Balu Santhanam

The discrete rotational Fourier transform

We define a discrete version of the angular Fourier transform and present the properties of the transform that show it to be a rotation in time-frequency space, this new transform is a generalization of the DFT. Efficient algorithms for its computation can then be based on the FFT and the eigenstructure of the DFT.

On the multiangle centered discrete fractional Fourier transform

Existing versions of the discrete fractional Fourier transform (DFRFT) are based on the discrete Fourier transform (DFT). These approaches need a full basis of DFT eigenvectors that serve as discrete versions of Hermite-Gauss functions. In this letter, we define a DFRFT based on a centered version of the DFT (CDFRFT) using eigenvectors derived from the Gru/spl uml/nbaum tridiagonal commutor that serve as excellent discrete approximations to the Hermite-Gauss functions. We develop a fast and efficient way to compute the multiangle version of the CDFRFT for a discrete set of angles using the FFT algorithm. We then show that the associated chirp-frequency representation is a useful analysis tool for multicomponent chirp signals.

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.

An improved spectrogram using the multiangle centered discrete fractional Fourier transform

The spectrogram is a useful tool for the time-frequency analysis of non stationary signals. 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. We present a modified version of the spectrogram which incorporates the centered discrete fractional Fourier transform and its multiangle version and which instead decomposes the signal over the analysis frame into multiple chirp signals. We present simulation results that study the efficiency of this improved spectrogram and its application to the analysis of harmonically related chirps and bat echolocation signals.

On discrete Gauss-Hermite functions and eigenvectors of the discrete Fourier transform

The problem of furnishing an orthogonal basis of eigenvectors for the discrete Fourier transform (DFT) is fundamental to signal processing and also a key step in the recent development of discrete fractional Fourier transforms with projected applications in data multiplexing, compression, and hiding. Existing solutions toward furnishing this basis of DFT eigenvectors are based on the commuting matrix framework. However, none of the existing approaches are able to furnish a commuting matrix where both the eigenvalue spectrum and the eigenvectors are a close match to corresponding properties of the continuous differential Gauss-Hermite (G-H) operator. Furthermore, any linear combination of commuting matrices produced by existing approaches also commutes with the DFT, thereby bringing up issues of uniqueness. In this paper, inspired by concepts from quantum mechanics in finite dimensions, we present an approach that furnishes a basis of orthogonal eigenvectors for both versions of the DFT. This approach furnishes a commuting matrix whose eigenvalue spectrum is a very close approximation to that of the G-H differential operator and in the process furnishes two generators of the group of matrices that commute with the DFT.

The DRFT-a rotation in time-frequency space

The continuous-time angular Fourier transformation (AFT) represents a rotation in continuous time-frequency space and also serves as an orthonormal signal representation for chirp signals. We present a discrete version of the AFT (DRFT) that represents a rotation in discrete time-frequency space and some properties of the transform that support its interpretation as a rotation. The transform is a generalization of the DFT. The eigenvalue structure of the DFT is then exploited to develop an efficient algorithm for the computation of this transform.

Segmentation of SAR images based on Markov random field model

In synthetic-aperture-radar (SAR) imaging, large volumes of data are normally processed and transported over airborne or space-based platforms. The development of fast and robust algorithms for processing and analysis of this type of data is therefore of great importance. It has been demonstrated recently that a Markov-random-field (MRF) model, based on the statistical properties of coherent imaging, provides an ideal framework to describe the spatial correlation within SAR imagery in the presence of speckle noise, which is present in all SAR imagery. When combined with Gibbs-energy-minimization techniques, the MRF-framework has also led to the development of effective and efficient speckle-reducing image restoration algorithms. In this work, the convexity of the Gibbs energy function for SAR imagery is established thereby facilitating the development of a novel image segmentation algorithm for speckled SAR imagery. The segmentation algorithm is too based on minimizing the Gibbs energy function, which is attained without the need for computationally intensive global optimization techniques such as simulated annealing. A comparative experimental analysis, using real SAR imagery, of the proposed segmentation algorithm against a statistical-thresholding approach is undertaken showing the advantage of the proposed approach in the presence of the speckle noise. Notably, unlike the thresholding technique, the proposed algorithm can be applied to speckled imagery directly without the need for preprocessing the imagery for speckle-noise reduction.

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.

Discrete Gauss-Hermite Functions and Eigenvectors of the Centered Discrete Fourier Transform

Existing approaches to furnishing a basis of eigenvectors for the discrete Fourier transform (DFT) are based upon defining tridiagonal operators that commute with the DFT. In this paper, motivated by ideas from quantum mechanics in finite dimensions, we define a symmetric matrix that commutes with the centered DFT, thereby furnishing a basis of eigenvectors for the DFT. We show that these eigenvectors in the limit converge to Gauss-Hermite (G-H) functions and that the eigenvalue spectrum of the commutor provides a very good discrete approximation to that of the continuous G-H differential operator.

ICA based blind adaptive MAI suppression in DS-CDMA systems

The performance of existing multiuser detectors is dependent on the available information about the structure of multiple access interference (MAI). The decorrelating detector requires complete knowledge of MAI. The minimum output energy (MOE) detector is a semiblind approach which exploits the information about the desired user only. The performance of the MOE detector is suboptimal compared to that of the decorrelator. The proposed approach is a semiblind code constrained-independent component analysis (CC-ICA) based approach that exploits the prior information about the signature code of the desired user to constrain the ICA vector to lie in the orthogonal complement of the interfering users data. Simulation results indicate that the performance of the proposed approach approaches that of the decorrelator using much less prior information.