. Bey, “Extraction of Signals Buried in Noise, Part II: Experimental Results,” Signal Processing, Vol. 86, No. 10, 2006, pp.2 994-3011. doi:10.1016/j.sigpro.2005.11.018
[26] N. Y. Bey, “Extraction of Signals Buried in Noise: Correlated Processes,” International Journal of Communications, Network and System Sciences, Vol. 3, No. 11, 2010, pp. 855-862.
[27] N. Yahya Bey, “Extraction of Signals Buried in Noise : Non-Ergodic Processes,” International Journal of Communications, Network and System Sciences, Vol. 3, No. 12, 2010, pp. 907-915. doi:10.4236/ijcns.2010.312124
[28] W. Heisenberg, “The Physical Principles of Quantum Theory,” University of Chicago Press, Chicago, 1930.
[29] D. Gabor, “Theory of Communication,” Journal of Institution of Electrical Engineers, Vol. 93, No. 26, 1946, pp. 429-441. doi:10.1049/ji-3-2.1946.0074
[30] C. L. Fefferman, “The Uncertainty Principle,” Bulletin of the American Mathematical Society, Vol. 9, No. 2, 1983, pp. 129-206.

comments powered by Disqus

Copyright © 2020 by authors and Scientific Research Publishing Inc.

Creative Commons License

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.