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Robust Direction-Finding Method for Sensor Gain and Phase Uncertainties in Non-uniform Environment

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Abstract

A direction-of-arrival (DOA) estimation algorithm, which is robust to sensor gain and phase uncertainties for completely uncalibrated arrays in a non-uniform noise environment, is proposed in this study. As a result of the sensor gain uncertainties or the shielding effects for some baffled arrays, the noise power may vary with sensors. Therefore, a non-uniform noise model is considered. A cost function established by the orthogonality of subspaces is accumulated along several rough space intervals surrounding the real angles of sources. After analyzing the influences of rough space intervals, an iterative refinement operation is carried out to improve the estimation performance of the DOA and sensor gain and phase responses. The Cramér–Rao bounds of the DOA and sensor gain and phase in the non-uniform noise model are derived. Simulations and experimental results show the superiority of the proposed method.

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Acknowledgements

This work was supported in part by the National Natural Science Foundation of China under Grants 11527809, 11604259, and 51679204, and in part by the China Postdoctoral Science Foundation under Grant 2019M653569.

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Authors and Affiliations

  1. National Laboratory of Radar Signal Processing, Xidian University, Xi’an, 710071, Shaanxi, China

    Long Yang & Guisheng Liao

  2. School of Marine Science and Technology, Northwestern Polytechnical University, Xi’an, 710072, China

    Yixin Yang & Yong Wang

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  1. Long Yang
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  2. Yixin Yang
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  3. Guisheng Liao
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  4. Yong Wang
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Correspondence to Yixin Yang.

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Appendix: The CRB in the Non-uniform Noise Environment

Appendix: The CRB in the Non-uniform Noise Environment

In this appendix, the CRB under the non-uniform noise is calculated. The parameter vector containing all unknown parameters is

$$ {\varvec{z}} = [{\boldsymbol{\theta }}^{\text{T}} ,{\varvec{g}}^{\text{T}} ,{\varvec{\varphi }}^{\text{T}} ,{\varvec{p}}^{\text{T}} ,{\boldsymbol{\sigma }}^{\text{T}} ]^{\text{T}} , $$
(23)

where \( {\boldsymbol{\theta }} = [\theta_{1} ,\theta_{2} , \ldots ,\theta_{K} ]^{\text{T}} \) is the direction of sources; \( {\varvec{g}} \), \( {\varvec{\varphi }} \), and \( {\varvec{p}} \) are the sensor gains, phases, and signal powers, respectively; and \( {\boldsymbol{\sigma }} = [\sigma_{1}^{{}} ,\sigma_{2}^{{}} , \ldots ,\sigma_{M}^{{}} ]^{\text{T}} \) includes the noise powers of each sensor.

When \( N \) independent samples of a zero-mean Gaussian process is given, the CRB equals the inverse of the Fisher information matrix (FIM) whose \( (m,n){\text{th}} \) entry is given by

$$ {\varvec{F}}_{mn} = N \times {\text{tr}}\left\{ {{\varvec{R}}^{ - 1} \frac{{\partial {\varvec{R}}}}{{\partial {\varvec{z}}_{m} }}{\varvec{R}}^{ - 1} \frac{{\partial {\varvec{R}}}}{{\partial {\varvec{z}}_{n} }}} \right\}, $$
(24)

where \( {\text{tr}}\left\{ \cdot \right\} \) denotes the trace operator and \( {\varvec{R}} \) is defined in (3). Then, the FIM can be expressed as a block matrix:

$$ {\varvec{F}} = \left[ {\begin{array}{*{20}c} {\begin{array}{*{20}c} {{\varvec{F}}_{{{\boldsymbol{\theta \theta }}}} } \\ {{\varvec{F}}_{{{\boldsymbol{\theta g}}}}^{\text{T}} } \\ \end{array} } & {\begin{array}{*{20}c} {{\varvec{F}}_{{{\boldsymbol{\theta g}}}} } \\ {{\varvec{F}}_{{{\varvec{gg}}}} } \\ \end{array} } & {\begin{array}{*{20}c} {{\varvec{F}}_{{{\boldsymbol{\theta \varphi }}}} } \\ {{\varvec{F}}_{{{\varvec{g\varphi }}}} } \\ \end{array} } & {\begin{array}{*{20}c} {\begin{array}{*{20}c} {{\varvec{F}}_{{{\boldsymbol{\theta p}}}} } \\ {{\varvec{F}}_{{{\varvec{gp}}}} } \\ \end{array} } & {\begin{array}{*{20}c} {{\varvec{F}}_{{{\boldsymbol{\theta \sigma }}}} } \\ {{\varvec{F}}_{{{\varvec{g\sigma }}}} } \\ \end{array} } \\ \end{array} } \\ {{\varvec{F}}_{{{\boldsymbol{\theta \varphi }}}}^{\text{T}} } & {{\varvec{F}}_{{{\varvec{g\varphi }}}}^{\text{T}} } & {{\varvec{F}}_{{{\varvec{\varphi \varphi }}}} } & {\begin{array}{*{20}c} {{\varvec{F}}_{{{\varvec{\varphi p}}}} } & {{\varvec{F}}_{{{\varvec{\varphi \sigma }}}} } \\ \end{array} } \\ {{\varvec{F}}_{{{\boldsymbol{\theta p}}}}^{\text{T}} } & {{\varvec{F}}_{{{\varvec{gp}}}}^{\text{T}} } & {{\varvec{F}}_{{{\varvec{\varphi p}}}}^{\text{T}} } & {\begin{array}{*{20}c} {{\varvec{F}}_{{{\varvec{pp}}}} } & {{\varvec{F}}_{{{\varvec{p\sigma }}}} } \\ \end{array} } \\ {{\varvec{F}}_{{{\boldsymbol{\theta \sigma }}}}^{\text{T}} } & {{\varvec{F}}_{{{\varvec{g\sigma }}}}^{\text{T}} } & {{\varvec{F}}_{{{\varvec{\varphi \sigma }}}}^{\text{T}} } & {\begin{array}{*{20}c} {{\varvec{F}}_{{{\varvec{p\sigma }}}}^{\text{T}} } & {{\varvec{F}}_{{{\boldsymbol{\sigma \sigma }}}} } \\ \end{array} } \\ \end{array} } \right]. $$
(25)

First, we define the matrices \( {\dot{\bar{\varvec{A}}}}_{\theta }^{{}} \), \( {\dot{\bar{\varvec{A}}}}_{g}^{{}} \), and \( {\dot{\bar{\varvec{A}}}}_{\varphi }^{{}} \) as

$$ {\dot{\bar{\varvec{A}}}}_{\theta }^{{}} = \sum\limits_{k = 1}^{K} {\frac{{\partial {\bar{\varvec{A}}}}}{{\partial \theta_{k} }}} ,{\dot{\bar{\varvec{A}}}}_{g}^{{}} = \sum\limits_{m = 1}^{M} {\frac{{\partial {\bar{\varvec{A}}}^{{}} }}{{\partial g_{m} }}} ,{\dot{\bar{\varvec{A}}}}_{\varphi }^{{}} = \sum\limits_{m = 1}^{M} {\frac{{\partial {\bar{\varvec{A}}}^{{}} }}{{\partial \varphi_{m} }}} , $$
(26)

where \( {\bar{\varvec{A}}} = {\text{diag}}\{ {\boldsymbol{\gamma }}\} {\varvec{A}} \) is the real steering vector and \( {\varvec{A}} \) is the nominal steering vector. A \( (M - M_{c} ) \times M \) selection matrix \( {\varvec{H}} \) is constituted by \( {\varvec{l}} \) rows of the identity matrix; the set \( {\varvec{l}} \) contains \( M - M_{c} \) uncalibrated sensors.

In consideration of the non-uniform noise model, the last column of the block matrices should be recalculated, and the other blocks remain unchanged, such as that in [8] and [23].

According to (3), we have

$$ \frac{{\partial {\varvec{R}}}}{{\partial {\boldsymbol{\theta }}_{p} }} = {\dot{\bar{\varvec{A}}}}_{\theta }^{{}} {\varvec{e}}_{p} {\varvec{e}}_{p}^{\text{T}} {\varvec{PA}}^{\text{H}} + {\varvec{APe}}_{p} {\varvec{e}}_{p}^{\text{T}} {\dot{\bar{\varvec{A}}}}_{\theta }^{\text{H}} , $$
(27a)
$$ \frac{{\partial {\varvec{R}}}}{{\partial {\varvec{g}}_{p} }} = {\varvec{e}}_{p} {\varvec{e}}_{p}^{\text{T}} {\dot{\bar{\varvec{A}}}}_{g}^{{}} {\varvec{PA}}^{\text{H}} + {\varvec{AP\dot{\bar{A}}}}_{g}^{\text{H}} {\varvec{e}}_{p} {\varvec{e}}_{p}^{\text{T}} , $$
(27b)
$$ \frac{{\partial {\varvec{R}}}}{{\partial {\varvec{\varphi }}_{p} }} = {\varvec{e}}_{p} {\varvec{e}}_{p}^{\text{T}} {\dot{\bar{\varvec{A}}}}_{\varphi }^{{}} {\varvec{PA}}^{\text{H}} + {\varvec{AP\dot{\bar{A}}}}_{\varphi }^{\text{H}} {\varvec{e}}_{p} {\varvec{e}}_{p}^{\text{T}} , $$
(27c)
$$ \frac{{\partial {\varvec{R}}}}{{\partial {\varvec{p}}_{p} }} = {\bar{\varvec{A}e}}_{p} {\varvec{e}}_{p}^{\text{T}} {\bar{\varvec{A}}}^{\text{H}} , $$
(27d)
$$ \frac{{\partial {\varvec{R}}}}{{\partial {\boldsymbol{\sigma }}_{p} }} = {\varvec{e}}_{p} {\varvec{e}}_{p}^{\text{T}} , $$
(27e)

where \( {\varvec{e}}_{p} \) is a vector with the \( p{\text{th}} \) element equal to 1 and the other equal to 0.

Substituting (27a)–(27e) into (24), we can, respectively, obtain \( {\varvec{F}}_{{{\boldsymbol{\theta \sigma }}}} \), \( {\varvec{F}}_{{{\varvec{g\sigma }}}} \), \( {\varvec{F}}_{{{\varvec{\varphi \sigma }}}} \), \( {\varvec{F}}_{{{\varvec{p\sigma }}}} \), and \( {\varvec{F}}_{{{\boldsymbol{\sigma \sigma }}}} \) as

$$ {\varvec{F}}_{{{\boldsymbol{\theta \sigma }}}} = 2\text{Re} \left\{ {\left( {{\varvec{P\bar{A}}}^{\text{H}} {\varvec{R}}^{ - 1} } \right) \circ \left( {{\varvec{R}}^{ - 1} {\dot{\bar{\varvec{A}}}}_{\theta } } \right)^{\text{T}} } \right\}, $$
(28a)
$$ {\varvec{F}}_{{{\varvec{g\sigma }}}} = 2\text{Re} \left\{ {{\varvec{H}}\left( {{\dot{\bar{\varvec{A}}}}_{g} {\varvec{P\bar{A}}}^{\text{H}} {\varvec{R}}^{ - 1} } \right) \circ \left( {{\varvec{R}}^{ - 1} } \right)^{\text{T}} } \right\}, $$
(28b)
$$ {\varvec{F}}_{{{\varvec{\varphi \sigma }}}} = 2\text{Re} \left\{ {{\varvec{H}}\left( {{\dot{\bar{\varvec{A}}}}_{\varphi } {\varvec{P\bar{A}}}^{\text{H}} {\varvec{R}}^{ - 1} } \right) \circ \left( {{\varvec{R}}^{ - 1} } \right)^{\text{T}} } \right\}, $$
(28c)
$$ {\varvec{F}}_{{{\varvec{p\sigma }}}} = \left( {{\bar{\varvec{A}}}^{\text{H}} {\varvec{R}}^{ - 1} } \right) \circ \left( {{\varvec{R}}^{ - 1} {\bar{\varvec{A}}}} \right)^{\text{T}} , $$
(28d)
$$ {\varvec{F}}_{{{\boldsymbol{\sigma \sigma }}}} = \left( {{\varvec{R}}^{ - 1} } \right) \circ \left( {{\varvec{R}}^{ - 1} } \right)^{\text{T}} . $$
(28e)

The other block matrices of FIM are listed as follows:

$$ {\varvec{F}}_{{{\boldsymbol{\theta \theta }}}} = 2\text{Re} \left\{ {\left( {{\varvec{P\bar{A}}}^{\text{H}} {\varvec{R}}^{ - 1} {\bar{\varvec{A}P}}} \right) \circ \left( {{\dot{\bar{\varvec{A}}}}_{\theta }^{\text{H}} {\varvec{R}}^{ - 1} {\dot{\bar{\varvec{A}}}}_{\theta } } \right)^{\text{T}} + \left( {{\varvec{P\bar{A}}}^{\text{H}} {\varvec{R}}^{ - 1} {\dot{\bar{\varvec{A}}}}_{\theta } } \right) \circ \left( {{\varvec{P\bar{A}}}^{\text{H}} {\varvec{R}}^{ - 1} {\dot{\bar{\varvec{A}}}}_{\theta } } \right)^{\text{T}} } \right\}, $$
$$ {\varvec{F}}_{{{\boldsymbol{\theta g}}}} = 2\text{Re} \left\{ {\left[ {\left( {{\varvec{P\bar{A}}}^{\text{H}} {\varvec{R}}^{ - 1} } \right) \circ \left( {{\dot{\bar{\varvec{A}}}}_{g} {\varvec{P\bar{A}}}^{\text{H}} {\varvec{R}}^{ - 1} {\dot{\bar{\varvec{A}}}}_{\theta } } \right)^{\text{T}} + \left( {{\varvec{P\bar{A}}}^{\text{H}} {\varvec{R}}^{ - 1} {\bar{\varvec{A}P\dot{\bar{A}}}}_{g}^{\text{H}} } \right) \circ \left( {{\varvec{R}}^{ - 1} {\dot{\bar{\varvec{A}}}}_{\theta } } \right)^{\text{T}} } \right]{\varvec{H}}^{\text{T}} } \right\}, $$
$$ {\varvec{F}}_{{{\boldsymbol{\theta \varphi }}}} = 2\text{Re} \left\{ {\left[ {\left( {{\varvec{P\bar{A}}}^{\text{H}} {\varvec{R}}^{ - 1} } \right) \circ \left( {{\dot{\bar{\varvec{A}}}}_{\varphi } {\varvec{P\bar{A}}}^{\text{H}} {\varvec{R}}^{ - 1} {\dot{\bar{\varvec{A}}}}_{\theta } } \right)^{\text{T}} + \left( {{\varvec{P\bar{A}}}^{\text{H}} {\varvec{R}}^{ - 1} {\bar{\varvec{A}P\dot{\bar{A}}}}_{\varphi }^{\text{H}} } \right) \circ \left( {{\varvec{R}}^{ - 1} {\dot{\bar{\varvec{A}}}}_{\theta } } \right)^{\text{T}} } \right]{\varvec{H}}^{\text{T}} } \right\}, $$
$$ {\varvec{F}}_{{{\boldsymbol{\theta p}}}} = 2\text{Re} \left\{ {\left( {{\varvec{P\bar{A}}}^{\text{H}} {\varvec{R}}^{ - 1} {\bar{\varvec{A}}}} \right) \circ \left( {{\bar{\varvec{A}}}^{\text{H}} {\varvec{R}}^{ - 1} {\dot{\bar{\varvec{A}}}}_{\theta } } \right)^{\text{T}} } \right\}, $$
$$ {\varvec{F}}_{{{\varvec{gg}}}} = 2\text{Re} \left\{ {{\varvec{H}}\left[ {\left( {{\dot{\bar{\varvec{A}}}}_{g} {\varvec{P\bar{A}}}^{\text{H}} {\varvec{R}}^{ - 1} } \right) \circ \left( {{\dot{\bar{\varvec{A}}}}_{g} {\varvec{P\bar{A}}}^{\text{H}} {\varvec{R}}^{ - 1} } \right)^{\text{T}} + \left( {{\dot{\bar{\varvec{A}}}}_{g} {\varvec{P\bar{A}}}^{\text{H}} {\varvec{R}}^{ - 1} {\bar{\varvec{A}P\dot{\bar{A}}}}_{g}^{\text{H}} } \right) \circ \left( {{\varvec{R}}^{ - 1} } \right)^{\text{T}} } \right]{\varvec{H}}^{\text{T}} } \right\}, $$
$$ {\varvec{F}}_{{{\varvec{g\varphi }}}} = 2\text{Re} \left\{ {{\varvec{H}}\left[ {\left( {{\dot{\bar{\varvec{A}}}}_{g} {\varvec{P\bar{A}}}^{\text{H}} {\varvec{R}}^{ - 1} } \right) \circ \left( {{\dot{\bar{\varvec{A}}}}_{\varphi } {\varvec{P\bar{A}}}^{\text{H}} {\varvec{R}}^{ - 1} } \right)^{\text{T}} + \left( {{\dot{\bar{\varvec{A}}}}_{\varphi } {\varvec{P\bar{A}}}^{\text{H}} {\varvec{R}}^{ - 1} {\bar{\varvec{A}P\dot{\bar{A}}}}_{g}^{\text{H}} } \right) \circ \left( {{\varvec{R}}^{ - 1} } \right)^{\text{T}} } \right]{\varvec{H}}^{\text{T}} } \right\}, $$
$$ {\varvec{F}}_{{{\varvec{gp}}}} = 2\text{Re} \left\{ {{\varvec{H}}\left[ {\left( {{\dot{\bar{\varvec{A}}}}_{g} {\varvec{P\bar{A}}}^{\text{H}} {\varvec{R}}^{ - 1} {\bar{\varvec{A}}}} \right) \circ \left( {{\bar{\varvec{A}}}^{\text{H}} {\varvec{R}}^{ - 1} } \right)^{\text{T}} } \right]{\varvec{H}}^{\text{T}} } \right\}, $$
$$ {\varvec{F}}_{{{\varvec{\varphi \varphi }}}} = 2\text{Re} \left\{ {{\varvec{H}}\left[ {\left( {{\dot{\bar{\varvec{A}}}}_{\varphi } {\varvec{P\bar{A}}}^{\text{H}} {\varvec{R}}^{ - 1} } \right) \circ \left( {{\dot{\bar{\varvec{A}}}}_{\varphi } {\varvec{P\bar{A}}}^{\text{H}} {\varvec{R}}^{ - 1} } \right)^{\text{T}} + \left( {{\dot{\bar{\varvec{A}}}}_{\varphi } {\varvec{P\bar{A}}}^{\text{H}} {\varvec{R}}^{ - 1} {\bar{\varvec{A}P\dot{\bar{A}}}}_{\varphi }^{\text{H}} } \right) \circ \left( {{\varvec{R}}^{ - 1} } \right)^{\text{T}} } \right]{\varvec{H}}^{\text{T}} } \right\}, $$
$$ {\varvec{F}}_{{{\varvec{\varphi p}}}} = 2\text{Re} \left\{ {{\varvec{H}}\left[ {\left( {{\dot{\bar{\varvec{A}}}}_{\varphi } {\varvec{P\bar{A}}}^{\text{H}} {\varvec{R}}^{ - 1} {\bar{\varvec{A}}}} \right) \circ \left( {{\bar{\varvec{A}}}^{\text{H}} {\varvec{R}}^{ - 1} } \right)^{\text{T}} } \right]{\varvec{H}}^{\text{T}} } \right\}, $$
$$ {\varvec{F}}_{{{\varvec{pp}}}} = \left( {{\bar{\varvec{A}}}^{\text{H}} {\varvec{R}}^{ - 1} {\bar{\varvec{A}}}} \right) \circ \left( {{\bar{\varvec{A}}}^{\text{H}} {\varvec{R}}^{ - 1} {\bar{\varvec{A}}}} \right)^{\text{T}} . $$

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Yang, L., Yang, Y., Liao, G. et al. Robust Direction-Finding Method for Sensor Gain and Phase Uncertainties in Non-uniform Environment. Circuits Syst Signal Process 39, 1943–1964 (2020). https://doi.org/10.1007/s00034-019-01237-4

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