Time-frequency Signature Sparse Reconstruction using Chirp Dictionary

This paper considers local sparse reconstruction of time-frequency signatures of windowed non-stationary radar returns. These signals can be considered instantaneously narrow-band, thus the local time-frequency behaviour can be recovered accurately with incomplete observations. The typically employed sinusoidal dictionary induces competing requirements on window length. It confronts converse requests on the number of measurements for exact recovery, and sparsity. In this paper, we use chirp dictionary for each window position to determine the signal instantaneous frequency laws. This approach can considerably mitigate the problems of sinusoidal dictionary, and enable the utilization of longer windows for accurate time-frequency representations. It also reduces the picket fence by introducing a new factor, the chirp rate . Simulation examples are provided, demonstrating the superior performance of local chirp dictionary over its sinusoidal counterpart

Super-Resolution Time of Arrival Estimation Using Random Resampling in Compressed Sensing

There is a strong demand for super-resolution time of arrival (TOA) estimation techniques for radar applications that can that can exceed the theoretical limits on range resolution set by frequency bandwidth. One of the most promising solutions is the use of compressed sensing (CS) algorithms, which assume only the sparseness of the target distribution but can achieve super-resolution. To preserve the reconstruction accuracy of CS under highly correlated and noisy conditions, we introduce a random resampling approach to process the received signal and thus reduce the coherent index, where the frequency-domain-based CS algorithm is used as noise reduction preprocessing. Numerical simulations demonstrate that our proposed method can achieve super-resolution TOA estimation performance not possible with conventional CS methods

Sparse Reconstruction of Time-Frequency Representation using the Fractional Fourier Transform

This paper describes a novel method to approximate instantaneous frequency of non-stationary signals through an application of fractional Fourier transform (FRFT). FRFT enables us to build a compact and accurate chirp dictionary for each windowed signal, thus the proposed approach offers improved computational efficiency, and good performance when compared with chirp atom method

Super-Resolution in Phase Space

This work considers the problem of super-resolution. The goal is to resolve a Dirac distribution from knowledge of its discrete, low-pass, Fourier measurements. Classically, such problems have been dealt with parameter estimation methods. Recently, it has been shown that convex-optimization based formulations facilitate a continuous time solution to the super-resolution problem. Here we treat super-resolution from low-pass measurements in Phase Space. The Phase Space transformation parametrically generalizes a number of well known unitary mappings such as the Fractional Fourier, Fresnel, Laplace and Fourier transforms. Consequently, our work provides a general super- resolution strategy which is backward compatible with the usual Fourier domain result. We consider low-pass measurements of Dirac distributions in Phase Space and show that the super-resolution problem can be cast as Total Variation minimization. Remarkably, even though are setting is quite general, the bounds on the minimum separation distance of Dirac distributions is comparable to existing methods.Comment: 10 Pages, short paper in part accepted to ICASSP 201

Time-Frequency Distributions: Approaches for Incomplete Non-Stationary Signals

There are many sources of waveforms or signals existing around us. They can be natural phenomena such as sound, light and invisible like elec- tromagnetic fields, voltage, etc. Getting an insight into these waveforms helps explain the mysteries surrounding our world and the signal spec- tral analysis (i.e. the Fourier transform) is one of the most significant approaches to analyze a signal. Nevertheless, Fourier analysis cannot provide a time-dependent spectrum description for spectrum-varying signals-non-stationary signal. In these cases, time-frequency distribu- tions are employed instead of the traditional Fourier transform. There have been a variety of methods proposed to obtain the time-frequency representations (TFRs) such as the spectrogram or the Wigner-Ville dis- tribution. The time-frequency distributions (TFDs), indeed, offer us a better signal interpretation in a two-dimensional time-frequency plane, which the Fourier transform fails to give. Nevertheless, in the case of incomplete data, the time-frequency displays are obscured by artifacts, and become highly noisy. Therefore, signal time-frequency features are hardly extracted, and cannot be used for further data processing. In this thesis, we propose two methods to deal with compressed observations. The first one applies compressive sensing with a novel chirp dictionary. This method assumes any windowed signal can be approximated by a sum of chirps, and then performs sparse reconstruction from windowed data in the time domain. A few improvements in computational com- plexity are also included. In the second method, fixed kernel as well as adaptive optimal kernels are used. This work is also based on the as- sumption that any windowed signal can be approximately represented by a sum of chirps. Since any chirp ’s auto-terms only occupy a certain area in the ambiguity domain, the kernel can be designed in a way to remove the other regions where auto-terms do not reside. In this manner, not only cross-terms but also missing samples’ artifact are mitigated signifi- cantly. The two proposed approaches bring about a better performance in the time-frequency signature estimations of the signals, which are sim- ulated with both synthetic and real signals. Notice that in this thesis, we only consider the non-stationary signals with frequency changing slowly with time. It is because the signals with rapidly varying frequency are not sparse in time-frequency domain and then the compressive sensing techniques or sparse reconstructions could not be applied. Also, the data with random missing samples are obtained by randomly choosing the samples’ positions and replacing these samples with zeros

The Discrete Linear Chirp Transform and its Applications

In many applications in signal processing, the discrete Fourier transform (DFT) plays a significant role in analyzing characteristics of stationary signals in the frequency domain. The DFT can be implemented in a very efficient way using the fast Fourier transform (FFT) algorithm. However, many actual signals by their nature are non--stationary signals which make the choice of the DFT to deal with such signals not appropriate. Alternative tools for analyzing non--stationary signals come with the development of time--frequency distributions (TFD). The Wigner--Ville distribution is a time--frequency distribution that represents linear chirps in an ideal way, but it has the problem of cross--terms which makes the analysis of such tools unacceptable for multi--component signals. In this dissertation, we develop three definitions of linear chirp transforms which are: the continuous linear chirp transform (CLCT), the discrete linear chirp transform (DLCT), and the discrete cosine chirp transform (DCCT). Most of this work focuses on the discrete linear chirp transform (DLCT) which can be considered a generalization of the DFT to analyze non--stationary signals. The DLCT is a joint frequency chirp--rate transformation, capable of locally representing signals in terms of linear chirps. Important properties of this transform are discussed and explored. The efficient implementation of the DLCT is given by taking advantage of the FFT algorithm. Since this novel transform can be implemented in a fast and efficient way, this would make the proposed transform a candidate to be used for many applications, including chirp rate estimation, signal compression, filtering, signal separation, elimination of the cross--terms in the Wigner--Ville distribution, and in communication systems. In this dissertation, we will explore some of these applications

Local Sparse Reconstructions of Doppler Frequency using Chirp Atoms

The paper considers sparse reconstruction of Doppler and microDoppler time-frequency (TF) signatures of radar returns of moving targets from limited or incomplete data. The typically employed sinusoidal dictionary, relating the windowed compressed measurements to the signal local frequency contents, induces competing requirements on the window size. In this paper, we use chirp dictionary for each window position to relax this adverse window length-sparsity interlocking. It is shown that local frequency reconstruction using chirp atoms better represents the approximate piece-wise chirp behavior of most Doppler TF signatures. This enables the utilization of longer windows for accurate time-frequency representations. Simulation examples are provided demonstrating the superior performance of local chirp dictionary over its sinusoidal counterpart