Jacobi-Anger Expansion Research Articles

A staggered time integrator for the linear acoustic wave equation using the Jacobi-Anger expansion

In this study, we introduce a staggered time integrator with the Fourier pseudospectral method to solve the first-order linear wave equation, which is accelerated by using the Jacobi-Anger expansion. The proposed method can reduce the computational cost by approximately half compared to the scheme whose temporal order of accuracy is extended by the Lax-Wendroff method under the equivalent modeling conditions, such as time step length, grid interval and maximum wave propagation speed. This is because the Jacobi-Anger expansion can effectively approximate the sinusoidal function, which in our case is the sine function in the wavenumber domain. Based on the wavenumber domain analysis of the proposed method, a strategy to optimally design the simulation parameters to maintain a preset level of accuracy is also introduced. According to the strategy, as the time step length increases, not only the computational cost of the simulation is reduced, but also the accuracy of the numerical solution is improved. Numerical simulations are also performed by using both homogeneous and heterogeneous models to validate the introduced strategy, which supports the practicality of the proposed method.

Estimation of Sinusoidal Frequency-Modulated Signal Parameters by Two Branches and Two Stages

Sinusoidal frequency-modulated (SFM) signal is widely applied in radar, and wireless communication. Therefore, there is significant importance in studying the estimation of SFM signal parameters. However, traditional estimation algorithms have high computational complexity while the corresponding estimation performance degrades when either modulation index or modulation frequency of SFM signals increases. To address these problems of computational complexity as well as the performance degradation, a two-branch two-stage estimation algorithm is proposed in this paper. In the first stage, received signals are passed through two different branches: one for the modulation index not greater than $\pi /2$ , and the other for that larger than $\pi /2$ . This two-branch strategy is derived from the spectrum properties of SFM signals with respect to modulation index, where the first branch utilizes the Jacobi-Anger expansion, and the second one employs an iterative estimation algorithm. In the second stage, precisions of the parameter estimates obtained from the two branches are compared, and the estimates with higher precision correspond to the final estimates of SFM signal parameters. Both theoretical analysis, and numerical results show that the proposed algorithm outperforms the current state-of-the-art algorithms, while yielding significantly lower computational complexities.

Efficient Implementations of First-Order Steerable Differential Microphone Arrays With Arbitrary Planar Geometry

We present a spatial filtering approach to first-order steerable Differential Microphone Arrays (DMAs) with arbitrary planar geometry. In particular, the design of the spatial filter is based on a recently proposed frequency-domain design methodology that approximates, in a least-square sense, a target beampattern using the Jacobi-Anger expansion involving Bessel functions. Despite the generality of that approach, however, its computational cost turns out to be excessive when working with limited processing resources. The beamforming technique proposed in this manuscript overcomes this issue by exploiting the fact that in DMAs the spacing between sensors is typically smaller than the smallest wavelength of audio signals of interest. This allows us to substitute zero- and first-order Bessel functions with their Taylor series approximation truncated to the first order. Moreover, we show that this approximation allows us to derive an efficient discrete-time-domain implementation of first-order steerable differential beamformers based on arrays with arbitrary geometries.

A Novel External Rate Sensing Algorithm for Gyrowheel Based on Jacobi-Anger Expansion

Gyrowheel is an attitude determination and control instrument for small spacecrafts. Large tilts of its rotor and de-tuned spin rates lead to complex and nonlinear dynamics as well as difficulties in measuring external angular rates. Thus, the problem of angular rate sensing with the gyrowheel is investigated, and a novel external rate sensing algorithm is developed in this paper. In an effort to avoid linearization errors of the traditional method, the low-frequency characteristics of the gyrowheel dynamics are extracted by utilizing the Jacobi-Anger expansion. A simplified low-frequency model is obtained based on the particle swarm optimization algorithm, and a practical rate sensing algorithm for the gyrowheel is constructed. Validation simulations and experiments on a gyrowheel prototype show that the proposed algorithm is significantly superior to the traditional one especially under large tilt conditions.

Insights Into Frequency-Invariant Beamforming With Concentric Circular Microphone Arrays

This paper studies the problem of frequency-invariant beamforming with concentric circular microphone arrays (CCMAs) and presents an approach to the design of frequency-invariant and symmetric beampatterns. We first apply the Jacobi-Anger expansion to each ring of the CCMA to approximate the beampattern. The beamformer is then designed by using all the expansions from different rings. In comparison with the existing work in the literature where a Jacobi-Anger expansion of the same order is applied to different rings, here in this contribution the order of the Jacobi-Anger expansion at a ring is related to its number of sensors and, as a result, the expansion order at different rings may be different. The developed approach is rather general. It is not only able to mitigate the deep nulls problem in the directivity factor and the white noise gain, that is common to circular microphone arrays (CMAs), and improve the steering flexibility, but is also flexible to use in practice where a smaller ring can have less microphones than a larger one. We discuss the conditions for the design of $N$ th-order symmetric beampatterns and examples of frequency-invariant beampatterns with commonly used array geometries such as CMAs, CMAs with a sensor at the center, and CCMAs. We show the advantage of adding one microphone at the center of either a CMA or a CCMA, i.e., circumventing the deep nulls problem caused by the 0th-order Bessel function.

Time-Domain Characterization of Digitized PWM Inverter With Dead-Time Effect

This paper presents an analytical solution to characterize harmonic behavior of digitally controlled H-bridge inverter output with two nonlinear effects: sample delay and dead-time. The effect of sampling frequency in uniformly sampled pulse-width modulation (PWM) is discussed, including spectral analysis and compared with the analog modulated PWM. The effect of dead-time on the rising and falling edge of digitized PWM inverter error output is derived in time-domain. The double Fourier integral solution is used to develop the accurate mathematical expressions for the switched output waveforms. The integral solutions are obtained using Jacobi–Anger expansions. The analytical results are verified with simulation using PSCAD and experimental results.

On the design of differential beamformers with arbitrary planar microphone array geometry.

This letter deals with the problem of differential beamforming with microphone arrays of arbitrary planar geometry. By approximating the beampattern with the Jacobi-Anger expansion, it develops an algorithm that can form any specified frequency-invariant beampattern with a microphone array of any planar geometry as long as the sensors' coordinates are given and the spacing between neighboring sensors is smaller than the smallest wavelength. This method is rather general and it can be used to design differential beamformers with linear, circular, and concentric circular differential microphone arrays as well as differential arrays of arbitrary planar geometry where sensors are placed in any specified positions.

Synchronous radio-frequency FM signal generator using direct digital synthesizers.

A novel Radio-Frequency Frequency-Modulated (RF-FM) signal generation method is introduced and a prototype circuit developed to evaluate its functionality and performance. The RF-FM signal generator uses a modulated, voltage-controlled time delay to correspondingly modulate the phase of a 10 MHz sinusoidal reference signal. This modulated reference signal is, in turn, used to clock a Direct Digital Synthesizer (DDS) circuit resulting in an FM signal at its output. The modulating signal that is input to the voltage-controlled time delay circuit is generated by another DDS that is synchronously clocked by the same 10 MHz sine wave signal before modulation. As a consequence, all of the digital components are timed from a single sine wave oscillator that forms the basis of all timing. The resultant output signal comprises a center, or carrier, frequency plus a series of phase-synchronized sidebands having exact integer harmonic frequency separation. In this study, carrier frequencies ranging from 10 MHz to 70 MHz are generated with modulation frequencies ranging from 10 kHz to 300 kHz. The captured spectra show that the FM signal characteristics, amplitude and phase, of the sidebands and the modulation depth are consistent with the Jacobi-Anger expansion for modulated harmonic signals.

Time‐domain characterisation of multicarrier‐based digital SPWM of multilevel VSI

The effect of sampling frequency on digital sinusoidal pulse width modulated (SPWM) output of phase-shifted carrier-based multilevel voltage source inverter (VSI) and the characteristics of output voltage and load current has been discussed here. The generalised equation of digitally controlled phase-shifted carrier-based multilevel VSI output voltage is derived using double Fourier integral solution. The Jacobi–Anger expansion has been used to obtain the integral solution. Time-domain analysis has been used to establish the relation between sampling frequency, carrier frequency, and fundamental frequency, at different modulation indices for n-level VSI. The load current of VSI is stepped in nature due to stepped variation in the output of the digital SPWM. Therefore, the low-frequency harmonic components closer to the fundamental frequency appear in the frequency spectrum. The variation of total harmonic distortion with respect to multisampling factor (sampling to carrier frequency ratio) has been discussed at different modulation indices for multicarrier-based multilevel inverter. The analytical results are verified through simulation and laboratory experimental results.

Generation of vector Bessel beams with diffractive phase elements based on the Jacobi-Anger expansion.

We report the design and optimization of a diffractive phase element whose phase modulation is derived from the Jacobi-Anger relation, which allows the simultaneous generation of multiple scalar Bessel beams of different integer orders. In addition, by the appropriate treatment of a couple of such Bessel beams, we generate a vector Bessel beam of arbitrary order m. This beam is constructed by the collinear superposition of the scalar Bessel beam modes of orders m-1 and m+1, with circular orthogonal polarizations. We demonstrate experimentally both the simultaneous generation of the multiple scalar Bessel beams and the generation of vector Bessel beams of orders m=0 and m=1. These tasks are performed in an optical setup based on a pixelated liquid-crystal spatial light modulator.

Decoupled 2-D Direction of Arrival Estimation in L-Shaped Array

In this letter, we propose a novel decoupled method for 2-D direction of arrival (2-D DOA) estimation in L-shaped array. Specifically, in the proposed method, the Jacobi-Anger expansion is utilized to decouple the elevation angle from the azimuth angle, which means that the reconstructed manifold of the L-shaped array is the product of the two matrices of elevation and azimuth angles. Although the traditional 2-D MUSIC algorithm can estimate the elevation and azimuth angles without pair matching, it requires the complex 2-D spectrum peaks search. In order to simplify the searching procedure, the new formulation of the array manifold is applied in the Root-MUSIC algorithm, in which only 1-D elevation searching is needed to estimate the 2-D DOA and the azimuth estimates can be paired automatically with the elevation estimates. Simulation results are presented to confirm the effectiveness of the proposed method, and demonstrate that our proposed method achieves a better tradeoff between the performance and complexity than other existing methods.

On the Design of Frequency-Invariant Beampatterns With Uniform Circular Microphone Arrays

This paper deals with two critical issues about uniform circular arrays (UCAs): frequency-invariant response and steering flexibility. It focuses on some optimal design of frequency-invariant beampatterns in any desired direction along the sensor plane. The major contributions are as follows. 1) We explain how to include the steering information in the desired directivity pattern. 2) We show that the optimal approximation of the beamformer's beampattern with a UCA from a least-squares error perspective is the Jacobi–Anger expansion. 3) We develop an approach to the design of any desired symmetric directivity pattern, where the deduced beampattern is almost frequency invariant and its main beam can be pointed to any wanted direction in the sensor plane. 4) With the proposed approach, we derive an explicit form of the white noise gain (WNG) and the directivity factor (DF), and explain clearly the white noise amplification problem at low frequencies and the DF degradation at high frequencies. The analysis also indicates that increasing the number of microphones can always improve the WNG. We show that the proposed method is a generalization of circular differential microphone arrays. The relationship between the proposed method and the so-called circular harmonics beamformers is also discussed.

Scaling Factor for Turn-Table Platform Stirring in Reverberation Chamber

The turn-table platform stirring technique has been used to improve the measurement accuracy of a reverberation chamber (RC), where half-wavelength spatial sampling on the platform is usually used as a rule of thumb. A smaller sampling distance (scaling factor) was suggested experimentally by curve fitting in a previous work. However, an analytical study of this scaling factor is still missing in the literature. By expanding the electromagnetic plane wave using the Jacobi-Anger expansion, the scaling factor is determined when the truncation error falls below certain threshold. The result is verified by measurements in a RC.

Sampling Effect Characterization of Digital SPWM of VSI in Time Domain

This paper characterizes the sampling effect in digital sinusoidal pulse width modulation (SPWM) of voltage source inverter (VSI). The time-domain analysis presented in this paper establishes the relation between the fundamental frequency, carrier frequency, and the sampling frequency at different modulation indices of the SPWM. The analysis investigates the effect of sampling frequency of the digital controller on the output of the VSI. It is shown that the low-frequency harmonic components appear in the frequency spectrum of the VSI output voltage due to sampling. As a result, the output current of digitally controlled VSI is stepped in nature. The double Fourier integral solution of the switched waveform for inner and outer integral limits has been used to derive the expression. The integral solution is obtained using Jacobi-Anger expansions. The generalized results developed in this paper are useful to investigate the sampling effect in high switching frequency DSPWM. The proposed sampling effect has been analyzed for SPWM of single-phase H-bridge inverter. The analytical results are verified using simulation and experimental results. The experimental results have been obtained with the use of field-programmable gate array (FPGA) as a digital controller.

Design of robust differential microphone arrays with the Jacobi–Anger expansion

Abstract Due to their small size, differential microphone arrays (DMAs) are very attractive. Moreover, they have been effective in combating noise and reverberation. Recently, a new class of DMAs of different orders have been developed with the MacLaurin’s series and the frequency-independent patterns. However, the MacLaurin’s series does not approximate well the exponential function, which appears in the general definition of the beampattern, when the intersensor spacing is not small enough. To circumvent this problem, we propose in this paper to approximate the exponential function with the Jacobi–Anger expansion. Based on this approximation and the frequency-independent Chebyshev patterns, we derive first-, second-, and third-order DMAs. Furthermore, in order to improve the robustness of DMAs against white noise amplification, we propose to use more microphones combined with minimum-norm filters. It is also shown that the Jacobi–Anger expansion is optimal from a mean-squared error perspective. Simulations are carried out to evaluate the performance of the proposed DMAs.

Geometry-Based One-Ring Models for MIMO Systems: Modeling Accuracy Assessment and Improvement

In this paper, we question the accuracies and validity ranges of conventional geometry-based one-ring models (GBORMs) and their variants whose correlation functions (CFs) are based on approximate total propagation distances (TPDs) under a small beamwidth (or small angular spread) assumption. To answer this, we use a reference GBORM for space-time-frequency (STF) correlated channels and analyze the accuracies and validity ranges of the conventional models. Our analysis shows that the conventional models become inaccurate for urban pico/micro/macrocells, vehicular-to-vehicular, and wideband channels, where large beamwidths and relative propagation delays are typical. In order to remedy these issues, we propose new TPDs and closed-form STF-CF using novel approximation methods with Jacobi-Anger expansion and show their superior accuracies and validity range. The new closed-form STF-CF represents or includes conventional closed-form CFs as special cases for small beamwidth. Also, it has favorable properties facilitating its use in a wide range of propagation channels. The applicability of the new solutions to conventional wideband models are tested and verified based on measured data.

Analytical investigation of optical vortices emitted from a collectively polarized dipole array.

Many approaches for producing optical vortices have been developed both for fundamental interests of science and for engineering applications. In particular, the approach with direct excitation of several emitters has a potential to control the topological charges with a control of the source conditions without any modifications of structures of the system. In this paper, we investigate the propagation properties of the optical vortices emitted from a collectively polarized electric dipole array as a simple model of the several emitters. Using an analytical approach based on the Jacobi-Anger expansion, we derive a relationship between the topological charge of the optical vortices and the source conditions of the emitter, and clarify and report our new finding; there exists an intrinsic split of the singular points in the electric field due to the spin-orbit interaction of the dipole fields.

Phase Regeneration of a BPSK Data Signal Using a Lithium Niobate Phase Modulator

We propose a scheme for phase regeneration of an optical binary phase shift keying (BPSK) data signal using a Lithium Niobate (LiNbO <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sub> ) phase modulator. The scheme is based on heterodyne detection of the BPSK data signal with a continuous wave local oscillator (CW-LO). Carrier recovery is then achieved in the electrical domain using a ×2 frequency-multiplier and a narrow-band filtering scheme. Subsequently, a superposition of the recovered carrier and the heterodyne detected data signal is used to modulate the CW-LO in a LiNbO3 phase modulator. The result is a parametric mixing process in the optical domain, leading to a phase-regenerated BPSK data signal by the coherent superposition with a phase-inverted copy. The proposed scheme constitutes a compact and stable setup, where active phase-stabilization of the electrical data- and carrier-paths can potentially be avoided. An analytical derivation of the working principle is provided, using Jacobi-Anger expansions to describe the phase-modulation. A proof-of-principle experiment is carried out, demonstrating regeneration of a 10 Gb/s NRZ-BPSK data signal degraded by a 5-GHz sinusoidal phase-noise tone. In the proof-of-principle demonstration, the decorrelated data- and LO-carriers are derived from the same CW source. A preliminary test with separate CW sources for data and LO, but without the required electrical narrow-band carrier filtering, is also included. Finally, numerical simulations of the regenerator performance in the presence of wideband phase- and amplitude-noise are performed.

Direction Finding for Bistatic MIMO Radar with Uniform Circular Array

A method of direction of arrival (DOA) and direction of departure (DOD) angle estimation based on polynomial rooting for bistatic multiple-input multiple-output (MIMO) radar with uniform circular array (UCA) configuration is proposed in this paper. The steering vector of the UCA is firstly transformed into a steering vector with a Vandermonde structure by using the Jacobi-Anger expansion. Then the null-spectrum function of the MIMO radar can be written as an expression in which the transmit and receive steering vectors are decoupled. Finally, a two-step polynomial rooting is used to estimate DOA and DOD of targets instead of two-dimensional multiple signal classification (MUSIC) search method for bistatic UCA MIMO radar. The angle estimation performance of the proposed method is similar to that of the MUSIC spectral search method, but the computation burden of the proposed polynomial rooting algorithm is much lower than that of the conventional MUSIC method. The simulation results of the proposed algorithm are presented and the performances are investigated and analyzed.

Higher order differential-integral microphone arrays

This paper develops theory to design higher order directional microphone arrays. The proposed higher order designs have similar inter sensor spacings as traditional first and second order differential arrays. The Jacobi-Anger expansion is used to exploit the underlying structure of microphone signals from pairs of closely spaced sensors. Specifically, the difference and sum of these microphone signals are processed to design the novel directional array.