Discrete Fourier-Invariant Signals: Design and Application

In this paper, two methods for the design of discrete Fourier-invariant signals are proposed. The direct design method provides splitting between independent and dependent signal parts and calculation of the dependent part for any given independent part. The iterative design method generates a family of discrete Fourier-invariant signals by a successive approach. Further we show how the proposed direct design method can be combined with the Gabor uncertainty principle to generate discrete Fourier-invariant signals with the minimum product of their bandwidth (B) and their time-width (T). We show that these signals as well as signal families generated with the iterative design method achieve the theoretical lower BT bound. Also, it is shown that the BT product of discrete Hermite-Gauss signals converges to the theoretical lower bound. Finally, possible applications are illustrated in the case of time-frequency spectral analysis using the obtained discrete Fourier-invariant signals as the window that provides isoresolution.

[1]  Cagatay Candan,et al.  The discrete fractional Fourier transform , 1999, 1999 IEEE International Conference on Acoustics, Speech, and Signal Processing. Proceedings. ICASSP99 (Cat. No.99CH36258).

[2]  Werner Dubitzky,et al.  A Practical Approach to Microarray Data Analysis , 2003, Springer US.

[3]  B. Dickinson,et al.  Eigenvectors and functions of the discrete Fourier transform , 1982 .

[4]  A. Messiah.  Quantum Mechanics , 1961 .

[5]  Balu Santhanam,et al.  Discrete Gauss-Hermite Functions and Eigenvectors of the Centered Discrete Fourier Transform , 2007, 2007 IEEE International Conference on Acoustics, Speech and Signal Processing - ICASSP '07.

[6]  Soo-Chang Pei,et al.  Generating matrix of discrete Fourier transform eigenvectors , 2009, 2009 IEEE International Conference on Acoustics, Speech and Signal Processing.

[7]  Shie Qian,et al.  Discrete Gabor transform , 1993, IEEE Trans. Signal Process..

[8]  Luis Mateus Rocha,et al.  Singular value decomposition and principal component analysis , 2003 .

[9]  Chien-Cheng Tseng,et al.  Eigenvalues and eigenvectors of generalized DFT, generalized DHT, DCT-IV and DST-IV matrices , 2002, IEEE Trans. Signal Process..

[10]  J. McClellan,et al.  Eigenvalue and eigenvector decomposition of the discrete Fourier transform , 1972 .

[11]  Soo-Chang Pei,et al.  Discrete Fractional Fourier Transform Based on New Nearly Tridiagonal Commuting Matrices , 2005, IEEE Transactions on Signal Processing.

[12]  Liang Tao,et al.  Novel DCT-Based Real-Valued Discrete Gabor Transform and Its Fast Algorithms , 2009, IEEE Transactions on Signal Processing.

[13]  D. Gabor,et al.  Theory of communication. Part 1: The analysis of information , 1946 .

[14]  J. Linnett,et al.  Quantum mechanics , 1975, Nature.

[15]  Magdy T. Hanna,et al.  Discrete fractional Fourier transform based on the eigenvectors of tridiagonal and nearly tridiagonal matrices , 2008, Digit. Signal Process..

[16]  Hélio M. de Oliveira,et al.  Fourier Eigenfunctions, Uncertainty Gabor Principle and Isoresolution Wavelets , 2015, Anais do XX Simpósio Brasileiro de Telecomunicações.

[17]  F. N. Kong,et al.  Analytic Expressions of Two Discrete Hermite–Gauss Signals , 2008, IEEE Transactions on Circuits and Systems II: Express Briefs.

[18]  T. Hughes,et al.  Signals and systems , 2006, Genome Biology.

[19]  P. Vaidyanathan.  Eigenfunctions of the Fourier Transform , 2008 .

[20]  Cagatay Candan,et al.  On the Eigenstructure of DFT Matrices [DSP Education] , 2011, IEEE Signal Processing Magazine.

[21]  Cagatay Candan,et al.  On Higher Order Approximations for Hermite–Gaussian Functions and Discrete Fractional Fourier Transforms , 2007, IEEE Signal Processing Letters.

[22]  Juan G. Vargas-Rubio,et al.  On the multiangle centered discrete fractional Fourier transform , 2005, IEEE Signal Processing Letters.