Browsing by Subject "Linear system of equations"

Now showing 1 - 5 of 5

Open Access
Calculation of the scalar diffraction field from curved surfaces by decomposing the three-dimensional field into a sum of Gaussian beams
(Optical Society of America, 2013) Şahin, E.; Onural, L.
We present a local Gaussian beam decomposition method for calculating the scalar diffraction field due to a twodimensional field specified on a curved surface. We write the three-dimensional field as a sum of Gaussian beams that propagate toward different directions and whose waist positions are taken at discrete points on the curved surface. The discrete positions of the beam waists are obtained by sampling the curved surface such that transversal components of the positions form a regular grid. The modulated Gaussian window functions corresponding to Gaussian beams are placed on the transversal planes that pass through the discrete beam-waist position. The coefficients of the Gaussian beams are found by solving the linear system of equations where the columns of the system matrix represent the field patterns that the Gaussian beams produce on the given curved surface. As a result of using local beams in the expansion, we end up with sparse system matrices. The sparsity of the system matrices provides important advantages in terms of computational complexity and memory allocation while solving the system of linear equations.
Open Access
Linear algebraic analysis of fractional Fourier domain interpolation
(IEEE, 2009) Öktem, Figen S.; Özaktaş, Haldun M.
In this work, we present a novel linear algebraic approach to certain signal interpolation problems involving the fractional Fourier transform. These problems arise in wave propagation, but the proposed approach to these can also be applicable to other areas. We see this interpolation problem as the problem of determining the unknown signal values from the given samples within some tolerable error. We formulate the problem as a linear system of equations and use the condition number as a measure of redundant information in given samples. By analyzing the effect of the number of known samples and their distributions on the condition number with simulation examples, we aim to investigate the redundancy and information relations between the given data.
Open Access
A novel technique for a linear system of equations applied to channel equalization
(IEEE, 2009) Pilancı, Mert; Arıkan, Orhan; Oǧuz, B.; Pınar, Mustafa Ç.
In many inverse problems of signal processing the problem reduces to a linear system of equations. Accurate and robust estimation of the solution with errors in both measurement vector and coefficient matrix is a challenging task. In this paper a novel formulation is proposed which takes into account the structure (e.g. Toeplitz, Hankel) and uncertainties of the system. A numerical algorithm is provided to obtain the solution. The proposed technique and other methods are compared in a channel equalization example which is a fundamental necessity in communication.
Open Access
Sparse solutions to an underdetermined system of linear equations via penalized Huber loss
(Springer, 2020) Kızılkale, C.; Pınar, Mustafa Çelebi
We investigate the computation of a sparse solution to an underdetermined system of linear equations using the Huber loss function as a proxy for the 1-norm and a quadratic error term à la Lasso. The approach is termed “penalized Huber loss”. The results of the paper allow to calculate a sparse solution using a simple extrapolation formula under a sign constancy condition that can be removed if one works with extreme points. Conditions leading to sign constancy, as well as necessary and sufficient conditions for computation of a sparse solution by penalized Huber loss, and ties among different solutions are presented.
Open Access
Structured least squares with bounded data uncertainties
(IEEE, 2009) Pilanci, Mert; Arıkan, Orhan; Oguz, B.; Pınar, Mustafa C.
In many signal processing applications the core problem reduces to a linear system of equations. Coefficient matrix uncertainties create a significant challenge in obtaining reliable solutions. In this paper, we present a novel formulation for solving a system of noise contaminated linear equations while preserving the structure of the coefficient matrix. The proposed method has advantages over the known Structured Total Least Squares (STLS) techniques in utilizing additional information about the uncertainties and robustness in ill-posed problems. Numerical comparisons are given to illustrate these advantages in two applications: signal restoration problem with an uncertain model and frequency estimation of multiple sinusoids embedded in white noise.