Appendix: Computation of Derivatives
In the “Appendix”, we provide the derivative results which can be used to compute the DFSs index in (19), the DF MSEs in (22) and (23) and ambiguity thresholds in (28) when using MUSIC algorithm.
1.1 Derivatives with respect to target direction \({\varvec{\Upsilon}}\)
The derivation of function \(F\left( {{\varvec{\Upsilon}} ;{\varvec{\Theta}} } \right)\) in (13) with respect to \({\varvec{\Upsilon}}\) is defined as following
$$F_{1} \left( {{{\varvec{\Upsilon}};{\varvec{\Theta}}}} \right) = \frac{{\partial F\left( {{{\varvec{\Upsilon}};{\varvec{\Theta}}}} \right)}}{{\partial {\varvec{\Upsilon}}}} = \left[ {\frac{{\partial F\left( {{{\varvec{\Upsilon}};{\varvec{\Theta}}}} \right)}}{{\partial \theta }},\frac{{\partial F\left( {{{\varvec{\Upsilon}};{\varvec{\Theta}}}} \right)}}{{\partial \varphi }}} \right]^{T} = \left[ {\frac{{\partial F\left( {{{\varvec{\Upsilon}};{\varvec{\Theta}}}} \right)}}{{\partial \left( {\varvec{\Upsilon}} \right)_{i} }}} \right] \in {\mathbb{C}}^{{2 \times 1}}$$
(29)
where \(i = 1,2\); Furthermore, \({\left( {\varvec{\Upsilon}} \right)_1} = \theta\) and \({\left( {\varvec{\Upsilon}} \right)_2} = \varphi\). And
$${{\partial F\left( {{\varvec{\Upsilon}} ;{\varvec{\Theta}} } \right)} \over {\partial {{\left( {\varvec{\Upsilon}} \right)}_i}}} = {{\partial {{{\varvec{A}}}^H}\left( {{\varvec{\Upsilon}} ;{{\varvec{\Theta}}_0}} \right)} \over {\partial {{\left( {\varvec{\Upsilon}} \right)}_i}}}P_{{\varvec{A}}}^\bot \left( {\varvec{\Theta}} \right)A\left( {{\varvec{\Upsilon}} ;{{\varvec{\Theta}}_0}} \right) + {{{\varvec{A}}}^H}\left( {{\varvec{\Upsilon}} ;{{\varvec{\Theta}}_0}} \right)P_{{\varvec{A}}}^\bot \left( {\varvec{\Theta}} \right){{\partial A\left( {{\varvec{\Upsilon}} ;{{\varvec{\Theta}}_0}} \right)} \over {\partial {{\left( {\varvec{\Upsilon}} \right)}_i}}} = 2{\mathop{\rm Re}\nolimits} \left\{ {{{{\varvec{A}}}^H}\left( {{\varvec{\Upsilon}} ;{{\varvec{\Theta}}_0}} \right)P_{{\varvec{A}}}^\bot \left( {\varvec{\Theta}} \right){{\partial A\left( {{\varvec{\Upsilon}} ;{{\varvec{\Theta}}_0}} \right)} \over {\partial {{\left( {\varvec{\Upsilon}} \right)}_i}}}} \right\}$$
(30)
With
$${{\partial A\left( {{\varvec{\Upsilon}} ;{{\varvec{\Theta}}_0}} \right)} \over {\partial {{\left( {\varvec{\Upsilon}} \right)}_i}}} = - j\left( {{{2\pi } \mathord{\left/ {\vphantom {{2\pi } \lambda }} \right. \kern-0pt} \lambda }} \right)diag\left( {{L^T}{{\dot k}_{{{\left( {\varvec{\Upsilon}} \right)}_i}}}} \right)A\left( {{\varvec{\Upsilon}} ;{{\varvec{\Theta}}_0}} \right)$$
(31)
And
$${\dot k_{{{\left( {\varvec{\Upsilon}} \right)}_i}}} = {{\partial k} \over {\partial {{\left( {\varvec{\Upsilon}} \right)}_i}}} = \left\{\begin{array}{l} {{{\left[ { - \cos \varphi \sin \theta ,\cos \varphi \cos \theta ,0} \right]}^T}} \quad {i = 1}\\ {{{\left[ { - \sin \varphi \cos \theta , - \sin \varphi \sin \theta ,\cos \varphi } \right]}^T}} \quad {{i = 2}} \end{array} \right.$$
(32)
In summary, (30) can be written by
$${{\partial F\left( {{\varvec{\Upsilon}} ;{\varvec{\Theta}} } \right)} \over {\partial {{\left( {\varvec{\Upsilon}} \right)}_i}}} = 2{\mathop{\rm Re}\nolimits} \left\{ {{{{\varvec{A}}}^H}\left( {{\varvec{\Upsilon}} ;{{\varvec{\Theta}}_0}} \right)P_{{\varvec{A}}}^\bot \left( {\varvec{\Theta}} \right)diag\left( { - j\left( {{{2\pi } \mathord{\left/ {\vphantom {{2\pi } \lambda }} \right. \kern-0pt} \lambda }} \right){L^T}{{\dot k}_{{{\left( {\varvec{\Upsilon}} \right)}_i}}}} \right)A\left( {{\varvec{\Upsilon}} ;{{\varvec{\Theta}}_0}} \right)} \right\}$$
(33)
The second derivation of function \(F\left( {{\varvec{\Upsilon}} ;{\varvec{\Theta}} } \right)\) in (13) with respect to \({\varvec{\Upsilon}}\) is defined as following
$$\frac{{\partial F_{1} \left( {{{\varvec{\Upsilon}};{\varvec{\Theta}}}} \right)}}{{\partial {\varvec{\Upsilon}}}} = \frac{{\partial ^{2} F\left( {{{\varvec{\Upsilon}};{\varvec{\Theta}}}} \right)}}{{\partial {\varvec{\Upsilon}}\partial {\varvec{\Upsilon}}^{T} }} = \left[ {\frac{{\partial ^{2} F\left( {{{\varvec{\Upsilon}};{\varvec{\Theta}}}} \right)}}{{\partial \left( {\varvec{\Upsilon}} \right)_{i} \partial \left( {\varvec{\Upsilon}} \right)_{{i'}} }}} \right] \in {\mathbb{C}}^{{2 \times 2}}$$
(34)
where \(i,i' = 1,2\), and
$$\begin{aligned} \frac{{\partial ^{2} F\left( {{{\varvec{\Upsilon}};{\varvec{\Theta}}}} \right)}}{{\partial \left( {\varvec{\Upsilon}} \right)_{i} \partial \left( {\varvec{\Upsilon}} \right)_{{i'}} }} &= \frac{{\partial ^{2} {{\varvec{A}}}^{H} \left( {{{\varvec{\Upsilon}};{\varvec{\Theta}}}_{0} } \right)}}{{\partial \left( {\varvec{\Upsilon}} \right)_{i} \partial \left( {\varvec{\Upsilon}} \right)_{{i'}} }}{{\varvec{P}}}_{{{\varvec{A}}}}^{ \bot } \left( {{\varvec{\Theta}}} \right){{\varvec{A}}}\left( {{{\varvec{\Upsilon}};{\varvec{\Theta}}}_{0} } \right) + \frac{{\partial {{\varvec{A}}}^{H} \left( {{{\varvec{\Upsilon}};{\varvec{\Theta}}}_{0} } \right)}}{{\partial \left( {\varvec{\Upsilon}} \right)_{i} }}{{\varvec{P}}}_{{{\varvec{A}}}}^{ \bot } \left( {{\varvec{\Theta}}} \right)\frac{{\partial {{\varvec{A}}}\left( {{{\varvec{\Upsilon}};{\varvec{\Theta}}}_{0} } \right)}}{{\partial \left( {\varvec{\Upsilon}} \right)_{{i'}} }} + \frac{{\partial {{\varvec{A}}}^{H} \left( {{{\varvec{\Upsilon}};{\varvec{\Theta}}}_{0} } \right)}}{{\partial \left( {\varvec{\Upsilon}} \right)_{{i'}} }}{{\varvec{P}}}_{{{\varvec{A}}}}^{ \bot } \left( {{\varvec{\Theta}}} \right)\frac{{\partial {{\varvec{A}}}\left( {{{\varvec{\Upsilon}};{\varvec{\Theta}}}_{0} } \right)}}{{\partial \left( {\varvec{\Upsilon}} \right)_{i} }} +{{\varvec{A}}} A^{H} \left( {{{\varvec{\Upsilon}};{\varvec{\Theta}}}_{0} } \right){{\varvec{P}}}_{{{\varvec{A}}}}^{ \bot } \left( {{\varvec{\Theta}}} \right)\frac{{\partial ^{2} {{\varvec{A}}}\left( {{{\varvec{\Upsilon}};{\varvec{\Theta}}}_{0} } \right)}}{{\partial \left( {\varvec{\Upsilon}} \right)_{i} \partial \left( {\varvec{\Upsilon}} \right)_{{i'}} }} \\ & = 2\text{Re} \left\{ {\frac{{\partial {{\varvec{A}}}^{H} \left( {{{\varvec{\Upsilon}};{\varvec{\Theta}}}_{0} } \right)}}{{\partial \left( {\varvec{\Upsilon}} \right)_{i} }}{{\varvec{P}}}_{{{\varvec{A}}}}^{ \bot } \left( {{\varvec{\Theta}}} \right)\frac{{\partial {{\varvec{A}}}\left( {{{\varvec{\Upsilon}};{\varvec{\Theta}}}_{0} } \right)}}{{\partial \left( {\varvec{\Upsilon}} \right)_{{i'}} }} + {{\varvec{A}}}^{H} \left( {{{\varvec{\Upsilon}};{\varvec{\Theta}}}_{0} } \right){{\varvec{P}}}_{{{\varvec{A}}}}^{ \bot } \left( {{\varvec{\Theta}}} \right)\frac{{\partial ^{2} {{\varvec{A}}}\left( {{{\varvec{\Upsilon}};{\varvec{\Theta}}}_{0} } \right)}}{{\partial \left( {\varvec{\Upsilon}} \right)_{i} \partial \left( {\varvec{\Upsilon}} \right)_{{i'}} }}} \right\} \\ \end{aligned}$$
(35)
where
$${{\partial A\left( {{\varvec{\Upsilon}} ;{{\varvec{\Theta}}_0}} \right)} \over {\partial {{\left( {\varvec{\Upsilon}} \right)}_i}\partial {{\left( {\varvec{\Upsilon}} \right)}_{i'}}}} = - \left[ {j\left( {{{2\pi } \mathord{\left/ {\vphantom {{2\pi } \lambda }} \right. \kern-0pt} \lambda }} \right)diag\left( {{L^T}\dot k_{{{\left( {\varvec{\Upsilon}} \right)}_i}{{\left( {\varvec{\Upsilon}} \right)}_{i'}}}^2} \right) + {{\left( {{{2\pi } \mathord{\left/ {\vphantom {{2\pi } \lambda }} \right. \kern-0pt} \lambda }} \right)}^2}diag\left( {{L^T}{{\dot k}_{{{\left( {\varvec{\Upsilon}} \right)}_i}}}} \right)diag\left( {{L^T}{{\dot k}_{{{\left( {\varvec{\Upsilon}} \right)}_{i'}}}}} \right)} \right]A\left( {{\varvec{\Upsilon}} ;{{\varvec{\Theta}}_0}} \right)$$
(36)
and
$$\dot k_{{{\left( {\varvec{\Upsilon}} \right)}_i}{{\left( {\varvec{\Upsilon}} \right)}_{i'}}}^2 = {{\partial k} \over {\partial {{\left( {\varvec{\Upsilon}} \right)}_i}\partial {{\left( {\varvec{\Upsilon}} \right)}_{i'}}}} = \left\{ {\begin{array}{ll} {{{\left[ { - \cos \varphi \cos \theta , - \cos \varphi \sin \theta ,0} \right]}^T}} & {i = i' = 1} \\ {{{\left[ {\sin \varphi \sin \theta , - \sin \varphi \cos \theta ,0} \right]}^T}} & {i \ne i'} \\ {{{\left[ { - \cos \varphi \cos \theta , - \cos \varphi \sin \theta , - \sin \varphi } \right]}^T}} & {i = i' = 2}\end{array}}\right.$$
(37)
Then (35) can be written by
$$\begin{aligned} \frac{{\partial ^{2} F\left( {{{\varvec{\Upsilon}};{\varvec{\Theta}}}} \right)}}{{\partial \left( {\varvec{\Upsilon}} \right)_{i} \partial \left( {\varvec{\Upsilon}} \right)_{{i'}} }} &= 2\text{Re} \left\{ {\frac{{\partial {{\varvec{A}}}^{H} \left( {{{\varvec{\Upsilon}};{\varvec{\Theta}}}_{0} } \right)}}{{\partial \left( {\varvec{\Upsilon}} \right)_{i} }}{{\varvec{P}}}_{{{\varvec{A}}}}^{ \bot } \left( {{\varvec{\Theta}}} \right)\frac{{\partial {{\varvec{A}}}\left( {{{\varvec{\Upsilon}};{\varvec{\Theta}}}_{0} } \right)}}{{\partial \left( {\varvec{\Upsilon}} \right)_{{i'}} }} + {{\varvec{A}}}^{H} \left( {{{\varvec{\Upsilon}};{\varvec{\Theta}}}_{0} } \right){{\varvec{P}}}_{{{\varvec{A}}}}^{ \bot } \left( {{\varvec{\Theta}}} \right)\frac{{\partial {{\varvec{A}}}\left( {{{\varvec{\Upsilon}};{\varvec{\Theta}}}_{0} } \right)}}{{\partial \left( {\varvec{\Upsilon}} \right)_{i} \partial \left( {\varvec{\Upsilon}} \right)_{{i'}} }}} \right\} \\ &= 2\text{Re} \left\{ \begin{array}{l} \left( {{{2\pi } \mathord{\left/ {\vphantom {{2\pi } \lambda }} \right. \kern-\nulldelimiterspace} \lambda }} \right)^{2} {{\varvec{A}}}^{H} \left( {{{\varvec{\Upsilon}};{\varvec{\Theta}}}_{0} } \right)\left( {diag\left( {{{\varvec{L}}}^{T} {\dot{{\varvec{k}}}}_{{\left( {\varvec{\Upsilon}} \right)_{i} }} } \right)} \right)^{H} {{\varvec{P}}}_{{{\varvec{A}}}}^{ \bot } \left( {{\varvec{\Theta}}} \right)diag\left( {{{\varvec{L}}}^{T} {\dot{{\varvec{k}}}}_{{\left( {\varvec{\Upsilon}} \right)_{{i'}} }} } \right){{\varvec{A}}}\left( {{{\varvec{\Upsilon}};{\varvec{\Theta}}}_{0} } \right) \\ - j\left( {{{2\pi } \mathord{\left/ {\vphantom {{2\pi } \lambda }} \right. \kern-\nulldelimiterspace} \lambda }} \right){{\varvec{A}}}^{H} \left( {{{\varvec{\Upsilon}};{\varvec{\Theta}}}_{0} } \right){{\varvec{P}}}_{{{\varvec{A}}}}^{ \bot } \left( {{\varvec{\Theta}}} \right)diag\left( {{{\varvec{L}}}^{T} {\dot{{\varvec{k}}}}_{{\left( {\varvec{\Upsilon}} \right)_{i} \left( {\varvec{\Upsilon}} \right)_{{i'}} }}^{2} } \right){{\varvec{A}}}\left( {{{\varvec{\Upsilon}};{\varvec{\Theta}}}_{0} } \right) \\ - \left( {{{2\pi } \mathord{\left/ {\vphantom {{2\pi } \lambda }} \right. \kern-\nulldelimiterspace} \lambda }} \right)^{2} {{\varvec{A}}}^{H} \left( {{{\varvec{\Upsilon}};{\varvec{\Theta}}}_{0} } \right){{\varvec{P}}}_{{{\varvec{A}}}}^{ \bot } \left( {{\varvec{\Theta}}} \right)diag\left( {{{\varvec{L}}}^{T} {\dot{{\varvec{k}}}}_{{\left( {\varvec{\Upsilon}} \right)_{i} }} } \right)diag\left( {{{\varvec{L}}}^{T} {\dot{{\varvec{k}}}}_{{\left( {\varvec{\Upsilon}} \right)_{{i'}} }} } \right){{\varvec{A}}}\left( {{{\varvec{\Upsilon}};{\varvec{\Theta}}}_{0} } \right) \\ \end{array} \right\} \\ \end{aligned}$$
(38)
1.2 Derivatives with Respect to SEs Parameter \({\varvec{\Theta}}\)
The second derivative of \(F\left( {{\varvec{\Upsilon}} ;{\varvec{\Theta}} } \right)\) with respect to \({\varvec{\Upsilon}}\) and \({\varvec{\Theta}}\) is a \(2 \times length\left( {\varvec{\Theta}} \right)\) matrix, defined by
$$\frac{{\partial F^{2} \left( {{{\varvec{\Upsilon}};{\varvec{\Theta}}}} \right)}}{{\partial {\varvec{\Upsilon}}\partial {{\varvec{\Theta}}}^{T} }} = \frac{{\partial F_{1} \left( {{{\varvec{\Upsilon}};{\varvec{\Theta}}}} \right)}}{{\partial {{\varvec{\Theta}}}^{{{{\bf T}}}} }} = \left[ {\frac{{\partial F^{2} \left( {{{\varvec{\Upsilon}};{\varvec{\Theta}}}} \right)}}{{\partial \left( {\varvec{\Upsilon}} \right)_{i} \partial \left( {{\varvec{\Theta}}} \right)_{k} }}} \right] \in {\mathbb{C}}^{{2 \times length\left( {{\varvec{\Theta}}} \right)}}$$
(39)
where \(i = 1,2;k = 1, \ldots ,length\left( {\varvec{\Theta}} \right)\),\({\left( {\varvec{\Theta}} \right)_k}\) is kth element of vector \({\varvec{\Theta}}\), we can easily get that
$$\begin{aligned} \frac{{\partial F^{2} \left( {{{\varvec{\Upsilon}};{\varvec{\Theta}}}} \right)}}{{\partial \left( {\varvec{\Upsilon}} \right)_{i} \partial \left( {{\varvec{\Theta}}} \right)_{k} }} &= \frac{{\partial {{\varvec{A}}}^{H} \left( {{{\varvec{\Upsilon}};{\varvec{\Theta}}}_{0} } \right)}}{{\partial \left( {\varvec{\Upsilon}} \right)_{i} }}\frac{{\partial {{\varvec{P}}}_{{{\varvec{A}}}}^{ \bot } \left( {{\varvec{\Theta}}} \right)}}{{\partial \left( {{\varvec{\Theta}}} \right)_{k} }}{{\varvec{A}}}\left( {{{\varvec{\Upsilon}};{\varvec{\Theta}}}_{0} } \right) + {{\varvec{A}}}^{H} \left( {{{\varvec{\Upsilon}};{\varvec{\Theta}}}_{0} } \right)\frac{{\partial {{\varvec{P}}}_{{{\varvec{A}}}}^{ \bot } \left( {{\varvec{\Theta}}} \right)}}{{\partial \left( {{\varvec{\Theta}}} \right)_{k} }}\frac{{\partial {{\varvec{A}}}\left( {{{\varvec{\Upsilon}};{\varvec{\Theta}}}_{0} } \right)}}{{\partial \left( {\varvec{\Upsilon}} \right)_{i} }} \\ & = 2\text{Re} \left\{ {{{\varvec{A}}}^{H} \left( {{{\varvec{\Upsilon}};{\varvec{\Theta}}}_{0} } \right)\frac{{\partial {{\varvec{P}}}_{{{\varvec{A}}}}^{ \bot } \left( {{\varvec{\Theta}}} \right)}}{{\partial \left( {{\varvec{\Theta}}} \right)_{k} }}diag\left( { - j\left( {{{2\pi } \mathord{\left/ {\vphantom {{2\pi } \lambda }} \right. \kern-\nulldelimiterspace} \lambda }} \right){{\varvec{L}}}^{T} {\dot{{\varvec{k}}}}_{{\left( {\varvec{\Upsilon}} \right)_{i} }} } \right){{\varvec{A}}}\left( {{{\varvec{\Upsilon}};{\varvec{\Theta}}}_{0} } \right)} \right\} \\ \end{aligned}$$
(40)
The third derivative of \(F\left( {{\varvec{\Upsilon}} ;{\varvec{\Theta}} } \right)\) with respect to \({\varvec{\Upsilon}}\) and \({\varvec{\Theta}}\) is a \(2 \times length\left( {\varvec{\Theta}} \right)\) matrix, defined by
$$\frac{{\partial ^{3} F\left( {{{\varvec{\Upsilon}};{\varvec{\Theta}}}} \right)}}{{\partial {\varvec{\Upsilon}}\partial {\varvec{\Upsilon}}^{T} \partial {{\varvec{\Theta}}}^{T} }} = \frac{{\partial F_{2} \left( {{\varvec{\Upsilon}}_{0} ;{\varvec{\Theta}}} \right)}}{{\partial {{\varvec{\Theta}}}}} = \left[ {\frac{{\partial ^{3} F\left( {{{\varvec{\Upsilon}};{\varvec{\Theta}}}} \right)}}{{\partial \left( {\varvec{\Upsilon}} \right)_{i} \partial \left( {\varvec{\Upsilon}} \right)_{{i'}} \partial \left( {{\varvec{\Theta}}} \right)_{k} }}} \right] \in {\mathbb{C}}^{{2 \times 2length\left( {{\varvec{\Theta}}} \right)}}$$
(41)
where \(i,i' = 1,2;k = 1, \ldots ,length\left( {\varvec{\Theta}} \right)\), and
$$\begin{aligned} \frac{{\partial ^{3} F\left( {{{\varvec{\Upsilon}};{\varvec{\Theta}}}} \right)}}{{\partial \left( {\varvec{\Upsilon}} \right)_{i} \partial \left( {\varvec{\Upsilon}} \right)_{{i'}} \partial \left( {{\varvec{\Theta}}} \right)_{k} }} &= \frac{{\partial ^{2} {{\varvec{A}}}^{H} \left( {{{\varvec{\Upsilon}};{\varvec{\Theta}}}_{0} } \right)}}{{\partial \left( {\varvec{\Upsilon}} \right)_{i} \partial \left( {\varvec{\Upsilon}} \right)_{{i'}} }}\frac{{\partial {{\varvec{P}}}_{{{\varvec{A}}}}^{ \bot } \left( {{\varvec{\Theta}}} \right)}}{{\partial \left( {{\varvec{\Theta}}} \right)_{k} }}{{\varvec{A}}}\left( {{{\varvec{\Upsilon}};{\varvec{\Theta}}}_{0} } \right) + \frac{{\partial {{\varvec{A}}}^{H} \left( {{{\varvec{\Upsilon}};{\varvec{\Theta}}}_{0} } \right)}}{{\partial \left( {\varvec{\Upsilon}} \right)_{i} }}\frac{{\partial {{\varvec{P}}}_{{{\varvec{A}}}}^{ \bot } \left( {{\varvec{\Theta}}} \right)}}{{\partial \left( {{\varvec{\Theta}}} \right)_{k} }}\frac{{\partial {{\varvec{A}}}\left( {{{\varvec{\Upsilon}};{\varvec{\Theta}}}_{0} } \right)}}{{\partial \left( {\varvec{\Upsilon}} \right)_{{i'}} }} \\ &\quad + \frac{{\partial {{\varvec{A}}}^{H} \left( {{{\varvec{\Upsilon}};{\varvec{\Theta}}}_{0} } \right)}}{{\partial \left( {\varvec{\Upsilon}} \right)_{{i'}} }}\frac{{\partial {{\varvec{P}}}_{{{\varvec{A}}}}^{ \bot } \left( {{\varvec{\Theta}}} \right)}}{{\partial \left( {{\varvec{\Theta}}} \right)_{k} }}\frac{{\partial {{\varvec{A}}}\left( {{{\varvec{\Upsilon}};{\varvec{\Theta}}}_{0} } \right)}}{{\partial \left( {\varvec{\Upsilon}} \right)_{i} }} + {{\varvec{A}}}^{H} \left( {{{\varvec{\Upsilon}};{\varvec{\Theta}}}_{0} } \right)\frac{{\partial {{\varvec{P}}}_{{{\varvec{A}}}}^{ \bot } \left( {{\varvec{\Theta}}} \right)}}{{\partial \left( {{\varvec{\Theta}}} \right)_{k} }}\frac{{\partial ^{2} {{\varvec{A}}}\left( {{{\varvec{\Upsilon}};{\varvec{\Theta}}}_{0} } \right)}}{{\partial \left( {\varvec{\Upsilon}} \right)_{{i'}} \partial \left( {\varvec{\Upsilon}} \right)_{i} }} \\ &= 2\text{Re} \left\{ {\left[ \begin{array}{l} j\left( {{{2\pi } \mathord{\left/ {\vphantom {{2\pi } \lambda }} \right. \kern-\nulldelimiterspace} \lambda }} \right)diag\left( {{{\varvec{L}}}^{T} {\dot{{\varvec{k}}}}_{{\left( {\varvec{\Upsilon}} \right)_{i} \left( {\varvec{\Upsilon}} \right)_{{i'}} }}^{2} } \right)^{H} \\ - \left( {{{2\pi } \mathord{\left/ {\vphantom {{2\pi } \lambda }} \right. \kern-\nulldelimiterspace} \lambda }} \right)^{2} diag\left( {{{\varvec{L}}}^{T} {\dot{{\varvec{k}}}}_{{\left( {\varvec{\Upsilon}} \right)_{i} }} } \right)^{H} diag\left( {{{\varvec{L}}}^{T} {\dot{{\varvec{k}}}}_{{\left( {\varvec{\Upsilon}} \right)_{{i'}} }} } \right)^{H} \\ \; + \left( {{{2\pi } \mathord{\left/ {\vphantom {{2\pi } \lambda }} \right. \kern-\nulldelimiterspace} \lambda }} \right)^{2} diag\left( {{{\varvec{L}}}^{T} {\dot{{\varvec{k}}}}_{{\left( {\varvec{\Upsilon}} \right)_{i} }} } \right)^{H} diag\left( {{{\varvec{L}}}^{T} {\dot{{\varvec{k}}}}_{{\left( {\varvec{\Upsilon}} \right)_{{i'}} }} } \right) \\ \end{array} \right]{{\varvec{A}}}^{H} \left( {{{\varvec{\Upsilon}};{\varvec{\Theta}}}_{0} } \right)\frac{{\partial {{\varvec{P}}}_{{{\varvec{A}}}}^{ \bot } \left( {{\varvec{\Theta}}} \right)}}{{\partial \left( {{\varvec{\Theta}}} \right)_{k} }}{{\varvec{A}}}\left( {{{\varvec{\Upsilon}};{\varvec{\Theta}}}_{0} } \right)\;} \right\} \\ \end{aligned}$$
(42)
The derivative of orthogonal projection matrix \(P_{{\varvec{A}}}^\bot \left( {\varvec{\Theta}} \right)\) with respect to \({\varvec{\Theta}}\), defined by
$$\begin{aligned} \frac{{\partial {{\varvec{P}}}_{{{\varvec{A}}}}^{ \bot } \left( {{\varvec{\Theta}}} \right)}}{{\partial \left( {{\varvec{\Theta}}} \right)_{k} }} &= - \frac{{\partial {{\varvec{A}}}\left( {{\varvec{\Theta}}} \right)}}{{\partial \left( {{\varvec{\Theta}}} \right)_{k} }}\left( {{{\varvec{A}}}^{H} \left( {{\varvec{\Theta}}} \right){{\varvec{A}}}\left( {{\varvec{\Theta}}} \right)} \right)^{{ - 1}} {{\varvec{A}}}^{H} \left( {{\varvec{\Theta}}} \right) - {{\varvec{A}}}\left( {{\varvec{\Theta}}} \right)\left( {{{\varvec{A}}}^{H} \left( {{\varvec{\Theta}}} \right){{\varvec{A}}}\left( {{\varvec{\Theta}}} \right)} \right)^{{ - 1}} \frac{{\partial {{\varvec{A}}}^{H} \left( {{\varvec{\Theta}}} \right)}}{{\partial \left( {{\varvec{\Theta}}} \right)_{k} }}\\ &\quad + {{\varvec{A}}}\left( {{\varvec{\Theta}}} \right)\left( {{{\varvec{A}}}^{H} \left( {{\varvec{\Theta}}} \right){{\varvec{A}}}\left( {{\varvec{\Theta}}}
\right)} \right)^{{ - 1}} \frac{{\partial {{\varvec{A}}}^{H} \left( {{\varvec{\Theta}}} \right)}}{{\partial \left( {{\varvec{\Theta}}} \right)_{k} }}{{\varvec{A}}}\left( {{\varvec{\Theta}}} \right)\left( {{{\varvec{A}}}^{H} \left( {{\varvec{\Theta}}} \right){{\varvec{A}}}\left( {{\varvec{\Theta}}} \right)} \right)^{{ - 1}} {{\varvec{A}}}^{H} \left( {{\varvec{\Theta}}} \right) \\ &\quad + {{\varvec{A}}}\left( {{\varvec{\Theta}}} \right)\left( {{{\varvec{A}}}^{H} \left( {{\varvec{\Theta}}} \right){{\varvec{A}}}\left( {{\varvec{\Theta}}} \right)} \right)^{{ - 1}} {{\varvec{A}}}^{H} \left( {{\varvec{\Theta}}} \right)\frac{{\partial {{\varvec{A}}}\left( {{\varvec{\Theta}}} \right)}}{{\partial \left( {{\varvec{\Theta}}} \right)_{k} }}\left( {{{\varvec{A}}}^{H} \left( {{\varvec{\Theta}}} \right){{\varvec{A}}}\left( {{\varvec{\Theta}}} \right)} \right)^{{ - 1}} {{\varvec{A}}}^{H} \left( {{\varvec{\Theta}}} \right) \\ & = - {{\varvec{P}}}_{{{\varvec{A}}}}^{ \bot } \left( {{\varvec{\Theta}}} \right)\frac{{\partial {{\varvec{A}}}\left( {{\varvec{\Theta}}} \right)}}{{\partial \left( {{\varvec{\Theta}}} \right)_{k} }}\left( {{{\varvec{A}}}^{H} \left( {{\varvec{\Theta}}} \right){{\varvec{A}}}\left( {{\varvec{\Theta}}} \right)} \right)^{{ - 1}} {{\varvec{A}}}^{H} \left( {{\varvec{\Theta}}} \right) - {{\varvec{A}}}\left( {{\varvec{\Theta}}} \right)\left( {{{\varvec{A}}}^{H} \left( {{\varvec{\Theta}}} \right){{\varvec{A}}}\left( {{\varvec{\Theta}}} \right)} \right)^{{ - 1}} \frac{{\partial {{\varvec{A}}}^{H} \left( {{\varvec{\Theta}}} \right)}}{{\partial \left( {{\varvec{\Theta}}} \right)_{k} }}{{\varvec{P}}}_{{{\varvec{A}}}}^{ \bot } \left( {{\varvec{\Theta}}} \right) \\ \end{aligned}$$
(43)
where \(k = 1, \ldots ,length\left( {\varvec{\Theta}} \right)\). For all the choices of \({\varvec{\Theta}}\) which are of interest in this paper, the derivative of \(A\left( {\varvec{\Theta}} \right)\) with respect to the elements of \({\varvec{\Theta}}\) have the following form
$${{\partial A\left({\varvec{\Theta}} \right)} \over {\partial {\varvec{\Theta}} }} = \left[ \begin{array}{c} {{{\partial A\left( {\varvec{\Theta}} \right)} \over {\partial {{\left( {\varvec{\Theta}} \right)}_1}}}} \\ \vdots \\ {{{\partial A\left( {\varvec{\Theta}} \right)} \over {\partial {{\left( {\varvec{\Theta}} \right)}_{length\left( {\varvec{\Theta}} \right)}}}}} \end{array} \right],\quad \frac{\partial A\left( {\varvec{\Theta}} \right)} {\partial {{\left( {\varvec{\Theta}} \right)}_k}} = {w_k} = \left[ \begin{array}{c} 0 \\ \vdots \\ {W_k} \\ \vdots \\ 0 \\ \end{array} \right]$$
(44)
For convenience, we define a matrix \(W\) which consists of these \({W_k}\) ‘s stacked as follows:
$$W = \left[ \begin{array}{c} {W_1} \cr \vdots \cr {{W_{length\left( {\varvec{\Theta}} \right)}}} \cr \end{array} \right]$$
(45)
\({\varvec{\Theta}}\) may refer to the sensor gains, phases, the sensor locations. Next we specialize the results for particular choices of \({\varvec{\Theta}}\).
(B.1) Derivatives with respect to phase, means \({\varvec{\Theta}} = \left[ {{\varPhi_t},{\varPhi_r}} \right] = {\left[ {{e^{j\phi_t^1}}, \ldots ,{e^{j\phi_t^{M_t}}},{e^{j\phi_r^1}}, \ldots ,{e^{j\phi_r^{M_r}}}} \right]^T}\) and \(length\left( {\varvec{\Theta}} \right) = {M_t} + {M_r}\). Note that \({{\partial A\left( {\varvec{\Theta}} \right)} \over {\partial {\varvec{\Theta}} }} = {\left[ {{{\partial A\left( {\varvec{\Theta}} \right)} \over {\partial \phi_t^T}},{{\partial A\left( {\varvec{\Theta}} \right)} \over {\partial \phi_r^T}}} \right]^T}\), and
$${{\partial A\left( {\varvec{\Theta}} \right)} \over {\partial \phi_t^T}} = {{\partial A_t\left( {\varvec{\Theta}} \right)} \over {\partial \phi_t^T}} \odot {A_r}\left( {\varvec{\Theta}} \right) = \left[ {{{\partial a_t\left( {{{\varvec{\Upsilon}}_1};{\varvec{\Theta}} } \right)} \over {\partial g_t^T}} \otimes {a_r}\left( {{{\varvec{\Upsilon}}_1};{\varvec{\Theta}} } \right), \ldots ,{{\partial a_t\left( {{{\varvec{\Upsilon}}_Q};{\varvec{\Theta}} } \right)} \over {\partial g_t^T}} \otimes {a_r}\left( {{{\varvec{\Upsilon}}_Q};{\varvec{\Theta}} } \right)} \right] = jA_t\left( {\varvec{\Theta}} \right) \odot {A_r}\left( {\varvec{\Theta}} \right) = jA\left( {\varvec{\Theta}} \right)$$
(46)
$${{\partial A\left( {\varvec{\Theta}} \right)} \over {\partial \phi_r^T}} = A_t\left( {\varvec{\Theta}} \right) \odot {{\partial {A_r}\left( {\varvec{\Theta}} \right)} \over {\partial \phi_r^T}} = \left[ {a_t\left( {{{\varvec{\Upsilon}}_1};{\varvec{\Theta}} } \right) \otimes {{\partial {a_r}\left( {{{\varvec{\Upsilon}}_1};{\varvec{\Theta}} } \right)} \over {\partial \phi_r^T}}, \ldots ,a_t\left( {{{\varvec{\Upsilon}}_Q};{\varvec{\Theta}} } \right) \otimes {{\partial {a_r}\left( {{{\varvec{\Upsilon}}_Q};{\varvec{\Theta}} } \right)} \over {\partial \phi_r^T}}} \right] = jA_t\left( {\varvec{\Theta}} \right) \odot {A_r}\left( {\varvec{\Theta}} \right) = jA\left( {\varvec{\Theta}} \right)$$
(47)
Finally, It is obvious that
$${W_{{\text{Phase}}}} = \left[ \begin{array}{c} {jA\left( {\varvec{\Theta}} \right)} \\ {jA\left( {\varvec{\Theta}} \right)} \end{array}\right]$$
(48)
(B.2) Derivatives with respect to gain, means \({\varvec{\Theta}} = \left[ {g_t^T,g_r^T} \right] = \left[ {g_t^1, \ldots ,g_t^{M_t},g_r^1, \ldots ,g_r^{M_r}} \right]\) and \(length\left( {\varvec{\Theta}} \right) = {M_t} + {M_r}\), Note that \({{\partial A\left( {\varvec{\Theta}} \right)} \over {\partial {\varvec{\Theta}} }} = {\left[ {{{\partial A\left( {\varvec{\Theta}} \right)} \over {\partial g_t^T}},{{\partial A\left( {\varvec{\Theta}} \right)} \over {\partial g_r^T}}} \right]^T}\), and
$${{\partial A\left( {\varvec{\Theta}} \right)} \over {\partial g_t^T}} = {{\partial A_t\left( {\varvec{\Theta}} \right)} \over {\partial g_t^T}} \odot {A_r}\left( {\varvec{\Theta}} \right) = \left[ {{{\partial a_t\left( {{{\varvec{\Upsilon}}_1};{\varvec{\Theta}} } \right)} \over {\partial g_t^T}} \otimes {a_r}\left( {{{\varvec{\Upsilon}}_1};{\varvec{\Theta}} } \right), \ldots ,{{\partial a_t\left( {{{\varvec{\Upsilon}}_Q};{\varvec{\Theta}} } \right)} \over {\partial g_t^T}} \otimes {a_r}\left( {{{\varvec{\Upsilon}}_Q};{\varvec{\Theta}} } \right)} \right] = \left( {{\varPhi_t}{{\tilde A}_t}\left( \beta \right)} \right) \odot \left( {{\varGamma_r}{\varPhi_r}{{\tilde A}_r}\left( \beta \right)} \right)$$
(49)
$${{\partial A\left( {\varvec{\Theta}} \right)} \over {\partial g_r^T}} = A_t\left( {\varvec{\Theta}} \right) \odot {{\partial {A_r}\left( {\varvec{\Theta}} \right)} \over {\partial g_r^T}} = \left[ {a_t\left( {{{\varvec{\Upsilon}}_1};{\varvec{\Theta}} } \right) \otimes {{\partial {a_r}\left( {{{\varvec{\Upsilon}}_1};{\varvec{\Theta}} } \right)} \over {\partial g_r^T}}, \ldots ,a_t\left( {{{\varvec{\Upsilon}}_Q};{\varvec{\Theta}} } \right) \otimes {{\partial {a_r}\left( {{{\varvec{\Upsilon}}_Q};{\varvec{\Theta}} } \right)} \over {\partial g_r^T}}} \right] = \left( {{\varGamma_t}{\varPhi_t}{{\tilde A}_t}\left( \beta \right)} \right) \odot \left( {{\varPhi_r}{{\tilde A}_r}\left( \beta \right)} \right)$$
(50)
Finally, It is obvious that
$${W_{{\text{Gain}}}} = \left[ \begin{array}{c} {\left( {{\varPhi_t}{{\tilde A}_t}\left( \beta \right)} \right) \odot \left( {{\varGamma_r}{\varPhi_r}{{\tilde A}_r}\left( \beta \right)} \right)} \\ {\left( {{\varGamma_t}{\varPhi_t}{{\tilde A}_t}\left( \beta \right)} \right) \odot \left( {{\varPhi_r}{{\tilde A}_r}\left( \beta \right)} \right)} \end{array} \right]$$
(51)
(B.3)Derivatives with respect to location, means \({\varvec{\Theta}} = \left[ {{\beta_t},{\beta_r}} \right] = {\left[ {{\bf{t}}_x^T,{\bf{t}}_y^T,{\bf{t}}_z^T,{\bf{r}}_x^T,{\bf{r}}_y^T,{\bf{r}}_z^T} \right]^T}\) and \(length\left( {\varvec{\Theta}} \right) = 3\left( {{M_t} + {M_r}} \right)\), Note that \({{\partial A\left( {\varvec{\Theta}} \right)} \over {\partial {\varvec{\Theta}} }} = {\left[ {{{\partial A\left( {\varvec{\Theta}} \right)} \over {\partial {\bf{t}}_x^T}},{{\partial A\left( {\varvec{\Theta}} \right)} \over {\partial {\bf{t}}_y^T}},{{\partial A\left( {\varvec{\Theta}} \right)} \over {\partial {\bf{t}}_z^T}},{{\partial A\left( {\varvec{\Theta}} \right)} \over {\partial {\bf{r}}_x^T}},{{\partial A\left( {\varvec{\Theta}} \right)} \over {\partial {\bf{r}}_y^T}},{{\partial A\left( {\varvec{\Theta}} \right)} \over {\partial {\bf{r}}_z^T}}} \right]^T}\), and
$${{\partial A\left( {\varvec{\Theta}} \right)} \over {\partial {\bf{t}}_p^T}} = {{\partial A_t\left( {\varvec{\Theta}} \right)} \over {\partial {\bf{t}}_p^T}} \odot {A_r}\left( {\varvec{\Theta}} \right) = \left[ {{{\partial a_t\left( {{{\varvec{\Upsilon}}_1};{\varvec{\Theta}} } \right)} \over {\partial {\bf{t}}_p^T}} \otimes {a_r}\left( {{{\varvec{\Upsilon}}_1};{\varvec{\Theta}} } \right), \ldots ,{{\partial a_t\left( {{{\varvec{\Upsilon}}_Q};{\varvec{\Theta}} } \right)} \over {\partial {\bf{t}}_p^T}} \otimes {a_r}\left( {{{\varvec{\Upsilon}}_Q};{\varvec{\Theta}} } \right)} \right] = - j\left( {{{2\pi } \mathord{\left/ {\vphantom {{2\pi } \lambda }} \right. \kern-0pt} \lambda }} \right)\left[ {\left( {A_t\left( {\varvec{\Theta}} \right)diag\left( {{{\dot k}_p}} \right)} \right) \odot {A_r}\left( {\varvec{\Theta}} \right)} \right]$$
(52)
$${{\partial A\left( {\varvec{\Theta}} \right)} \over {\partial {\bf{r}}_p^T}} = A_t\left( {\varvec{\Theta}} \right) \odot {{\partial {A_r}\left( {\varvec{\Theta}} \right)} \over {\partial {\bf{r}}_p^T}} = \left[ {a_t\left( {{{\varvec{\Upsilon}}_1};{\varvec{\Theta}} } \right) \otimes {{\partial {a_r}\left( {{{\varvec{\Upsilon}}_1};{\varvec{\Theta}} } \right)} \over {\partial {\bf{r}}_p^T}}, \ldots ,a_t\left( {{{\varvec{\Upsilon}}_Q};{\varvec{\Theta}} } \right) \otimes {{\partial {a_r}\left( {{{\varvec{\Upsilon}}_Q};{\varvec{\Theta}} } \right)} \over {\partial {\bf{r}}_p^T}}} \right] = - j\left( {{{2\pi } \mathord{\left/ {\vphantom {{2\pi } \lambda }} \right. \kern-0pt} \lambda }} \right)\left[ {A_t\left( {\varvec{\Theta}} \right) \odot \left( {{A_r}\left( {\varvec{\Theta}} \right)diag\left( {{{\dot k}_p}} \right)} \right)} \right]$$
(53)
$${\dot k_x} = {\left[ {\cos {\varphi_1}\cos {\theta_1}, \ldots ,\cos {\varphi_Q}\cos {\theta_Q}} \right]^T}$$
(54a)
$${\dot k_y} = {\left[ {\cos {\varphi_1}\sin {\theta_1}, \ldots ,\cos {\varphi_Q}\sin {\theta_Q}} \right]^T}$$
(54b)
$${\dot k_z} = {\left[ {\sin {\varphi_1}, \ldots ,\sin {\varphi_Q}} \right]^T}$$
(54c)
Finally, It is obvious that
$${W_{{\text{Location}}}} = \left[ \begin{array}{c} { - j\left( {{{2\pi } \mathord{\left/ {\vphantom {{2\pi } \lambda }} \right. \kern-0pt} \lambda }} \right)\left[ {\left( {A_t\left( {\varvec{\Theta}} \right)diag\left( {{{\dot k}_x}} \right)} \right) \odot {A_r}\left( {\varvec{\Theta}} \right)} \right]} \\ { - j\left( {{{2\pi } \mathord{\left/ {\vphantom {{2\pi } \lambda }} \right. \kern-0pt} \lambda }} \right)\left[ {\left( {A_t\left( {\varvec{\Theta}} \right)diag\left( {{{\dot k}_y}} \right)} \right) \odot {A_r}\left( {\varvec{\Theta}} \right)} \right]} \\ { - j\left( {{{2\pi } \mathord{\left/ {\vphantom {{2\pi } \lambda }} \right. \kern-0pt} \lambda }} \right)\left[ {\left( {A_t\left( {\varvec{\Theta}} \right)diag\left( {{{\dot k}_z}} \right)} \right) \odot {A_r}\left( {\varvec{\Theta}} \right)} \right]} \\ { - j\left( {{{2\pi } \mathord{\left/ {\vphantom {{2\pi } \lambda }} \right. \kern-0pt} \lambda }} \right)\left[ {A_t\left( {\varvec{\Theta}} \right) \odot \left( {{A_r}\left( {\varvec{\Theta}} \right)diag\left( {{{\dot k}_x}} \right)} \right)} \right]} \\ { - j\left( {{{2\pi } \mathord{\left/ {\vphantom {{2\pi } \lambda }} \right. \kern-0pt} \lambda }} \right)\left[ {A_t\left( {\varvec{\Theta}} \right) \odot \left( {{A_r}\left( {\varvec{\Theta}} \right)diag\left( {{{\dot k}_y}} \right)} \right)} \right]} \\ { - j\left( {{{2\pi } \mathord{\left/ {\vphantom {{2\pi } \lambda }} \right. \kern-0pt} \lambda }} \right)\left[ {A_t\left( {\varvec{\Theta}} \right) \odot \left( {{A_r}\left( {\varvec{\Theta}} \right)diag\left( {{{\dot k}_z}} \right)} \right)} \right]} \end{array} \right]$$
(55)
What’s more, assume all the antennas and the targets on one planar. By comparing the norms of \({{\partial A\left( {\varvec{\Theta}} \right)} \over {\partial {\varvec{\Theta}} }}\) for the location derivatives and the phase derivatives, it is straightforward to show that in this case
$${\left\| {{{\partial A\left( {\varvec{\Theta}} \right)} \over {\partial {\varvec{\Theta}} }}} \right\|_{{\varvec{\Theta}} =location}} = {{2\pi } \over \lambda }{\left\| {{{\partial A\left( {\varvec{\Theta}} \right)} \over {\partial {\varvec{\Theta}} }}} \right\|_{{\varvec{\Theta}} =phase}}$$
(56)
Therefore, through simple calculation, we can get finally that the sensitivity parameter \(\delta _{{MUSIC}}\) for location errors is \({{2\pi } \mathord{\left/ {\vphantom {{2\pi } \lambda }} \right. \kern-0pt} \lambda }\) times the sensitivity parameter for phase errors. Similarly, it can be shown that the failure threshold \({\delta_{fail}}\) for location errors is \({\lambda \mathord{\left/ {\vphantom {\lambda {2\pi }}} \right. \kern-0pt} {2\pi }}\) times the failure threshold for phase errors. In mathematical notation
$${\delta_{MUSIC{\text{-}}location}} = \left( {{{2\pi } \mathord{\left/ {\vphantom {{2\pi } \lambda }} \right. \kern-0pt} \lambda }} \right){\delta_{MUSIC{\text{-}}phase}}$$
(57a)
$${\delta_{fail{\text{-}}location}} = \left( {{\lambda \mathord{\left/ {\vphantom {\lambda {2\pi }}} \right. \kern-0pt} {2\pi }}} \right){\delta_{fail{\text{-}}phase}}$$
(57b)
