Fast Frequency Estimation by Zero Crossings of Differential Spline Wavelet Transform

Zero crossings or extrema of a wavelet transform constitute important signatures for signal analysis with the advantage of great simplicity. In this paper, we introduce a fast frequency-estimation method based on zero-crossing counting in the transform domain of a family of differential spline wavelets. The resolution and order of the vanishing moments of the chosen wavelets have a close relation with the frequency components of a signal. Theoretical results on estimating the highest and the lowest frequency components are derived, which are particularly useful for frequency estimation of harmonic signals. The results are illustrated with the help of several numerical examples. Finally, we discuss the connection of this approach with other frequency estimation methods, with the high-order level-crossing analysis in statistics, and with the scaling theorem in computer vision.

[1]  Max A. Viergever,et al.  Scale-Space Theory in Computer Vision , 1997 .

[2]  Yu-Ping Wang,et al.  Scale-Space Derived From B-Splines , 1998, IEEE Trans. Pattern Anal. Mach. Intell..

[3]  Yu-Ping Wang,et al.  Image representations using multiscale differential operators , 1999, IEEE Trans. Image Process..

[4]  Richard E. Passarelli,et al.  The Autocorrelation Function and Doppler Spectral Moments: Geometric and Asymptotic Interpretations , 1983 .

[5]  L. J. Hadjileontiadis.  Discrimination analysis of discontinuous breath sounds using higher-order crossings , 2006, Medical and Biological Engineering and Computing.

[6]  Michael Unser,et al.  The L2-Polynomial Spline Pyramid , 1993, IEEE Trans. Pattern Anal. Mach. Intell..

[7]  D. B. Preston.  Spectral Analysis and Time Series , 1983 .

[8]  Ta-Hsin Li,et al.  Monotone gain, first-order autocorrelation and zero-crossing rate , 1991 .

[9]  Patrick Flandrin,et al.  Wavelet analysis and synthesis of fractional Brownian motion , 1992, IEEE Trans. Inf. Theory.

[10]  Naoki Saito,et al.  Multiresolution representations using the autocorrelation functions of compactly supported wavelets , 1993, IEEE Trans. Signal Process..

[11]  T. Lindeberg,et al.  Scale-Space Theory : A Basic Tool for Analysing Structures at Different Scales , 1994 .

[12]  Yu-Ping Wang,et al.  Multiscale curvature-based shape representation using B-spline wavelets , 1999, IEEE Trans. Image Process..

[13]  Stéphane Mallat,et al.  Zero-crossings of a wavelet transform , 1991, IEEE Trans. Inf. Theory.

[14]  Benjamin Kedem,et al.  On the Sinusoidal Limit of Stationary Time Series , 1984 .

[15]  Ingrid Daubechies,et al.  Ten Lectures on Wavelets , 1992 .

[16]  B. Kedem,et al.  Spectral analysis and discrimination by zero-crossings , 1986, Proceedings of the IEEE.

[17]  Tony Lindeberg,et al.  Scale-Space for Discrete Signals , 1990, IEEE Trans. Pattern Anal. Mach. Intell..

[18]  R. Laha.  Probability Theory , 1979 .

[19]  Tony Lindeberg,et al.  Scale-Space Theory in Computer Vision , 1993, Lecture Notes in Computer Science.

[20]  Richard Kronland-Martinet,et al.  Asymptotic wavelet and Gabor analysis: Extraction of instantaneous frequencies , 1992, IEEE Trans. Inf. Theory.

[21]  Benjamin Kedem,et al.  A Stochastic Characterization of the Sine Function , 1986 .

[22]  Henry C. Thode.  Applied Probability for Engineers and Scientists , 1999, Technometrics.

[23]  S. Rice.  Mathematical analysis of random noise , 1944 .

[24]  N Radhakrishnan,et al.  A fast algorithm for detecting contractions in uterine electromyography. , 2000, IEEE engineering in medicine and biology magazine : the quarterly magazine of the Engineering in Medicine & Biology Society.

[25]  B. Logan.  Information in the zero crossings of bandpass signals , 1977, The Bell System Technical Journal.

[26]  Bart M. ter Haar Romeny,et al.  Scale-Space Theory in Computer Vision: First International Conference, Scale-Space '97, Utrecht, The Netherlands, July 2 - 4, 1997, Proceedings , 1997 .

[27]  Alan L. Yuille,et al.  Scaling Theorems for Zero Crossings , 1987, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[28]  C. R. Rao,et al.  The simultaneous estimation of the number of signals and frequencies of multiple sinusoids when some observations are missing: I. Asymptotics. , 1999, Proceedings of the National Academy of Sciences of the United States of America.

[29]  Robert M. Haralick,et al.  Digital Step Edges from Zero Crossing of Second Directional Derivatives , 1984, IEEE Transactions on Pattern Analysis and Machine Intelligence.