Synthetic aperture processing for wireless communication signals with passive moving array

Abstract

In this paper, we investigate the use of synthetic aperture technique in communication signal processing with passive moving arrays. First, demodulation preprocessing schemes proper to distinctive situations are provided to increase the coherence time of target signal. Then demodulated signal sequence is resampled and reconstructed for synthetic array data production. Secondly we consider the practical implementation of direction of arrival estimation and present approaches to adjust for synthetic array data. The proposed technique incorporates Doppler information caused by the moving of platform into spatial processing, leading to significant enhancement in achievable array performance. Both theoretical analysis and numerical simulations are presented to illustrate the effectiveness of the proposed methods.

Appendix

To derive (23), the differential of $\varvec{c}(\theta )$ is firstly computed as

$$ \begin{aligned} \frac{{\partial \varvec{c}(\theta )}}{\partial \theta } & = j2\pi M\cos \theta \left[ {0,\frac{{df_{c} }}{c}e^{{j2\pi M\frac{{df_{c} }}{c}\sin \theta }} } \right.,{ \ldots },\left( {N - 1} \right)\frac{{df_{c} }}{c}e^{{j2\pi M\left( {N - 1} \right)\frac{{df_{c} }}{c}\sin \theta }} , \\ & \quad \frac{{vf_{c} }}{{cf_{s} }}e^{{j2\pi M\frac{{vf_{c} }}{{cf_{s} }}\sin \theta }} ,{ \ldots },\left. {\left( {\frac{{\left( {N - 1} \right)d}}{c} + \frac{{\left( {L - 1} \right)v}}{{cf_{s} }}} \right)f_{c} e^{{j2\pi M\left( {\frac{{\left( {N - 1} \right)df_{c} }}{c} + \frac{{\left( {L - 1} \right)vf_{c} }}{{cf_{s} }}} \right)\sin \theta }} } \right] \\ \end{aligned} $$

(24)

Then the inner product of ${{\partial \varvec{c}(\theta )} \mathord{\left/ {\vphantom {{\partial \varvec{c}(\theta )} {\partial \theta }}} \right. \kern-0pt} {\partial \theta }}$ leads to

$$ \frac{{\partial {\mathbf{c}}^{H} (\theta )}}{\partial \theta }\frac{{\partial {\mathbf{c}}(\theta )}}{\partial \theta } = NL\left( {2\pi M\cos \theta \frac{{f_{c} }}{c}} \right)^{2} \left( {\frac{{\left( {L - 1} \right)\left( {2L - 1} \right)}}{6}\left( {\frac{v}{{f_{s} }}} \right)^{2} } \right.\left. { + \frac{{\left( {N - 1} \right)\left( {2N - 1} \right)}}{6}d^{2} + \frac{{\left( {L - 1} \right)\left( {N - 1} \right)}}{2}\frac{vd}{{f_{s} }}} \right) $$

(25)

Similarly, the inner product of vector $\varvec{c}(\theta )$ and ${{\partial \varvec{c}(\theta )} \mathord{\left/ {\vphantom {{\partial \varvec{c}(\theta )} {\partial \theta }}} \right. \kern-0pt} {\partial \theta }}$ becomes

$$ \frac{{\partial \varvec{c}^{H} (\theta )}}{\partial \theta }\varvec{c}(\theta ) = - j2\pi NLM\cos \theta \frac{{f_{c} }}{c}\left( {\frac{{\left( {L - 1} \right)}}{2}\frac{v}{{f_{s} }} + \frac{{\left( {N - 1} \right)}}{2}d} \right) $$

(26)

With (25) and (26), the mean square error defined in (22) can be written as

$$ E\left[ {\left| {\hat{\theta } - \theta } \right|^{2} } \right] = \frac{{6\sigma_{n}^{2} \left( {1 + \frac{{\sigma_{n}^{2} }}{{LN\sigma_{\gamma }^{2} }}} \right)}}{{NLP\sigma_{\gamma }^{2} \left( {2\pi M\cos \theta \frac{{f_{c} }}{c}} \right)^{2} }}\left( {\left( {L^{2} - 1} \right)\left( {\frac{v}{{f_{s} }}} \right)^{2} + \left( {N^{2} - 1} \right)d^{2} } \right)^{ - 1} $$

(27)

Denoting ${{LN\sigma_{\gamma }^{2} } \mathord{\left/ {\vphantom {{LN\sigma_{\gamma }^{2} } {\sigma_{n}^{2} }}} \right. \kern-0pt} {\sigma_{n}^{2} }}$ as the array SNR (ASNR) and supposing $L^{2} \gg 1\;,\;N^{2} \gg 1\;$, $E\left[ {\left| {\hat{\theta } - \theta } \right|^{2} } \right]$ can be approximated as (23).