A NEW PRONY-BASED CIRCULAR-HYPERBOLIC DECOMPOSITION

• Published in: Journal of sound and vibration Vol. 242; no. 1; pp. 85 - 102
• Main Authors: FAHEY, S.O'F., SULEIMAN, B.M., NAYFEH, A.H., HAJJ, M.R.
• Format: Journal Article
• Language: English
• Published: London Elsevier Ltd 19.04.2001; Elsevier
• Subjects: Acoustics; Exact sciences and technology; Fundamental areas of phenomenology (including applications); Noise: its effects and control; Physics; Complex function; Noise reduction; Decomposition method; Discrete Fourier transformation; Closed form equation; Exponential function; Modeling; Implementation; Experimental result; Gaussian noise; Covariance; Cosine function; Least squares method; White noise; Sine function; Prony method; Power spectrum
• ISSN: 0022-460X; 1095-8568
• DOI: 10.1006/jsvi.2000.3361

We introduce a new closed-form decomposition technique for estimating the model parameters of an evenly sampled signal known to be composed of circular and hyperbolic sine and cosine functions in the presence of Gaussian white noise. The techniqe is closely related to Prony's method and hereditary algorithms that fit complex exponential functions to evenly sampled data. The circular and hyperbolic sine and cosine functions are obtained by adding constraints that limit the form of the characteristic polynomial coefficients. It avoids the leakage effects associated with the discrete fourier transform (DFT) for circular sine and cosine functions. When the signal contains frequency components that are not rational multiples of each other, the proposed decomposition yields amplitude and phase parameters that are more accurate than those obtained with the DFT in moderate levels of noise. First, we review Prony's method and one hereditary algorithm (the complex exponential algorithm). Then, we detail three implementation procedures of the new technique. The first is a two-stage least-squares approach. The second utilises a novel concept of noise reduction which is attributed to Pisarenko. The last provides additional means of noise reduction through a covariance formulation that avoids zero-lag terms. Experimental and numerical examples of the application of the circular-hyperbolic decomposition (CHD) are given.