Direct Position Determination of Multiple Noncircular Sources with a Moving Array

Abstract

Compared with conventional two-step localization methods, direct position determination (DPD) is a promising technique that offers superior performance under low signal-to-noise ratio conditions. However, existing DPD methods mainly focus on complex circular sources without considering noncircular signals, which can be exploited to enhance the localization accuracy. This study proposes an improved subspace data fusion (SDF)-based DPD algorithm for multiple noncircular sources with a moving array. By constructing and decomposing the extended covariance matrices, extended noise subspaces are obtained for all positions of the moving array. The source positions are then directly estimated by fusing the extended noise subspaces without computing the intermediate parameters, thereby avoiding the data association problem inherent in two-step methods. Our proposed DPD algorithm combines the low complexity of SDF with the high robustness to noise and sensor errors that comes from exploiting signal noncircularity. Specifically, a closed-form expression for the localization mean square error (MSE) of the algorithm and the stochastic Cramér–Rao bound for strict-sense noncircular signals are derived. Simulation results validate our theoretical prediction for MSE and also demonstrate that the proposed algorithm outperforms other localization methods in terms of accuracy and capacity to resolve noncircular sources.

Appendices

Appendix 1

Proof of (29):

Let $p_i$ be the ith entry of the position vector ${\varvec{p}}\in {\mathbb {R}}^{D\times 1}$. To derive ${\hat{{\varvec{v}}}}^{1( {\varvec{p}} )} (\varvec{\bar{{p}}}_q )$, we need to start from the first-order partial derivative of $\hat{{V}}({\varvec{p}})$ (see 20) with respect to $p_i$. This is given by

$$\begin{aligned} \hat{{v}}^{( {p_i } )} ({\varvec{p}})= & {} \frac{\partial \hat{{V}}({\varvec{p}})}{\partial p_i } \nonumber \\= & {} \sum _{k=1}^K {\hat{{g}}_{k1}^{( {p_i } )} ({\varvec{p}})\hat{{g}}_{k1} ({\varvec{p}})} +\sum _{k=1}^K {\hat{{g}}_{k1}^{( {p_i } )} ({\varvec{p}})\hat{{g}}_{k1} ({\varvec{p}})} -\sum _{k=1}^K {\hat{{g}}_{k2}^{( {p_i } )} ({\varvec{p}})\hat{{g}}_{k2}^*({\varvec{p}})} \nonumber \\&-\sum _{k=1}^K {\hat{{g}}_{k2}^{( {p_i } )*} ({\varvec{p}})\hat{{g}}_{k2} ({\varvec{p}})} \nonumber \\= & {} \sum _{k=1}^K {2\hat{{g}}_{k1}^{( {p_i } )} ({\varvec{p}})\hat{{g}}_{k1} ({\varvec{p}})} -\sum _{k=1}^K {2\hbox {Re}\left\{ {\hat{{g}}_{k2}^{( {p_i } )} ({\varvec{p}})\hat{{g}}_{k2}^*({\varvec{p}})} \right\} } , \end{aligned}$$

(48)

where the fact that $\hat{{g}}_{k1} ({\varvec{p}})$ has a real value for its Hermitian form is used. $\hat{{g}}_{k1}^{( {p_i } )} ({\varvec{p}})$ and $\hat{{g}}_{k2}^{( {p_i } )} ({\varvec{p}})$ are the first-order partial derivatives of $\hat{{g}}_{k1} ({\varvec{p}})$ and $\hat{{g}}_{k2} ({\varvec{p}})$ with respect to $p_i$ and can be written as

$$\begin{aligned} \hat{{g}}_{k1}^{( {p_i } )} ({\varvec{p}})= & {} 2\hbox {Re}\left\{ {{\varvec{a}}_k^{( {p_i } ) {\mathrm{H}}}({\varvec{p}}){\hat{\varvec{{\varPi }}}}_{k1} {\varvec{a}}_k ({\varvec{p}})} \right\} , \nonumber \\ \hat{{g}}_{k2}^{( {p_i } )} ({\varvec{p}})= & {} 2{\varvec{a}}_k^{({p_i}) {\mathrm{H}}} ({\varvec{p}}) {\hat{\varvec{{\varPi }}}}_{k2} {\varvec{a}}_k^*({\varvec{p}}), \end{aligned}$$

(49)

in which ${\varvec{a}}_k^{( {p_i } )} ({\varvec{p}})$ denotes the first-order partial derivative of ${\varvec{a}}_k ({\varvec{p}})$ with respect to $p_i$. Inserting (49) into (48) and replacing ${\varvec{p}}$ with the true position $\varvec{\bar{{p}}}_q$ leads to

$$\begin{aligned} \hat{{v}}^{( {p_i } )} (\varvec{\bar{{p}}}_q )= & {} \sum _{k=1}^K {4\hbox {Re}\left\{ {{\varvec{a}}_k^{( {p_i }) {\mathrm{H}}} (\varvec{\bar{{p}}}_q ){\hat{\varvec{{\varPi }}}}_{k1} {\varvec{a}}_k (\varvec{\bar{{p}}}_q ){\varvec{a}}_k^{\mathrm{H}}(\varvec{\bar{{p}}}_q ){\hat{\varvec{{\varPi }}}}_{k1} {\varvec{a}}_k (\varvec{\bar{{p}}}_q )} \right\} } \nonumber \\&+\sum _{k=1}^K {4\hbox {Re}\left\{ {{\varvec{a}}_k^{( {p_i } ) {\mathrm{H}}} (\varvec{\bar{{p}}}_q ){\hat{\varvec{{\varPi }}}}_{k2} {\varvec{a}}_k^{*}(\varvec{\bar{{p}}}_q ){\varvec{a}}_k^{\mathrm{T}}(\varvec{\bar{{p}}}_q ){\hat{\varvec{{\varPi }}}}_{k2}^*{\varvec{a}}_k (\varvec{\bar{{p}}}_q )} \right\} } . \end{aligned}$$

(50)

After substituting ${\hat{\varvec{{\varPi }}}}_{k1} =\varvec{{\varPi }}_{k1} +\varvec{\delta \hat{{{\varPi }}}}_{k1} +o( {\Vert {\varvec{\delta \hat{{{\varPi }}}}_{k1} } \Vert _{\mathrm{F}} })$ and ${\hat{\varvec{{\varPi }}}}_{k2} =\varvec{{\varPi }}_{k2} +\varvec{\delta \hat{{{\varPi }}}}_{k2} +o( {\Vert {\varvec{\delta \hat{{{\varPi }}}}_{k2} } \Vert _{\mathrm{F}}})$ into (50), we have

$$\begin{aligned} \hat{{v}}^{( {p_i } )} (\varvec{\bar{{p}}}_q )= & {} v^{( {p_i } )} (\varvec{\bar{{p}}}_q )+ \sum _{k=1}^K {4\hbox {Re}\left\{ {{\varvec{a}}_k^{( {p_i }) {\mathrm{H}}} (\varvec{\bar{{p}}}_q )\varvec{\delta \hat{{{\varPi }}}}_{k1} {\varvec{a}}_k (\varvec{\bar{{p}}}_q ){\varvec{a}}_k^{\mathrm{H}}(\varvec{\bar{{p}}}_q )\varvec{{\varPi }}_{k1} {\varvec{a}}_k (\varvec{\bar{{p}}}_q )} \right\} } \nonumber \\&+\sum _{k=1}^K {4\hbox {Re}\left\{ {{\varvec{a}}_k^{( {p_i }) {\mathrm{H}}} (\varvec{\bar{{p}}}_q )\varvec{{\varPi }}_{k1} {\varvec{a}}_k (\varvec{\bar{{p}}}_q ){\varvec{a}}_k^{\mathrm{H}}(\varvec{\bar{{p}}}_q )\varvec{\delta \hat{{{\varPi }}}}_{k1} {\varvec{a}}_k (\varvec{\bar{{p}}}_q )} \right\} } \nonumber \\&-\sum _{k=1}^K {4\hbox {Re}\left\{ {{\varvec{a}}_k^{( {p_i } ) {\mathrm{H}}} (\varvec{\bar{{p}}}_q )\varvec{\delta \hat{{{\varPi }}}}_{k2} {\varvec{a}}_k^{*}(\varvec{\bar{{p}}}_q ){\varvec{a}}_k^{\mathrm{T}}(\varvec{\bar{{p}}}_q )\varvec{{\varPi }}_{k2}^*{\varvec{a}}_k (\varvec{\bar{{p}}}_q )} \right\} } \nonumber \\&-\sum _{k=1}^K {4\hbox {Re}\left\{ {{\varvec{a}}_k^{( {p_i }) {\mathrm{H}}} (\varvec{\bar{{p}}}_q )\varvec{{\varPi }}_{k2} {\varvec{a}}_k^{*}(\varvec{\bar{{p}}}_q ){\varvec{a}}_k^{\mathrm{T}}(\varvec{\bar{{p}}}_q )\varvec{\delta \hat{{{\varPi }}}}_{k2}^*{\varvec{a}}_k (\varvec{\bar{{p}}}_q )} \right\} }\nonumber \\&+o\left( {\Vert {\varvec{\delta \hat{{{\varPi }}}}_{\mathrm{NC},k} } \Vert _{\mathrm{F}} } \right) . \end{aligned}$$

(51)

Now, we recall the ith component of the matrix equation in (26):

$$\begin{aligned} \hat{{v}}^{( {p_i } )} (\varvec{\bar{{p}}}_q )=v^{( {p_i } )} (\varvec{\bar{{p}}}_q )+\;\hat{{v}}^{1( {p_i } )} (\varvec{\bar{{p}}}_q )+o\left( {\Vert {\varvec{\delta \hat{{{\varPi }}}}_{\mathrm{NC},k} } \Vert _{\mathrm{F}} } \right) . \end{aligned}$$

(52)

By comparing the terms in (51) with those in (52), we obtain an expression for $\hat{{v}}^{1( {p_i } )} (\varvec{\bar{{p}}}_q)$ as

$$\begin{aligned} \hat{{v}}^{1( {p_i } )} (\varvec{\bar{{p}}}_q )= & {} \sum _{k=1}^K {4\hbox {Re}\left\{ {{\varvec{a}}_k^{( {p_i }) {\mathrm{H}}} (\varvec{\bar{{p}}}_q )\varvec{\delta \hat{{{\varPi }}}}_{k1} {\varvec{a}}_k (\varvec{\bar{{p}}}_q ){\varvec{a}}_k^{\mathrm{H}}(\varvec{\bar{{p}}}_q )\varvec{{\varPi }}_{k1} {\varvec{a}}_k (\varvec{\bar{{p}}}_q )} \right\} } \nonumber \\&+\sum _{k=1}^K {4\hbox {Re}\left\{ {{\varvec{a}}_k^{( {p_i }) {\mathrm{H}}} (\varvec{\bar{{p}}}_q )\varvec{{\varPi }}_{k1} {\varvec{a}}_k (\varvec{\bar{{p}}}_q ){\varvec{a}}_k^{\mathrm{H}}(\varvec{\bar{{p}}}_q )\varvec{\delta \hat{{{\varPi }}}}_{k1} {\varvec{a}}_k (\varvec{\bar{{p}}}_q )} \right\} } \nonumber \\&-\sum _{k=1}^K {4\hbox {Re}\left\{ {{\varvec{a}}_k^{( {p_i }) {\mathrm{H}}} (\varvec{\bar{{p}}}_q )\varvec{\delta \hat{{{\varPi }}}}_{k2} {\varvec{a}}_k^{*}(\varvec{\bar{{p}}}_q ){\varvec{a}}_k^{\mathrm{T}}(\varvec{\bar{{p}}}_q )\varvec{{\varPi }}_{k2}^*{\varvec{a}}_k (\varvec{\bar{{p}}}_q )} \right\} } \nonumber \\&-\sum _{k=1}^K {4\hbox {Re}\left\{ {{\varvec{a}}_k^{( {p_i}) {\mathrm{H}}} (\varvec{\bar{{p}}}_q) \varvec{{\varPi }}_{k2} {\varvec{a}}_k^{*} (\varvec{\bar{{p}}}_q) {\varvec{a}}_k^{\mathrm{T}}(\varvec{\bar{{p}}}_q )\varvec{\delta \hat{{{\varPi }}}}_{k2}^*{\varvec{a}}_k (\varvec{\bar{{p}}}_q )} \right\} } . \end{aligned}$$

(53)

Using $\hbox {vec}[{\varvec{XYZ}}]=( {{\varvec{Z}}^{\mathrm{T}}\otimes {\varvec{X}}} )\hbox {vec}[{\varvec{Y}}]$ (see [27]), (53) becomes

$$\begin{aligned} \hat{{v}}^{1( {p_i } )} (\varvec{\bar{{p}}}_q )= & {} \sum _{k=1}^K 4\hbox {Re}\left\{ \left\{ {\varvec{a}}_k^{\mathrm{H}} (\varvec{\bar{{p}}}_q )\varvec{{\varPi }}_{k1} {\varvec{a}}_k (\varvec{\bar{{p}}}_q )\left( {{\varvec{a}}_k^{\mathrm{T}} (\varvec{\bar{{p}}}_q )\otimes {\varvec{a}}_k^{( {p_i }) {\mathrm{H}}} (\varvec{\bar{{p}}}_q )} \right) \right. \right. \\&\left. \left. +{\varvec{a}}_k^{( {p_i}) {\mathrm{H}}} (\varvec{\bar{{p}}}_q )\varvec{{\varPi }}_{k1} {\varvec{a}}_k (\varvec{\bar{{p}}}_q )\left( {{\varvec{a}}_k^{\mathrm{T}} (\varvec{\bar{{p}}}_q )\otimes {\varvec{a}}_k^{\mathrm{H}} (\varvec{\bar{{p}}}_q )} \right) \right\} \times \hbox {vec} \left[ {\varvec{\delta \hat{{{\varPi }}}}_{k1} }\right] \right\} \nonumber \\&-\sum _{k=1}^K {4\hbox {Re}\left\{ {{\varvec{a}}_k^{\mathrm{T}} (\varvec{\bar{{p}}}_q )\varvec{{\varPi }}_{k2}^*{\varvec{a}}_k (\varvec{\bar{{p}}}_q )\left( {{\varvec{a}}_k^{\mathrm{H}} (\varvec{\bar{{p}}}_q )\otimes {\varvec{a}}_k^{({p_i}) {\mathrm{H}}} (\varvec{\bar{{p}}}_q )}\right) \times \hbox {vec}\left[ {\varvec{\delta \hat{{{\varPi }}}}_{k2} }\right] } \right\} } \nonumber \\&-\sum _{k=1}^K {4\hbox {Re}\left\{ {{\varvec{a}}_k^{( {p_i }) {\mathrm{H}}} (\varvec{\bar{{p}}}_q )\varvec{{\varPi }}_{k2} {\varvec{a}}_k^{*}(\varvec{\bar{{p}}}_q )\left( {{\varvec{a}}_k^{\mathrm{T}} (\varvec{\bar{{p}}}_q )\otimes {\varvec{a}}_k^{\mathrm{T}} (\varvec{\bar{{p}}}_q )} \right) \times \hbox {vec}\left[ {\varvec{\delta \hat{{{\varPi }}}}_{k2}^*}\right] } \right\} }. \nonumber \end{aligned}$$

(54)

As $\hat{{v}}^{1( {p_i } )} (\varvec{\bar{{p}}}_q )$ is the ith element of ${\hat{{\varvec{v}}}}^{1( {\varvec{p}} )} (\varvec{\bar{{p}}}_q )$, the expression for ${\hat{{\varvec{v}}}}^{1( {\varvec{p}} )} (\varvec{\bar{{p}}}_q )$ in (29) is given by (54).

Appendix 2

Proof of (31):

For the expression ${\varvec{H}}_q =\mathop {\hbox {lim}}\limits _{L\rightarrow \infty } {\hat{{\varvec{V}}}}^{( {\varvec{p,p}} )} (\varvec{\bar{{p}}}_q )$, we first derive the i, jth entry. Based on (48), the second-order partial derivative of $\hat{{V}}({\varvec{p}})$ can be further calculated as

$$\begin{aligned} \hat{{V}}^{( {p_i ,p_j } )} ({\varvec{p}})= & {} \frac{\partial ^{2}\hat{{V}}({\varvec{p}})}{\partial p_i \partial p_j } \nonumber \\= & {} \sum _{k=1}^K {2\left\{ {\hat{{g}}_{k1}^{( {p_i ,p_j } )} ({\varvec{p}})\hat{{g}}_{k1} ({\varvec{p}})+\hat{{g}}_{k1}^{( {p_i } )} ({\varvec{p}})\hat{{g}}_{k1}^{( {p_j } )} ({\varvec{p}})} \right\} } \nonumber \\&-\sum _{k=1}^K {2\hbox {Re}\left\{ {\hat{{g}}_{k2}^{( {p_i ,p_j } )} ({\varvec{p}})\hat{{g}}_{k2}^*({\varvec{p}})+\hat{{g}}_{k2}^{( {p_i } )} ({\varvec{p}})\hat{{g}}_{k2}^{( {p_j } )*} ({\varvec{p}})} \right\} } , \end{aligned}$$

(55)

where $\hat{{g}}_{k1}^{( {p_i } )} ({\varvec{p}})$, $\hat{{g}}_{k1}^{( {p_j } )} ({\varvec{p}})$, $\hat{{g}}_{k2}^{( {p_i } )} ({\varvec{p}})$, and $\hat{{g}}_{k2}^{( {p_j } )} ({\varvec{p}})$ have similar expressions as those in (49). $\hat{{g}}_{k1}^{( {p_i ,p_j } )} ({\varvec{p}})$ and $\hat{{g}}_{k2}^{( {p_i ,p_j } )} ({\varvec{p}})$ represent the second-order partial derivatives of $\hat{{g}}_{k1} ({\varvec{p}})$ and $\hat{{g}}_{k2} ({\varvec{p}})$, respectively, and can be expressed as

$$\begin{aligned} \left\{ {\begin{array}{l} \hat{{g}}_{k1}^{( {p_i ,p_j } )} ({\varvec{p}})=2\hbox {Re}\left\{ {{\varvec{a}}_k^{( {p_i}) {\mathrm{H}}} ({\varvec{p}}){\hat{\varvec{{\varPi }}}}_{k1} {\varvec{a}}_k^{({p_j } )} ({\varvec{p}})+{\varvec{a}}_k^{({p_i, p_j}) {\mathrm{H}}} ({\varvec{p}}){\hat{\varvec{{\varPi }}}}_{k1} {\varvec{a}}_k ({\varvec{p}})} \right\} , \\ \hat{{g}}_{k2}^{( {p_i ,p_j } )} ({\varvec{p}})=2{\varvec{a}}_k^{({p_i}) {\mathrm{H}}} ({\varvec{p}}){\hat{\varvec{{\varPi }}}}_{k2} {\varvec{a}}_k^{( {p_j}) {*}} ({\varvec{p}})+2{\varvec{a}}_k^{( {p_i ,p_j}) {\mathrm{H}}} ({\varvec{p}}){\hat{\varvec{{\varPi }}}}_{k2} {\varvec{a}}_k^{*}({\varvec{p}}). \\ \end{array}}\right. \end{aligned}$$

(56)

Here, ${\varvec{a}}_k^{( {p_i ,p_j } )} ({\varvec{p}})=\frac{\partial ^{2}{\varvec{a}}_k ({\varvec{p}})}{\partial p_i \partial p_j }$ denotes the second-order partial derivative of ${\varvec{a}}_k ({\varvec{p}})$.

According to (10), we have $\varvec{{\varPi }}_{k1} {\varvec{a}}_k (\varvec{\bar{{p}}}_q )+\varvec{{\varPi }}_{k2} {\varvec{a}}_k^*(\varvec{\bar{{p}}}_q) e^{-\hbox {j}\bar{{{\phi }}}_q }={\varvec{0}}$ ($\bar{{{\phi }}}_q$ is the true value of ${\phi }_q$), following which it can be proved that $\varvec{{\varPi }}_{k1} {\varvec{a}}_k (\varvec{\bar{{p}}}_q ){\varvec{a}}_k^{\mathrm{H}}(\varvec{\bar{{p}}}_q )\varvec{{\varPi }}_{k1} =\varvec{{\varPi }}_{k2} {\varvec{a}}_k^*(\varvec{\bar{{p}}}_q ){\varvec{a}}_k^{\mathrm{T}}(\varvec{\bar{{p}}}_q )\varvec{{\varPi }}_{k2}^*$ (see [1]). Then, using this result and substituting (56) into (55) with ${\varvec{p}}=\varvec{\bar{{p}}}_q$, we obtain

$$\begin{aligned} \mathop {\lim }\limits _{L\rightarrow \infty } \hat{{V}}^{( {p_i ,p_j } )} (\varvec{\bar{{p}}}_q )= & {} \sum _{k=1}^K 4\hbox {Re}\left\{ {\varvec{a}}_k^{({p_i}) {\mathrm{H}}} (\varvec{\bar{{p}}}_q ){\varvec{a}}_k^{\mathrm{H}}(\varvec{\bar{{p}}}_q )\varvec{{\varPi }}_{k1} {\varvec{a}}_k (\varvec{\bar{{p}}}_q )\varvec{{\varPi }}_{k1} {\varvec{a}}_k^{( {p_j } )} (\varvec{\bar{{p}}}_q ) \right. \nonumber \\&-{\varvec{a}}_k^{( {p_i}) {\mathrm{H}}} (\varvec{\bar{{p}}}_q )\varvec{{\varPi }}_{k2} {\varvec{a}}_k^{*}(\varvec{\bar{{p}}}_q ){\varvec{a}}_k^{\mathrm{T}}(\varvec{\bar{{p}}}_q )\varvec{{\varPi }}_{k2}^*{\varvec{a}}_k^{( {p_j } )} (\varvec{\bar{{p}}}_q )\nonumber \\&+{\varvec{a}}_k^{({p_i}) {\mathrm{H}}} (\varvec{\bar{{p}}}_q )\varvec{{\varPi }}_{k1} {\varvec{a}}_k (\varvec{\bar{{p}}}_q ){\varvec{a}}_k^{\mathrm{T}}(\varvec{\bar{{p}}}_q )\varvec{{\varPi }}_{k1}^*{\varvec{a}}_k^{( {p_j}) {*}} (\varvec{\bar{{p}}}_q ) \nonumber \\&\left. -{\varvec{a}}_k^{( {p_i}) {\mathrm{H}}} (\varvec{\bar{{p}}}_q ){\varvec{a}}_k^{\mathrm{T}}(\varvec{\bar{{p}}}_q )\varvec{{\varPi }}_{k2}^*{\varvec{a}}_k (\varvec{\bar{{p}}}_q )\varvec{{\varPi }}_{k2} {\varvec{a}}_k^{({p_j}) {*}} (\varvec{\bar{{p}}}_q )\right\} . \end{aligned}$$

(57)

For simplicity, we let ${\varvec{W}}_{k1} (\varvec{\bar{{p}}}_q )={\varvec{a}}_k^{\mathrm{H}}(\varvec{\bar{{p}}}_q )\varvec{{\varPi }}_{k1} {\varvec{a}}_k (\varvec{\bar{{p}}}_q )\varvec{{\varPi }}_{k1} -\varvec{{\varPi }}_{k2} {\varvec{a}}_k^{*}(\varvec{\bar{{p}}}_q ){\varvec{a}}_k^{\mathrm{T}}(\varvec{\bar{{p}}}_q )\varvec{{\varPi }}_{k2}^*$ and ${\varvec{W}}_{k2} (\varvec{\bar{{p}}}_q )={\varvec{a}}_k^{\mathrm{T}}(\varvec{\bar{{p}}}_q )\varvec{{\varPi }}_{k2}^*{\varvec{a}}_k (\varvec{\bar{{p}}}_q )\varvec{{\varPi }}_{k2} -\varvec{{\varPi }}_{k1} {\varvec{a}}_k (\varvec{\bar{{p}}}_q ){\varvec{a}}_k^{\mathrm{T}}(\varvec{\bar{{p}}}_q )\varvec{{\varPi }}_{k1}^*$. Consequently, the expression in (57) reduces to

$$\begin{aligned}&\mathop {\hbox {lim}}\limits _{L\rightarrow \infty } \hat{{V}}^{( {p_i ,p_j } )} (\varvec{\bar{{p}}}_q ) \\&\quad =\sum _{k=1}^K {4\hbox {Re}\left\{ {{\varvec{a}}_k^{( {p_i}) {\mathrm{H}}} (\varvec{\bar{{p}}}_q ){\varvec{W}}_{k1} (\varvec{\bar{{p}}}_q ){\varvec{a}}_k^{( {p_j } )} (\varvec{\bar{{p}}}_q )} {-{\varvec{a}}_k^{({p_i}) {\mathrm{H}}} (\varvec{\bar{{p}}}_q ){\varvec{W}}_{k2} (\varvec{\bar{{p}}}_q ){\varvec{a}}_k^{( {p_j}) {*}} (\varvec{\bar{{p}}}_q )} \right\} }. \nonumber \end{aligned}$$

(58)

Finally, the expression for ${\varvec{H}}_q$ in (31) can be deduced from (58) by noting that ${\varvec{a}}_k^{( {p_i } )} (\varvec{\bar{{p}}}_q )$ and ${\varvec{a}}_k^{( {p_j } )} (\varvec{\bar{{p}}}_q )$ are the ith andjth columns of ${\varvec{A}}_k^{( {\varvec{p}} )} (\varvec{\bar{{p}}}_q )$ for $k=1,2,\ldots ,K$.

Appendix 3

Proof of (34):

Inserting $\Delta {\varvec{p}}_q =-{\varvec{H}}_q^{-1}{\hat{{\varvec{v}}}}^{1( {\varvec{p}} )} (\varvec{\bar{{p}}}_q )$ into ${\varvec{C}}_{{\varvec{p}}_q } =\hbox {E}[{\Delta {\varvec{p}}_q \Delta {\varvec{p}}_q^{\mathrm{T}}}]$ yields

$$\begin{aligned} {\varvec{C}}_{{\varvec{p}}_q } ={\varvec{H}}_q^{-1}\hbox {E}\left[ {{\hat{{\varvec{v}}}}^{1( {\varvec{p}} )} (\varvec{\bar{{p}}}_q ){\hat{{\varvec{v}}}}^{1( {\varvec{p}} )\hbox {T}} (\varvec{\bar{{p}}}_q )} \right] {\varvec{H}}_q^{-1}. \end{aligned}$$

(59)

Recall that the batches of data in K time slots are assumed to be statistically independent. Therefore, the perturbation vectors for K time slots are uncorrelated, leading to

$$\begin{aligned} \left\{ {\begin{array}{l} \hbox {E}\left[ {\hbox {vec}[{\varvec{\delta {\varPi }}_{ki} }]\hbox {vec}[{\varvec{\delta {\varPi }}_{hj} }]^{\mathrm{H}}}\right] ={\varvec{O}}, \\ \hbox {E}\left[ {\hbox {vec}[{\varvec{\delta {\varPi }}_{ki} }]\hbox {vec}[{\varvec{\delta {\varPi }}_{hj} }]^{\mathrm{T}}}\right] ={\varvec{O}}, \\ \end{array}}\right. \qquad i,j=1,2;\forall k\ne h. \end{aligned}$$

(60)

Based on (60), using $\hbox {Re}\{ {\varvec{x}} \} \hbox {Re}\{ {{\varvec{x}}^{\mathrm{T}}} \} = \frac{1}{2} \hbox {Re}\{ {\varvec{xx}^{\mathrm{T}}+\varvec{xx}^{\mathrm{H}}} \}$ and the expression for ${\hat{{\varvec{v}}}}^{1( {\varvec{p}} )} (\varvec{\bar{{p}}}_q )$ given in (29), we can write the covariance of ${\hat{{\varvec{v}}}}^{1( {\varvec{p}} )} (\varvec{\bar{{p}}}_q )$ as

$$\begin{aligned}&\hbox {E}\left[ {{\hat{{\varvec{v}}}}^{1( {\varvec{p}} )} (\varvec{\bar{{p}}}_q ){\hat{{\varvec{v}}}}^{1( {\varvec{p}} )\hbox {T}} (\varvec{\bar{{p}}}_q )}\right] \nonumber \\&\quad =\frac{8}{L}\sum _{k=1}^K \hbox {E}\left[ \hbox {Re}\left\{ {\varvec{F}}_k (\varvec{\bar{{p}}}_q )\times \sqrt{L}\varvec{\delta \hat{{\pi }}}( {\sqrt{L}\varvec{\delta \hat{{\pi }}}} )^{\mathrm{T}}\times {\varvec{F}}_k^{\mathrm{T}}(\varvec{\bar{{p}}}_q )+{\varvec{F}}_k (\varvec{\bar{{p}}}_q ) \right. \right. \nonumber \\&\qquad \qquad \quad \qquad \left. \left. \times \sqrt{L}\varvec{\delta \hat{{\pi }}}( {\sqrt{L}\varvec{\delta \hat{{\pi }}}} )^{\mathrm{H}}\times {\varvec{F}}_k^{\mathrm{H}}(\varvec{\bar{{p}}}_q ) \right\} \right] \nonumber \\&\quad =\frac{8}{L}\sum _{k=1}^K \hbox {Re}\left\{ {\varvec{F}}_k (\varvec{\bar{{p}}}_q )\times \hbox {E}\left\{ {\sqrt{L}\varvec{\delta \hat{{\pi }}}( {\sqrt{L}\varvec{\delta \hat{{\pi }}}} )^{\mathrm{T}}} \right\} \times {\varvec{F}}_k^{\mathrm{T}}(\varvec{\bar{{p}}}_q )+{\varvec{F}}_k (\varvec{\bar{{p}}}_q ) \right. \nonumber \\&\qquad \qquad \qquad \qquad \left. \times \hbox {E}\left\{ {\sqrt{L}\varvec{\delta \hat{{\pi }}}( {\sqrt{L}\varvec{\delta \hat{{\pi }}}} )^{\mathrm{H}}} \right\} \times {\varvec{F}}_k^{\mathrm{H}}(\varvec{\bar{{p}}}_q ) \right\} \\&\quad =\frac{8}{L}\sum _{k=1}^K {\hbox {Re}\left\{ {{\varvec{F}}_k (\varvec{\bar{{p}}}_q )\varvec{{C}}_{\varvec{{\varPi }}_{k1} ,\varvec{{\varPi }}_{k2} ,\varvec{{\varPi }}_{k2}^*}^{\prime } {\varvec{F}}_k^{\mathrm{T}}(\varvec{\bar{{p}}}_q )+{\varvec{F}}_k (\varvec{\bar{{p}}}_q ){\varvec{C}}_{\varvec{{\varPi }}_{k1} ,\varvec{{\varPi }}_{k2} ,\varvec{{\varPi }}_{k2}^*} {\varvec{F}}_k^{\mathrm{H}}(\varvec{\bar{{p}}}_q )} \right\} }, \nonumber \end{aligned}$$

(61)

where $\varvec{\delta \hat{{\pi }}}=[{\hbox {vec}[{\varvec{\delta \hat{{{\varPi }}}}_{k1} }]^{\mathrm{T}}\hbox {,vec}[{\varvec{\delta \hat{{{\varPi }}}}_{k2} }]^{\mathrm{T}}\hbox {,vec}[{\varvec{\delta \hat{{{\varPi }}}}_{k2}^*}]^{\mathrm{T}}} ]^{\mathrm{T}}$. ${\varvec{C}}_{\varvec{{\varPi }}_{k1} ,\varvec{{\varPi }}_{k2} ,\varvec{{\varPi }}_{k2}^*}$ and $\varvec{{C}}_{\varvec{{\varPi }}_{k1} ,\varvec{{\varPi }}_{k2} ,\varvec{{\varPi }}_{k2}^*}^{\prime }$ are the covariance matrix and unconjugated covariance matrix of $\sqrt{L}\varvec{\delta \hat{{\pi }}}$, respectively. Then, because $\hbox {Re}\{ {\varvec{X}} \}=\frac{1}{2}( {{\varvec{X}}+{\varvec{X}}^{*}})$, (61) becomes

$$\begin{aligned}&\hbox {E}\left[ {{\hat{{\varvec{v}}}}^{1( {\varvec{p}} )} (\varvec{\bar{{p}}}_q ){\hat{{\varvec{v}}}}^{1( {\varvec{p}} )\hbox {T}} (\varvec{\bar{{p}}}_q )}\right] \nonumber \\&\quad =\frac{4}{L}\sum _{k=1}^K \left\{ {{\varvec{F}}_k (\varvec{\bar{{p}}}_q )\varvec{{C}}_{\varvec{{\varPi }}_{k1} ,\varvec{{\varPi }}_{k2} ,\varvec{{\varPi }}_{k2}^*}^{\prime } {\varvec{F}}_k^{\mathrm{T}} (\varvec{\bar{{p}}}_q )+{\varvec{F}}_k^{*} (\varvec{\bar{{p}}}_q )\varvec{{C}}_{\varvec{{\varPi }}_{k1} ,\varvec{{\varPi }}_{k2} ,\varvec{{\varPi }}_{k2}^*}^{\prime *} {\varvec{F}}_k^{\mathrm{H}} (\varvec{\bar{{p}}}_q )} \right. \nonumber \\&\qquad \left. +{\varvec{F}}_k (\varvec{\bar{{p}}}_q ){\varvec{C}}_{\varvec{{\varPi }}_{k1} ,\varvec{{\varPi }}_{k2} ,\varvec{{\varPi }}_{k2}^*} {\varvec{F}}_k^{\mathrm{H}} (\varvec{\bar{{p}}}_q )+{\varvec{F}}_k^{*} (\varvec{\bar{{p}}}_q ){\varvec{C}}_{\varvec{{\varPi }}_{k1} ,\varvec{{\varPi }}_{k2} ,\varvec{{\varPi }}_{k2}^*}^*{\varvec{F}}_k^{\mathrm{T}} (\varvec{\bar{{p}}}_q ) \right\} \\&\quad =\frac{4}{L}\sum _{k=1}^K {\tilde{{\varvec{F}}}_k (\varvec{\bar{{p}}}_q) \varvec{{C}}_{\tilde{\varvec{{\varPi }}}_k}^{\prime } \tilde{{\varvec{F}}}_k^{\mathrm{T}} (\varvec{\bar{{p}}}_q )}. \nonumber \end{aligned}$$

(62)

where $\varvec{{C}}_{\tilde{\varvec{{\varPi }}}_k}^{\prime }$ are defined in (33). After substituting (62) into (59), the proof of (34) is complete.

Appendix 4

Derivation of $\frac{\partial \tilde{{\varvec{y}}}}{\partial \tilde{{\varvec{p}}}^{\mathrm{T}}}$ and $\frac{\partial \tilde{{\varvec{y}}}}{\partial \varvec{\eta }^{\mathrm{T}}}$:

To further derive the expressions for $\frac{\partial \tilde{{\varvec{y}}}}{\partial \tilde{{\varvec{p}}}^{\mathrm{T}}}$ and $\frac{\partial \tilde{{\varvec{y}}}}{\partial \varvec{\eta }^{\mathrm{T}}}$ in (45), we first attempt to compute $\frac{\partial \varvec{{{\tilde{y}}}}_k^{\prime }}{\partial \tilde{{\varvec{p}}}^{\mathrm{T}}}$, $\frac{\partial \varvec{{{\tilde{y}}}}_k^{\prime }}{\partial {\varvec{{\phi }}}^{\mathrm{T}}}$, $\frac{\partial \varvec{{{\tilde{y}}}}_k^{\prime }}{\partial \varvec{\omega }^{\mathrm{T}}}$, and $\frac{\partial \varvec{{{\tilde{y}}}}_k^{\prime }}{\partial \sigma _{\mathrm{n}}^2}$ for

$$\begin{aligned} \frac{\partial \tilde{{\varvec{y}}}}{\partial \tilde{{\varvec{p}}}^{\mathrm{T}}}=\left[ {\begin{array}{c} \frac{\partial \varvec{{{\tilde{y}}}}_1^{\prime }}{\partial \tilde{{\varvec{p}}}^{\mathrm{T}}} \\ \frac{\partial \varvec{{{\tilde{y}}}}_2^{\prime }}{\partial \tilde{{\varvec{p}}}^{\mathrm{T}}} \\ \vdots \\ \frac{\partial \varvec{{{\tilde{y}}}}_K^{\prime }}{\partial \tilde{{\varvec{p}}}^{\mathrm{T}}} \\ \end{array}}\right] , \quad \frac{\partial \tilde{{\varvec{y}}}}{\partial \varvec{\eta }^{\mathrm{T}}}= \left[ {{\begin{array}{ccc} \frac{\partial \varvec{{{\tilde{y}}}}_1^{\prime }}{\partial {\varvec{{\phi }}}^{\mathrm{T}}} &{} \frac{\partial \varvec{{{\tilde{y}}}}_1^{\prime }}{\partial \varvec{\omega }^{\mathrm{T}}} &{} \frac{\partial \varvec{{{\tilde{y}}}}_1^{\prime }}{\partial \sigma _{\mathrm{n}}^2 } \\ \frac{\partial \varvec{{{\tilde{y}}}}_2^{\prime }}{\partial {\varvec{{\phi }}}^{\mathrm{T}}} &{} \frac{\partial \varvec{{{\tilde{y}}}}_2^{\prime }}{\partial \varvec{\omega }^{\mathrm{T}}} &{} \frac{\partial \varvec{{{\tilde{y}}}}_2^{\prime }}{\partial \sigma _{\mathrm{n}}^2 } \\ \vdots &{} \vdots &{} \vdots \\ \frac{\partial \varvec{{{\tilde{y}}}}_K^{\prime }}{\partial {\varvec{{\phi }}}^{\mathrm{T}}} &{} \frac{\partial \varvec{{{\tilde{y}}}}_K^{\prime }}{\partial \varvec{\omega }^{\mathrm{T}}} &{} \frac{\partial \varvec{{{\tilde{y}}}}_K^{\prime }}{\partial \sigma _{\mathrm{n}}^2 } \\ \end{array} }}\right] . \end{aligned}$$

(63)

Applying the formula $\hbox {vec}[{\varvec{XYZ}}]= ({{\varvec{Z}}^{\mathrm{T}}\otimes {\varvec{X}}}) \hbox {vec}[{\varvec{Y}}]$ (see [27]) to $\tilde{{\varvec{y}}}_k =\hbox {vec}[{\tilde{{\varvec{R}}}_k }]$ yields

$$\begin{aligned} \tilde{{\varvec{y}}}_k =\left( {{\varvec{A}}_{\mathrm{NC},k}^*\otimes {\varvec{A}}_{\mathrm{NC},k} } \right) {\varvec{r}}_k^{\mathrm{s}} +\sigma _{\mathrm{n}}^2 \hbox {vec}[{{\varvec{I}}_{2M} }], \end{aligned}$$

(64)

where ${\varvec{r}}_k^{\mathrm{s}} =\hbox {vec} [{{\varvec{R}}_k^{\mathrm{s}} }]$.

Let $p_{qi} ( {i=1,2,\ldots ,D} )$ be the ith element of the position vector ${\varvec{p}}_q ( q=1,2,\ldots ,Q)$. Then, we define the vector $\tilde{{\varvec{p}}}_i =[p_{1i} ,p_{2i} , \ldots ,p_{Qi} ]^{\mathrm{T}}$ and the following two derivative matrices:

$$\begin{aligned} \left\{ {\begin{array}{l} {\varvec{A}}_{\mathrm{NC},k}^{( {\tilde{{\varvec{p}}}_i } )} =\left[ {\frac{\partial {\varvec{a}}_{\mathrm{NC},k} ({\varvec{p}}_1 ,{\phi }_1 )}{\partial p_{1i} }\,,\frac{\partial {\varvec{a}}_{\mathrm{NC},k} ({\varvec{p}}_2 ,{\phi }_2 )}{\partial p_{2i} }\;,\ldots ,\frac{\partial {\varvec{a}}_{\mathrm{NC},k} ({\varvec{p}}_Q ,{\phi }_Q )}{\partial p_{Qi} }}\right] , \quad i=1,2,\ldots ,D, \\ {\varvec{A}}_{\mathrm{NC},k}^{( {\varvec{{\phi }}} )} =\left[ {\frac{\partial {\varvec{a}}_{\mathrm{NC},k} ({\varvec{p}}_1 ,{\phi }_1 )}{\partial {\phi }_1 }\;,\frac{\partial {\varvec{a}}_{\mathrm{NC},k} ({\varvec{p}}_2 ,{\phi }_2 )}{\partial {\phi }_2 } \, ,\ldots ,\frac{\partial {\varvec{a}}_{\mathrm{NC},k} ({\varvec{p}}_Q ,{\phi }_Q )}{\partial {\phi }_Q }}\right] , \\ \end{array}}\right. \end{aligned}$$

(65)

for $k=1,2,\ldots ,K$. As the sources are assumed to be uncorrelated, ${\varvec{R}}_k^{\mathrm{s}}$ is a diagonal matrix. Using this result, a series of manipulations based on (64) lead to

$$\begin{aligned} \left\{ {\begin{array}{l} \frac{\partial \tilde{{\varvec{y}}}_k }{\partial \tilde{{\varvec{p}}}_i^{\mathrm{T}} }=\left( {{\varvec{A}}_{\mathrm{NC},k} {\varvec{R}}_k^{\mathrm{s}} } \right) ^{*}{*}{\varvec{A}}_{\mathrm{NC},k}^{( {\tilde{{\varvec{p}}}_i } )} +{\varvec{A}}_{\mathrm{NC},k}^{( {\tilde{{\varvec{p}}}_i } )*} {*}\left( {{\varvec{A}}_{\mathrm{NC},k} {\varvec{R}}_k^{\mathrm{s}} } \right) , \qquad i=1,2\ldots D, \\ \frac{\partial \tilde{{\varvec{y}}}_k }{\partial {\varvec{{\phi }}}^{\mathrm{T}}}=\left( {{\varvec{A}}_{\mathrm{NC},k} {\varvec{R}}_k^{\mathrm{s}} } \right) ^{*}{*}{\varvec{A}}_{\mathrm{NC},k}^{( {\varvec{{\phi }}} )} +{\varvec{A}}_{\mathrm{NC},k}^{({\varvec{{\phi }}} )*} {*}\left( {{\varvec{A}}_{\mathrm{NC},k} {\varvec{R}}_k^{\mathrm{s}} } \right) , \\ \frac{\partial \tilde{{\varvec{y}}}_k }{\partial \varvec{\omega }^{\mathrm{T}}}=\left[ {{\varvec{O}},\ldots {\varvec{O}},{\varvec{A}}_{\mathrm{NC},k}^*{*}{\varvec{A}}_{\mathrm{NC},k} ,{\varvec{O}},\ldots {\varvec{O}}}\right] , \\ \frac{\partial \tilde{{\varvec{y}}}_k }{\partial \sigma _{\mathrm{n}}^2 }=\hbox {vec}[{{\varvec{I}}_{2M} }], \\ \end{array}}\right. \end{aligned}$$

(66)

for $k=1,2,\ldots ,K$. (Here, ${\varvec{A}}_{\mathrm{NC},k}$ denotes ${\varvec{A}}_{\mathrm{NC},k} (\tilde{{\varvec{p}}}, {\varvec{{\phi }}})$.) Combining (66) and $\varvec{{{\tilde{y}}}}_k^{\prime } =( {\tilde{{\varvec{R}}}_k^{-\mathrm{T}/2}\otimes \tilde{{\varvec{R}}}_k^{-1/2}} )\tilde{{\varvec{y}}}_k$, we employ the formulas $( {{\varvec{B}}\otimes {\varvec{C}}}) ({{\varvec{X}}*{\varvec{Y}}}) = \varvec{BX}*\varvec{CY}$ and $\hbox {vec}[{\varvec{XYZ}}] = ({{\varvec{Z}}^{\mathrm{T}}\otimes {\varvec{X}}} )\hbox {vec}[{\varvec{Y}}]$ (see [27]) to obtain the partial derivatives of $\varvec{{{\tilde{y}}}}_k^{\prime }$:

$$\begin{aligned} \left\{ \begin{aligned} \frac{\partial \varvec{{{\tilde{y}}}}_k^{\prime }}{\partial \tilde{{\varvec{p}}}_i^{\mathrm{T}} }=&\left( {\tilde{{\varvec{R}}}_k^{-1/2} {\varvec{A}}_{\mathrm{NC},k} {\varvec{R}}_k^{\mathrm{s}} } \right) ^{\mathrm{*}}{*}\tilde{{\varvec{R}}}_k^{-1/2} {\varvec{A}}_{\mathrm{NC},k}^{( {\tilde{{\varvec{p}}}_i } )} +\left( {\tilde{{\varvec{R}}}_k^{-1/2} {\varvec{A}}_{\mathrm{NC},k}^{( {\tilde{{\varvec{p}}}_i } )} } \right) ^{\mathrm{*}}*\tilde{{\varvec{R}}}_k^{-1/2} {\varvec{A}}_{\mathrm{NC},k} {\varvec{R}}_k^{\mathrm{s}}, \\&\quad i=1,2,\ldots ,D, \\ \frac{\partial \varvec{{{\tilde{y}}}}_k^{\prime }}{\partial {\varvec{{\phi }}}^{\mathrm{T}}}=&\left( {\tilde{{\varvec{R}}}_k^{-1/2} {\varvec{A}}_{\mathrm{NC},k} {\varvec{R}}_k^{\mathrm{s}} } \right) ^{\mathrm{*}}{*}\tilde{{\varvec{R}}}_k^{-1/2} {\varvec{A}}_{\mathrm{NC},k}^{( {\varvec{{\phi }}} )} +\left( {\tilde{{\varvec{R}}}_k^{-1/2} {\varvec{A}}_{\mathrm{NC},k}^{({\varvec{{\phi }}} )} } \right) ^{\mathrm{*}}*\tilde{{\varvec{R}}}_k^{-1/2} {\varvec{A}}_{\mathrm{NC},k} {\varvec{R}}_k^{\mathrm{s}} , \\ \frac{\partial \varvec{{{\tilde{y}}}}_k^{\prime }}{\partial \varvec{\omega }^{\mathrm{T}}}=&\left[ {{\varvec{O}},\ldots {\varvec{O}},\tilde{{\varvec{R}}}_k^{-\hbox {T/2}} {\varvec{A}}_{\mathrm{NC},k}^*{*}\tilde{{\varvec{R}}}_k^{-1/2} {\varvec{A}}_{\mathrm{NC},k} ,{\varvec{O}},\ldots {\varvec{O}}}\right] , \\ \frac{\partial \varvec{{{\tilde{y}}}}_k^{\prime }}{\partial \sigma _{\mathrm{n}}^2 }=&\hbox {vec}\left[ {\tilde{{\varvec{R}}}_k^{-1}}\right] . \\ \end{aligned}\right. \nonumber \\ \end{aligned}$$

(67)

Now, $\frac{\partial \tilde{{\varvec{y}}}}{\partial \tilde{{\varvec{p}}}^{\mathrm{T}}}$ and $\frac{\partial \tilde{{\varvec{y}}}}{\partial \varvec{\eta }^{\mathrm{T}}}$ can be constituted by inserting the above expressions into (63).