Harish Parthasarathy

Dynamics of a stochastically perturbed two-body problem

In classical mechanics, the two-body problem has been well studied. The governing equations form a system of two-coupled second-order nonlinear differential equations for the radial and angular coordinates. The perturbation induced by the astronomical disturbance like ‘dust’ is normally not considered in the orbit dynamics. Distributed dust produces an additional random force on the orbiting particle, which can be modelled as a random force having ‘Gaussian statistics’. The estimation of accurate positioning of the orbiting particle is not possible without accounting for the stochastic perturbation felt by the orbiting particle. The objective of this paper is to use the stochastic differential equation (SDE) formalism to study the effect of such disturbances on the orbiting body. Specifically, in this paper, we linearize SDEs about the mean of the state vector. The linearization operation performed above, transforms the system of SDEs into another system of SDEs that resembles a bilinear system, as described in signal processing and control literature. However, the mean trajectory of the resulting bilinear stochastic differential model does not preserve the perturbation effect felt by the orbiting particle; only the variance trajectory includes the perturbation effect. For this reason, the effectiveness of the dust-perturbed model is examined on the basis of the bilinear and second-order approximations of the system nonlinearity. The bilinear and second-order approximations of the system nonlinearity allow substantial simplifications for the numerical implementation and preserve some of the properties of the original stochastically perturbed model. Most notably, this paper reveals that the Brownian motion process is accurate to model and study the effect of dust perturbation on the orbiting particle. In addition, analytical findings are supported with finite difference method-based numerical simulations.

ANALYSIS OF A RECTANGULAR WAVEGUIDE USING FINITE ELEMENT METHOD

The characteristics impedance of the fundamental mode in a rectangular waveguide is computed using finite element method. The method is validated by comparison with the theoretical results. In addition to this, we have considered the problem of determining the modes of propagation of electromagnetic waves in a rectangular waveguide for the simple homogeneous dielectric case. The starting point is Maxwells equations with an assumed exponential dependence of the fields on the Z-coordinates. From these equations we have arrived at the Helmholtz equation for the homogeneous case. Finite- element-method has been used to derive approximate values of the possible propagation constant for each frequency. The finite element method (FEM) has been widely used during the last two decades in the analysis of waveguide components. With this method propagation characteristics of arbitrarily shaped waveguides of different composition are easily attainable. The finite element method is based on a spatial discretization (1). This approximation allows one to handle waveguide cross section geometries which are very similar to the real structures employed in practical devices. These complex structures do not lend themselves to analytical solutions. As a consequence, the FEM constitutes a promising tool to characterize such problems (2, 11). Modern phased array radars imply the requirements for polarization agility of wideband array elements. One possible choice for a radiating element with this property is the rectangular waveguide. In this paper a formulation is proposed to solve waveguides problems. A numerically efficient finite element formulation is presented that shows propagation modes and which may be used to analyze problems

A MUSIC-like method for estimating quadratic phase coupling

In this paper, a two-dimensional version of the well-known MUSIC algorithm for estimating the quadratically coupled frequency pairs (QC pairs) in a noise-corrupted complex harmonic process is proposed. It is shown that the algorithm can also, with minor modifications, be used for estimating the bispectrum of a general third-order stationary harmonic process. The algorithm involves arranging the complex third-order cumulants of the noisy harmonic process in the form of a matrix having a normal structure, i.e., a matrix with an orthonormal eigenbasis. It is shown that a necessary and sufficient condition for an ordered pair (θ1, θ2) in the two-dimensional frequency plane to be a QC pair is that the Kronecker product between steering vectors associated with the two frequencies θ1 and θ2 lies in the signal subspace of this matrix. By exploiting this result and the orthogonality between the signal and noise subspaces of the above matrix, a symmetric search function of two frequency variables (termed as the MUSIC pseudo-bispectrum estimator) is constructed using the signal eigenvectors. This function is shown to peak precisely at the QC pair locations in the two-dimensional frequency plane. Simulation results are presented, and are shown to testify to the high-resolution performance of this estimator.

Conditions for third-order stationarity and ergodicity of a harmonic random process

The finite data estimates of the complex third-order moments of a signal consisting of random harmonics are analysed. Conditions for the third-order stationarity and ergodicity are obtained. Explicit formulas for the estimation error and its variance, as well as their limiting large sample values are derived. A special case relevant to quadratic phase coupling is considered, and these results are stated for this case. The variance is shown to comprise an ergodic and a nonergodic part. >

Third-order approximate Kushner filter for a non-linear dynamical system

The analytical and numerical solutions of the non-linear exact Kushner filter are not possible, since the mean and variance evolutions are infinite dimensional and require the knowledge of the higher-order moment evolutions. The approximate filters seem to preserve some of the qualitative characteristics of the exact filter. In this paper, evolutions of conditional mean and conditional covariance of the third-order approximate filter for estimating the states of a non-linear dynamical system, especially accounting state-dependent and state-independent noise perturbations, are derived. In this analysis, we make a comparison of this filter with second-order Gaussian filter discussed in standard textbooks on non-linear filtering. This paper discusses Duffing filter, by taking up two different non-linear observation equations to demonstrate the effectiveness of the higher-order filters, i.e. third order, and second-order filters. Most notably, this paper is about examining the ability of the higher-order filters for estimating the states of the stochastically perturbed non-linear dynamical systems. In addition, the analytical findings are supported with numerical work generated by a simple, but effective, finite difference scheme.

Perturbation Approach to Ebers–Moll Equations for Transistor Circuit Analysis

In this paper, a perturbation theoretic technique has been used to analyze an Ebers–Moll modeled transistor amplifier circuit. The main advantage of the proposed method is that the use of the perturbation technique helps to obtain more accurate closed form Volterra series. These expressions can be used to derive corrections to the behavior of the amplifier when the input swing is not small enough. The expressions derived in this paper can also be used for transistor parameter estimation, as variations in transistor parameters affect the circuit performance critically.

An adaptive non-linear PDE-based speckle reduction technique for ultrasound images

In this paper, an adaptive nonlinear complex diffusion based filter is proposed to reduce the speckle noise from ultrasound images. The adaptive and modified version of a fourth order PDE is also proposed and examined for its suitability and efficacy for speckle reduction from ultrasound images. The performance of the proposed method is compared with constraint driven anisotropic diffusion based method, modified fourth order PDE based method and other methods such as Speckle Reducing Anisotropic Diffusion (SRAD) filter, Lee filter, Frost filter and Kuan filter in terms of various performance metrics such as Mean Square Error (MSE), Peak Signal-to-Noise Ratio (PSNR), Correlation Parameter (CP) and Mean Structure Similarity Index Map (MSSIM). The obtained results justify the applicability of the proposed scheme.

Wavelet-Based Transistor Parameter Estimation

In this paper a wavelet-based parameter estimation method has been proposed for the common emitter transistor amplifier circuit and compared with the least squares method. As the maximal precision of simulation requires the modeling of electronic circuits in terms of device parameters and circuit components, the Volterra model of the common emitter amplifier circuit derived using the Ebers–Moll model and perturbation technique has been used for parameter estimation. The advantage of the proposed method is a smaller data storage requirement and accurate parameter estimation as compared to the least squares method because the wavelet method is adapted to time-frequency resolution.

A perturbation-based model for rectifier circuits

A perturbation-theoretic analysis of rectifier circuits is presented. The governing differential equation of the half-wave rectifier with capacitor filter is analyzed by expanding the output voltage as a Taylor series with respect to an artificially introduced parameter in the nonlinearity of the diode characteristic as is done in quantum theory. The perturbation parameter introduced in the analysis is independent of the circuit components as compared to the method presented by multiple scales. The various terms appearing in the perturbation series are then modeled in the form of an equivalent circuit. This model is subsequently used in the analysis of full-wave rectifier. Matlab simulation results are included which confirm the validity of the theoretical formulations. Perturbation analysis acts a helpful tool in analyzing time-varying systems and chaotic systems.

Optimal nonlinear system identification using fractional delay second-order volterra system

The aim of this work is to design a fractional delay second order Volterra filter that takes a discrete time sequence as input and its output is as close as possible to the output of a given nonlinear unknown system which may have higher degree nonlinearities in the least square sense. The basic reason for such a design is that rather than including higher than second degree nonlinearities in the designed system, we use the fractional delay degrees of freedom to approximate the given system. The advantage is in terms of obtaining a better approximation of the given nonlinear system than is possible by using only integer delays U+0028 since we are giving more degrees of freedom via the fractional delays U+0029 and simultaneously it does not require to incorporate higher degree nonlinearities than two. This work hinges around the fact that if the input signal is a decimated version of another signal by a factor of M, then fractional delays can be regarded as delays by integers less than M. Using the well known formula for calculating the discrete time Fourier transform U+0028 DTFT U+0029 of a decimated signal, we then arrive at an expression for the DTFT of the output of a fractional delay system in terms of the unknown first and second order Volterra system coefficients and the fractional delays. The final energy function to be minimized is the norm square of the difference between the DTFT of the given output and the DTFT of the output of the fractional delay system. Minimization over the filter coefficients is a linear problem and thus the final problem is to minimize a highly nonlinear function of the fractional delays which is accomplished using search techniques like the gradient-search and nature inspired optimization algorithms. The effectiveness of the proposed method is demonstrated using two nonlinear benchmark systems tested with five different input signals. The accuracy of the stated models using the globally convergent metaheuristic, cuckoo-search algorithm U+0028 CSA U+0029 are observed to be superior when compared with other techniques such as real-coded genetic algorithm U+0028 RGA U+0029, particle swarm optimization U+0028 PSO U+0029 and gradient-search U+0028 GS U+0029 methods. Finally, statistical analysis affirms the potential of the proposed designs for its successful implementation.