Non-convex optimization techniques for radar waveform design under practical constraints
Open Access
- Author:
- Al Hujaili, Khaled
- Graduate Program:
- Electrical Engineering
- Degree:
- Doctor of Philosophy
- Document Type:
- Dissertation
- Date of Defense:
- February 10, 2020
- Committee Members:
- Vishal Monga, Dissertation Advisor/Co-Advisor
Vishal Monga, Committee Chair/Co-Chair
John F Doherty, Committee Member
Julio Urbina, Committee Member
Hosam Fathy, Outside Member
Kultegin Aydin, Program Head/Chair - Keywords:
- MIMO radar
wideband beampattern
waveform design
constant modulus
complex circle manifold
manifolds
cognitive radar
PDR
Ambiguity function
QGD
Spectral Constraint - Abstract:
- Radar technology has improved significantly over the last few decades. One of the key aspects of improvement is enhanced transmission technology. Adaptive transmission, usually referred to in the literature as the waveform design, denotes the ability of the radar system to alter its transmit waveform in response to many factors such as the changes in its surrounding environment. The transmit waveform is one of many fundamental features of radar systems that influences many of the radar performance criteria. It has been shown in the literature that a proper waveform selection enables the radar to improve the signal-to-interference-plus-noise ratio (SINR), obtain good range and Doppler resolutions, and efficiently utilize the signal energy in spatial and frequency domains. Along with this selection of the transmit waveform to fulfill the improvement in the performance of radar systems, some critical characteristics of the waveform should be considered to accommodate hardware requirements. In the open literature, the waveform design process boils down to an optimization problem that consists of 1) an objective function that represents the targeted performance criteria, and 2) a set of constraints that represent the aforementioned characteristics of the waveform. In general, incorporating these constraints in the optimization process leads to hard non-convex problems and is therefore known to be a longstanding open challenge. Many significant practical constraints of the designed waveform have attracted the interest of radar researchers. Examples include (but are not limited to) modulus, similarity, orthogonality, and spectral constraints. The family of modulus constraints includes maximum modulus, energy, and constant modulus constraints. A principally important constraint among these constraints is the constant modulus constraint (CMC). The CMC is desirable in the design process due to the presence of nonlinear amplifiers in the radar systems, which must operate in saturation mode and cannot be efficiently utilized without CMC. In the presence of this constraint, conducting the design problem is profoundly challenging due to the non-convexity of this constraint. The past literature about this constraint shows a stiff trade-off between the performance accuracy and problem tractability. In this dissertation, we propose new frameworks that deal with the constant modulus constraint and break this classical trade-off. In the first part of the dissertation, we develop a novel gradient-based method to address this non-convex constraint by invoking principles of optimization over manifolds. That is, we propose a new numerical algorithm that involves direct optimization over the constant modulus constraint or equivalently the non-convex complex circle manifold. Consequently, we apply this new framework to address the problem of designing the transmit beampattern for Multiple-input multiple-output (MIMO) radar under the CMC. We derive a new Projection, Descent, and Retraction (PDR) update strategy that allows for monotonic cost function improvement while maintaining feasibility. For quadratic cost functions (as is the case with beampattern design), we provide analytical guarantees and formally show the monotonicity of the cost function along with proof of convergence to local minima. We evaluate the PDR algorithm against other candidates’ MIMO beampattern design methods and show that PDR can outperform the competing beampattern design methods while being computationally less expensive. Moreover, this framework of optimizing over the complex circle manifold enables us to incorporate orthogonality across antennas by adding a penalty term to the beampattern cost function. Using the aforementioned framework for the CMC, we addressed another important problem in the radar signal processing literature. In particular, we utilize this framework to control or shape the Slow-Time Ambiguity Function (STAF) for Single-Input-Single-Output (SISO) radar by designing the transmit waveform under the constant modulus constraint. In this problem, the design mechanism relies on minimizing the average values of the ambiguity function over the clutter locations. This is equivalent to minimizing the disturbance in the output of the matched filter, which is, in turn, formulated as a complex quartic polynomial optimization problem. Gradient-based methods remain elusive for this problem due to the quartic nature of the cost and to the non-convexity of CMC. We derive a new update strategy (Quartic-Gradient-Descent (QGD)) that computes an exact gradient of the quartic cost and invokes principles of optimization over manifolds towards an iterative procedure with guarantees of monotonic cost function decrease and convergence. Experimentally, QGD outperforms state of the art approaches for shaping STAF under the CMC with less computational complexity. The final part of the dissertation addresses the waveform design in spectrally crowded environments where co-existence between radar and other wireless systems is desired. That is, in addition to the CMC, a spectral interference constraint (SpecC) must be enforced. The PDR approaches exclusively focus on CMC and are hence not applicable. Due to this limitation, we address the CMC in a different novel approach to integrate SpecC with CMC in a tractable form, we redefine the CMC as an intersection of two sets (one convex and other non-convex) by exploiting its geometrical structure. This new perspective is used to design the transmit beampattern of MIMO radar (spatial compatibility) under spectral constraint (spectral compatibility) via a tractable method called Iterative Beampattern with Spectral design (IBS). The proposed IBS algorithm develops and solves a sequence of convex problems such that constant modulus is achieved at convergence. Crucially, we show that at convergence the obtained solution satisfies the Karush-Kuhn-Tucker (KKT) conditions of the aforementioned non-convex problem. Finally, we evaluate the proposed algorithm over challenging simulated scenarios, and show that it outperforms the state-of-the-art competing methods. In summary, this dissertation develops new approaches to deal with the practical, non- convex constant modulus constraint for different waveform design problems for radar systems. These approaches provide solutions for longstanding open challenges and problems and offer a favorable performance complexity trade-off.