Igor Djurovic

Fractional Fourier transform as a signal processing tool: An overview of recent developments

Fractional Fourier transform (FRFT) is a generalization of the Fourier transform, rediscovered many times over the past 100 years. In this paper, we provide an overview of recent contributions pertaining to the FRFT. Specifically, the paper is geared toward signal processing practitioners by emphasizing the practical digital realizations and applications of the FRFT. It discusses three major topics. First, the manuscripts relates the FRFT to other mathematical transforms. Second, it discusses various approaches for practical realizations of the FRFT. Third, we overview the practical applications of the FRFT. From these discussions, we can clearly state that the FRFT is closely related to other mathematical transforms, such as time-frequency and linear canonical transforms. Nevertheless, we still feel that major contributions are expected in the field of the digital realizations and its applications, especially, since many digital realizations of the FRFT still lack properties of the continuous FRFT. Overall, the FRFT is a valuable signal processing tool. Its practical applications are expected to grow significantly in years to come, given that the FRFT offers many advantages over the traditional Fourier analysis.

Watermarking in the space/spatial-frequency domain using two-dimensional Radon-Wigner distribution

A two-dimensional (2-D) signal with a variable spatial frequency is proposed as a watermark in the spatial domain. This watermark is characterized by a linear frequency change. It can be efficiently detected by using 2-D space/spatial-frequency distributions. The projections of the 2-D Wigner distribution--the 2-D Radon-Wigner distribution, are used in order to emphasize the watermark detection process. The watermark robustness with respect to some very important image processing attacks, such as for example, the translation, rotation, cropping, JPEG compression, and filtering, is demonstrated and tested by using Stirmark 3.1.

Digital watermarking in the fractional Fourier transformation domain

An application of the fractional Fourier transform for the multimedia copyright protection is proposed in the paper. The watermark robustness as well as statistical performance are considered.

Separation of target rigid body and micro-doppler effects in ISAR imaging

Micro-Doppler (m-D) effect is caused by moving parts of the radar target. It can cover rigid parts of a target and degrade the inverse synthetic aperture radar (ISAR) image. Separation of the patterns caused by stationary parts of the target from those caused by moving (rotating or vibrating) parts is the topic of this paper. Two techniques for separation of the rigid part from the rotating parts have been proposed. The first technique is based on time-frequency (TF) representation with sliding window and order statistics techniques. The first step in this technique is recognition of rigid parts in the range/cross-range plane. In the second step, reviewed TF representation and order statistics setup are employed to obtain signals caused by moving parts. The second technique can be applied in the case of very emphatic m-D effect. In the first step the rotating parts are recognized, based on the inverse Radon transform (RT). After masking these patterns, a radar image with the rigid body reflection can be obtained. The proposed methods are illustrated by examples

Micro-Doppler-based target detection and feature extraction in indoor and outdoor environments

Abstract In many cases, a target or a structure on a target may have micro-motions, such as vibrations or rotations. Micro-motions of structures on a target may introduce frequency modulation on the returned radar signal and generate sidebands on the Doppler frequency shift of the targets body. The modulation due to micro-motion is called the micro-Doppler (m-D) phenomenon. In this paper, we present an effective quadratic time–frequency S-method-based approach in conjunction with the Viterbi algorithm to extract m-D features. For target recognition applications, mainly those in military surveillance and reconnaissance operations, m-D features have to be extracted quickly so that they can be used for real-time target identification. The S-method is computationally simple, requiring only slight modifications to the existing Fourier transform-based algorithm. The effectiveness of the S-method in extracting m-D features is demonstrated through the application to indoor and outdoor experimental data sets such as rotating fan and human gait. The Viterbi algorithm for the instantaneous frequency estimation is used to enhance the weak human m-D features in relatively high noise environments. As such, this paper contributes additional experimental m-D data and analysis, which should help in developing a better picture of the human gait m-D research and its applications to indoor and outdoor imaging and automatic gait recognition systems.

Integrated Cubic Phase Function for Linear FM Signal Analysis

In this paper, an integrated cubic phase function (ICPF) is introduced for the estimation and detection of linear frequency-modulated (LFM) signals. The ICPF extends the standard cubic phase function (CPF) to handle cases involving low signal-to-noise ratio (SNR) and multi-component LFM signals. The asymptotic mean squared error (MSE) of an ICPF-based estimator as well as the output SNR of an ICPF-based detector are derived in closed form and verified by computer simulation. Comparison with several existing approaches is also included, which shows that the ICPF serves as a good candidate for LFM signal analysis.

An algorithm for the Wigner distribution based instantaneous frequency estimation in a high noise environment

Estimation of the instantaneous frequency (IF) in a high noise environment, by using the Wigner distribution (WD), is considered. In this case the error is of impulse nature. An algorithm for the IF estimation, which combines the nonparametric method based on the WD maxima with the minimization of the IF variations between consecutive points, is proposed. The off-line and on-line realizations are presented. The on-line realization is an instance of the (generalized) Viterbi algorithm. Application of this algorithm on the monocomponent and multicomponent frequency modulated signals is demonstrated. For multicomponent signals, the algorithm is applied on other (reduced interference) distributions. Numerical examples, including statistical study of the algorithm performance, are given.

Robust L-estimation based forms of signal transforms and time-frequency representations

The L-estimation based signal transforms and time-frequency (TF) representations are introduced by considering the corresponding minimization problems in the Huber (1981, 1998) estimation theory. The standard signal transforms follow as the maximum likelihood solutions for the Gaussian additive noise environment. For signals corrupted by an impulse noise, the median-based transforms produce robust estimates of the non-noisy signal transforms. When the input noise is a mixture of Gaussian and impulse noise, the L-estimation-based signal transforms can outperform other estimates. In quadratic and higher order TF analysis, the resulting noise is inherently a mixture of the Gaussian input noise and an impulse noise component. In this case, the L-estimation-based signal representations can produce the best results. These transforms and TF representations give the standard and the median-based forms as special cases. A procedure for parameter selection in the L-estimation is proposed. The theory is illustrated and checked numerically.

A Hybrid CPF-HAF Estimation of Polynomial-Phase Signals: Detailed Statistical Analysis

In this paper, we consider parameter estimation of high-order polynomial-phase signals (PPSs). We propose an approach that combines the cubic phase function (CPF) and the high-order ambiguity function (HAF), and is referred to as the hybrid CPF-HAF method. In the proposed method, the phase differentiation is first applied on the observed PPS to produce a cubic phase signal, whose parameters are, in turn, estimated by the CPF. The performance analysis, carried out in the paper, considers up to the tenth-order PPSs, and is supported by numerical examples revealing that the proposed approach outperforms the HAF in terms of the accuracy and signal-to-noise-ratio threshold. Extensions to multicomponent and multidimensional PPSs are also considered, all supported by numerical examples. Specifically, when multicomponent PPSs are considered, the product version of the CPF-HAF outperforms the product HAF (PHAF) that fails to estimate parameters of components whose PPS order exceeds three.

Generalized High-Order Phase Function for Parameter Estimation of Polynomial Phase Signal

The high-order phase function (HPF) has been introduced recently to estimate the parameters of a polynomial phase signal (PPS). In this correspondence, we generalize the standard HPF by introducing multiple time instants. Thus, the standard HPF can be treated as a special example of the generalized HPF with identical time instants. We propose a procedure for finding time instants minimizing the mean-square error (MSE). The proposed method achieves better performances than the high-order ambiguity function (HAF) and polynomial Wigner-Ville distribution (PWVD). The theoretical analysis as well as the Monte Carlo simulations verify the advantages such as lower MSE and lower SNR threshold for the PPS.