Universal analytic properties of noise: introducing the J-matrix formalism

• Published in: Journal of physics. A, Mathematical and theoretical Mathematical and theoretical, 2009-09, Vol.42 (36), p.365202-365202 (15)
• Main Authors: Bessis, Daniel, Perotti, Luca
• Format: Article
• Language: English
• Subjects: Exact sciences and technology; Physics
• ISSN: 1751-8121; 1751-8113; 1751-8121
• DOI: 10.1088/1751-8113/42/36/365202

We propose a new method in the spectral analysis of noisy time-series data for damped oscillators. From the Jacobi three terms recursive relation for the denominators of the Pade approximations built on the well-known Z-transform of an infinite time series, we build a Hilbert space operator, a J-operator, where each bound state (inside the unit circle in the complex plane) is simply associated with one damped oscillator while the essential spectrum of the J-operator, which lies on the unit circle itself, is shown to represent the noise. Signal and noise are thus clearly separated in the complex plane. For a finite time series of length 2N, the J-operator is replaced by a finite order J-matrix JN, having N eigenvalues which are time reversal covariant. Different classes of input noise, such as blank (white and uniform), Gaussian and pink, are discussed in detail, the J-matrix formalism allowing us to efficiently calculate hundreds of poles of the Z-transform. Evidence of a universal behavior in the final statistical distribution of the associated poles and zeros of the Z-transform is shown. In particular, the poles and zeros tend, when the length of the time series goes to infinity, to a uniform angular distribution on the unit circle. Therefore at finite order, the roots of unity in the complex plane appear to be noise attractors. We show that the Z-transform presents the exceptional feature of allowing lossless undersampling and how to make use of this property. A few basic examples are given to suggest the power of the proposed method.