Instantaneous frequency estimation of multicomponent non- stationary signals using Fourier Bessel series and Time-Varying Auto Regressive Model

In this paper, we propose a novel technique for Instantaneous frequency (IF) estimation of multi component non stationary signals using Fourier Bessel Series and Time–Varying Auto Regressive (FB-TVAR) model. In the proposed technique, the Fourier-Bessel (FB) expansion decomposes the multicomponent non stationary signal into a number of monocomponent signals and TVAR model is used to model each monocomponent signal. In TVAR modeling approach the time varying parameters are expanded as a linear combination of basis functions. In this paper, the TVAR parameters are expanded by a discrete cosine basis functions. The maximum likelihood estimation algorithm for model order selection in TVAR models is also discussed. The Instantaneous frequency (IF) is extracted from the time-varying parameters by calculating the angles of the estimation error filter polynomial roots. The estimation of the TVAR parameters of a multicomponent signal requires the inversion of a large covariance matrix, while the projected technique (FB-TVAR) requires the inversion of a number of comparatively small covariance matrices with better numerical stability properties. Simulation results are presented for three component discrete Amplitude and Frequency modulated(AM-FM)signa

Financial LPPL Bubbles with Mean-Reverting Noise in the Frequency Domain

The log-periodic power law (LPPL) is a model of asset prices during endogenous bubbles. A major open issue is to verify the presence of LPPL in price sequences and to estimate the LPPL parameters. Estimation is complicated by the fact that daily LPPL returns are typically orders of magnitude smaller than measured price returns, suggesting that noise obscures the underlying LPPL dynamics. However, if noise is mean-reverting, it would quickly cancel out over subsequent measurements. In this paper, we attempt to reject mean-reverting noise from price sequences by exploiting frequency-domain properties of LPPL and of mean reversion. First, we calculate the spectrum of mean-reverting \ou noise and devise estimators for the noise's parameters. Then, we derive the LPPL spectrum by breaking it down into its two main characteristics of power law and of log-periodicity. We compare price spectra with noise spectra during historical bubbles. In general, noise was strong also at low frequencies and, even if LPPL underlied price dynamics, LPPL would be obscured by noise

Spectral analysis for nonstationary audio

A new approach for the analysis of nonstationary signals is proposed, with a focus on audio applications. Following earlier contributions, nonstationarity is modeled via stationarity-breaking operators acting on Gaussian stationary random signals. The focus is on time warping and amplitude modulation, and an approximate maximum-likelihood approach based on suitable approximations in the wavelet transform domain is developed. This paper provides theoretical analysis of the approximations, and introduces JEFAS, a corresponding estimation algorithm. The latter is tested and validated on synthetic as well as real audio signal.Comment: IEEE/ACM Transactions on Audio, Speech and Language Processing, Institute of Electrical and Electronics Engineers, In pres

Map-making in small field modulated CMB polarisation experiments: approximating the maximum-likelihood method

Map-making presents a significant computational challenge to the next generation of kilopixel CMB polarisation experiments. Years worth of time ordered data (TOD) from thousands of detectors will need to be compressed into maps of the T, Q and U Stokes parameters. Fundamental to the science goal of these experiments, the observation of B-modes, is the ability to control noise and systematics. In this paper, we consider an alternative to the maximum-likelihood method, called destriping, where the noise is modelled as a set of discrete offset functions and then subtracted from the time-stream. We compare our destriping code (Descart: the DEStriping CARTographer) to a full maximum-likelihood map-maker, applying them to 200 Monte-Carlo simulations of time-ordered data from a ground based, partial-sky polarisation modulation experiment. In these simulations, the noise is dominated by either detector or atmospheric 1/f noise. Using prior information of the power spectrum of this noise, we produce destriped maps of T, Q and U which are negligibly different from optimal. The method does not filter the signal or bias the E or B-mode power spectra. Depending on the length of the destriping baseline, the method delivers between 5 and 22 times improvement in computation time over the maximum-likelihood algorithm. We find that, for the specific case of single detector maps, it is essential to destripe the atmospheric 1/f in order to detect B-modes, even though the Q and U signals are modulated by a half-wave plate spinning at 5-Hz.Comment: 18 pages, 17 figures, MNRAS accepted v2: content added (inc: table 2), typos correcte

Cramer–Rao lower bounds for change points in additive and multiplicative noise

The paper addresses the problem of determining the Cramer–Rao lower bounds (CRLBs) for noise and change-point parameters, for steplike signals corrupted by multiplicative and/or additive white noise. Closed-form expressions for the signal and noise CRLBs are first derived for an ideal step with a known change point. For an unknown change-point, the noise-free signal is modeled by a sigmoidal function parametrized by location and step rise parameters. The noise and step change CRLBs corresponding to this model are shown to be well approximated by the more tractable expressions derived for a known change-point. The paper also shows that the step location parameter is asymptotically decoupled from the other parameters, which allows us to derive simple CRLBs for the step location. These bounds are then compared with the corresponding mean square errors of the maximum likelihood estimators in the pure multiplicative case. The comparison illustrates convergence and efficiency of the ML estimator. An extension to colored multiplicative noise is also discussed

Five-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Data Processing, Sky Maps, and Basic Results

We present new full-sky temperature and polarization maps in five frequency bands from 23 to 94 GHz, based on data from the first five years of the WMAP sky survey. The five-year maps incorporate several improvements in data processing made possible by the additional years of data and by a more complete analysis of the instrument calibration and in-flight beam response. We present several new tests for systematic errors in the polarization data and conclude that Ka band data (33 GHz) is suitable for use in cosmological analysis, after foreground cleaning. This significantly reduces the overall polarization uncertainty. With the 5 year WMAP data, we detect no convincing deviations from the minimal 6-parameter LCDM model: a flat universe dominated by a cosmological constant, with adiabatic and nearly scale-invariant Gaussian fluctuations. Using WMAP data combined with measurements of Type Ia supernovae and Baryon Acoustic Oscillations, we find (68% CL uncertainties): Omega_bh^2 = 0.02267 \pm 0.00059, Omega_ch^2 = 0.1131 \pm 0.0034, Omega_Lambda = 0.726 \pm 0.015, n_s = 0.960 \pm 0.013, tau = 0.084 \pm 0.016, and Delta_R^2 = (2.445 \pm 0.096) x 10^-9. From these we derive: sigma_8 = 0.812 \pm 0.026, H_0 = 70.5 \pm 1.3 km/s/Mpc, z_{reion} = 10.9 \pm 1.4, and t_0 = 13.72 \pm 0.12 Gyr. The new limit on the tensor-to-scalar ratio is r < 0.22 (95% CL). We obtain tight, simultaneous limits on the (constant) dark energy equation of state and spatial curvature: -0.14 < 1+w < 0.12 and -0.0179 < Omega_k < 0.0081 (both 95% CL). The number of relativistic degrees of freedom (e.g. neutrinos) is found to be N_{eff} = 4.4 \pm 1.5, consistent with the standard value of 3.04. Models with N_{eff} = 0 are disfavored at >99.5% confidence.Comment: 46 pages, 13 figures, and 7 tables. Version accepted for publication, ApJS, Feb-2009. Includes 5-year dipole results and additional references. Also available at http://lambda.gsfc.nasa.gov/product/map/dr3/map_bibliography.cf

Are all RR Lyrae stars modulated?

We analyzed 151 variables previously classified as fundamental mode RR Lyrae stars from Campaigns 01-04 of the Kepler two wheel (K2) archive. By employing a method based on the application of systematics filtering with the aid of co-trending light curves in the presence of the large amplitude signal component, we searched for additional Fourier signals in the close neighborhood of the fundamental period. We found only 13 stars without such components, yielding the highest rate of 91% of modulated (Blazhko) stars detected so far. A detection efficiency test suggests that this occurrence rate likely implies a 100% underlying rate. Furthermore, the same test performed on a subset of the Large Magellanic Cloud RR Lyrae stars from the MACHO archive shows that the conjecture of high true occurrence rate fits well to the low observed rate derived from this database.Comment: Submitted to Astronomy & Astrophysics (6 pages, 8 figures