Abstract
This chapter introduces a new method for frequency determination that employs the authors’ discrete Hermite transform. Particularly for an input signal that is a linear combination of general sinusoids, this method provides highly accurate estimations of both frequencies and amplitudes of those sinusoids. The method is based primarily on the property of the discrete Hermite functions (DHf) being eigenvectors of the centered Fourier matrix, analogous to the classical result that the continuous Hermite functions (CHf) are eigenfunctions of the Fourier transform. Using this method for frequency determination, a new Hermite transform-based time-frequency representation, the HDgram, is developed that can provide clearer interpretations of frequency and amplitude content of a signal than corresponding spectrograms or scalograms.
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References
Clary, S., Mugler, D.H.: Shifted Fourier matrices and their tridiagonal commutors. SIAM J. Matrix Anal. 24(3), 809–821 (2003)
Clary, S., Mugler, D.H.: Eigenvectors for a class of discrete cosine and sine transforms. Sampl. Theory Signal Image Process. 3, 83–94, (2004)
Escalante-Ramrez, B., Martens, J.B.: Noise reduction in computerized tomography images by means of polynomial transforms. J. Visual Comm. Image Rep. 3, 272–285 (1992)
Gopalikrishnan, R., Mugler, D.H.: The evolution of Hermite transform in biomedical applications. In: Shukla, A., Tiwari, R. (eds.) Intelligent Medical Technologies and Biomedical Engineering: Tools and Applications, pp. 260–278. IGI Global, Hershey (2010)
Gopalikrishnan, R., Acharya, S., Mugler, D.H.: Real time monitoring of ischemic changes in electrocardiograms using discrete Hermite functions. In: Proceedings of 26th International Conference of the IEEE, Engineering in Medicine and Biology Society, pp. 438–441. IEEE, Piscataway, New Jersey (2004)
Mahadevan, A., Acharya, S., Sheffer, D., Mugler, D.H.: Ballistocardiogram artifact removal in EEG-fMRI signals using Discrete Hermite transforms. IEEE J. Sel. Top. Signal Process. (Special Issue on: fMRI Analysis for Human Brain Mapping) 2(6), 839–853 (2008)
Martens, J.B.: The Hermite transform-theory. IEEE Trans. Acoust. Speech Signal Process. 38(9), 1595–1606 (1990)
Martens, J.B.: The Hermite transform-applications. IEEE Trans. Acoust. Speech Signal Process. 38(9), 1607–1618 (1990)
Mugler, D.H., Clary, S.: Discrete Hermite functions. In: Proceedings of the International Conference on Scientific Computing and Mathematical Modeling, IMACS 2000, pp. 318–321. Milwankee, Wisconsin (2000)
Mugler, D.H., Clary, S.: Discrete Hermite functions and the fractional Fourier transform. In: Proceedings of the International Workshop on Sampling Theory, pp. 303–308. IEEE, Piscataway, New Jersey (2001)
Mugler, D.H., Mahadevan, A.: Multiscale signal processing with discrete Hermite functions. In: Shen, X., Zayed, A.I. (eds.) Multiscale Signal Analysis and Modeling, pp. 257–274. Springer, New York (2012)
Mugler, D.H., Clary S., Wu, Y.: Discrete Hermite expansion of digital signals: applications to ECG signals. In: Proceedings of the IEEE Signal Processing Society 10th DSP Workshop, pp. 271–276. Georgia (2002)
Poularikas, A.D.: The Handbook of Formulas and Tables for Signal Processing. CRC Press, Boca Raton (1999)
Refregier, A.: Shapelets I. A method for image analysis. Mon. Not. R. Astron. Soc. 338, 35–47 (2003)
Rodieck, R.W.: Quantitative Analysis of cat retinal ganglion cell response to visual stimuli. Vision Res. 5, 583–601 (1965)
Silván-Cárdenas, J.L., Escalante-Ramrez, B.: The multiscale Hermite transform for local orientation analysis. IEEE Trans. Image Process. 15, 1236–1253 (2006)
Van Rullen, R., Thorpe, S.J.: Rate coding versus temporal order coding: what the retinal ganglion cells tell the visual cortex. Neural Comput. 13, 1255–1283 (2001)
Acknowledgments
The authors are grateful to Jacob Trombetta, master’s student of the first-named author, for some initial development.
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Mugler, D.H., Clary, S. (2014). Frequency Determination Using the Discrete Hermite Transform. In: Zayed, A., Schmeisser, G. (eds) New Perspectives on Approximation and Sampling Theory. Applied and Numerical Harmonic Analysis. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-08801-3_16
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DOI: https://doi.org/10.1007/978-3-319-08801-3_16
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