Proc. SPIE. 6697, Advanced Signal Processing Algorithms, Architectures, and Implementations XVII
KEYWORDS: Fourier transforms, Heart, Computer simulations, Biological research, Signal generators, Physical phenomena, Signal analyzers, Signal detection, Time-frequency analysis, Information operations
We discuss the application of time-frequency analysis to biomechanical-type signals, and in particular
to signals that would be encountered in the study of rotation rates of bicycle pedaling. We simulate a
number of such signals and study how well they are represented by various time-frequency methods. We
show that time-frequency representations track very well the instantaneous frequency even when there
are very fast changes. In addition, we do a correlation analysis between time-series whose instantaneous
frequency is changing and show that the traditional correlation coeficient is insuffcient to characterize
the correlations. We instead show that the correlation coeficient should be evaluated directly from the
instantaneous frequencies of the time series, which can be easily estimated from their time-frequency
We present a brief review of time-varying spectral analysis and we discuss the applicability of the methods to the case of x-ray bursts where it is known that there are time-varying frequency components. A preliminary analysis is done on x-ray burst the experimental data. Two methods are presented to estimat the instanteous frequency and both methods give approximately the same results.
We define the transient spectrum as the time-frequency spectrum of a random system undergoing a transient behavior. We show that the transient spectrum approaches the classical frequency spectrum when time goes to infinity. We prove that it is always possible to decompose the transient spectrum into the sum of a stationary spectrum and a decaying spectrum. The stationary spectrum is, up to a constant, the classical power spectrum, while the decaying spectrum accounts for the nonstationary behavior of the transient. All the results are valid for random LTI systems defined by stochastic differential equations of n-th order. The Langevin equation is studied as an example.
Atomic clocks are ultra-precise time references, and because of this fact in the past 20 years they have found a fundamental application in navigation problems. The error in the localization of the user is highly dependent on the clock stability: variations of few nanoseconds in the clock phase result in an increase of the localization error by a factor of meters. Unfortunately atomic clocks are nonstationary, and their stability changes with time. We have recently proposed the DAVAR, or dynamic Allan variance, a representation of the instantaneous stability of an atomic clock. In this paper we will discuss the definition of the DAVAR and we will apply it to simulated nonstationary data, to prove its validity in clock noise characterization.
Proc. SPIE. 6313, Advanced Signal Processing Algorithms, Architectures, and Implementations XVI
KEYWORDS: Defense and security, Signal to noise ratio, Stars, Fourier transforms, Interference (communication), Numerical simulations, Computer simulations, Signal processing, Signal detection, Time-frequency analysis
We argue that the standard definition of signal to noise ratio may be misleading when the signal
or noise are nonstationary. We introduce a new measure that we call local signal to noise ratio
(LSNR) which is well suited to take into account nonstationary situations. The advantage of our
measure is that it is a local property unlike the standard SNR which is a single number representing
the total duration of the signal. We simulated a number of cases to show that our measure is more
indicative of the noise and signal level for nonstationary situations.
We use time-frequency distributions to define local stationarity of a random process. We argue that local stationarity is achieved when the Wigner spectrum is approximately factorable. We show that when that is the case the autocorrelation function is the one considered by Silverman in 1957. Other time-frequency representations are also considered.
Many man made and natural noises in nature are nonstationary, however most methods that have been devised to study noises, theoretically or experimentally, have been devised for the stationary situations. We have developed a number of methods to study noises that are nonstationary. We use the Wigner spectrum and other time-frequency representations to represent time-varying noises. These representations can be thought of as a generalization of the standard power spectrum. In this paper we study simple models of nonstationary noises and obtain the Wigner spectrum numerically from realizations of the noises and also by direct calculation. We show that for our test cases the Wigner spectrum clearly shows the nonstationarities of the noise. We also present a method to generate nonstationary white noise that has very different behavior than the standard white Gaussian noise.
Clouds have a nonstationary nature in that their local spectrum changes with position. We model this nonstationarity by extending the classical 1/f<sup>γ</sup> type spectrum. We make γ a function of position and we show that with this choice we can generate nonstationary clouds. Our model can be used to improve denoising algorithms.
Proc. SPIE. 5910, Advanced Signal Processing Algorithms, Architectures, and Implementations XV
KEYWORDS: Defense and security, Signal to noise ratio, Stars, Detection and tracking algorithms, Sensors, Fourier transforms, Interference (communication), Signal processing, Signal detection, Time-frequency analysis
When one calculates a time-frequency distribution of white noise there sometimes appear transients of short duration. Superficially, these transients appear to be real signals but they are not. This comes about by random chance in the noise and also because particular types of distributions do not resolve components well in time. These fictitious signals can be misclassified by detectors and hence it is important to understand their origin and statistical properties. We present experimental studies regarding these false transients, and by simulation we statistically quantify their duration for various distributions. We compare the number and duration of the false transients when different distributions are used.
<i>Image fusion</i> methods provide an enhanced image from a set of source images which present regions with different spatial degradation patterns. Here within a fusion procedure is presented, based on the use of a new <i>defocusing</i> pixel-level measure. Such measure is defined through a 1-D Pseudo Wigner Distribution Function (PWD) applied to non-overlapping N-pixel window slices of the original image. The process is repeated up to cover the full image size. By taking a low resolution image as a reference image, which can be defined i.e. by averaging and blurring the two source images, a pixel-level distance measure of the defocus degree can be obtained from the PWD of each image. This procedure makes possible choosing from a <i>focusing</i> point of view, the <i>in-focus</i> pixels from each one of the given source images. The method is illustrated with different examples. The image fusion approach that we proposed here can work for any source and number of images available. Also, evaluation measures, such as mean square error or percentage of correct decisions, show that our framework can outperform the current approaches for the analyzed cases. One additional advantage of the present approach is its reduced computational cost in comparison with other methods based on a full 2-D implementation of the PWD.
We address the issue of cloud removal from images. Typically a cloud on an image is not uniform and we develop methods that do denoising on a local level. In this paper we present preliminary studies of such methods and also a method for image fusion. The procedure is based on the use of a <i>denoising </i>pixel-level measure. The measure is defined through a 1-D Pseudo Wigner Distribution (PWD) applied to non-overlapping N-pixel window slices of the original
image. The method is illustrated with different set of artificial and natural cloudy or foggy images, which are partially occluded by clouds in different regions. Another advantage of the present approach is its reduced computational cost in comparison with other methods based on a full 2-D implementation of the PWD.
We obtain the Wigner-Ville spectrum of the Wiener process with arbitrary initial conditions. Two different approaches are presented leading to the same result. The solution allows one to study the stationary part, the transient part, and the initial condition dependence.
It is generally accepted that cloud like images have an 1/<i>f</i><sup>γ</sup> power spectra. We investigate whether other spectra also produce cloud like images, and we show by numerical simulation that the hypothesis is true. Also, we show how systems defined by fractional differential equations can generate such spectra.
We have previously presented a method to write equations for the Wigner distribution corresponding to the solution of a linear differential equation. We extend this method to arbitrary time-frequency distributions, the Short Time Fourier Transform, and the wavelet transform. Also, we apply the method to stochastic signals and address a number of applications including Brownian motion. We obtain the Wigner distribution for Brownian motion by solving the
Langevin equation and also by using our method. We obtain exact solutions.
We present an effective method for texture segmentation and analysis using a local spectral method. The method combines the advantages of a high spectral resolution of a joint representation given by the Pseudo-Wigner distribution with an effective adaptive principal component analysis. Performance of the method is evaluated using fabric samples with defects, medical images, and crack detection in metallic surfaces. The examples demonstrate the discrimination power of the present method for detecting even very subtle changes in the homogeneity of textures.
Using artificially generated clouds we study the spectral phase and
amplitude contribution to the cloud image. This is done by reconstructing the cloud image from spectral amplitude and/or phase only. Also, images are reconstructed from partial phase and amplitude in such a way that one may control the relative contribution of the phase and amplitude. We conclude that both phase and amplitude contribute to the cloud like appearance.
We present a method for writing the differential equation for the smoothed Wigner distribution that corresponds to the solution of an ordinary linear differential equation. The method can be applied on any linear ordinary differential equation with constant or time-varying coefficients.
We study time series of the X-ray intensity of the binary XTE-J1550-564 with the goal of estimating its instantaneous power spectrum. We develop a method that, from the initial sequence of photon arrival times, is able to estimate the time-frequency spectrum in conjunction with noise reduction techniques. This method clearly highlights the presence of a quasi-periodic oscillation (QPO), a spectral component the frequency of which changes in time. Furthermore, the QPO is extracted by using signal processing methods in the time-frequency plane. The method is also validated using a synthetic signal to show the quality and reliability of its performance.
We present a general procedure for obtaining equations of motion for the Wigner distribution of functions that are governed by ordinary and partial differential equations. For the case of fields we show that in general one must consider Wigner distribution of the four variables, position, momentum, time and frequency. We also show that in general one cannot write an equation of motion for position and momentum however it can be done in some cases, the Schrodinger equation being one such case. Our method leads to an equation of motion for the Schrodinger equation with time dependent potentials in contrast to the result obtained by Wigner and Moyal which was for time independent potentials.
In this paper, a method for signal component separation, operating in the Time-Frequency (TF) plane and employing a Turbo Estimation Algorithm (TEA), is described. A novel 2D distribution is proposed, named Two Window Spectrogram (TWS), which is free from crossterms and able to yield good time and frequency resolution. Then, a set of parameters is defined in the time-frequency plane, which are able to carry the relevant information on the signal components. An algorithm of estimation of these parameters is proposed, making use of a TEA scheme to yield improved performance. The algorithm has been tested by simulation, yielding very encouraging performance.