Universal analytic properties of noise. Introducing the J-Matrix formalism

📝 Abstract

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 Pad'e Approximations built on the well-known Z-transform of an infinite time-series, we build an Hilbert space operator, a J-Operator, where each bound state (inside the unit circle in the complex plane) is simply associated to one damped oscillator while the continuous 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 J_N, 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 behaviour 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.

💡 Analysis

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 Pad'e Approximations built on the well-known Z-transform of an infinite time-series, we build an Hilbert space operator, a J-Operator, where each bound state (inside the unit circle in the complex plane) is simply associated to one damped oscillator while the continuous 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 J_N, 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 behaviour 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.

📄 Content

arXiv:0807.3101v2 [physics.data-an] 13 May 2009 Universal analytic properties of noise. Introducing the J-Matrix formalism Daniel Bessis and Luca Perotti Department of Physics, Texas Southern University, Houston, Texas 77004 USA (Dated: November 18, 2018) 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 Pad´e Approximations built on the well-known Z-transform of an infinite time-series, we build an Hilbert space operator, a J-Operator, where each bound state (inside the unit circle in the complex plane) is simply associated to one damped oscillator while the continuous 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 dis- cussed in detail, the J-Matrix formalism allowing us to efficiently calculate hundreds of poles of the Z-transform. Evidence of a universal behaviour 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. PACS numbers: 07.05.Kf 2 I. INTRODUCTION Spectral analysis of highly noisy time-series data impacts such diverse fields as gravitational wave detection, Nuclear Magnetic Resonance Spectroscopy as applied to nuclear waste, brain/breast cancer detection, oil detection and other similar areas of application. Experimental time-series are always affected by the presence of noise. As long as the signal to noise ratio is not too poor, several filters are available to denoise the data within the framework of Fourier analysis and its variants. All such techniques, on the other hand, have drawbacks. Fourier analysis, being linear, treats noise on the same foot as the signal, and therefore is per se unable to distinguish the two. In the case of stationary uncorrelated noise, Weiner-Khintchine theorem can be used to denoise the data [1], but it is unable to distinguish signal from correlated or non stationary noise. Wavelets denoising methods require some knowledge of the signal to be found in order to avoid smoothing out the signal itself [2]. Prony’s method is tailored to analyze signals composed of damped oscillators, but it is known to be unstable in the presence of noise. This is because -at least in its original formulation- it assumes an “all-pole” system; in other words, it is equivalent to the construction of a Pad´e approximant (rational approximation) with constant numerator. Having no zeros it is unable to model noise, as we shall see in section II B 3. Modifications have been proposed to stabilize it [3], but, apart from being cumbersome, no effort has been made to classify the poles so as to distinguish the noise ones from the signal ones. The matched filtering technique, which calculates cross-correlations between noisy detector outputs and reference waveforms from a “library”, is only useful when the waveforms are well predicted theoretically [4]. Still, all the above techniques fail when the (average power) signal to noise ratio approaches 1. It is exactly this very high noise case we deal with here, presenting a denoising method based on the analytic properties of the Z-transform (generating function) of the data [5]. The present paper is thus organized: From a given infinite time series, we build -by way of Pad´e Approximations of its Z-transform- a tridiagonal Hilbert space operator, a J-Operator, which, for a finite time series of length 2N, is replaced by a finite order J-Matrix JN, having N time reversal covariant eigenvalues. Such eigenvalues correspond to the poles of the Z-transform itself. We then introduce a specific class of signals made of an arbitrary number of damped oscillators. The poles of the Z-transform of such signals are known to be inside the unit circle [5]. It is on the other hand known that a Taylor series with random coefficients has the unit circle as natural boundary [11]. For a time series of noisy damped oscillators, the spectrum of our J-Operator will therefore comprise a discrete spectrum where each bound state (inside the unit circle) is simply associated to one damped oscillator; and a continuous spectrum which represents the noise, and lies on the unit circle in the complex plane. It is known that

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