No abstract

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.

Previously investigated multicomponent AM-FM demodulation techniques either assume that the individual component signals are spectrally isolated from each other or that the components can be isolated by linear time-invariant filtering techniques and, consequently, break down in the case where the components overlap spectrally or when one of the components is stronger than the other. In this paper, we present a nonlinear algorithm for the separation and demodulation of discrete-time multicomponent AM-FM signals. Our approach divides the demodulation problem into two independent tasks: algebraic separation of the components based on periodicity assumptions and then monocomponent demodulation of each component by instantaneously tracking and separating its source energy into its amplitude and frequency parts. The proposed new algorithm avoids the shortcomings of previous approaches and works well for extremely small spectral separations of the components and for a wide range of relative amplitude/power ratios. We present its theoretical analysis and experimental results and outline its application to demodulation of cochannel FM voice signals.

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 Grünbaum 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.Index Terms-Chirp rate estimation, discrete Fourier transform (DFT), discrete fractional Fourier transform (DFRFT), eigenvalues, eigenvectors, fast Fourier transform (FFT), fractional matrix power, Hermite-Gauss functions, multicomponent chirp signals.

The problem of furnishing an orthogonal basis of eigenvectors for the discrete Fourier transform (DFT) is fundamental to signal processing. Recent developments in the area of discrete fractional Fourier analysis also rely upon the ability to furnish a basis of eigenvectors for the DFT or its centralized version. 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 operator. Furthermore, any linear combination of commuting matrices produced by existing approaches also commutes with the DFT, thereby bringing up the question 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 also furnishes a commuting matrix whose eigenvalue spectrum is a very close approximation to that of the Gauss-Hermite differential operator and consequently a framework for a unique definition of the discrete Gauss-Hermite operator.