Joint Estimation of Time Delay and Clock Error in the Incoherent Reception Systems

Abstract

The assumption that observed signals are ideally received is commonly used in the literature on time-delay estimation. However, for practical digital receivers, the received radio-frequency signals are usually downconverted to the baseband and digitized by analog-to-digital converters. Because inaccurate and unstable system clocks are used, frequency and phase offsets are usually induced by the frequency-mixing procedure, whereas a time stretch may ensue from the sampling procedure. The aforementioned problems are almost inevitable for incoherent reception systems. Therefore, the neglect of imperfect reception for incoherent systems may reduce the performance of conventional time-delay estimation methods. In this paper, employing refined signal models developed specifically for incoherent time-delay estimation in active and passive systems, the corresponding joint maximum likelihood estimates of time delay and system-clock frequency error are proposed. The Cramér–Rao lower bounds (CRLBs) on time-delay and clock frequency error estimations are also derived. The performance of the proposed time-delay estimators can be improved for frequency/phase offsets and time stretches, approaching the performance of the CRLBs in scenarios of moderate and high signal-to-noise ratios. Furthermore, both CRLBs analysis and simulation results verify that the accuracy of the proposed time-delay estimator is unaffected by the performance of the system clock in moderate scenarios.

Appendices

Appendix 1: Derivation of the Objective Function

Maximizing the log value of the likelihood function given by (15) is equivalent to maximizing

$$\begin{aligned}&- \left( {\mathbf{x}}_{\mathrm{a}} - {\mathbf{u}}_{\mathrm{a}}({\varvec{\theta }} )\right) ^H {\mathbf{C}}_{\mathrm{a}}^{ - 1} \left( {\mathbf{x}}_{\mathrm{a}} - {\mathbf{u}}_{\mathrm{a}}({\varvec{\theta }} )\right) \nonumber \\&\quad = 2{\mathop {\mathfrak {R}}\nolimits } \left\{ {\mathbf{x}}_{\mathrm{a}}^H {\mathbf{C}}_{\mathrm{a}}^{ - 1} {\mathbf{u}}_{\mathrm{a}}({\varvec{\theta }})\right\} - {\mathbf{x}}_{\mathrm{a}}^H {\mathbf{C}}_{\mathrm{a}}^{ - 1} {\mathbf{x}}_{\mathrm{a}} - {\mathbf{u}}_{\mathrm{a}}^H ({\varvec{\theta }} ){\mathbf{C}}_{\mathrm{a}}^{ - 1} {\mathbf{u}}_{\mathrm{a}}({\varvec{\theta }} ) \end{aligned}$$

(39)

with respect to $\varvec{\theta }$. The second item on the right-hand side of (39) is independent of the unknown vector $\varvec{\theta }$. In addition, invoking Assumption 5, we obtain

$$\begin{aligned} {\mathbf{s}}_{D,\beta }^H {\mathbf{s}}_{D,\beta } = {\mathbf{s}}^H {\mathbf{s}}. \end{aligned}$$

(40)

Further, according to (13), (14), and (40), the third term on the right-hand side of (39) may be rewritten as

$$\begin{aligned} {\mathbf{u}}_{a}^H ({\varvec{\theta }} ) {\mathbf{C}}_{\mathrm{a}}^{ - 1} {\mathbf{u}}_{a}({\varvec{\theta }} ) = a^2 {\sigma }_{\mathrm{a}} ^{-2} {\mathbf{s}}^H {\mathbf{s}}. \end{aligned}$$

(41)

That is, this term is unrelated to the unknown vector $\varvec{\theta }$. Hence, the ML estimate seeks to maximize

$$\begin{aligned} {\mathop {\mathfrak {R}}\nolimits } \left\{ {\mathbf{x}}_{\mathrm{a}}^H {\mathbf{C}}_{\mathrm{a}}^{ - 1} {\mathbf{u}}_{\mathrm{a}}({\varvec{\theta }})\right\} = a {\sigma }_{\mathrm{a}} ^{-2} {\mathop {\mathfrak {R}}\nolimits } \left\{ \hbox {e}^{j\varphi } {\mathbf{x}}_{\mathrm{a}}^H {\varvec{\Phi }} {\mathbf{s}}_{D,\beta }\right\} . \end{aligned}$$

(42)

After ignoring real constants a and ${\sigma }_{\mathrm{a}} ^{-2}$ in (42), maximizing ${\mathop {\mathfrak {R}}\nolimits } \{{\mathbf{x}}_{\mathrm{a}}^H {\mathbf{C}}_{\mathrm{a}}^{ - 1} {\mathbf{u}}_{\mathrm{a}}({\varvec{\theta }} )\}$ is equivalent to maximizing ${\mathop {\mathfrak {R}}\nolimits } \{ \hbox {e}^{j\varphi } {\mathbf{x}}_{\mathrm{a}}^H {\varvec{\Phi }} {\mathbf{s}}_{D,\beta } \}$. Hence, the objective function $L_{\mathrm{a}}({D,\beta ,\varphi })$ is as given in (16).

Appendix 2: Derivation of the CRLB for an Active System

The FIM ${\mathbf{J}}_{\mathrm{a}}$ for the estimation of a vector of real-valued parameters $\varvec{\theta }$ from the Gaussian vector ${\mathbf{x}}_{\mathrm{a}}$, when only the mean value ${\mathbf{u}}_{\mathrm{a}}$ depends on $\varvec{\theta }$, is given by Kay [11]

$$\begin{aligned} {\mathbf{J}}_{\mathrm{a}} = 2{\mathop {\mathfrak {R}}\nolimits } \left\{ {\mathbf{G}}^H_{\mathrm{a}} ({\varvec{\theta }} ) {\mathbf{C}}_{\mathrm{a}}^{ - 1} {\mathbf{G}}_{\mathrm{a}} ({\varvec{\theta }} ) \right\} = 2{\sigma }_{\mathrm{a}}^{-2} {\mathop {\mathfrak {R}}\nolimits } \left\{ {\mathbf{G}}_{\mathrm{a}}^H({\varvec{\theta }} ) {\mathbf{G}}_{\mathrm{a}} ({\varvec{\theta }} )\right\} , \end{aligned}$$

(43)

where the Jacobian matrix ${\mathbf{G}}_{\mathrm{a}} ({\varvec{\theta }} )$ is defined as ${\mathbf{G}}_{\mathrm{a}}({\varvec{\theta }} ) \buildrel \varDelta \over = \frac{{\partial {{\mathbf{u}}_{\mathrm{a}}}({\varvec{\theta }} )}}{{\partial {\varvec{\theta }} }}$. With $[\cdot ]_{i,l}$ denoting the ith row and lth column element of the matrix, the matrix entries of ${\mathbf{G}}_{\mathrm{a}}^H {\mathbf{G}}_{\mathrm{a}}$ are given as

$$\begin{aligned} {\left[ {\mathbf{G}}_{\mathrm{a}}^H {\mathbf{G}}_{\mathrm{a}}\right] _{1,1}}&= a^2 {{\dot{\mathbf{s}}}}^H {{\dot{\mathbf{s}}}}, \nonumber \\ {\left[ {\mathbf{G}}_{\mathrm{a}}^H {\mathbf{G}}_{\mathrm{a}}\right] _{1,2}}&= a^2 \left( {{\dot{\mathbf{s}}}}^H {\mathbf{N}\dot{\mathbf{s}}} + j2\pi f_\mathrm{m} T_\mathrm{s} {{\dot{\mathbf{s}}}}^H {\mathbf{N}} {\mathbf{s}}_{D,\beta }\right) , \nonumber \\ {\left[ {\mathbf{G}}_{\mathrm{a}}^H {\mathbf{G}}_{\mathrm{a}}\right] _{1,3}}&= - ja^2 {{\dot{\mathbf{s}}}}^H {\mathbf{s}}_{D,\beta },\nonumber \\ {\left[ {\mathbf{G}}_{\mathrm{a}}^H {\mathbf{G}}_{\mathrm{a}}\right] _{1,4}}&= - a{{\dot{\mathbf{s}}}}^H {\mathbf{s}}_{D,\beta },\nonumber \\ {\left[ {\mathbf{G}}_{\mathrm{a}}^H {\mathbf{G}}_{\mathrm{a}}\right] _{2,2}}&= a^2 \left( {{\dot{\mathbf{s}}}}^H {\mathbf{N}}^2 {{\dot{\mathbf{s}}}} + j2\pi f_\mathrm{m} T_\mathrm{s} {{\dot{\mathbf{s}}}}^H {\mathbf{N}}^2 {\mathbf{s}}_{D,\beta },\right. \nonumber \\&\left. \quad - j2\pi f_\mathrm{m} T_\mathrm{s} {\mathbf{s}}_{D,\beta }^H {\mathbf{N}}^2 {{\dot{\mathbf{s}}}} + 4\pi ^2 f_\mathrm{m}^2 T_\mathrm{s}^2 {\mathbf{s}}_{D,\beta }^H {\mathbf{N}}^2 {\mathbf{s}}_{D,\beta }\right) , \nonumber \\ {\left[ {\mathbf{G}}_{\mathrm{a}}^H {\mathbf{G}}_{\mathrm{a}}\right] _{2,3}}&= - a^2 (j{{\dot{\mathbf{s}}}}^H {\mathbf{N}}{\mathbf{s}}_{D,\beta } + 2\pi f_\mathrm{m} T_\mathrm{s} {\mathbf{s}}_{D,\beta }^H {\mathbf{N}}{\mathbf{s}}_{D,\beta } ), \nonumber \\ {\left[ {\mathbf{G}}_{\mathrm{a}}^H {\mathbf{G}}_{\mathrm{a}}\right] _{2,4}}&= -a\left( {{\dot{\mathbf{s}}}}^H {\mathbf{N}}{\mathbf{s}}_{D,\beta } + j2\pi f_\mathrm{m} T_\mathrm{s} {\mathbf{s}}_{D,\beta }^H {\mathbf{N}}{\mathbf{s}}_{D,\beta }\right) , \nonumber \\ {\left[ {\mathbf{G}}_{\mathrm{a}}^H {\mathbf{G}}_{\mathrm{a}}\right] _{3,3}}&= a^2 {\mathbf{s}}_{D,\beta }^H {\mathbf{s}}_{D,\beta },\nonumber \\ {\left[ {\mathbf{G}}_{\mathrm{a}}^H {\mathbf{G}}_{\mathrm{a}}\right] _{3,4}}&= - ja{\mathbf{s}}_{D,\beta }^H {\mathbf{s}}_{D,\beta },\nonumber \\ {\left[ {\mathbf{G}}_{\mathrm{a}}^H {\mathbf{G}}_{\mathrm{a}}\right] _{4,4}}&= {\mathbf{s}}_{D,\beta }^H {\mathbf{s}}_{D,\beta }, \end{aligned}$$

(44)

where

$$\begin{aligned} {\mathbf{N}}&\buildrel \varDelta \over =&{\mathrm{diag}} \{ \mathbf{n} \}, \end{aligned}$$

(45)

$$\begin{aligned} {{\dot{\mathbf{s}}}}&\buildrel \varDelta \over =&\frac{{\partial {\mathbf{s}}_{D,\beta } }}{{\partial n}} = j \frac{2 \pi }{N} {\mathbf{F}}^H {\mathbf{N}} {\mathbf{F}} {\mathbf{s}}_{D,\beta }. \end{aligned}$$

(46)

The vector $\mathbf{n}$ and the $N \times N$ unitary matrix ${\mathbf{F}}$ for the discrete Fourier transform are defined as

$$\begin{aligned} {\mathbf{n}}&\buildrel \varDelta \over =&\left[ 0,1, \ldots , N - 1\right] ^\mathrm{T}, \end{aligned}$$

(47)

$$\begin{aligned} {\mathbf{F}}&\buildrel \varDelta \over =&\frac{1}{{\sqrt{N} }}\exp \left( { - j\frac{{2\pi }}{N}{\mathbf{nn}}^\mathrm{T} } \right) . \end{aligned}$$

(48)

According to (46), the partial derivative may be obtained by taking the Fourier transform.⁹

Clearly ${\mathop {\mathfrak {R}}\nolimits }\{[{\mathbf{G}}_{\mathrm{a}}^H {\mathbf{G}}_{\mathrm{a}}]_{3,4}\}=0$, and moreover that (see the proof in “Appendix 4”)

$$\begin{aligned} {\mathop {\mathfrak {R}}\nolimits } \left\{ \left[ {\mathbf{G}}_{\mathrm{a}}^H {\mathbf{G}}_{\mathrm{a}}\right] _{1,4}\right\}= & {} 0, \end{aligned}$$

(49)

$$\begin{aligned} {\mathop {\mathfrak {R}}\nolimits } \left\{ \left[ {\mathbf{G}}_{\mathrm{a}}^H {\mathbf{G}}_{\mathrm{a}}\right] _{2,4}\right\}= & {} 0; \end{aligned}$$

(50)

i.e.,

$$\begin{aligned} {\mathop {\mathfrak {R}}\nolimits } \left\{ \left[ {\mathbf{G}}_{\mathrm{a}}^H {\mathbf{G}}_{\mathrm{a}}\right] _{i,4}\right\} = 0,\quad i = 1,2,3. \end{aligned}$$

(51)

In addition, from the Hermiticity of ${\mathbf{G}}_\mathrm{a}^H {\mathbf{G}}_\mathrm{a}$, the matrix ${\mathbf{J}}_{\mathrm{a}}$ is given by

$$\begin{aligned} {\mathbf{J}}_{\mathrm{a}} = \left[ {\begin{array}{*{20}c} {\left[ {\mathbf{J}}_{\mathrm{a}}\right] _{1:3,1:3}} &{}\quad 0\\ 0 &{}\quad {\left[ {\mathbf{J}}_{\mathrm{a}}\right] _{4,4}},\\ \end{array}} \right] \end{aligned}$$

(52)

where $[{\mathbf{J}}_{\mathrm{a}}]_{1:3,1:3}$ denotes the upper-left $3 \times 3$ block matrix of ${\mathbf{J}}_{\mathrm{a}}$. As

$$\begin{aligned} \left[ {\begin{array}{*{20}c} {\mathbf{A}} &{}\quad {\mathbf{0}} \\ {\mathbf{0}} &{}\quad a\\ \end{array}} \right] ^{ - 1}= \left[ {\begin{array}{*{20}c} {{\mathbf{A}}^{ - 1} } &{}\quad {\mathbf{0}}\\ {\mathbf{0}} &{}\quad {a^{ - 1} }\\ \end{array}} \right] , \end{aligned}$$

(53)

the CRLB on the unbiased estimation of ${\varvec{\theta }}_1\buildrel \varDelta \over = [D,\beta , \varphi ]^\mathrm{T}$ is given by diagonal elements of the matrix $[{\mathbf{J}}_{\mathrm{a}}] ^{-1} _{1:3,1:3}$. Hence, the CRLBs on the estimates of D and $\beta$ are given, respectively, by

$$\begin{aligned} {\mathrm{CRLB}}_{\mathrm{a},1} \left( \hat{D}\right)= & {} \frac{{\det \left\{ \left[ {\mathbf{J}}_{\mathrm{a}}\right] _{2:3,2:3}\right\} }}{{\det \left\{ \left[ {\mathbf{J}}_{\mathrm{a} } \right] _{1:3,1:3}\right\} }} = \frac{{ \sigma _{\mathrm{a}}^2 }}{{2a^2 }} \cdot \frac{{\gamma }}{{\gamma {{\dot{\mathbf{s}}}}^H {{\dot{\mathbf{s}}}} - \kappa }}, \end{aligned}$$

(54)

$$\begin{aligned} {\mathrm{CRLB}}_{\mathrm{a},1} \left( \hat{\beta }\right)= & {} \frac{{\det \left\{ \left[ {\mathbf{J}}_{\mathrm{a}}\right] _{1:2:3,1:2:3}\right\} }}{{\det \left\{ \left[ {\mathbf{J}}_{\mathrm{a} }\right] _{ 1:3,1:3}\right\} }} = \frac{{ \sigma _{\mathrm{a}}^2 }}{{2a^2 }} \cdot \frac{{\nu }}{{\gamma {{\dot{\mathbf{s}}}}^H {{\dot{\mathbf{s}}}} - \kappa }}, \end{aligned}$$

(55)

with

$$\begin{aligned} \left[ {\mathbf{J}}_{\mathrm{a}}\right] _{1:2:3,1:2:3} \buildrel \varDelta \over = \left[ {\begin{array}{*{20}c} {\left[ {\mathbf{J}}_{\mathrm{a}}\right] _{1,1} } &{}\quad {\left[ {\mathbf{J}}_{\mathrm{a}}\right] _{1,3} }\\ {\left[ {\mathbf{J}}_{\mathrm{a}}\right] _{3,1} } &{}\quad {\left[ {\mathbf{J}}_{\mathrm{a}}\right] _{3,3} }\\ \end{array}} \right] \end{aligned}$$

(56)

where $\gamma$, $\kappa$ and $\nu$ are defined, respectively, by

$$\begin{aligned} \gamma= & {} \left\| {\left( {{\dot{\mathbf{s}}}}^H {\mathbf{N}} - 2\pi f_\mathrm{m} T_\mathrm{s} {\mathbf{s}}_{D,\beta }^H {\mathbf{N}}\right) } \right\| ^2 {\mathbf{s}}_{D,\beta }^H {\mathbf{s}}_{D,\beta }\nonumber \\&-\,\left( {{\mathop {\mathfrak {I}}\nolimits } \left\{ {{\dot{\mathbf{s}}}}^H {\mathbf{N}} {\mathbf{s}}_{D,\beta }\right\} - 2\pi f_\mathrm{m} T_\mathrm{s} {\mathbf{s}}_{D,\beta }^H {\mathbf{N}} {\mathbf{s}}_{D,\beta }}\right) ^2, \end{aligned}$$

(57)

$$\begin{aligned} \kappa= & {} \left\| {\mathop {\mathfrak {I}}\nolimits } \left\{ {{\dot{\mathbf{s}}}}^H {\mathbf{s}}_{D,\beta } \right\} \left( {{\dot{\mathbf{s}}}}^H {\mathbf{N}} - 2\pi f_\mathrm{m} T_\mathrm{s} {\mathbf{s}}_{D,\beta }^H {\mathbf{N}}\right) \right. \nonumber \\&\left. -\, {\mathbf{s}}_{D,\beta }^H \left( {{\dot{\mathbf{s}}}}^H {\mathbf{N}\dot{\mathbf{s}}} - 2\pi f_\mathrm{m} T_\mathrm{s} {\mathop {\mathfrak {I}}\nolimits } \left\{ {{\dot{\mathbf{s}}}}^H {\mathbf{N}} {\mathbf{s}}_{D,\beta } \right\} \right) \right\| ^2, \end{aligned}$$

(58)

$$\begin{aligned} \nu= & {} {{\dot{\mathbf{s}}}}^H {{\dot{\mathbf{s}}}}{\mathbf{s}}_{D,\beta }^H {\mathbf{s}}_{D,\beta } - {\mathop {\mathfrak {I}}\nolimits } \{ {{\dot{\mathbf{s}}}}^H {\mathbf{s}}_{D,\beta } \} {\mathop {\mathfrak {I}}\nolimits } \{ {{\dot{\mathbf{s}}}}^H {\mathbf{s}}_{D,\beta } \}. \end{aligned}$$

(59)

Appendix 3: Derivation of the CRLB for a Passive System

The FIM ${\mathbf{J}}_{\mathrm{p}}$ for the estimation of a real-valued parameter vector $\varvec{\vartheta }$ from a Gaussian vector ${\mathbf{x}}_{\mathrm{p}}$, when only the mean ${\mathbf{u}}_{\mathrm{p}}$ depends on $\varvec{\vartheta }$, is given by Kay [11]

$$\begin{aligned} {\mathbf{J}}_{\mathrm{p}} = 2{\mathop {\mathfrak {R}}\nolimits } \left\{ {\left[ {\frac{{\partial {\mathbf{u}}_{\mathrm{p}}({\varvec{\vartheta }} )}}{{\partial {\varvec{\vartheta }} }}} \right] ^H {\mathbf{C}}_{\mathrm{p}}^{ - 1} \left[ {\frac{{\partial {\mathbf{u}}_{\mathrm{p}}({\varvec{\vartheta }} )}}{{\partial {\varvec{\vartheta }} }}} \right] } \right\} . \end{aligned}$$

(60)

From the derivative of ${\mathbf{u}}_{\mathrm{p}}$ with respect to $\varvec{\vartheta }$, we get the $2N \times (2N+4)$ Jacobian matrix

$$\begin{aligned} {\mathbf{H}}({\varvec{\vartheta }} ) \buildrel \varDelta \over = \frac{{\partial {\mathbf{u}}_{\mathrm{p}}({\varvec{\vartheta }} )}}{{\partial {\varvec{\vartheta }} }} = \left[ {\begin{array}{*{20}c} {\mathbf{I}} &{}\quad {j {\mathbf{I}}} &{}\quad {\mathbf{0}}\\ {\frac{{\partial {\mathbf{u}}_{{\mathrm{p}},2} ({\varvec{\vartheta }} )}}{{\partial {\mathop {\mathfrak {R}}\nolimits } \{ {\mathbf{s}}\} }}} &{}\quad {\frac{{\partial {\mathbf{u}}_{{\mathrm{p}},2} ({\varvec{\vartheta }} )}}{{\partial {\mathop {\mathfrak {I}}\nolimits } \{ {\mathbf{s}}\} }}} &{}\quad {\frac{{\partial {\mathbf{u}}_{{\mathrm{p}},2} ({\varvec{\vartheta }} )}}{{\partial {\varvec{\theta }} }}}\\ \end{array}} \right] , \end{aligned}$$

(61)

where ${\mathbf{u}}_{{\mathrm{p}},2} ({\varvec{\vartheta }} ) \buildrel \varDelta \over = a\hbox {e}^{j\varphi } {\varvec{\Phi }} {\mathbf{s}}_{D,\beta }$, and $\mathbf{0}$ denotes the $N \times 4$ null matrix. In addition, because $\beta$ is usually small,¹⁰ ${\mathbf{s}}_{D,\beta }$ approximates to the product

$$\begin{aligned} {\mathbf{s}}_{D,\beta } \approx {\mathbf{Q}} {\mathbf{s}}, \end{aligned}$$

(62)

where the unitary matrix ${\mathbf{Q}}$ is defined as

$$\begin{aligned} {\mathbf{Q}} \buildrel \varDelta \over = {\mathbf{F}}^H {\varvec{\Phi }}_D {\mathbf{F}} \end{aligned}$$

(63)

with the diagonal matrix ${\varvec{\Phi }}_{\mathrm{D}}$ given by

$$\begin{aligned} {\varvec{\Phi }}_\mathrm{D}\buildrel \varDelta \over = {\mathrm{diag}} \left\{ {\exp \left( { - j\frac{{2\pi }}{N}{} \textit{DT}_\mathrm{s} {\mathbf{n}}} \right) }\right\} , \end{aligned}$$

(64)

and $\mathbf{n}$ and ${\mathbf{F}}$ are defined by (47) and (48), respectively.

Hence, from the definitions of ${\mathbf{u}}_{{\mathrm{p}},2} ({\varvec{\vartheta }} )$ and (62), ${\frac{{\partial {\mathbf{u}}_{{\mathrm{p}},2} ({\varvec{\vartheta }} )}}{{\partial {\mathop {\mathfrak {R}}\nolimits } \{ {\mathbf{s}}\} }}}$ and ${\frac{{\partial {\mathbf{u}}_{{\mathrm{p}},2} ({\varvec{\vartheta }} )}}{{\partial {\mathop {\mathfrak {I}}\nolimits } \{ {\mathbf{s}}\} }}}$ may be approximated as

$$\begin{aligned} \frac{{\partial {\mathbf{u}}_{{\mathrm{p}},2} (\varvec{\vartheta } )}}{{\partial {\mathop {\mathfrak {R}}\nolimits } \{ {\mathbf{s}}\} }} \approx \frac{{\partial a\hbox {e}^{j\varphi } {\varvec{\Phi }} {\mathbf{Q}} {\mathbf{s}}}}{{\partial {\mathop {\mathfrak {R}}\nolimits } \{ {\mathbf{s}}\} }}= & {} a \hbox {e}^{j\varphi } {\varvec{\Phi }} {\mathbf{Q}}, \end{aligned}$$

(65)

$$\begin{aligned} \frac{{\partial {\mathbf{u}}_{{\mathrm{p}},2} (\varvec{\vartheta } )}}{{\partial {\mathop {\mathfrak {I}}\nolimits } \{ {\mathbf{s}}\} }} \approx \frac{{\partial a\hbox {e}^{j\varphi } {\varvec{\Phi }} {\mathbf{Q}} {\mathbf{s}}}}{{\partial {\mathop {\mathfrak {I}}\nolimits } \{ {\mathbf{s}}\} }}= & {} ja\hbox {e}^{j\varphi } {\varvec{\Phi }} {\mathbf{Q}}. \end{aligned}$$

(66)

The Jacobian matrix ${\mathbf{H}}(\varvec{\vartheta })$, under the above approximations, is the same as the Jacobian matrix of the joint TDOA and FDOA estimation found in the literature [20]. Fortunately, we are only interested in the bound on the parameters $\varvec{\theta }$ rather than $\varvec{\vartheta }$. This bound is given by the lower-right $4\times 4$ block of the inverse of ${\mathbf{J}}_{\mathrm{p}}$. Hence, as deduced in [20], the FIM corresponding to the unknown parameters $\varvec{\theta }$ is given by

$$\begin{aligned} {\mathbf{J}}_{{\varvec{\theta }} }= \frac{2}{{a^2 \sigma _{\mathrm{p},1}^2+ \sigma _{\mathrm{p},2}^2 }}{\mathop {\mathfrak {R}}\nolimits } \left\{ {{\mathbf{G}}_{\mathrm{p}}^H {\mathbf{G}}_{\mathrm{p}}} \right\} \end{aligned}$$

(67)

where ${\mathbf{G}}_{\mathrm{p}} \buildrel \varDelta \over = \frac{{\partial {\mathbf{u}}_{\mathrm{p},2} ({\varvec{\vartheta }} )}}{{\partial {\varvec{\theta }} }}$. It is obvious that ${\mathbf{G}}_{\mathrm{p}}$ is identical to ${\mathbf{G}}_{\mathrm{a}}$. Hence, ${\mathbf{G}}_{\mathrm{p}}^H {\mathbf{G}}_{\mathrm{p}}$ is identical to ${\mathbf{G}}_{\mathrm{a}}^H {\mathbf{G}}_{\mathrm{a}}$. Similar to the derivation for an active system, the CRLBs on the estimates of D and $\beta$ in incoherent reception are given by

$$\begin{aligned} {\mathrm{CRLB}}_{\mathrm{p},1} \left( \hat{D}\right)= & {} \frac{{\det \left\{ \left[ {\mathbf{J}}_{\varvec{\theta } }\right] _{2:3,2:3}\right\} }}{{\det \left\{ \left[ {\mathbf{J}}_{\varvec{\theta }}\right] _{1:3,1:3}\right\} }} = \frac{{a^2 \sigma _{\mathrm{p},1}^2+ \sigma _{\mathrm{p},2}^2 }}{{2a^2 }} \cdot \frac{{\gamma }}{{\gamma {{\dot{\mathbf{s}}}}^H {{\dot{\mathbf{s}}}} - \kappa }} \end{aligned}$$

(68)

$$\begin{aligned} {\mathrm{CRLB}}_{\mathrm{p},1} \left( \hat{\beta }\right)= & {} \frac{{\det \left\{ \left[ {\mathbf{J}}_{\varvec{\theta } }\right] _{1:2:3,1:2:3}\right\} }}{{\det \left\{ \left[ {\mathbf{J}}_{\varvec{\theta } }\right] _{ 1:3,1:3}\right\} }} = \frac{{a^2 \sigma _{\mathrm{p},1}^2+ \sigma _{\mathrm{p},2}^2 }}{{2a^2 }} \cdot \frac{{\nu }}{{\gamma {{\dot{\mathbf{s}}}}^H {{\dot{\mathbf{s}}}} - \kappa }}, \end{aligned}$$

(69)

where $\gamma$, $\kappa$ and $\nu$ are defined as for the active system, and $[{\mathbf{J}}_{\varvec{\theta } }]_{1:2:3,1:2:3}$ is similar to the definition (56), and $\dot{\mathbf{s}}^H \dot{\mathbf{s}}$ may be given by

$$\begin{aligned} \dot{\mathbf{s}}^H \dot{\mathbf{s}} = \frac{4\pi ^2}{N^2} {{\mathbf{s}}_{D,\beta }^H} {\mathbf{F}}^H {\mathbf{N}}^2 {\mathbf{F}} {{\mathbf{s}}_{D,\beta }}. \end{aligned}$$

(70)

Note, $\dot{\mathbf{s}}^H \dot{\mathbf{s}}$ denotes the energy of $\dot{\mathbf{s}}$ defined in (46).¹¹

Appendix 4: Proof of (49) and (50)

According to (46), we obtain

$$\begin{aligned} {{\dot{\mathbf{s}}}}^H {\mathbf{s}}_{D,\beta } = - j\frac{{2\pi }}{N}\left( {\mathbf{Fs}}_{D,\beta }\right) ^H {\mathbf{N}}\left( {\mathbf{Fs}}_{D,\beta }\right) , \end{aligned}$$

(71)

where $\mathbf{N}$ is defined by (45); i.e., $\mathbf{N}$ is a diagonal matrix with real diagonal elements. Thus, ${{\dot{\mathbf{s}}}}^H {\mathbf{s}}_{D,\beta }$ is imaginary valued. Thus, (49) is proved.

$$\begin{aligned} \mathfrak {R}\left\{ -a {{\dot{\mathbf{s}}}}^H {\mathbf{s}}_{D,\beta }\right\} = 0. \end{aligned}$$

(72)

From Assumption 5, we have

$$\begin{aligned} {{\dot{\mathbf{s}}}}^H {\mathbf{Ns}}_{D,\beta } = \sum \limits _{n = 0}^{N - 1} {\left[ jm[n]\hbox {e}^{jm[n]}\cdot n\right] ^* \hbox {e}^{jm[n]} } = - j \sum \limits _{n = 0}^{N - 1} {n \cdot m[n]}. \end{aligned}$$

(73)

Therefore, (50) is proved.

Appendix 5: Proof of $\gamma {{\dot{\mathbf{s}}}}^H {{\dot{\mathbf{s}}}} - \kappa \ge 0$

The FIM $[{\mathbf{J}}_{\mathrm{a}}]_{1:3,1:3}$ may be decomposed as

$$\begin{aligned}{}[{\mathbf{J}}_{\mathrm{a}}]_{1:3,1:3}= \frac{{2{a^2}}}{{\sigma _{\mathrm{a}}^2}} {\mathbf{G}_{\mathrm{a}}} {\mathbf{G}}_{\mathrm{a}}^H, \end{aligned}$$

(74)

where the $3\times N$ matrix $\mathbf{G}_{\mathrm{a}}$ is given by

$$\begin{aligned} {\mathbf{G}_{\mathrm{a}}} = \left[ \begin{array}{l} {{\dot{\mathbf{s}}}}^H\\ \left( {\mathbf{N}\dot{\mathbf{s}}} - 2\pi f_{\mathrm{m}} T_{\mathrm{s}} {\mathbf{Ns}}_{D,\beta }\right) ^H\\ {\mathbf{s}}_{D,\beta }^H\\ \end{array} \right] . \end{aligned}$$

(75)

Hence, $\forall \mathbf{x} \ne 0$, where $\mathbf{x}$ is an N-dimensional column vector, $\exists {\mathbf{x}}^H {\mathbf{G}_{\mathrm{a}}} {\mathbf{G}}_{\mathrm{a}}^H {\mathbf{x}} = ( {\mathbf{G}}_{\mathrm{a}}^H {\mathbf{x}} )^H {\mathbf{G}}_{\mathrm{a}}^H {\mathbf{x}} \ge 0$. From (74), it is known that $[{\mathbf{J}}_{\mathrm{a}}]_{1:3,1:3}$ is a positive definite matrix. Therefore,

$$\begin{aligned} \det \left\{ [{\mathbf{J}}_{\mathrm{a}}]_{1:3,1:3}\right\} \ge 0. \end{aligned}$$

(76)

In addition, using (57) and (58), it follows that

$$\begin{aligned} \det \left\{ [{\mathbf{J}}_{\mathrm{a}}]_{1:3,1:3}\right\} = \left( \frac{{2a^2 }}{{ \sigma _{\mathrm{a}}^2 }} \right) ^3 \left( \gamma {{\dot{\mathbf{s}}}}^H {{\dot{\mathbf{s}}}} - \kappa \right) . \end{aligned}$$

(77)

From (76) and (77), we obtain

$$\begin{aligned} \gamma {{\dot{\mathbf{s}}}}^H {{\dot{\mathbf{s}}}} - \kappa \ge 0. \end{aligned}$$

(78)