Signal design for active sensing

Recent advances in hardware technology across the active sensing spectra, from RF to optical, enable the construction of sophisticated excitation patterns that can be varied across time, space, frequency, wavenumber, and polarization. In the RF band, modern radars are increasingly being equipped with arbitrary waveform generators that allow the transmission of different waveforms across multiple degrees of freedoms simultaneously. The emergence of multiple-input multiple-output (MIMO) phased-array radars, with multiple degrees of freedom at transmitter and receiver, brings the promise of improved surveillance and tracking performance. In the optical band, the advent of spatial light modulators and digital light processing devices allows us to construct structured excitation patterns for illuminating objects.;These advances open up exciting possibilities for design of illumination patterns and signal processing algorithms. In this dissertation, we develop new signal design and processing methods for a subset of active sensing problems in radar and optical imaging. In addition, we exploit the Kalman filter as an efficient signal processing approach for sensing systems with dynamical state update and measurement acquisition.;Radar imaging: Broadly speaking, signal designs for radar imaging can be divided into two categories: designs for desired ambiguity functions and designs for interference rejection. The first category typically involves designing radar waveforms of a given time-bandwidth product, such that their ambiguity functions have narrow mainlobes and small sidelobes in desired regions in the range-Doppler plane. The designs in this category are typically suited for resolving point scatters in white noise. The second category typically considers the joint design of radar transmit and receive filters, to detect or estimate a point or extended target in the presence of interference and clutter.;In conventional designs in both categories, typically, the focus has been on designing a single transmit filter or a single transmit-receive pair to achieve a design goal. But recent hardware advances enable the utilization of banks of transmit-receive filters across multiple degrees of freedom, and give rise to new opportunities for radar waveform design and signal processing. In this dissertation, we take advantage of these new capabilities to develop novel signal design principles for MIMO radar in the first of the aforementioned categories. Our contributions are as follows:;Doppler resilient illuminations: In radar range detection, typically, localization in range is performed by matched filtering the received signal with the transmitted waveform. The output of the matched filter would ideally be an impulse at the desired delay. Therefore, waveforms with impulse-like autocorrelation functions are of great value in these applications. Such waveforms are typically constructed through phase coding a narrow pulse shape with appropriate unimodular codes. Unimodularity is desired due to constraints set by power amplifiers used in radar transmitters. In the absence of Doppler, the near ideal auto-correlation property of such waveforms enables separation of closely-spaced targets in range. However, all phase-coded waveforms are sensitive to Doppler effect; off the zero Doppler axis, the magnitude of range sidelobe of a phase-coded waveform's ambiguity function is typically large. These Doppler-induced range sidelobes can in turn result in masking of a weak target that is located in range near a strong reflector with a different Doppler frequency.;As part of this dissertation, we develop a general framework for designing Doppler re-silient radar illuminations through proper waveform coordination across time, frequency, and aperture. The building blocks of our Doppler resilient illuminations are phase-coded wave- forms constructed from unimodular codes such as Golay complementary codes. We first show that by properly coordinating these complementary waveforms across time, we can annihilate the range sidelobe of the corresponding pulse train's ambiguity function inside a modest Doppler around the zero-Doppler axis. This in turn enables us to extract weak targets that are situated near strong reflectors. However, this Doppler resilience comes at the expense of Doppler response. We characterize the tradeoff between the two for time-coordinated transmissions. We then extend our design to the coordination of complementary waveforms across both time and frequency. The added degrees of freedom for transmission (frequency) allow us to improve Doppler response without reducing resilience to Doppler. Finally, we extend our work to the design of Doppler resilient paraunitary waveform matrices. Construction of paraunitary illuminations has received significant attention from the MIMO radar community. However, all designs suffer from sensitivity to Doppler. Our approach provides a way to maintain the paraunitary property even in the presence of Doppler.;Optical imaging: The invention of charge-coupled devices and two-dimensional arrays revolutionize optical imaging, by shifting the measurement collection paradigm in optics from serial collection of light intensities at the detector plane to the parallel recording of light intensities. This translates to potentially several fold increase in imaging speed in many imaging systems operating in mid infrared to soft X-ray band. However, CCDs are not readily available in the Terahertz to far infrared region, and optical systems in these bands (including confocal microscopes) still rely on single pixel detectors. This has led to investigation of techniques that employ structured illumination to eliminate the need for pixel-wise scanning. A popular approach, based on compressive sensing theory, has been to use random illumination patterns along with sparse reconstruction algorithms to reconstruct the full object intensity from a small number of intensity measurements. This approach has been mostly investigate under ideal imaging conditions, with no or very little optical aberrations.;As part of this dissertation, we investigate the viability of such compressive sensing approaches for high resolution optical microscopy under more practical conditions. In particular, we analyze the sensitivity of compressive optical microscopy to misfcous effects, which are inevitable in imaging most specimens. Our analysis indicates that compressive imaging is highly sensitive to misfcous effects at high magnifications factors, which are typical in microscopy.;Kalman filtering: In the past few decades the invention of Kalman filters (KFs) and their variations has led to improved adaptive signal processing performance of various sensing systems whose state evolution and measurement acquisition can be characterized by dynamical linear model equations. valued state and measurements.;Complex-valued signals are ubiquitous in science and engineering. A random complex-valued vector x is said to be improper, if x is correlated to its complex conjugate, i.e., the complementary covariance ExxT is nonzero. The impropriety of complex-valued sig-nal exists in many application including communication, smart grid, optical imaging, and acoustic imaging. Since conventional statistical signal processing essentially treats complex-valued signals as real-valued signals and ignores the complementary covariances, it becomes necessary to revisit the theory. Compared to conventional strictly linear processing, the widely linear processing is proven to be an efficient signal processing technique for resolving improper complex signals.;As part of this dissertation, we are motivated to make use of widely linear processing to develop novel complex KFs and their nonlinear versions for improper complex states. We show that complementary covariance of improper states may be used to develop widely linear complex KFs (WLCKFs) and Unscented WLCKFs. We show that, compared to the conventional complex KFs and Unscented complex KFs which ignore the complementary covariances, the WLCKFs and Unscented WLCKFs can significantly reduce the mean square error of state estimation, by utilizing full first and second order statistical information of improper complex states.