Sensor Array Calibration in Presence of Mutual Coupling and Gain/Phase Errors by Combining the Spatial-Domain and Time-Domain Waveform Information of the Calibration Sources

Abstract

This paper is concerned with the maximum-likelihood (ML) calibration methods tailored to the antenna arrays whose spatial responses are perturbed by mutual coupling effects and unknown sensor gain/phase responses. Unlike the existing work, the proposed methods are capable of jointly exploiting the spatial-domain information and time-domain waveform information of the calibration sources. Two kinds of numerical optimization algorithm are devised dependent on different array geometries. One is suitable for arbitrary irregular array manifold, while the other applies to some particular uniform arrays. Additionally, based on the maximum a posteriori probability (MAP) criterion, we extend the two algorithms to the scenario where the true values of the calibration source azimuths deviate slightly from the nominal ones with a priori known Gaussian distribution. The Cramér–Rao bound (CRB) expressions for the unknowns are derived in the absence and presence of the azimuth deviations, respectively. Simulation results support that the performances of the proposed algorithms are preferable to the ones which merely employs the spatial-domain information of the calibration sources, and are able to attain the corresponding CRB.

Appendices

Appendix A: Proof of (22)–(24)

In light of the first-order derivation operator of the orthogonal projection matrix, it follows that

(86)

The substitution of (86) into (20) yields

(87)

which implies that

(88)

Similarly, it follows that

$$ \frac{\partial \phi^{( \mathrm{II} )}[ \boldsymbol{c} ]}{\partial ( \bar{\boldsymbol{I}}_{K}\boldsymbol{c}^{( \mathrm{i} )} )} = 2 \cdot\operatorname{Im} \bigl\{ \boldsymbol{h}^{( \mathrm{II} )}[ \boldsymbol{c} ] \bigr\} $$

(89)

On the other hand, after some algebraic manipulations we can approximately obtain

(90)

which results in

(91)

Through similar derivation, it can also be concluded that

$$ \everymath{\displaystyle} \left \{ \begin{array} {l} \frac{\partial^{2}\phi^{( \mathrm{II} )}[ \boldsymbol{c} ]}{\partial ( \bar{\boldsymbol{I}}_{K}\boldsymbol{c}^{( \mathrm{r} )} ) \cdot \partial ( \bar{\boldsymbol{I}}_{K}\boldsymbol{c}^{( \mathrm{i} )} )^{\mathrm{T}}} \approx 2 \cdot\operatorname{Re} \bigl\{ \mathrm{i} \cdot \boldsymbol{H}^{( \mathrm{II} )}[ \boldsymbol{c} ] \bigr\} = - 2 \cdot \operatorname{Im} \bigl\{ \boldsymbol{H}^{( \mathrm{II} )}[ \boldsymbol{c} ] \bigr\} \\\noalign{\vspace{5pt}} \frac{\partial^{2}\phi^{( \mathrm{II} )}[ \boldsymbol{c} ]}{\partial ( \bar{\boldsymbol{I}}_{K}\boldsymbol{c}^{( \mathrm{i} )} ) \cdot \partial ( \bar{\boldsymbol{I}}_{K}\boldsymbol{c}^{( \mathrm{r} )} )^{\mathrm{T}}} \approx 2 \cdot\operatorname{Re} \bigl\{ - \mathrm{i} \cdot \boldsymbol{H}^{( \mathrm{II} )}[ \boldsymbol{c} ] \bigr\} = 2 \cdot\operatorname{Im} \bigl\{ \boldsymbol{H}^{( \mathrm{II} )}[ \boldsymbol{c} ] \bigr\} \\\noalign{\vspace{5pt}} \frac{\partial^{2}\phi^{( \mathrm{II} )}[ \boldsymbol{c} ]}{\partial ( \bar{\boldsymbol{I}}_{K}\boldsymbol{c}^{( \mathrm{i} )} ) \cdot \partial ( \bar{\boldsymbol{I}}_{K}\boldsymbol{c}^{( \mathrm{i} )} )^{\mathrm{T}}} \approx 2 \cdot\operatorname{Re} \bigl\{ \boldsymbol{H}^{( \mathrm{II} )}[ \boldsymbol{c} ] \bigr\} \end{array} \right . $$

(92)

Based on the above analysis, (22)–(24) hold true.

Appendix B: Proof of (33)–(37)

Following the same procedure as in Appendix A leads to

(93)

which implies that

(94)

On the other hand, making some algebraic effort we can approximately obtain

(95)

which yields

(96)

Based on the above analysis, equalities (33)–(37) are proved.

Appendix C: Proof of (40)–(43)

Follow along the line of the derivation in Appendix B results in

(97)

which implies that

(98)

On the other hand, after some algebraic manipulations we can approximately obtain

(99)

which yields

(100)

At this point, the proof of (40)–(43) is finished.