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This report is reviewing both linear and quadratic time-frequency representations. The linear representations discussed are Short-Time Fourier Transform and S-transform. The quadratic representation discussed is Wigner distribution. We outline the motivations, interpretations, mathematical fundamentals, properties, and applications of these linear and quadratic time-frequency representations. We also compare these three different time-frequency analysis techniques and show that each technique has its strengths and drawbacks. The simulated data sets have been used for the comparison. The choice of the particular time-frequency representation depends upon the specific area of application and what we aim to achieve with a local frequency analysis. We show that time-frequency analysis methods should enable us to classify signals with a considerably greater interpretation of the physical situation than can be achieved by the conventional Fourier Transform method alone.

This work addresses two related questions. The first question is what joint time-frequency energy representations are most appropriate for auditory signals, in particular, for speech signals in sonorant regions. The quadratic transf for the representation: (1) shift-invariance, (2) positivity, (3) superposition, (4) locality, and (5) smoothness. The second question addressed is how to obtain a rich, symbolic description of the phonetically relevant features om these time-frequency energy surfaces, the so-called schematic spectrogram Time-frequency ridges, the 2-D analog of spectral peaks, are one feature that is proposed. If non-oriented kernels are used for the energy representation, then the ridge tops can be identified with zero-crossings in the inner project of the gradient vector and the direction of greatest downward curvature. If oriented kernels are used, the method can be generalized to give better orientation selectivity (e.g., intersecting ridges) at the cost of poorer time-frequency locality.

Radar Cross Section imaging of a non-wide sense stationary signal poses significant problems in identifying scattering centers in the post processed radar-generated image. A non-wide sense stationary RCS is typically encountered when moving parts on the target impress a phase shift into the backscatter signal that is uncorrelated to the previous return pulse. When the Fourier transform of the phase shifted complex signal is taken, range and cross range information on scattering centers are misplaced. Time Frequency Representations must be used to help locate these scattering centers so they can be properly treated to reduce the target's RCS and increase its survivability. This thesis analyzes and compares various TFRs on non-wide sense stationary signals in hope of providing test centers with better methods for locating scattering centers under time variant conditions.

We present a novel family of continuous linear time-frequency transforms adapted to a multitude of (nonlinear) frequency scales. Similar to classical time-frequency or time-scale representations, the representation coefficients are obtained as inner products with the elements of a continuously indexed family of time-frequency atoms. These atoms are obtained from a single prototype function, by means of modulation, translation and warping. By warping we refer to the process of nonlinear evaluation according to a bijective, increasing function, the warping function. Besides showing that the resulting integral transforms fulfill certain basic, but essential properties, such as continuity and invertibility, we will show that a large subclass of warping functions gives rise to families of generalized coorbit spaces, i.e. Banach spaces of functions whose representations possess a certain localization. Furthermore, we obtain sufficient conditions for subsampled warped time-frequency systems to form atomic decompositions and Banach frames. To this end, we extend results previously presented by Fornasier and Rauhut to a larger class of function systems via a simple, but crucial modification. The proposed method allows for great flexibility, but by choosing particular warping functions we also recover classical time-frequency representations, e.g. $F(t) = ct$ provides the short-time Fourier transform and $F(t)=\log_a(t)$ provides wavelet transforms. This is illustrated by a number of examples provided in the manuscript.

In this paper, we study the Plancherel measure of a class of non-connected nilpotent groups which is of special interest in Gabor theory. Let $G$ be a time-frequency group. More precisely, that is $G=\left\langle T_{k},M_{l}:k\in\mathbb{Z}^{d},l\in B\mathbb{Z}^{d}\right\rangle ,$ $T_{k}$, $M_{l}$ are translations and modulations operators acting in $L^{2}(\mathbb{R}^{d}),$ and $B$ is a non-singular matrix. We compute the Plancherel measure of the left regular representation of $G\ $which is denoted by $L.$ The action of $G$ on $L^{2}(\mathbb{R}^{d})$ induces a representation which we call a Gabor representation. Motivated by the admissibility of this representation, we compute the decomposition of $L$ into direct integral of irreducible representations by providing a precise description of the unitary dual and its Plancherel measure. As a result, we generalize Hartmut F\"uhr's results which are only obtained for the restricted case where $d=1$, $B=1/L,L\in\mathbb{Z}$ and $L>1.$ Even in the case where $G$ is not type I, we are able to obtain a decomposition of the left regular representation of $G$ into a direct integral decomposition of irreducible representations when $d=1$. Some interesting applications to Gabor theory are given as well. For example, when $B$ is an integral matrix, we are able to obtain a direct integral decomposition of the Gabor representation of $G.$

Talk given for the Redwood Center for Theoretical Neuroscience on February 13, 2008. Speaker is Marcelo Magnasco of Rockefeller University. Abstract. Auditory neurons preserve exquisite temporal information about sound features, but we do not know how the brain uses this information to parse the rapidly changing sounds of the natural world. A simple argument for making effective use of temporal information in the auditory nerve leads us to consider the reassignment class of time-frequency representations as a potential model of auditory processing. We show that these representations are sparse even for spectrally dense signals. Many details of complex sounds that are virtually undetectable in standard sonograms are readily perceptible and visible in reassignment; as the only known class of time-frequency representations that is always ‘‘in focus’’ this methodology may help explain the remarkable acuity of auditory perception. We also consider how to determine, experimentally, when a neural code embeds information in the detailed timing of spikes. We show that standard ``spike-triggered'' receptive field constructions are inadequate to extract this level of information and present a new method, ``differential reverse correlations'', based on correlating small changes in spike timing due to small changes to the stimulus.

In this article, we consider the joint estimation of direction-of-departure (DOD) and direction-of- arrival (DOA) information of maneuvering targets in a bistatic multiple-input multiple-output (MIMO) radar system that exploits spatial time-frequency distribution (STFD). STFD has been found useful in solving various array processing problems, such as direction finding and blind source separation, where nonstationary signals with time-varying spectral characteristics are encountered. The STFD approach to array processing has been primarily limited to conventional problems for passive radar platform that deals with signal arrivals, while its use in a MIMO radar configuration has received much less attention. This paper examines the use of STFD in MIMO radar systems with application to direction finding of moving targets with nonstationary signatures. Within this framework, we consider the use of joint transmit and receive apertures for the improved estimation of both target time-varying Doppler signatures and joint DOD/DOA. It is demonstrated that the STFD is an effective tool in MIMO radar processing when moving targets produce Doppler signatures that are highly localized in the time-frequency domain. Keywords: radar signal processing; MIMO radar; direction finding; joint DOD/DOA estimation; time- frequency analysis; moving target tracking.

Traditionally, radar signals have been analyzed in either the time or the frequency domain. Joint time-frequency representations characterize signals over a time-frequency plane. They thus combine time-domain and frequency-domain analyses to yield a potentially more revealing picture of the temporal localization of a signal's spectral components. Therefore, for air target returns with time-varying frequency content, the joint time-frequency representations offer a powerful analysis tool. A concise review of time-frequency transforms is provided as background and is needed to appreciate how time-frequency processing methods can improve conventional time or frequency processing methods. The report then describes and illustrates the advantages of using joint time-frequency techniques to analyze a multi-component signal, a noisy signal, and experimental aircraft data. Finally, we use time-frequency analysis techniques for the detection of maneuvering aircraft using HF radar in heavily cluttered regions. We compare the ability of different time-frequency transforms to resolve several experimental aircraft returns. The results clearly demonstrate that time-frequency analysis techniques can significantly improve the detection performance of the HF radar and add considerable physical insight over what can be achieved by conventional Fourier transform methods currently used by HF radars.

We consider signals and operators in finite dimension which have sparse time-frequency representations. As main result we show that an $S$-sparse Gabor representation in $\mathbb{C}^n$ with respect to a random unimodular window can be recovered by Basis Pursuit with high probability provided that $S\leq Cn/\log(n)$. Our results are applicable to the channel estimation problem in wireless communications and they establish the usefulness of a class of measurement matrices for compressive sensing.

With the aim of measuring the sparsity of a real signal, Donoho and Logan introduced the concept of maximum Nyquist density, and used it to extend Bombieri's principle of the large sieve to bandlimited functions. This led to several recovery algorithms based on the minimization of the $L_{1}$-norm. In this paper we introduce the concept of {\ planar maximum} Nyquist density, which measures the sparsity of the time-frequency distribution of a function. We obtain a planar large sieve principle which applies to time-frequency representations with a gaussian window, or equivalently, to Fock spaces, $\mathcal{F}_{1}\left( \mathbb{C}\right) $, allowing for perfect recovery of the short-Fourier transform (STFT) of functions in the modulation space $M_{1}$ (also known as Feichtinger's algebra $S_{0}$) corrupted by sparse noise and for approximation of missing STFT data in $M_{1}$, by $L_{1}$-minimization.

Classical and recent results on uncertainty principles for functions on finite Abelian groups relate the cardinality of the support of a function to the cardinality of the support of its Fourier transforms. We use these results and their proofs to obtain similar results relating the support sizes of functions and their short--time Fourier transforms. Further, we discuss applications of our results. For example, we use our results to construct a class of equal norm tight Gabor frames that are maximally robust to erasures and we discuss consequences of our findings to the theory of recovering and storing signals which have sparse time--frequency representations.

The research presented in this report outlines the time-frequency characterization of mammalian click and whistle sounds. Analysis of the group delay structure of the mammalian vocal communication signals was matched to the appropriate quadratic time-frequency class for proper signal processing with minimal skewing of the results. The presented material includes a discussion of the recorded mammalian data recordings from the Woods Hole Oceanographic Institution, and a mathematical introduction into quadratic time-frequency classes, properties, and mathematical structure. This report determines the computational formulations appropriate for analyzing underwater mammalian vocal sounds using quadratic time-frequency representations. Once the proper quadratic time-frequency representations were identified for use in the analysis of clicks and whistles, a large selection of sound files was analyzed to characterize the time-frequency structure. The time duration and bandwidth of the clicks and whistles of various genera of dolphins and whales are presented in this report.

Mobile communication channels are often modeled as linear time-varying filters or, equivalently, as time-frequency integral operators with finite support in time and frequency. Such a characterization inherently assumes the signals are narrowband and may not be appropriate for wideband signals. In this paper time-scale characterizations are examined that are useful in wideband time-varying channels, for which a time-scale integral operator is physically justifiable. A review of these time-frequency and time-scale characterizations is presented. Both the time-frequency and time-scale integral operators have a two-dimensional discrete characterization which motivates the design of time-frequency or time-scale rake receivers. These receivers have taps for both time and frequency (or time and scale) shifts of the transmitted signal. A general theory of these characterizations which generates, as specific cases, the discrete time-frequency and time-scale models is presented here. The interpretation of these models, namely, that they can be seen to arise from processing assumptions on the transmit and receive waveforms is discussed. Out of this discussion a third model arises: a frequency-scale continuous channel model with an associated discrete frequency-scale characterization.

This paper addresses the problem of audio scenes classification and contributes to the state of the art by proposing a novel feature. We build this feature by considering histogram of gradients (HOG) of time-frequency representation of an audio scene. Contrarily to classical audio features like MFCC, we make the hypothesis that histogram of gradients are able to encode some relevant informations in a time-frequency {representation:} namely, the local direction of variation (in time and frequency) of the signal spectral power. In addition, in order to gain more invariance and robustness, histogram of gradients are locally pooled. We have evaluated the relevance of {the novel feature} by comparing its performances with state-of-the-art competitors, on several datasets, including a novel one that we provide, as part of our contribution. This dataset, that we make publicly available, involves $19$ classes and contains about $900$ minutes of audio scene recording. We thus believe that it may be the next standard dataset for evaluating audio scene classification algorithms. Our comparison results clearly show that our HOG-based features outperform its competitors

This dissertation examines the use of linear time frequency representations in the detection and classification of transient signals. In particular, the Gabor transform and short time Fourier transform (STFT) are shown to be effective tools in detecting and classifying signals that are accurately modeled by linear subspaces. Transient signal detection is studied first, within a framework that expresses transient signals as linear combinations of time frequency shifted, one sided exponential window functions. For the case where signal components have known locations in the time-frequency plane, it is shown that a generalized likelihood ratio test (GLRT) detector based on the oversampled Gabor transform is more robust to mismatch than GLRT detectors based on the critically sampled Gabor transform and critically sampled STFT. For the case where signal component locations are not precisely known, it is shown that, for a given transform, a GLRT detector which does not make assumptions about component location information is more robust to component location mismatch than a GLRT detector which does make those assumptions. When the oversampled Gabor transform is used for data reduction, one of its main drawbacks is its lack of stability: small variations in a signal can cause large variations in the magnitudes of the Gabor coefficients. Thus, several modifications designed to improve the stability of the over-sampled Gabor transform has been widely used, and the transform is stable in this form. However, it is shown here that there are several serious problems with the expanded form of the oversampled Gabor transform that make it unsuitable for use in transient signal detection.