arXiv subjects

David A. Hague

Publications and source records attributed to David A. Hague.

At least 19 recordsLinked to original sources

arXiv2026

Wideband MIMO Beampattern Synthesis using Adaptive Frequency Modulated Waveforms

Multiple-Input Multiple-Output (MIMO) arrays transmit unique waveforms on each of their elements increasing the degrees of freedom available to synthesize novel transmit beampatterns. The wideband MIMO beampattern shape is determined by the structure of the MIMO Cross Spectral Density Matrix (CSDM) whose elements are the cross-spectra between each waveform in the set as a function of frequency. This paper demonstrates a model for synthesizing wideband MIMO beampatterns using Multi-Tone Sinusoidal Frequency Modulated (MTSFM) waveform sets. The MTSFM waveform's instantaneous phase is a finite Fourier series. The Fourier coefficients are modified to synthesize constant modulus and spectrally compact waveforms whose CSDM closely approximates a desired wideband MIMO transmit beampattern. An optimization routine is formulated that synthesizes MTSFM waveform sets that possess the necessary CSDM structure and is demonstrated using an illustrative design example.

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arXiv2025

Multi-Directional Periodic Driving of a Two-Level System beyond Floquet Formalism

In this manuscript, we introduce an exact expression for the response of a semi-classical two-level quantum system subject to arbitrary periodic driving. Determining the transition probabilities of a two-level system driven by an arbitrary periodic waveform necessitates numerical calculations through methods such as Floquet theory, requiring the truncation of an infinite matrix. However, such truncation can lead to a loss of significant interference information, hindering quantum sensors or introducing artifacts in quantum control. To alleviate this issue, we use the $\star$-resolvent formalism with the path-sum theorem to determine the exact series solution to Schrödinger's equation, therefore providing the exact transition probability. The resulting series solution is generated from a compact kernel expression containing all of the information of the periodic drive and then expanded in a non-harmonic Fourier series basis given by the divided difference of complex exponentials with coefficients corresponding to products of generalized Bessel functions. The present method provides an analytical formulation for quantum sensors and control applications.

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arXiv2025

Equiripple MIMO Beampattern Synthesis using Chebyshev Approximation

This letter presents a method for synthesizing equiripple MIMO transmit beampatterns using Chebyshev approximation. The MIMO beampattern is represented as a non-negative real-valued trigonometric polynomial where the $\ell^{\text{th}}$ order polynomial coefficient is the sum of the $\ell^{\text{th}}$ order diagonal of the waveform correlation matrix. The optimal coefficients for a given equiripple beampattern design is then posed as a Chebyshev approximation problem which is efficiently solved using the Parks-McClellan algorithm from optimal Finite Impulse Response (FIR) filter design theory. The unique advantage of this synthesis method is that it provides a closed form method to generating MIMO correlation matrices that realize the desired equiripple beampattern. This correspondingly facilitates the design of waveform sets that closely approximate those correlation matrices. This method is demonstrated via two illustrative design examples; the first using traditional partial signal correlation methods and the second using transmit beamspace processing. Both examples realize equiripple beampatterns using constant envelope and spectrally compact waveform sets.

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arXiv2024

MIMO Beampattern Synthesis using Adaptive Frequency Modulated Waveforms

This paper demonstrates a method that synthesizes narrowband Multiple-Input Multiple-Output (MIMO) beampatterns using the Multi-Tone Sinusoidal Frequency Modulated (MTSFM) waveform model. MIMO arrays transmit unique waveforms on each of their elements which increases the degrees of freedom available to synthesize novel transmit beampatterns. The MIMO beampattern shape is determined by the structure of the MIMO correlation matrix whose entries are the inner products between the waveforms transmitted on each element. The MTSFM waveform possesses an instantaneous phase that is represented as a finite Fourier series. The Fourier coefficients are modified to synthesize sets of waveforms whose correlation matrix realizes a desired MIMO transmit beampattern. The MIMO correlation matrix for a MTSFM waveform set has an analytical form expressed in terms of Generalized Bessel Functions. These mathematical properties are utilized to develop an optimization routine that synthesizes MTSFM waveform sets to approximate a desired MIMO transmit beampattern. The performance of this optimization routine is then demonstrated via an illustrative design example.

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arXiv2023

Characterizing the Narrowband Ambiguity Function of Multi-Tone Sinusoidal Frequency Modulated Waveforms

This paper characterizes the mainlobe and sidelobe structure of the Multi-Tone Sinusoidal Frequency Modulated (MTSFM) transmit waveform's narrowband Ambiguity Function (AF) for active sonar applications. The MTSFM waveform's modulation function is represented as a Fourier series. The Fourier coefficients form a discrete set of parameters that are modified to synthesize waveforms with novel characteristics. The contour of the AF's mainlobe is well approximated as a coupled ellipse known as the Ellipse of Ambiguity (EOA). The EOA parameters determine whether a waveform is Doppler sensitive or Doppler tolerant. This paper derives exact closed form expressions for the EOA parameters of the MTSFM's AF. The MTSFM's design coefficients allow for fine control of its AF mainlobe width in range and Doppler as well as its RDCF. This fine control facilitates designing waveforms that can smoothly trade-off between possessing Doppler sensitive and Doppler tolerant characteristics. Additionally, this paper introduces a method to control the sidelobe structure of the MTSFM's Auto Correlation Function (ACF) while maintaining the waveform's AF mainlobe shape. This is achieved using a numerical optimization technique that minimizes the ratio of $\ell_2$-norms of the ACF mainlobe and sidelobe regions subject to constraints on the EOA parameters. Simulations demonstrate the effectiveness of this optimization technique.

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arXiv2023

Gradient-Descent Based Optimization of Multi-Tone Sinusoidal Frequency Modulated Waveforms

This paper describes a gradient-descent based optimization algorithm for synthesizing Multi-Tone Sinusoidal Frequency Modulated (MTSFM) waveforms with low Auto-Correlation Function (ACF) sidelobes in a specified region of time delays while preserving the ACF mainlobe width. The algorithm optimizes the Generalized Integrated Sidelobe Level (GISL) which controls the mainlobe and sidelobe structure of the waveform's ACF. This optimization is performed subject to nonlinear constraints on the waveform's RMS bandwidth which directly controls the ACF mainlobe width. Since almost all of the operations of the algorithm utilize the Fast Fourier Transform (FFT), it is substantially more computationally efficient than previous methods that synthesized MTSFM waveforms with low ACF sidelobes. The computational efficiency of this new algorithm facilitates the design of larger dimensional and correspondingly larger time-bandwidth product MTSFM waveform designs. The algorithm is demonstrated through several illustrative MTSFM design examples.

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arXiv2023

Characterizing the Ambiguity Function of Constant-Envelope OFDM Waveforms

This paper investigates the radar Ambiguity Function (AF) properties of Constant Envelope Orthogonal Frequency Division Multiplexing (CE-OFDM) waveforms employing Phase-Shift Keying (PSK). The CE-OFDM is in fact a special case of the Multi-Tone Sinusoidal Frequency Modulated (MTSFM) waveform which allows for applying many of the same mathematical techniques of the MTSFM model to the CE-OFDM model. This results in novel compact closed-form expressions for the spectrum, AF, and Auto-Correlation Function (ACF) of the CE-OFDM waveform. The mainlobe structure of the CE-OFDM's AF is characterized by the Ellipse of Ambigiuity (EOA) model. This produces precise closed-form expressions for the CE-OFDM's Root-Mean Square (RMS) bandwidth and the degree of range-Doppler coupling present in the waveform's AF mainlobe. These expressions show that a CE-OFDM waveform employing PSK as the symbol encoding scheme will possess a fixed RMS bandwidth for fixed modulation index $h$ and number of sub-carriers $L$. Additionally, we show that the EOA model predicts that a CE-OFDM waveform employing PSK encoding will almost always possess a ``Thumbtack-Like'' AF shape.

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arXiv2023

Gradient-Descent Based Optimization of Constant Envelope OFDM Waveforms

This paper describes a gradient-descent based optimization algorithm for synthesizing Constant Envelope Orthogonal Frequency Division Multiplexing (CE-OFDM) waveforms with low Auto-Correlation Function (ACF) sidelobes in a specified region of time-delays. The algorithm optimizes the Generalized Integrated Sidelobe Level (GISL) which controls the mainlobe and sidelobe structure of the waveform's ACF. The operations of this Gradient-Descent GISL (GD-GISL) algorithm are FFT-based making it computationally efficient. This computational efficiency facilitates the design of large dimensional waveform design problems. Simulations demonstrate the GD-GISL algorithm on CE-OFDM waveforms employing Phase-Shift Keying (PSK) symbols that take on a continuum of values (i.e, $M_{\text{PSK}} = \infty$). Results from these simulations show that the GD-GISL algorithm can indeed reduce ACF sidelobes in a desired region of time-delays. However, truncating the symbols to finite M-ary alphabets introduces perturbations to the waveform's instantaneous phase which increases the waveform's ACF sidelobe levels.

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arXiv2022

Continuous Phase Modulation of Phase Coded Transmit Waveforms using Multi-Tone Sinusoidal Frequency Modulation

Phase Coded (PC) waveforms possess desirable Auto-Correlation Function (ACF) properties for use in radar and sonar systems. However, their spectra possess high spectral leakage due to the abrupt phase transitions between the chips in the waveform. This paper describes a method of Continuous Phase Modulation (CPM) to reduce a PC waveform's spectral leakage using the Multi-Tone Sinusoidal Frequency Modulation (MTSFM) model. The MTSFM-CPM model represents the PC waveform's instantaneous phase as a finite Fourier series. This representation smooths the abrupt phase transitions between chips resulting in a spectrally compact waveform. This smoothing of the PC waveform's instantaneous phase introduces perturbations to the waveform's ACF mainlobe and sidelobe structure. Adjusting the MTSFM-CPM waveform's parameters refines its ACF mainlobe and sidelobe structure while also preserving its compact spectral shape.

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arXiv2021

A Generalized Lerche-Newberger Formula

The Lerche-Newberger formula simplifies harmonic sums of Bessel functions and has seen application in plasma physics and frequency modulated quantum systems. In this paper, we rigorously prove the formula and extend the classical result to a family of multi-dimensional extensions of the single variable Bessel functions called generalized Bessel functions. Since prevailing definitions of these functions do not accommodate arbitrary complex order, we use an auxiliary family of functions called generalized Anger functions and show that the single-variable result holds in multiple dimensions for a certain selection of parameters. We conclude by applying these results to physical systems.

math.CA↗

arXiv2021

Adaptive Transmit Waveform Design

Recent research efforts in the Anti-Submarine Warfare (ASW) community have focused on developing sonar systems that adapt to their acoustic environment, referred to as "cognitive" sonars. Cognitive active sonar systems utilize principles of the perception action cycle of cognition to leverage information gathered from earlier sensing interactions with the underwater acoustic environment. This in turn informs the selection of system parameters to optimize target detection, classification, localization, and tracking performance in that acoustic environment. Of the many system parameters such a cognitive sonar system could potentially adapt, the acoustic signal transmitted into the medium, also known as the transmit waveform, has a profound impact on system performance. Many of the physical characteristics of the acoustic environment are contained in the return echo signal that is composed of amplitude scaled (target strength), time-delayed (target range) and Doppler shifted (target range-rate) echoes of the transmit waveform. This paper briefly describes a spectrally compact adaptive FM waveform model using Multi-Tone Sinusoidal Frequency Modulation (MTSFM). The MTSFM waveform's frequency and phase modulation functions are composed of a finite set of weighted sinusoidal harmonics. The weights for each harmonic are utilized as a discrete set of design coefficients. Adjusting these coefficients results in constant amplitude, spectrally compact FM waveforms with unique characteristics. The adaptability of the MTSFM combined with its transmitter friendly properties make it an attractive waveform type for a variety of active sonar applications and may provide a cognitive sonar system the ability to generate a complementary set of finely tuned waveforms for the novel scenarios and environments that it may encounter.

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arXiv2021

Exploring the Interdependencies Between Transmit Waveform Ambiguity Function Shape and Off-Axis Bearing Estimation

The frequency dependent beampatterns of an active sonar projector filters the acoustic signal that is transmitted into the medium, also known as the transmit waveform. This filtering encodes information about the target's bearing relative to the main response axis. For any given projector and transmit waveform spectrum, there exists an optimal angle of operation which maximizes the Fisher Information (FI) of the target bearing estimate. Previous investigations into this phenomena show that for narrowband (i.e, high $Q$) Linear Frequency Modulated (LFM) waveforms, the angle of maximum FI is solely determined by its center frequency $f_c$. Steering the region of maximum bearing estimation precision is then achieved by appropriate selection of the LFM waveform's center frequency $f_c$. This fine bearing estimation is accomplished without steering the projector's main response axis. In addition to LFM waveforms, a wide variety of other active sonar waveform types exist that possess distinct spectral characteristics. These other waveforms possess different Ambiguity Function (AF) shapes from the LFM and are typically utilized to suite the range-Doppler resolution requirements of the active sonar system. This paper investigates the transmit waveform impact on off-axis bearing estimation performance and the spectral filtering impact on the waveform's AF shape. High $Q$ waveforms perform similarly to the LFM for off-axis bearing estimation while the transducer's spectral filtering perturbs the waveform's AF shape.

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arXiv2021

Matched Illumination Waveforms using Multi-Tone Sinusoidal Frequency Modulation

This paper explores the design of constant modulus Matched-Illumination (MI) waveforms using the Multi-Tone Sinusoidal Frequency Modulation (MTSFM) waveform model. MI waveforms are optimized for detecting targets in known noise and clutter Power Spectral Densities (PSDs). There exist well-defined information theoretic methods that describe the design of MI waveforms for a myriad of target/noise/clutter models. However, these methods generally only produce the magnitude square of the MI waveform's spectrum. Additionally, the waveform's time-series is not guaranteed to be constant modulus. The MTSFM is a constant modulus waveform model with a discrete set of design coefficients. The coefficients are adjusted to synthesize constant modulus waveforms that approximate the ideal MI waveform's spectrum. Simulations demonstrate that the MTSFM's detection performance closely approximates an ideal MI waveform spectrum and generally outperforms flat spectrum waveforms across a range of transmit energies when the noise and clutter PSDs vary greatly across the operational band.

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arXiv2021

Identities and Properties of Multi-Dimensional Generalized Bessel Functions

The Generalized Bessel Function (GBF) extends the single variable Bessel function to several dimensions and indices in a nontrivial manner. Two-dimensional GBFs have been studied extensively in the literature and have found application in laser physics, crystallography, and electromagnetics. In this article, we document several properties of $m$-dimensional GBFs including an underlying partial differential equation structure, asymptotics for simultaneously large order and argument, and analysis of generalized Neumann, Kapteyn, and Schlömilch series. We extend these results to mixed-type GBFs where appropriate.

math.GM↗

arXiv2020

Adaptive Transmit Waveform Design using Multi-Tone Sinusoidal Frequency Modulation

This paper presents an adaptive waveform design method using Multi-Tone Sinusoidal Frequency Modulation (MTSFM). The MTSFM waveform's modulation function is represented as a finite Fourier series expansion. The Fourier coefficients are utilized as a discrete set of design parameters that may be modified to adapt the waveform's properties. The MTSFM's design parameters are adjusted to shape the spectrum, Auto-Correlation Function (ACF), and Ambiguity Function (AF) shapes of the waveform. The MTSFM waveform model naturally possesses the constant envelope and spectrally compact waveforms that make it well suited for transmission on practical radar/sonar transmitters which utilize high power amplifiers. The MTSFM has an exact mathematical definition for its time-series using Generalized Bessel Functions which allow for deriving closed-form analytical expressions for its spectrum, AF, and ACF. These expressions allow for establishing well-defined optimization problems that finely tune the MTSFM's properties. This adaptive waveform design model is demonstrated by optimizing MTSFM waveforms that initially possess a "thumbtack-like" AF shape. The resulting optimized designs possess substantially improved sidelobe levels over specified regions in the range-Doppler plane without increasing the Time-Bandwidth Product (TBP) that the initialized waveforms possessed. Simulations additionally demonstrate that the optimized thumbtack-like MTSFM waveforms are competitive with thumbtack-like phase-coded waveforms derived from design algorithms available in the published literature.

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arXiv2020

Generating Waveform Families using Multi-Tone Sinusoidal Frequency Modulation

This paper presents a method for generating a family of waveforms with low in-band Auto/Cross-Correlation Function (ACF/CCF) properties using the Multi-Tone Sinusoidal Frequency Modulated (MTSFM) waveform model. The MTSFM waveform's modulation function is represented using a Fourier series expansion. The Fourier coefficients are utilized as a set of discrete parameters that can be modified to optimize the waveform family's properties. The waveforms' ACF/CCF properties are optimized utilizing a multi-objective optimization problem. Each objective function is weighted to place emphasis on either low ACF or CCF sidelobes. The resulting optimized MTSFM waveforms each possess a thumbtack-like Ambiguity Function in addition to the specifically designed ACF/CCF properties. Most importantly, the resulting MTSFM waveform families possess both ideally low Peak-to-Average Power Ratios (PAPR) and high Spectral Efficiency (SE) making them well suited for transmission on practical radar transmitters.

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arXiv2019

Spectral Perturbations of the Line Graph Laplacian

The graph Laplacian is an important tool in Graph Signal Processing (GSP) as its eigenvalue decomposition acts as an analogue to the Fourier transform and is known as the Graph Fourier Transform (GFT). The line graph has a GFT that is a direct analogue to the Discrete Cosine Transform Type II (DCT-II). Leveraging Fourier transform properties, one can then define processing operations on this graph structure that is loosely analogous to processing operations encountered in Digital Signal Processing (DSP) theory. This raises the question of whether well defined DSP-like operations can be derived from the GFT for more complex graph structures. One potential approach to this problem is to perturb simple graph structures and study the perturbation's impact on the graph Laplacian. This paper explores this idea by examining the eigenvalue decomposition of the Laplacian of undirected line graphs that undergo a single edge weight perturbation. This single perturbation can perturb either an existing edge weight or create new edge between distant unconnected vertices. The eigenvalue bounds are expressed in closed form and agree with simulated examples. The theory can be extended to include multiple perturbations such that the GFT can be defined for a more general graph structure.

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arXiv2019

Waveform Design using Multi-Tone Feedback Frequency Modulation

This paper introduces a waveform design method using Multi-Tone Feedback Frequency Modulation (MT-FFM), a generalization of the single oscillator feedback FM method developed by [Tomisawa, 1981]. The MT-FFM utilizes a collection of $K$ harmonically related oscillators each governed by a design parameter $z_k$ which are utilized as a discrete set of parameters that may be modified to generate a richer set of modulation functions than in the single oscillator case. The resulting modulation function is represented using a form of Kapteyn series composed of Generalized Bessel Functions. This paper describes the structure of the MT-FFM waveform, derives the Kapteyn series representation of the waveform's modulation function, and demonstrates the design method with a waveform design example.

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