A fundamental assumption when applying Synthetic Aperture Radar (SAR) to a ground scene is that all targets are motionless. If a target is not stationary, but instead vibrating in the scene, it will introduce a non-stationary phase modulation, termed the micro-Doppler effect, into the returned SAR signals. Previously, the authors proposed a pseudosubspace method, a modification to the Discrete Fractional Fourier Transform (DFRFT), which demonstrated success for estimating the instantaneous accelerations of vibrating objects. However, this method may not yield reliable results when clutter in the SAR image is strong. Simulations and experimental results have shown that the DFRFT method can yield reliable results when the signal-to-clutter ratio (SCR) > 8 dB. Here, we provide the capability to determine a target's frequency and amplitude in a low SCR environment by presenting two methods that can perform vibration estimations when SCR < 3 dB. The first method is a variation and continuation of the subspace approach proposed previously in conjunction with the DFRFT. In the second method, we employ the dual-beam SAR collection architecture combined with the extended Kalman filter (EKF) to extract information from the returned SAR signals about the vibrating target. We also show the potential for extending this SAR-based capability to remotely detect and classify objects housed inside buildings or other cover based on knowing the location of vibrations as well as the vibration histories of the vibrating structures that house the vibrating objects.
In synthetic-aperture radar (SAR) returned signals, ground-target vibrations introduce a phase modulation that
is linearly proportional to the vibration displacement. Such modulation, termed the micro-Doppler effect, introduces
ghost targets along the azimuth direction in reconstructed SAR images that prevents SAR from forming
focused images of the vibrating targets. Recently, a discrete fractional Fourier transform (DFrFT) based method
was developed to estimate the vibration frequencies and instantaneous vibration accelerations of the vibrating
targets from SAR returned signals. In this paper, a demodulation-based algorithm is proposed to reconstruct
focused SAR images of vibrating targets by exploiting the estimation results of the DFrFT-based vibration
estimation method. For a single-component harmonic vibration, the history of the vibration displacement is
first estimated from the estimated vibration frequency and the instantaneous vibration accelerations. Then a
reference signal whose phase is modulated by the estimated vibration displacement with a delay of 180 degree is
constructed. After that, the SAR phase history from the vibration target is multiplied by the reference signal and
the vibration-induced phase modulation is canceled. Finally, the SAR image containing the re-focused vibration
target is obtained by applying the 2-D Fourier transform to the demodulated SAR phase history. This algorithm
is applied to simulated SAR data and successfully reconstructs the SAR image containing the re-focused
In synthetic-aperture radar (SAR), ground-target vibrations introduce a phase modulation in the returned signals,
a phenomenon often referred to as the micro-Doppler effect. Earlier work has shown that the problem of
estimating common ground-target vibrations can be transformed into the problem of successively estimating
chirp parameters of the returned signal in properly sized subapertures. Recently, a method based on the discrete
fractional Fourier transform (DFRFT) was proposed, in conjunction with the subaperture framework, to estimate
target vibrations in the absence of noise. In this paper a pseudo-subspace approach is employed to extend the
applicability of the DFRFT-based vibration-estimation method to signals that are corrupted by white noise.
The new algorithm first calculates the inverse discrete Fourier transform of row and column projections of
the magnitude of the DFRFT spectrum of the SAR returned signal to obtain two vectors. Next, covariance
matrices are estimated from the sample covariance matrices of the two vectors. A pseudo-subspace approach is
then applied to the covariance matrices to yield the pseudo-spectra. The chirp rate of the signal is estimated
by finding the principle frequency component in the corresponding pseudo-spectrum. Monte-Carlo simulations
demonstrate that the proposed method generally offers improved mean-square-error performance in the presence
of noise compared to the direct DFRFT-based method.
Recent reports on the effects of vibrating targets on synthetic-aperture radar (SAR) imagery and the potential
of SAR to extract non-stationary signatures have drawn significant interest from the remote-sensing community.
SAR returned signals are the superposition of the transmitted pulses modulated by both static and non-static
targets in both amplitude and phase. More precisely, the vibration of a target causes a small sinusoid-like frequency
modulation along the synthetic aperture (slow time), whereby the phase deviation is proportional to
the displacement of the vibrating object. By looking at successive small segments in slow time, each frequency
modulated pulse can be tracked and further approximated as a piecewise-linear frequency-modulated signal. The
discrete-time fractional Fourier transform (DFRFT) is an analysis tool geared toward such signals containing linear
frequency modulated components. Within each segment, the DFRFT transforms each frequency-modulated
component into a peak in the DFRFT plane, and the peak position corresponds to the frequency modulation rate.
A series of such measurements provides the instantaneous-acceleration history and its spectrum bears the vibrating
signature of the target. Additionally, when the chirp z-transform (CZT) is incorporated into the DFRFT,
vibration-induced modulations can be identified with high resolution. In this work, the interplay amongst SAR
system parameters, vibration parameters, the DFRFT's window size, and the CZT's zoom-in factor is characterized
analytically for the proposed SAR-vibrometry approach. Simulations verify the analysis showing that the
detection of vibration using the slow-time approach has significantly higher fidelity than that of the previously
reported fast-time approach.
A novel signal-processing approach is reported for vibrometry in synthetic aperture radar (SAR) imaging systems.
The approach exploits the conventional deramp process; however, in place of Fourier-transform processing we
utilize the fractional Fourier transform (FRFT) as a processing tool. The FRFT is geared toward non-stationary
signals and chirped sinusoids particularly. A simplified mathematical expression is developed to describe the
reflectivity of the aimed patch of ground containing vibrating targets as a function of space and time. Under the
approximation that the velocities of vibrating point targets are constant during each probing chirped pulse, it is
shown that the returned echo after the deramp process is a superposition of sinusoids that are chirped according to
the Doppler effects induced by the vibrating point targets. By applying the multiangle centered discrete fractional
Fourier transform (MA-CDFRFT) to the demodulated echoes, the instantaneous velocities of the vibrating point
targets are estimated from the two coordinates of each peak in the MA-CDFRFT's frequency-angle plane. By
repeating this process where a sequence of successive pulses are used to interrogate the vibrating targets, the
velocities of the targets are estimated in each pulse, thereby generating a piecewise-linear estimate of the history
of the vibration velocity in time. Theoretical performance evaluation of the proposed technique is carried out
using real SAR-system parameters and simulated vibrating targets. The interplay amongst minimum detectable
velocity, maximum detectable vibration frequency, pulse duration and chirp rate is determined analytically.