Scalable antinoise-robustness estimation of frequency and direction of arrival based on the relaxed sparse array

Abstract

In order to realize low-cost frequency and direction of arrival (DOA) estimation in the harsh temporal-spatial undersampling condition, this paper proposed a joint estimator with scalable antinoise robustness. Essentially, owing to that two basic problems are well solved, the proposed estimator concurrently achieves high array sparsity, low hardware complexity and scalable antinoise robustness. On one hand, in the sparse sensor arrangement, a relaxed coprime array was proposed, which is distinguished with low hardware cost (only $L=3$ sensors in low sampling rates are employed) and high array sparsity (the inter-element spacings are much greater than the Nyquist spacings). On the other hand, in the design of the frequency and DOA reconstruction algorithm, a series of techniques (including the reconstruction towards robustness in residue number system, spectrum correction and phase difference adjustment) are organically integrated, which endows the proposed estimator with scalable antinoise robustness. Numerical results confirmed the above advantages, which present the proposed joint estimator vast potentials in the radar, remote sensing and other passive sensing related applied fields.

Appendix: Ratio spectrum correction

Zhang et al. (2001) pointed out that, given an M-length exponential sequence $x(n)=A_0e^{j(2\pi {f}_{0}n+\varphi _0)},n=0,\ldots ,M-1$, $f_0=(k^*+\delta )/M,k^*\in {\mathbb {Z}}^+,\delta \in [-0.5, 0.5)$. The following procedure provides the frequency estimate ${\hat{f}}_0$, the amplitude estimate ${\hat{A}}_0$ and the phase estimate ${\hat{\varphi }} _0$ from the hanning windowed DFT result $X_f(k), k=0,\ldots , M-1$.

Step 1 :

Calculate the amplitude ratio v between the peak DFT bin $X_f(k_{p})$ and its sub-peak neighbor, i.e.,

$$\begin{aligned} v =\frac{ |X_f( k^{*})|}{\max \{ | X_f( k^{*}-1)|, | X_f( k^{*}+1) |\} } \end{aligned}$$

(43)

Step 2 :

The absolute value u of the frequency offset $\delta$ is estimated as

$$\begin{aligned} u=(2-v )/(1+v ), \end{aligned}$$

(44)

and the frequency offset is estimated as

$$\begin{aligned} {\hat{\delta }} = \left\{ \begin{array}{ll} u, &{}\hbox {if} \ |X_f( k^{*}+1)|> |X_f( k^{*}-1)|\\ -u, &{}\hbox {else} \end{array} \right. \end{aligned}$$

(45)

Step 3 :

Calculate the frequency estimate ${\hat{f}}_0$,the amplitude estimate ${\hat{A}}_0$ and the phase estimate ${{\hat{\varphi }}} _0$ as

$$\begin{aligned} {\hat{f}}_{0}= & {} (k^{*}+{\hat{\delta }} ) /M, \end{aligned}$$

(46)

$$\begin{aligned} {{\hat{A}}}_{0}= & {} \pi {\hat{\delta }} (1-{\hat{\delta }} ^{2}) \cdot |X_{f}(k^{*})|/\sin (\pi {\hat{\delta }} ), \end{aligned}$$

(47)

$$\begin{aligned} {{\hat{\varphi }}}_0= & {} \hbox {ang}[X_f(k^{*})]-\pi {\hat{\delta }} (M-1)/M, \end{aligned}$$

(48)

where $\hbox { ang}(\cdot$) refers to the angle operation.