Self-similar Signals Research Articles

Spatial Signal Analysis Based on Wave-Spectral Fractal Scaling: A Case of Urban Street Networks

A number of mathematical methods have been developed to make temporal signal analyses based on time series. However, no effective method for spatial signal analysis, which are as important as temporal signal analyses for geographical systems, has been devised. Nonstationary spatial and temporal processes are associated with nonlinearity, and cannot be effectively analyzed by conventional analytical approaches. Fractal theory provides a powerful tool for exploring spatial complexity and is helpful for spatio-temporal signal analysis. This paper is devoted to developing an approach for analyzing spatial signals of geographical systems by means of wave-spectrum scaling. The traffic networks of 10 Chinese cities are taken as cases for positive studies. Fast Fourier transform (FFT) and ordinary least squares (OLS) regression methods are employed to calculate spectral exponents. The results show that the wave-spectrum density distribution of all these urban traffic networks follows scaling law, and that the spectral scaling exponents can be converted into fractal dimension values. Using the fractal parameters, we can make spatial analyses for the geographical signals. The wave-spectrum scaling methods can be applied to both self-similar fractal signals and self-affine fractal signals in the geographical world. This study has implications for the further development of fractal-based spatiotemporal signal analysis in the future.

Information Theory for Non-Stationary Processes with Stationary Increments

We describe how to analyze the wide class of non-stationary processes with stationary centered increments using Shannon information theory. To do so, we use a practical viewpoint and define ersatz quantities from time-averaged probability distributions. These ersatz versions of entropy, mutual information, and entropy rate can be estimated when only a single realization of the process is available. We abundantly illustrate our approach by analyzing Gaussian and non-Gaussian self-similar signals, as well as multi-fractal signals. Using Gaussian signals allows us to check that our approach is robust in the sense that all quantities behave as expected from analytical derivations. Using the stationarity (independence on the integration time) of the ersatz entropy rate, we show that this quantity is not only able to fine probe the self-similarity of the process, but also offers a new way to quantify the multi-fractality.

Analysis of the nonlinear relationship between commodity prices in the last two decades

We propose the construction of a network to study the correlation among price indices of different commodities, by using the Multifractal Cross-Correlation method proposed by Podobnik and Stanley. This estimator, based on the method Multifractal Detrended Fluctuation Analysis, is effective for self-similar signals with characteristics such as those we here analyze. We construct different networks for time periods between 1991 and 2012. Each node represents a commodity group and the links are the cross-correlation between nodes. We study the evolution of these networks from January 1991 to April 2012. The results show that after 2007, high connectivity arises between the nodes of the matrix. We conjecture that this is a consequence of the cash flow from equities and real estate markets to the commodity market due to the subprime mortage crisis.

A Study of Wavelet Analysis and Data Extraction from Second-Order Self-Similar Time Series

Statistical analysis and synthesis of self-similar discrete time signals are presented. The analysis equation is formally defined through a special family of basis functions of which the simplest case matches the Haar wavelet. The original discrete time series is synthesized without loss by a linear combination of the basis functions after some scaling, displacement, and phase shift. The decomposition is then used to synthesize a new second-order self-similar signal with a different Hurst index than the original. The components are also used to describe the behavior of the estimated mean and variance of self-similar discrete time series. It is shown that the sample mean, although it is unbiased, provides less information about the process mean as its Hurst index is higher. It is also demonstrated that the classical variance estimator is biased and that the widely accepted aggregated variance-based estimator of the Hurst index results biased not due to its nature (which is being unbiased and has minimal variance) but to flaws in its implementation. Using the proposed decomposition, the correct estimation of theVariance Plotis described, as well as its close association with the popularLogscale Diagram.

Fractional Order Signal Processing of Electrochemical Noises

The corrosion processes of stainless steel under different solutions were examined using electrochemical noise (ECN). Using rescaled range analysis, we demonstrated that ECN signals produced by corrosion processes have non-stationary and self-similar properties. The comparison and analysis of ECN signals in both the time and frequency domains showed that conventional methods failed to sufficiently distinguish between the ECN signals obtained under different solutions. Therefore, we introduced the use of fractional Fourier transforms, a powerful tool for the time-frequency analysis of self-similar signals, to process ECN signals that can better describe the corrosion behaviours of the electrode in different solutions.

Vehicular traffic through a self-similar sequence of traffic lights

We study the dynamical behavior of vehicular traffic through a sequence of traffic lights positioned self-similarly on a highway, where all traffic lights turn on and off simultaneously with cycle time T s . The signals are positioned self-similarly by Cantor set. The nonlinear-map model of vehicular traffic controlled by self-similar signals is presented. The vehicle exhibits the complex behavior with varying cycle time. The tour time is much lower such that signals are positioned periodically with the same interval. The arrival time T( x) at position x scales as ( T ( x ) - x ) ∝ x d f , where d f is the fractal dimension of Cantor set. The landscape in the plot of T( x)− x against cycle time T s shows a self-affine fractal with roughness exponent α=1− d f .

A new class of fractal ultra-wideband signals

INTRODUCTIONThe variety of communication signals that existsnowadays makes it possible to successfully solve tele-communication, information, and other important prac-tical problems. The requirements for the qualitativeproperties of the corresponding systems are continuallygrowing. New types of signals exhibiting wider poten-tialities were actively searched for in [1–3]. Therefore,it seems reasonable to develop a novel class of fractalultra-wideband (FUWB) signals possessing all theadvantages of both existing ultra-wideband (UWB) sig-nals and fractal signals.MODELS OF FRACTAL UWB SIGNALSWe call FUWB signals those UWB signals thatexhibit the self-affinity property and the fractionaldimension. We now consider self-similar FUWB sig-nals. In order to denote their dimension, we also use theMinkowski dimension d

Detrending moving average algorithm: A closed-form approximation of the scaling law

The Hurst exponent H of long range correlated series can be estimated by means of the detrending moving average ( DMA) method. The computational tool, on which the algorithm is based, is the generalized variance σ DMA 2 = 1 / ( N - n ) ∑ i = n N [ y ( i ) - y ˜ n ( i ) ] 2 , with y ˜ n ( i ) = 1 / n ∑ k = 0 n y ( i - k ) being the average over the moving window n and N the dimension of the stochastic series y ( i ) . The ability to yield H relies on the property of σ DMA 2 to vary as n 2 H over a wide range of scales [E. Alessio, A. Carbone, G. Castelli, V. Frappietro, Eur. J. Phys. B 27 (2002) 197]. Here, we give a closed form proof that σ DMA 2 is equivalent to C H n 2 H and provide an explicit expression for C H . We furthermore compare the values of C H with those obtained by applying the DMA algorithm to artificial self-similar signals.

A new approach for estimation of statistically matched wavelet

This paper presents a new approach for the estimation of wavelets that is matched to a given signal in the statistical sense. Based on this approach, a number of new methods to estimate statistically matched wavelets are proposed. The paper first proposes a new method for the estimation of statistically matched two-band compactly supported biorthogonal wavelet system. Second, a new method is proposed to estimate statistically matched semi-orthogonal two-band wavelet system that results in compactly supported or infinitely supported wavelet. Next, the proposed method of estimating two-band wavelet system is generalized to M-band wavelet system. Here, the key idea lies in the estimation of analysis wavelet filters from a given signal. This is similar to a sharpening filter used in image enhancement. The output of analysis highpass filter branch is viewed to be equivalent to an error in estimating the middle sample from the neighborhood. To minimize this error, a minimum mean square error (MMSE) criterion is employed. Since wavelet expansion acts like Karhunen-Loe/spl grave/ve-type expansion for generalized 1/f/sup /spl beta// processes, it is assumed that the given signal is a sample function of an mth-order fractional Brownian motion. Therefore, the autocorrelation structure of a generalized 1/f/sup /spl beta// process is used in the estimation of analysis filters using the MMSE criterion. We then present methods to design a finite impulse response/infinite impulse response (FIR/IIR) biorthogonal perfect reconstruction filterbank, leading to the estimation of a compactly supported/infinitely supported statistically matched wavelet. The proposed methods are very simple. Simulation results to validate the proposed theory are presented for different synthetic self-similar signals as well as music and speech clips. Estimated wavelets for different signals are compared with standard biorthogonal 9/7 and 5/3 wavelets for the application of compression and are shown to have better results.

A Generalized Signal Flow Graph Approach for Hybrid Acquisition of Ultra-Wideband Signals

This paper establishes a general framework for the study of various search procedures in which a group of observers is searching for a group of items in some arbitrary manner. A generalized signal flow graph approach yields a complete statistical description of the search time as well as the probability of correctly terminating the search. The motivation for this study comes from the acquisition of an ultra-wideband (UWB) signal in a dense multipath channel. In this particular problem many resolvable paths tend to cluster at a receiver containing multiple correlators allowing for a hybrid serial/parallel search. This hybrid search, as well as a bound on a sorted hybrid search, is analyzed using the generalized signal flow graph approach. Rapid code and frame acquisition at the receiver is desired and it is shown that a specific nonconsecutive search, the bit reversal search, yields an optimum hybrid search procedure under certain conditions. The acquisition of UWB signals also leads to the concept of a self-similar signal flow graph, in which a new search is initiated upon the completion of the previous one.

Discrete-time models for statistically self-similar signals

Wide-sense statistical self-similarity in continuous-time random processes is defined through invariance of its first-order and second-order statistics to scaling in time. Since scaling has an unambiguous definition in continuous-time but not in discrete-time, researchers have provided various definitions of discrete-time self-similarity without reference to scaling. This paper proposes a discrete-time continuous-dilation scaling operator and develops a framework based on it for formulating statistical self-similarity from first principles in a manner analogous to the continuous-time development. Relationship between the resulting model and fractional order transfer function systems is presented. The potential for using this model in applications involving long-range dependent phenomena is explored.

Analysis of non-linear systems using wavelets

Analysis of non-linear systems is an essential part of engineering structural dynamics. A number of methods have been developed in recent years. Classical Fourier-based methods have been extended to the use of phase plane, combined time-frequency, time-scale approaches and multidimensional spectra. This paper is an attempt to collate in one place some of the recent advances in wavelet analysis for the study of non-linear systems. This includes methods related to system identification based on wavelet ridges and skeletons, damping estimation procedures, wavelet-based frequency response functions, cross-wavelet analysis, self-similar signals, coherent structures and chaos.

Macroscopic 1/f behavior of fractal signals generated by additive substitution

We analyze the structure of the discrete Fourier transform of a particular class of self-similar discrete signals which are iteratively generated by an additive substitution procedure, starting from a finite generator sequence. The amplitude spectra of such signals obtained from different generators have a similar form. The amplitude spectrum contains several affine branches, the number of which depends on the iteration level, on the size of the generator, and on the spectrum's symmetry. It has a macroscopic 1/f-characteristic.

Signal Processing with Fractals: A Wavelet Based Approach

Wavelet transmission statistically self-similar signals detection and estimation with 1/processes deterministically self-similar signals fractal modulation linear self-similar signals.

Long Memory and Self-Similar Signals: A New Recursive Maximum Likelihood Estimator

This paper is devoted to the estimation problem of linear non-stationary systems. The basic point of view is the maximum likelihood technic. We design the global (ML) and recursive (RML) algorithms for a class of non-stationary AutoRegressive models. We apply those algorithms to a class of self-similar and long memory signals. To prevent divergence of the (RML) algorithm it is necessary to compute the analytic gradient and hessien of the criteria. Finally we present some results issued from simulations with comparisons between (ML) and (RML) methods.

Self-similar set identification in the time-scale domain

We study the wavelet transform of deterministic self-similar signals and derive their properties, as well as a new algorithm for identification of the self-similarity parameter. We also include such applications as characterization and analysis of real chaotic signals in the presence of additive noise.

Wavelet-based representations for a class of self-similar signals with application to fractal modulation

A potentially important family of self-similar signals based upon a deterministic scale-invariance characterization is introduced. These signals, which are referred to as 'dy-homogeneous' signals because they generalize the well-known homogeneous functions, have highly convenient representations in terms of orthonormal wavelet bases. In particular, wavelet representations can be exploited to construct orthonormal self-similar bases for these signals. The spectral and fractal characteristics of dy-homogeneous signals make them appealing candidates for use in a number of applications. As one potential example, their use in a communications-based context is considered. Specifically, a strategy for embedding information into a dy-homogeneous waveform on multiple time-scales is developed. This multirate modulation strategy, called fractal modulation, is potentially well-suited for use with noisy channels of simultaneously unknown duration and bandwidth. >