On the Grunbaum Commutor Based Discrete Fractional Fourier Transform

The basis functions of the continuous fractional Fourier transform (FRFT) are linear chirp signals that are suitable for time-frequency analysis of signals with chirping time-frequency content. In the continuous--time case, analytical results linking the chirp rate of the signal to a specific angle where the FRET of the chirp signal is an impulse exist. Recent efforts towards developing a discrete and computable version of the fractional Fourier transform (DFRFT) have focussed on furnishing a orthogonal set of eigenvectors for the DFT that serve as discrete versions of the Gauss--Hermite functions in the hope of replicating this property. In the discrete case, however, no analytical results connecting the chirp rate of the signal to the angle at which we obtain an impulse exist. Defined via the fractional matrix power of the centered version of the DFT, computation of this transform has been constrained due to the need for computing an eigenvalue decomposition. Analysis of the centered version of the DFRFT obtained from Grunbaum\u27s tridiagonal commuter and the kernel associated with it reveals the presence of both amplitude and frequency modulation in contrast to just frequency modulation seen in the continuous case. Furthermore, the instantaneous frequency of the basis functions of the DFRFT are sigmoidal rather than linear. In this report, we define a centered version of the DFRFT based on the Grunbaum commutor and investigate its capabilities towards representing and concentrating chirp signals in a few transform coefficients. We then propose a fast algorithm using the FFT for efficient computation of the multiangle version of the CDFRFT (MA-CDFRFT) using symmetries in the computed eigenvectors to reduce the size of the eigenvalue problem. We further develop approximate empirical relations that will enable us to estimate the chirp rate of the multicomponent chirp signals from the peaks of the computed MA-CDFRFT. This MA-CDFRFT also lays the ground work for a novel chirp rate Vs. frequency signal representation that is more suitable for the time-frequency analysis of multicomponent chirp signals

STFT with Adaptive Window Width Based on the Chirp Rate

An adaptive time-frequency representation (TFR) with higher energy concentration usually requires higher complexity. Recently, a low-complexity adaptive short-time Fourier transform (ASTFT) based on the chirp rate has been proposed. To enhance the performance, this method is substantially modified in this paper: i) because the wavelet transform used for instantaneous frequency (IF) estimation is not signal-dependent, a low-complexity ASTFT based on a novel concentration measure is addressed; ii) in order to increase robustness to IF estimation error, the principal component analysis (PCA) replaces the difference operator for calculating the chirp rate; and iii) a more robust Gaussian kernel with time-frequency-varying window width is proposed. Simulation results show that our method has higher energy concentration than the other ASTFTs, especially for multicomponent signals and nonlinear FM signals. Also, for IF estimation, our method is superior to many other adaptive TFRs in low signal-to-noise ratio (SNR) environments.Comment: Accepted by IEEE Transactions on Signal Processin

Breaking the Limits -- Redefining the Instantaneous Frequency

The Carson and Fry (1937) introduced the concept of variable frequency as a generalization of the constant frequency. The instantaneous frequency (IF) is the time derivative of the instantaneous phase and it is well-defined only when this derivative is positive. If this derivative is negative, the IF creates problem because it does not provide any physical significance. This study proposes a mathematical solution and eliminate this problem by redefining the IF such that it is valid for all monocomponent and multicomponent signals which can be nonlinear and nonstationary in nature. This is achieved by using the property of the multivalued inverse tangent function. The efforts and understanding of all the methods based on the IF would improve significantly by using this proposed definition of the IF. We also demonstrate that the decomposition of a signal, using zero-phase filtering based on the well established Fourier and filter theory, into a set of desired frequency bands with proposed IF produces accurate time-frequency-energy (TFE) distribution that reveals true nature of signal. Simulation results demonstrate the efficacy of the proposed IF that makes zero-phase filter based decomposition most powerful, for the TFE analysis of a signal, as compared to other existing methods in the literature.Comment: 12 pages, 15 figures. arXiv admin note: text overlap with arXiv:1604.0499

Compressed Sensing for Time-Frequency Gravitational Wave Data Analysis

The potential of compressed sensing for obtaining sparse time-frequency representations for gravitational wave data analysis is illustrated by comparison with existing methods, as regards i) shedding light on the fine structure of noise transients (glitches) in preparation of their classification, and ii) boosting the performance of waveform consistency tests in the detection of unmodeled transient gravitational wave signals using a network of detectors affected by unmodeled noise transientComment: 16 pages + 17 figure

Optimal Scale Invariant Wigner Spectrum Estimation of Gaussian Locally Self-Similar Processes Using Hermite Functions

This paper investigates the mean square error optimal estimation of scale invariant Wigner spectrum for the class of Gaussian locally self-similar processes, by the multitaper method. In this method, the spectrum is estimated as a weighted sum of scale invariant windowed spectrograms. Moreover, it is shown that the optimal multitapers are approximated by the quasi Lamperti transformation of Hermite functions, which is computationally more efficient. Finally, the performance and accuracy of the estimation is studied via simulation.Comment: 17 pages, 4 figure

SAR-Based Vibration Estimation Using the Discrete Fractional Fourier Transform

A vibration estimation method for synthetic aperture radar (SAR) is presented based on a novel application of the discrete fractional Fourier transform (DFRFT). Small vibrations of ground targets introduce phase modulation in the SAR returned signals. With standard preprocessing of the returned signals, followed by the application of the DFRFT, the time-varying accelerations, frequencies, and displacements associated with vibrating objects can be extracted by successively estimating the quasi-instantaneous chirp rate in the phase-modulated signal in each subaperture. The performance of the proposed method is investigated quantitatively, and the measurable vibration frequencies and displacements are determined. Simulation results show that the proposed method can successfully estimate a two-component vibration at practical signal-to-noise levels. Two airborne experiments were also conducted using the Lynx SAR system in conjunction with vibrating ground test targets. The experiments demonstrated the correct estimation of a 1-Hz vibration with an amplitude of 1.5 cm and a 5-Hz vibration with an amplitude of 1.5 mm

Time-Frequency analysis via the Fourier Representation

The nonstationary nature of signals and nonlinear systems require the time-frequency representation. In time-domain signal, frequency information is derived from the phase of the Gabor's analytic signal which is practically obtained by the inverse Fourier transform. This study presents time-frequency analysis by the Fourier transform which maps the time-domain signal into the frequency-domain. In this study, we derive the time information from the phase of the frequency-domain signal and obtain the time-frequency representation. In order to obtain the time information in Fourier domain, we define the concept of `frequentaneous time' which is frequency derivative of phase. This is very similar to the group delay, which is also defined as frequency derivative of phase and it provide physical meaning only when it is positive. The frequentaneous time is always positive or negative depending upon whether signal is defined for only positive or negative times, respectively. If a signal is defined for both positive and negative times, then we divide the signal into two parts, signal for positive times and signal for negative times. The proposed frequentaneous time and Fourier transform based time-frequency distribution contains only those frequencies which are present in the Fourier spectrum. Simulations and numerical results, on many simulated as well as read data, demonstrate the efficacy of the proposed method for the time-frequency analysis of a signal.Comment: 7 pages, 6 figure

The Fourier-based Synchrosqueezing Transform

The short-time Fourier transform (STFT) and continuous wavelet transform (CWT) are intensively used to analyze and process multicomponent signals, ie superpositions of mod- ulated waves. The synchrosqueezing is a post-processing method which circumvents the uncertainty relations, inherent to these linear transforms, by reassigning the coefficients in scale or frequency. Originally introduced in the setting of the continuous wavelet transform, it provides a sharp, con- centrated representation, while remaining invertible. This technique received a renewed interest with the recent publi- cation of an approximation result, which provides guarantees for the decomposition of a multicomponent signal. This paper adapts the formulation of the synchrosqueezing to the STFT, and states a similar theoretical result. The emphasis is put on the differences with the CWT-based synchrosqueezing, and all the content is illustrated through numerical experiments

A Novel Approach for Ridge Detection and Mode Retrieval of Multicomponent Signals Based on STFT

Time-frequency analysis is often used to study non stationary multicomponent signals, which can be viewed as the surperimposition of modes, associated with ridges in the TF plane. To understand such signals, it is essential to identify their constituent modes. This is often done by performing ridge detection in the time-frequency plane which is then followed by mode retrieval. Unfortunately, existing ridge detectors are often not enough robust to noise therefore hampering mode retrieval. In this paper, we therefore develop a novel approach to ridge detection and mode retrieval based on the analysis of the short-time Fourier transform of multicomponent signals in the presence of noise, which will prove to be much more robust than state-of-the-art methods based on the same time-frequency representation

A Chirplet Transform-based Mode Retrieval Method for Multicomponent Signals with Crossover Instantaneous Frequencies

In nature and engineering world, the acquired signals are usually affected by multiple complicated factors and appear as multicomponent nonstationary modes. In such and many other situations, it is necessary to separate these signals into a finite number of monocomponents to represent the intrinsic modes and underlying dynamics implicated in the source signals. In this paper, we consider the mode retrieval of a multicomponent signal which has crossing instantaneous frequencies (IFs), meaning that some of the components of the signal overlap in the time-frequency domain. We use the chirplet transform (CT) to represent a multicomponent signal in the three-dimensional space of time, frequency and chirp rate and introduce a CT-based signal separation scheme (CT3S) to retrieve modes. In addition, we analyze the error bounds for IF estimation and component recovery with this scheme. We also propose a matched-filter along certain specific time-frequency lines with respect to the chirp rate to make nonstationary signals be further separated and more concentrated in the three-dimensional space of CT. Furthermore, based on the approximation of source signals with linear chirps at any local time, we propose an innovative signal reconstruction algorithm, called the group filter-matched CT3S (GFCT3S), which also takes a group of components into consideration simultaneously. GFCT3S is suitable for signals with crossing IFs. It also decreases component recovery errors when the IFs curves of different components are not crossover, but fast-varying and close to one and other. Numerical experiments on synthetic and real signals show our method is more accurate and consistent in signal separation than the empirical mode decomposition, synchrosqueezing transform, and other approache