Quasi-maximum-Likelihood Estimator of PPS on the Uniform Linear Array

Abstract

In this paper, an idea from the quasi-maximum-likelihood (QML) algorithm has been applied to estimation of the direction of arrival (DOA) and parameters of the polynomial phase signals (PPS) impinging on the uniform linear array. The algorithm uses a search over a set of possible DOA values. For considered value, the DOA is compensated and resulted signal parameters are estimated using the original QML approach. In this way, several sets of potential estimates are obtained. Optimal DOA and PPS parameters belong to set that maximizes the proposed cost function. In the proposed approach, parameters are estimated using 2-D search over DOA and window widths used for calculation of the short Fourier transform in the QML that is an important advantage with respect to existing strategies like, for example, the polynomial beamformer requiring multi-dimensional or metaheuristic search strategies. Numerical validation of the proposed approach for both mono- and multi-component signals is provided in the paper. Both far- and near-field scenarios are considered.

Appendix

1.1 CRLB Derivation

The CRLB evaluation for signal model considered in the paper is addressed briefly. The CRLB derivations for case of a simplified array signal model are available in [18].

Our interest is to calculate the CRLB of parameter vector $\mathbf {E} =[a_{0},a_{1},\dots ,a_{P}$, $A,\tau ]$. Elements of vector $\mathbf {E}$ are denoted by $\mathbf {E}_{i}=a_{i}$, for $i\in 0,\dots ,P$, $E_{P+1}=A$, $E_{P+2}=\tau$. The CRLB of $\mathbf {E}_{i}$ is a diagonal element of the inverse Fisher information matrix $\mathbf {F}$ on position $\mathbf {F}_{i,i}$

$$\begin{aligned} \text {CRLB}\{{\mathbf {E}}_{i}\}=(\mathbf {F}^{-1})_{i,i}. \end{aligned}$$

(12)

Elements of the Fisher information matrix are second-order partial derivatives of the log-likelihood function. For details related to the Fisher matrix determination, refer to [18, 30, 32]. In the following, we give summary of our derivations related to this matrix with $F_{i,j}$ as the (i, j)th element of $(P+3)\times (P+3)$ matrix $\mathbf {F}$:

$$\begin{aligned} F_{i,j}= & {} 2\frac{A^{2}}{\sigma ^{2}}\nonumber \\&\times \left\{ \begin{array} [c]{cc} NM, &{} i=j=P+1\\ 0, &{} i=P+1\oplus j=P+1\\ \sum \limits _{n=0}^{N-1}\sum \limits _{k=1}^{M-1}\left( \sum \nolimits _{p=1}^{P} a_{p}pk(n-k\tau )^{p-1}(\varDelta t)^{p}\right) ^{2}, &{} i=j=P+2\\ \begin{array} [c]{c} \sum \limits _{n=0}^{N-1}\sum \limits _{k=1}^{M-1}(n-k\tau )^{\min (i,j)}(\varDelta t)^{\min (i,j)}\\ \times \,\sum \limits _{p=1}^{P}a_{p}pk(n-k\tau )^{p-1}(\varDelta t)^{p}, \end{array} &{} \begin{array} [c]{c} (i=P+2,j\le P)\\ \vee (i\le P,j=P+2) \end{array} \\ (\varDelta t)^{(i+j)}\sum \limits _{n=0}^{N-1}\sum \limits _{k=1}^{M-1} (n-k\tau )^{(i+j)}, &{} i\le P,\quad j\le P. \end{array} \right. \nonumber \\ \end{aligned}$$

(13)

Due to terms involving the ISD, we have evaluated the CRLB from (12) and (13) numerically.

1.2 NMS Method

Parameters of the proposed algorithm are refined in the final step using the NMS method. The NMS method finds minimum of the objective function using operations such as reflection, expansion, contraction and shrinking. Since our goal is function maximization, optimization function passed to the NMS method is $\varGamma (\mathbf {g})=1/J(\upsilon ;b_{1},\dots ,b_{P}),$ where $\mathbf {g}=[\upsilon ,b_{1},\dots ,b_{P}]$. The NMS method is summarized in Algorithm 4.

Standard values for the reflection, expansion, contraction and shrink coefficients are $\alpha =1$, $\gamma =2$, $\rho =0.5$ and $\varsigma =0.5$. Note that fminsearch function is MATLAB realization of the NMS method.