Analysis of Cross-Correlation Peak Distortion Caused by Antenna Array Space-Time Adaptive Processing

Abstract

Space-time adaptive processing (STAP) combines information available from both spatial and temporal domains, which can effectively enhance the GNSS receivers’ performance of interference suppression. But STAP has the potential to distort cross-correlation peak, which would deteriorate acquisition and tracking performance. In this paper, the cross-correlation peak distortion under space-time power inversion (PI) criterion and minimum variance distortionless response (MVDR) criterion are analyzed by theoretical analysis and simulations. Analysis and simulation show that the spatial correlation coefficient between interference and desired signal, taps number of Tapped Delay Line (TDL) and taps delay can directly affect the cross-correlation peak. The conclusions can be used to optimize the antenna array design and space-time adaptive algorithm for reducing cross-correlation peak distortion caused by STAP.

Appendix A

The space-time correlation matrix of the input signal, $\varvec{x^{\prime}}\left( t \right)$, can be written as

$$ \varvec{R}_{xx} = E\left[ {\varvec{x^{\prime}}\left( t \right)\left( {\varvec{x^{\prime}}\left( t \right)} \right)^{H} } \right] $$

(17)

In practice, interference power and noise power are much higher than the desired signal power, so the desired signal can be ignored in Eq. (17). Assuming the interference and the noise are not related. One gets

$$ \varvec{R}_{xx} = \varvec{R}_{jj} + \delta_{n}^{2} \varvec{I} $$

(18)

where $\varvec{R}_{jj}$ is the space-time correlation matrix of interference, $\delta_{n}^{2}$ denotes the noise variance, $\varvec{I}$ is a $MN \times MN$ unit matrix. Under narrowband assumption, the interference space-time correlation matrix can be written as

$$ \varvec{R}_{jj} = \varvec{R}_{J} \otimes \left( {\varvec{a}_{J} \varvec{a}_{J}^{H} } \right) $$

(19)

where $\varvec{R}_{J}$ represents the temporal correlation matrix, $\otimes$ denotes the Kronecker tensor product.

Because $\varvec{R}_{J}$ is a Hermite matrix, its eigen value decomposition can be expressed as

$$ \varvec{R}_{J} = \sum\limits_{i = 1}^{M} {\lambda_{i} u_{i} u_{i}^{H} } \quad \quad \left( {i = 1,2, \cdots ,M} \right) $$

(20)

where $\lambda_{i}$ is the eigen value, $u_{i}$ is the corresponding eigen vector. $\varvec{a}_{J} \varvec{a}_{J}^{H}$ is also a Hermite matrix with only one nonzero eigen value which is $N$. According to the property of Hermite matrix, the nonzero eigen values of $\varvec{R}_{jj}$ are $N\lambda_{i}$ and the corresponding eigen vectors are $\varvec{\xi}_{i} = \varvec{u}_{i} \otimes \frac{{\varvec{a}_{J} }}{\sqrt N },\left( {i = 1,2, \cdots ,M} \right)$. According to Schmidt orthogonalization, the eigen vectors $\varvec{\xi}_{i} = \varvec{u}_{i} \otimes \frac{{\varvec{a}_{J} }}{\sqrt N },\left( {i = 1,2, \cdots ,M} \right)$ can be extended to $MN$ standard orthogonal eigen vectors. Hence, the eigen value decomposition of $\varvec{R}_{xx}$ can be written as

$$ \begin{aligned} \varvec{R}_{xx} & = \sum\limits_{i = 1}^{M} {N\lambda_{i} \left( {\varvec{\xi}_{i}\varvec{\xi}_{i}^{H} } \right)} + \delta_{n}^{2} \sum\limits_{i = 1}^{MN} {\left( {\varvec{\xi}_{i}\varvec{\xi}_{i}^{H} } \right)} \\ & = \sum\limits_{i = 1}^{M} {\left( {N\lambda_{i} + \delta_{n}^{2} } \right)\left( {\varvec{\xi}_{i}\varvec{\xi}_{i}^{H} } \right)} + \delta_{n}^{2} \sum\limits_{i = M + 1}^{MN} {\left( {\varvec{\xi}_{i}\varvec{\xi}_{i}^{H} } \right)} \\ \end{aligned} $$

(21)

The inverse of $\varvec{R}_{xx}$ can be obtained as

$$ \begin{aligned} \varvec{R}_{xx}^{ - 1} & = \sum\limits_{i = 1}^{M} {\frac{1}{{N\lambda_{i} + \delta_{n}^{2} }}\left( {\varvec{\xi}_{i}\varvec{\xi}_{i}^{H} } \right)} + \frac{1}{{\delta_{n}^{2} }}\sum\limits_{i = M + 1}^{MN} {\left( {\varvec{\xi}_{i}\varvec{\xi}_{i}^{H} } \right)} \\ & = \frac{1}{{\delta_{n}^{2} }}\sum\limits_{i = 1}^{MN} {\left( {\varvec{\xi}_{i}\varvec{\xi}_{i}^{H} } \right)} - \frac{1}{{\delta_{n}^{2} }}\sum\limits_{i = 1}^{M} {\frac{{N\lambda_{i} }}{{N\lambda_{i} + \delta_{n}^{2} }}\left( {\varvec{\xi}_{i}\varvec{\xi}_{i}^{H} } \right)} \\ & = \frac{1}{{\delta_{n}^{2} }}\left[ {\varvec{I} - \sum\limits_{i = 1}^{M} {\frac{{\lambda_{i} }}{{N\lambda_{i} + \delta_{n}^{2} }}\left( {\varvec{u}_{i} \otimes \varvec{a}_{J} } \right)} \left( {\varvec{u}_{i} \otimes \varvec{a}_{J} } \right)^{H} } \right] \\ \end{aligned} $$

(22)