On approximate least squares estimators of parameters of one-dimensional chirp signal

Abstract

In this paper, we address the problem of parameter estimation of a 2-D chirp model under the assumption that the errors are stationary. We extend the 2-D periodogram method for the sinusoidal model, to find initial values to use in any iterative procedure to compute the least squares estimators (LSEs) of the unknown parameters, to the 2-D chirp model. Next we propose an estimator, known as the approximate least squares estimator (ALSE), that is obtained by maximising a periodogram-type function and is observed to be asymptotically equivalent to the LSE. Moreover the asymptotic properties of these estimators are obtained under slightly mild conditions than those required for the LSEs. For the multiple component 2-D chirp model, we propose a sequential method of estimation of the ALSEs, that significantly reduces the computational difficulty involved in reckoning the LSEs and the ALSEs. We perform some simulation studies to see how the proposed method works and a data set has been analysed for illustrative purposes.

An efficient methodology to estimate the parameters of a two-dimensional chirp signal model

“…However, this replacement simplifies the estimation process to a great extent as the evaluation of periodogram-type functions does not involve matrix inversion. This relationship is analogous to the one that was first proposed by Walker [2] for the sinusoidal model and later Grover et al [17,18] extended the same for 1-D and 2-D chirp models. The estimators obtained by maximising a periodogram function [2] or a periodogram-type function [17,18] are called the approximate least squares estimators (ALSEs).…”

“…This relationship is analogous to the one that was first proposed by Walker [2] for the sinusoidal model and later Grover et al [17,18] extended the same for 1-D and 2-D chirp models. The estimators obtained by maximising a periodogram function [2] or a periodogram-type function [17,18] are called the approximate least squares estimators (ALSEs). In fact, Grover et al [17,18] showed that the ALSEs are strongly consistent and asymptotically equivalent to the corresponding LSEs.…”

“…We propose a sequential procedure to estimate these parameters. The main idea supporting the proposed sequential procedure is same as that behind the ones proposed by Prasad et al [9] for a sinusoidal model and Lahiri et al [13] and Grover et al [17] for a chirp model− the orthogonality of different regressor vectors. Along with the computationally efficiency, the sequential method provides estimators with the same rates of convergence as the LSEs.…”

“…. y(M, N ) T is the observed data vector, and the matrix W(α 1 , β1 , γ1 , δ1 ) can be obtained by replacing α, β, γ and δ by α1 , β1 , γ1 and δ1 respectively in (17).…”

Chirp-like model and its parameters estimation

“…For comparison of the chirp-like model with the chirp model, we re-analyse these data sets by fitting chirp model to each of them (for methodology, see Lahiri et al [11] and Grover et al [4]). In the following table, we report the number of components required to fit the chirp model and the chirp-like model to each of the data sets and in the subsequent figures, we plot the original data along with the estimated signals obtained by fitting a chirp model and a chirp-like model to these data.…”