A note on the angular Fourier transformation

It is demonstrated that an angular Fourier transformation is obtained by making a rotation around the non-compact axis of So(2,1), the Lorentz group in three dimensions

Clutter Suppression via Hankel Rank Reduction for DFrFT-Based Vibrometry Applied to SAR

Hankel rank reduction (HRR) is a method that, by prearranging the data in a Hankel matrix and performing rank reduction via singular value decomposition, suppresses the noise of a time-history vector comprised of the superposition of a finite number of sinusoids. In this letter, the HRR method is studied for performing clutter suppression in synthetic aperture radar (SAR)-based vibrometry. Specifically, three different applications of the HRR method are presented. First, resembling the SAR slow-time signal model, the HRR method is utilized for separating a chirp signal immersed in a sinusoidal clutter. Second, using simulated airborne SAR data with 10 dB of signal-to-clutter ratio, the HRR method is applied to perform target isolation and to improve the results of an SAR-based vibration estimation algorithm. Finally, the vibrometry approach combined with the HRR method is validated using actual airborne SAR data

On the Grunbaum Commutor Based Discrete Fractional Fourier Transform,”

ABSTRACT 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 timefrequency content. Efforts to develop a discrete 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 GaussHermite functions. Analysis of the DFRFT obtained from Grunbaum's 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

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

Reduction of Vibration-Induced Artifacts in Synthetic Aperture Radar Imagery

Target vibrations introduce nonstationary phase modulation, which is termed the micro-Doppler effect, into returned synthetic aperture radar (SAR) signals. This causes artifacts, or ghost targets, which appear near vibrating targets in reconstructed SAR images. Recently, a vibration estimation method based on the discrete fractional Fourier transform (DFrFT) has been developed. This method is capable of estimating the instantaneous vibration accelerations and vibration frequencies. In this paper, a deghosting method for vibrating targets in SAR images is proposed. For single-component vibrations, this method first exploits the estimation results provided by the DFrFT-based vibration estimation method to reconstruct the instantaneous vibration displacements. A reference signal, whose phase is modulated by the estimated vibration displacements, is then synthesized to compensate for the vibration-induced phase modulation in returned SAR signals before forming the SAR image. The performance of the proposed method with respect to the signal-to-noise and signalto-clutter ratios is analyzed using simulations. Experimental results using the Lynx SAR system show a substantial reduction in ghosting caused by a 1.5-cm 0.8-Hz target vibration in a true SAR image

On a Sturm-Liouville Framework for Continuous and Discrete Frequency Modulation

Abstract-It is well known that purely sinusoidal signals satisfy a linear second-order constant coefficient differential equation. It is also well known that a broad class of orthogonal special functions such as the Legendre and Hermite polynomials satisfy the second-order Sturm-Liouville differential equation. Both sinusoidal and AM-FM models have been used for analysis and synthesis of speech signals. In this paper, we present a SturmLiouville differential and difference equation approach to both continuous and discrete time frequency modulation. Orthogonal modes of frequency modulation that are not distorted by the Sturm-Liouville operator are described

Wideband partial response CPM demodulation via multirate frequency transformations and decision feedback equalization

Abstract ■■■ Continuous phase modulation (CPM) is a popular frequency modulation technique used in mobile communications due to its power efficiency and constant modulus properties. Conventional narrowband CPM demodulation employs the Viterbi algorithm after phase demodulation and requires that the phase states be rational and contain additive white noise. The complexity of the Viterbi approach further increases with the number of phase states. Frequency discrimination approaches that estimate the instantaneous frequency provide a simpler suboptimal approach but are primarily for full response CPM and are not well known for wideband partial response CPM. In this paper, we investigate an approach that combines multirate frequency transformations for wideband CPM demodulation with decision feedback equalization for memory removal. This combined approach avoids the problems of complexity and restrictive requirements of the Viterbi approach. Simulation results are used to demonstrate the validity of the combined approach

An Improved Spectrogram Using the Multiangle Centered Discrete Fractional Fourier Transform,” IEEE-ICASSP conference

ABSTRACT The spectrogram is a useful time-frequency analysis tool for non stationary signal analysis. This tool however, is based upon a multicomponent sinusoidal model over a signal analysis frame and is not suitable when for example the frequency content over the frame is chirping. Recently the centered version of the discrete Fractional Fourier Transform was shown to possess the capability to concentrate a linear chirp signal in a few transform coefficients. In this paper, we present a modified version of the spectrogram that incorporates the centered discrete Fractional Fourier transform and its multiangle version that instead decomposes the signal over the analysis frame into multiple chirp signals. Simulation results that study the efficiency of this improved spectrogram and its application to the analysis of harmonically related chirps and bat echolocation signals are presented

Scene Estimation from Speckled Synthetic Aperture Radar Imagery: Markov-Random-Field Approach

A novel Markov-random-field model for speckled synthetic aperture radar (SAR) imagery is derived according to the physical, spatial statistical properties of speckle noise in coherent imaging. A convex Gibbs energy function for speckled images is derived and utilized to perform speckle-compensating image estimation. The image estimation is formed by computing the conditional expectation of the noisy image at each pixel given its neighbors, which is further expressed in terms of the derived Gibbs energy function. The efficacy of the proposed technique, in terms of reducing speckle noise while preserving spatial resolution, is studied by using both real and simulated SAR imagery. Using a number of commonly used metrics, the performance of the proposed technique is shown to surpass that of existing speckle-noise-filtering methods such as the Gamma MAP, the modified Lee, and the enhanced Frost

Speckle Reduction of SAR Images Based on a Combined Markov Random Field and Statistical Optics Approach (Version1)

One of the major factors plaguing the performance of synthetic aperture radar (SAR) imagery is the presence of signal-dependent, speckle noise. Grainy in appearance, speckle noise is primarily due to the phase fluctuations of the electromagnetic returned signals. Since the inherent spatial-correlation characteristics of speckle in SAR images are not exploited in existing multiplicative models for speckle noise, a new approach is proposed here that provides a new mathematical framework for modeling and mitigation of speckle noise. The contribution of this paper is twofold. First, a novel model for speckled SAR imaging is introduced based on Markov random fields (MRFs) in conjunction with statistical optics. Second, utilizing the model, a global energy-minimization algorithm based on simulated annealing (SA), is introduced for speckle reduction. In particular, the joint conditional probability density function (cpdf) of the intensity of any two points in the speckled image and the associated correlation function are used to derive the cpdf of any center pixel intensity given its four neighbors. The Hammersley-Clifford theorem is then used to derive the energy function associated with the MRF. The SA, built on the Metropolis sampler, is employed for speckle reduction. Four metrics are used to assess the quality of the speckle reduction: the mean-square error, SNR, an edge-preservation parameter and the equivalent number of looks. A comparative study using both simulations and real SAR images indicates that the proposed approach performs better in comparison to filtering techniques such as the Gamm Map, the modified Lee and the enhanced Frost algorithms