Joint range and angle estimation for an integrated system combining MIMO radar with OFDM communication

Abstract

To perform the integration of radar and communication in waveform, we design an integrated system combining multiple-input multiple-output radar with orthogonal frequency division multiplexing (OFDM) communication. In this system, each antenna transmits the integrated waveform with a nonoverlapping block sub-frequency band. The utilized waveform is a variation of the classical OFDM communication waveform. In order to sufficiently exploit the entire system bandwidth and array aperture, a joint time and space processing approach is proposed, and hence the range and angle estimations with high resolution are obtained, whereas the range and angle are coupled. Moreover, the loss in processing gain and the Cramér–Rao bounds of range and angle estimates based on integrated waveform are derived, respectively. Theoretical analysis validates that the designed system is capable of implementing the radar and communication functions simultaneously. Finally, numerical results are presented to verify the effectiveness of the proposed approach.

Appendices

Appendix A

1.1 Calculation of $\tilde{r}_{i,p} ( \tau)$

The $\tilde{r}_{i,p} ( \tau)$ can be equivalently defined as $\tilde{r}_{i,p} ( \tau ) = \int_{{\left( {p - 1} \right)T_{\text{r}} + T_{\text{p}} }}^{{\left({p + 1} \right)T_{\text{r}} - T_{\text{p}} }} {\tilde{s}_{i,p} (t)\tilde{s}_{i,p}^{*} \left( {t - \tau }\right){\text{d}}t}$. Consider $p = 0$, hence

$$ \tilde{r}_{i,p} ( \tau) = \int_{{T_{\text{p}} - T_{\text{r}} }}^{{T_{\text{r}} - T_{\text{p}}}} {\left\{ \begin{aligned} &\sum\limits_{n = 0}^{{N_{\text{s}} -1}} {\left\{ \begin{aligned} {\text{rect}}\left[ {{{\left( {t -nT_{\text{s}} - pT_{\text{r}} } \right)} \mathord{\left/ {\vphantom {{\left( {t - nT_{\text{s}} - pT_{\text{r}} } \right)} {T_{\text{s}}}}} \right. \kern-0pt} {T_{\text{s}} }}} \right]\exp \left\{ { - j2\pi \left( {i - 1} \right)B_{\text{a}} nT_{\text{s}} } \right\} \hfill \\ \sum\limits_{m = 0}^{{N_{\text{c}} - 1}} {a_{i} \left({m,n,p} \right)\exp \left\{ {j2\pi m\Delta f\left( {t -nT_{\text{s}} } \right)} \right\}} \hfill \\ \end{aligned} \right.} \hfill \\ &\cdot \sum\limits_{{n^{\prime} = 0}}^{{N_{\text{s}} - 1}} {\left\{ \begin{aligned} {\text{rect}}\left[ {{{\left( {t -n^{\prime}T_{\text{s}} - pT_{\text{r}} - \tau } \right)} \mathord{\left/ {\vphantom {{\left( {t - n^{\prime}T_{\text{s}} - pT_{\text{r}} - \tau } \right)} {T_{\text{s}} }}} \right. \kern-0pt} {T_{\text{s}} }}} \right]\exp \left\{ {j2\pi \left( {i - 1} \right)B_{\text{a}} n^{\prime}T_{\text{s}} } \right\} \hfill \\ \sum\limits_{{m^{\prime} = 0}}^{{N_{c} - 1}} {a_{i}^{*} \left({m^{\prime},n^{\prime},p} \right)\exp \left\{ { - j2\pi m^{\prime}\Delta f\left( {t - \tau - n^{\prime}T_{\text{s}} } \right)} \right\}} \hfill \\ \end{aligned} \right.} \hfill \\ \end{aligned} \right.} {\text{d}}t $$

(A.1)

Case 1

For $T_{\text{p}} \le \left| \tau \right| \le T_{\text{r}} - T_{\text{p}}$, it is obvious that

$$ \tilde{r}_{i,p} ( \tau) = 0 $$

(A.2a)

Case 2

For $- T_{\text{p}} < \tau \le 0$ and $\left\lfloor {{\tau \mathord{\left/ {\vphantom {\tau {T_{\text{s}} }}} \right. \kern-0pt} {T_{\text{s}} }}} \right\rfloor = n_{0}$ with $\left\lfloor x \right\rfloor$ denoting the integer part of $x$, simplifying (A.1) yields

$$ \begin{aligned} \tilde{r}_{i,p} ( \tau) & = - \left( {\left| {n_{0} } \right|T_{\text{s}} + \tau } \right)\sum\limits_{n = 1}^{{N_{\text{s}} - 1 - \left| {n_{0} } \right|}} {\sum\limits_{m = 0}^{{N_{\text{c}} - 1}} {\sum\limits_{{m^{\prime} = 0}}^{{N_{\text{c}} - 1}} {\left\{ \begin{aligned} & a_{i} \left( {m,n - 1,p} \right)a_{i}^{*} \left( {m^{\prime } ,n + \left| {n_{0} } \right|,p} \right) \\ & \cdot \exp \left\{ {j\pi \Delta f\left[ {\left( {m + m^{\prime } } \right)\left| {n_{0} } \right| + 2m} \right]T_{\text{s}} } \right\} \\ & \cdot \exp \left\{ {j2\pi \left( {i - 1} \right)\left( {\left| {n_{0} } \right| + 1} \right)B_{\text{a}} T_{\text{s}} } \right\} \\ & \cdot \exp \left\{ {j\pi \left( {m + m^{\prime } } \right)\Delta f\tau } \right\} \\ & \cdot {\text{sinc}}\left\{ {\pi \left( {m - m^{\prime } } \right)\Delta f\left( { - \left| {n_{0} } \right|T_{\text{s}} - \tau } \right)} \right\} \\ \end{aligned} \right.} } } \\ & \quad + \left[ {\left( {\left| {n_{0} } \right| + 1} \right)T_{\text{s}} + \tau } \right]\sum\limits_{n = 0}^{{N_{\text{s}} - 1 - \left| {n_{0} } \right|}} {\sum\limits_{m = 0}^{{N_{\text{c}} - 1}} {\sum\limits_{{m^{\prime} = 0}}^{{N_{\text{c}} - 1}} {\left\{ \begin{aligned} & a_{i} \left( {m,n,p} \right)a_{i}^{*} \left( {m^{\prime } ,n + \left| {n_{0} } \right|,p} \right) \\ & \cdot \exp \left\{ {j2\pi \left( {i - 1} \right)\left| {n_{0} } \right|B_{\text{a}} T_{\text{s}} } \right\} \\ & \cdot \exp \left\{ {j\pi \left( {m + m^{\prime } } \right)\Delta f\tau } \right\} \\ & \cdot \exp \left\{ {j\pi \Delta f\left[ {\left( {m + m^{\prime } } \right)\left| {n_{0} } \right| + \left( {m - m^{\prime } } \right)} \right]T_{\text{s}} } \right\} \\ & \cdot {\text{sinc}}\left\{ {\pi \left( {m - m^{\prime } } \right)\Delta f\left[ {\left( {\left| {n_{0} } \right| + 1} \right)T_{\text{s}} + \tau } \right]} \right\} \\ \end{aligned} \right.} } } \\ \end{aligned} $$

(A.2b)

where ${\text{sinc}}\left( x \right) = \sin x/x \cdot$

Case 3

For $0 < \tau < T_{\text{p}}$ and $\left\lfloor {{\tau \mathord{\left/ {\vphantom {\tau {T_{\text{s}} }}} \right. \kern-0pt} {T_{\text{s}} }}} \right\rfloor = n_{0}$, we can obtain

$$ \begin{aligned} \tilde{r}_{i,p} ( \tau) & = \left( {\tau - \left| {n_{0} } \right|T_{\text{s}} } \right)\sum\limits_{n = 1}^{{N_{\text{s}} - 1 - \left| {n_{0} } \right|}} {\sum\limits_{m = 0}^{{N_{\text{c}} - 1}} {\sum\limits_{{m^{\prime} = 0}}^{{N_{\text{c}} - 1}} {\left\{ \begin{aligned} & a_{i} \left( {m,n + \left| {n_{0} } \right|,p} \right)a_{i}^{*} \left( {m^{\prime } ,n - 1,p} \right) \\ & \cdot \exp \left\{ { - j\pi \Delta f\left[ {\left( {m + m^{\prime}} \right)\left| {n_{0} } \right| + 2m^{\prime } } \right]T_{\text{s}} } \right\} \\ & \cdot \exp \left\{ {j\pi \left( {m + m^{\prime } } \right)\Delta f\tau } \right\} \\ & \cdot \exp \left\{ { - j2\pi \left( {i - 1} \right)\left( {\left| {n_{0} } \right| + 1} \right)B_{\text{a}} T_{\text{s}} } \right\} \\ & {\text{sinc}}\left\{ {\pi \left( {m - m^{\prime } } \right)\Delta f\left( { - \left| {n_{0} } \right|T_{\text{s}} + \tau } \right)} \right\} \\ \end{aligned} \right.} } } \\ & \quad + \left[ {\left( {\left| {n_{0} } \right| + 1} \right)T_{\text{s}} - \tau } \right]\sum\limits_{n = 0}^{{N_{\text{s}} - 1 - \left| {n_{0} } \right|}} {\sum\limits_{m = 0}^{{N_{\text{c}} - 1}} {\sum\limits_{{m^{\prime} = 0}}^{{N_{\text{c}} - 1}} {\left\{ \begin{aligned} & a_{i} \left( {m,n + \left| {n_{0} } \right|,p} \right)a_{i}^{*} \left( {m^{\prime } ,n,p} \right) \\ & \cdot \exp \left\{ {j\pi \Delta f\left[ {\left( {m - m^{\prime } } \right) - \left( {m + m^{\prime } } \right)\left| {n_{0} } \right|} \right]T_{\text{s}} } \right\} \\ & \cdot \exp \left\{ {j2\pi \left( {m + m^{\prime } } \right)\Delta f\frac{\tau }{2}} \right\} \\ & \cdot \exp \left\{ { - j2\pi \left( {i - 1} \right)\left| {n_{0} } \right|B_{\text{a}} T_{\text{s}} } \right\} \\ & {\text{sinc}}\left\{ {\pi \left( {m - m^{\prime } } \right)\Delta f\left[ {\left( {\left| {n_{0} } \right| + 1} \right)T_{\text{s}} - \tau } \right]} \right\} \\ \end{aligned} \right.} } } \\ \end{aligned} $$

(A.2c)

Since $a_{i} \left( {m,n,p} \right)$ varies with the communication information, the expected value of $\tilde{r}_{i,p} ( \tau)$ is interested. Assume that $a_{i} \left( {m,n,p} \right)$ is random phase code with uniform distribution. The following can be achieved

$$ {\text{E}}\left[ {a_{i} \left( {m,n,p} \right)a_{i}^{*} \left( {m^{\prime } ,n^{\prime } ,p^{\prime } } \right)} \right] = \left\{ {\begin{array}{*{20}c} {1,} & {m = m^{\prime } ,n = n^{\prime } ,p = p^{\prime } } \\ {0,} & {\text{else}} \\ \end{array} } \right. $$

(A.3)

For $- T_{\text{p}} < \tau \le 0$, according to (A.3), ${\text{E}}\left[ {\tilde{r}_{i,p} ( \tau)} \right]$ can be expressed as

$$ {\text{E}}\left[ {\tilde{r}_{i,p} ( \tau)} \right] = \left\{ {\begin{array}{*{20}l} {N_{\text{s}} \left[ {T_{\text{s}} + \tau } \right]\exp \left\{ {j\pi \left( {N_{\text{c}} - 1} \right)\Delta f\tau } \right\}\frac{{\sin \left( {\pi N_{\text{c}} \Delta f\tau } \right)}}{{\sin \left( {\pi \Delta f\tau } \right)}},} \hfill & { - T_{\text{s}} < \tau } \hfill \\ {0,} \hfill & {\text{else}} \hfill \\ \end{array} } \right. $$

(A.4a)

For $0 < \tau < T_{\text{p}}$, ${\text{E}}\left[ {\tilde{r}_{i,p} ( \tau)} \right]$ can be described as

$$ {\text{E}}\left[ {\tilde{r}_{i,p} ( \tau)} \right] = \left\{ {\begin{array}{*{20}l} {N_{\text{s}} \left[ {T_{\text{s}} - \tau } \right]\exp \left\{ {j\pi \left( {N_{\text{c}} - 1} \right)\Delta f\tau } \right\}\frac{{\sin \left( {\pi N_{\text{c}} \Delta f\tau } \right)}}{{\sin \left( {\pi \Delta f\tau } \right)}},} \hfill & {\tau \le T_{\text{s}} } \hfill \\ {0,} \hfill & {\text{else}} \hfill \\ \end{array} } \right. $$

(A.4b)

The (A.2a), (A.4a) and (A.4b) are summarized as

$$ {\text{E}}\left[ {\tilde{r}_{i,p} ( \tau)} \right] = N_{\text{s}} \left[ {T_{\text{s}} - \tau } \right]\exp \left\{ {j\pi \left( {N_{\text{c}} - 1} \right)\Delta f\tau } \right\}\frac{{\sin \left( {\pi N_{\text{c}} \Delta f\tau } \right)}}{{\sin \left( {\pi \Delta f\tau } \right)}}{\text{rect}}\left[ {\frac{{T_{\text{s}} + \tau }}{{2T_{\text{s}} }}} \right] $$

(A.5)

For $\tilde{r^{\prime}}_{i,p} ( \tau) = {{\tilde{r}_{i,p} ( \tau)} \mathord{\left/ {\vphantom {{\tilde{r}_{i,p} ( \tau)} {r_{i,p} \left( 0 \right)}}} \right. \kern-0pt} {r_{i,p} \left( 0 \right)}}$, we can obtain that

$$ \begin{aligned} {\text{E}}\left[ {\tilde{r^{\prime}}_{i,p} ( \tau)} \right] & = \frac{{N_{\text{s}} \left[ {T_{\text{s}} - \tau } \right]}}{{N_{\text{c}} N_{\text{s}} T_{\text{s}} }}\exp \left\{ {j\pi \left( {N_{\text{c}} - 1} \right)\Delta f\tau } \right\}\frac{{\sin \left( {\pi N_{\text{c}} \Delta f\tau } \right)}}{{\sin \left( {\pi \Delta f\tau } \right)}}{\text{rect}}\left[ {\frac{{T_{\text{s}} + \tau }}{{2T_{\text{s}} }}} \right], \\ & \quad i = 1,2, \ldots ,M;\quad p = 0,1, \ldots ,N_{\text{p}} - 1 \\ \end{aligned} $$

(A.6)

Appendix B

2.1 CRB derivation for range and angle

Assume that the sample rate is $f_{\text{s}}$, each pule is sampled by $N$ times, the received signal at each antenna is divided into $M$ channels and the band-pass of the $q$th channel is $\left[ {\left( {q - 1} \right)N_{\text{c}} \Delta f\;,\;qN_{\text{c}} \Delta f} \right]$, for $q = 1,2, \ldots ,M$. The received signals of the $p$th pulse, $q$th channel, and $i$th antenna can be expressed as

$$ \begin{aligned} {\mathbf{y}}_{i,q,p} & = {\mathbf{s}}_{q,p} \sigma^{\prime } e^{{j\frac{2\pi }{\lambda }\left( {q - 1} \right)d\sin \theta_{\text{t}} }} e^{{j\frac{2\pi }{\lambda }\left( {i - 1} \right)d\sin \theta_{\text{r}} }} + {\mathbf{n}}_{i,q,p} , \\ & \quad i = 1,2, \ldots \quad ,M;\quad q = 1,2, \ldots ,M;\quad p = 0,1, \ldots ,N_{p} - 1 \\ \end{aligned} $$

(B.1)

where ${\mathbf{y}}_{i,q,p} = \left[ {y_{i,q,p} \left( 1 \right)y_{i,q,p} \left( 2 \right) \ldots y_{i,q,p} \left( N \right)} \right]^{\text{T}}$ is an $N \times 1$ vector containing the samples of echoes; $\theta_{\text{r}} = \theta_{\text{t}} = \theta$; ${\mathbf{n}}_{i,q,p} = \left[ {n_{i,q,p} \left( 1 \right) \ldots n_{i,q,p} \left( 2 \right) \ldots n_{i,q,p} \left( N \right)} \right]^{\text{T}}$ is an $N \times 1$ Gaussian noise vector; and ${\mathbf{s}}_{q,p} = \left[ {\begin{array}{*{20}c} {s_{q,p} \left( 1 \right)} & {s_{q,p} \left( 2 \right)} & \cdots & {s_{q,p} \left( N \right)} \\ \end{array} } \right]^{\text{T}}$ is an $N \times 1$ signal vector with

$$ \begin{aligned} s_{q,p} \left( l \right) & = \sum\limits_{n = 0}^{{N_{\text{s}} - 1}} {\left\{ \begin{aligned} & {\text{rect}}\left[ {\frac{l}{{f_{\text{s}} T_{\text{s}} }} - \frac{{nT_{\text{s}} + pT_{\text{r}} }}{{T_{\text{s}} }}} \right]\exp \left\{ { - j2\pi f_{\text{c}} \frac{{l_{\tau } }}{{f_{\text{s}} }}} \right\} \\ & \exp \left\{ {j2\pi \left( {q - 1} \right)N_{\text{c}} \Delta f\left( {\frac{l}{{f_{\text{s}} }} - nT_{\text{s}} - pT_{\text{r}} } \right)} \right\} \\ & \sum\limits_{m = 0}^{{N_{\text{c}} - 1}} {a\left( {m,n,p} \right)\exp \left\{ {j2\pi m\Delta f\left( {\frac{l}{{f_{\text{s}} }} - nT_{\text{s}} - pT_{\text{r}} } \right)} \right\}} \\ \end{aligned} \right.} \\ & \quad l = 1,2, \ldots N;\quad q = 1,2, \ldots ,M;\quad p = 1,2, \ldots ,N_{p} \\ \end{aligned} $$

(B.2)

where $l_{\tau } = \left\lfloor {{{2Rf_{\text{s}} } \mathord{\left/ {\vphantom {{2Rf_{\text{s}} } c}} \right. \kern-0pt} c}} \right\rfloor$.Stacking received data of $M^{2}$ channels into a vector yields

$$ {\mathbf{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{y} }}_{p} \left( l \right) = \sigma^{\prime } {\mathbf{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{s} }}_{p} \left( {l - l_{\tau } } \right) \odot {\mathbf{a}}_{\text{tr}} \left( \theta \right) + {\mathbf{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{n} }}_{p} \left( l \right),\quad p = 0,1, \ldots ,N_{p} - 1 $$

(B.3)

where ${\mathbf{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{s} }}_{p} \left( l \right) = \left[ {s_{1,p} \left( l \right)s_{2,p} \left( l \right)\; \ldots \;s_{M,p} \left( l \right)} \right]^{\text{T}} \otimes {\mathbf{1}}_{M}$ is an $M^{2} \times 1$ vector containing the transmitted signals; ${\mathbf{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{y} }}_{p} \left( l \right) = \left[ {y_{1,1,p} \left( l \right)\; \ldots y_{M,1,p} \left( l \right) \ldots y_{M,M,p} \left( l \right)} \right]^{\text{T}}$ is an $M^{2} \times 1$ vector containing the received signal; and ${\mathbf{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{n} }}_{p} \left( l \right) = \left[ {n_{1,1,p} \left( l \right) \ldots n_{2,1,p} \left( l \right) \ldots n_{M,1,p} \left( l \right) \ldots \ldots } \right]^{\text{T}}$ is an $M^{2} \times 1$ zero-mean complex random vector with the covariance matrix $\sigma^{2} {\mathbf{I}}_{{M^{2} }}$. $\sigma^{2} = N_{0} f_{\text{s}}$ is noise power and $N_{0}$ is flat power spectral density of white noise. Assume that the $N_{\text{p}}$ random vectors ${\mathbf{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{n} }}_{0} \left( l \right),{\mathbf{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{n} }}_{1} \left( l \right), \ldots ,{\mathbf{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{n} }}_{{N_{\text{p}} - 1}} \left( l \right)$ are statistical independent.

The probability density function of the received signals ${\mathbf{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{y} }}_{0} \left( {\tau ,\theta } \right),{\mathbf{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{y} }}_{1} \left( {\tau ,\theta } \right), \ldots ,{\mathbf{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{y} }}_{{N_{\text{p}} - 1}} \left( {\tau ,\theta } \right)$ is

$$ \begin{aligned} p\left( {{\mathbf{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{y} }}_{0} ,{\mathbf{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{y} }}_{1} , \ldots ,{\mathbf{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{y} }}_{{N_{\text{p}} - 1}} \left| {\tau ,\theta } \right.} \right) & = \frac{1}{{\pi^{{N + N_{\text{p}} + M^{2} }} \left| {\sigma^{2} {\mathbf{I}}_{{M^{2} }} } \right|^{{N + N_{\text{p}} }} }} \\ & \exp \left\{ {\frac{ - 1}{{\sigma^{2} }}\sum\limits_{p = 0}^{{N_{\text{p}} - 1}} {\sum\limits_{l = 1}^{N} {\left\{ \begin{aligned} \left[ {{\mathbf{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{y} }}_{p} \left( l \right) - \sigma^{\prime } {\mathbf{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{s} }}_{p} \left( {l - l_{\tau } } \right) \odot {\mathbf{a}}_{\text{tr}} \left( \theta \right)} \right]^{\text{H}} \hfill \\ \left[ {{\mathbf{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{y} }}_{p} \left( l \right) - \sigma^{\prime } {\mathbf{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{s} }}_{p} \left( {l - l_{\tau } } \right) \odot {\mathbf{a}}_{\text{tr}} \left( \theta \right)} \right] \hfill \\ \end{aligned} \right.} } } \right\} \\ \end{aligned} $$

(B.4)

The log-likelihood function of $p\left( {{\mathbf{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{y} }}_{0} ,{\mathbf{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{y} }}_{1} , \ldots ,{\mathbf{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{y} }}_{{N_{\text{p}} - 1}} \left| {\tau ,\theta } \right.} \right)$ is

$$ \begin{aligned} L\left( {\tau ,\theta } \right) & = - \ln \left( {\pi^{{N + N_{\text{p}} + M^{2} }} \left| {\sigma^{2} {\mathbf{I}}_{{M^{2} }} } \right|^{{N + N_{\text{p}} }} } \right) \\ & \quad - \frac{1}{{\sigma^{2} }}\sum\limits_{p = 0}^{{N_{\text{p}} - 1}} {\sum\limits_{l = 1}^{N} {\left\{ \begin{aligned} \left[ {{\mathbf{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{y} }}_{p} \left( l \right) - \sigma^{\prime } {\mathbf{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{s} }}_{p} \left( {l - l_{\tau } } \right) \odot {\mathbf{a}}_{\text{tr}} \left( \theta \right)} \right]^{\text{H}} \hfill \\ \left[ {{\mathbf{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{y} }}_{p} \left( l \right) - \sigma^{\prime } {\mathbf{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{s} }}_{p} \left( {l - l_{\tau } } \right) \odot {\mathbf{a}}_{\text{tr}} \left( \theta \right)} \right] \hfill \\ \end{aligned} \right.} } \\ \end{aligned} $$

(B.5)

To obtain the CRB, the Fisher information matrix ${\mathbf{J}}\left( {\tau ,\theta } \right)$ is required to be calculated, which is defined as (Scharf 1991)

$$ {\mathbf{J}}\left( {\tau ,\theta } \right) = - \left[ {\begin{array}{*{20}c} {{\text{E}}\left\{ {\frac{{\partial^{2} L\left( {\tau ,\theta } \right)}}{{\partial \tau^{2} }}} \right\}} & {{\text{E}}\left\{ {\frac{{\partial^{2} L\left( {\tau ,\theta } \right)}}{\partial \tau \partial \theta }} \right\}} \\ {{\text{E}}\left\{ {\frac{{\partial^{2} L\left( {\tau ,\theta } \right)}}{\partial \theta \partial \tau }} \right\}} & {{\text{E}}\left\{ {\frac{{\partial^{2} L\left( {\tau ,\theta } \right)}}{{\partial \theta^{2} }}} \right\}} \\ \end{array} } \right] $$

(B.6)

As the sample interval $\Delta t = 1 /f_{\text{s}} \to 0$, the terms in Fisher information matrix are

$$ J_{11} = - {\text{E}}\left\{ {\frac{{\partial^{2} L\left( {\tau ,\theta } \right)}}{{\partial \tau^{2} }}} \right\} = \frac{2}{{N_{0} }}\sum\limits_{p = 0}^{{N_{\text{p}} - 1}} {\left| {\sigma^{\prime } } \right|^{2} {\varvec{\upomega}}_{p}^{\text{H}} {\mathbf{1}}_{{M^{2} }} } $$

(B.7a)

$$ J_{12} = - {\text{E}}\left\{ {\frac{{\partial^{2} L\left( {\tau ,\theta } \right)}}{\partial \tau \partial \theta }} \right\} = \frac{2}{{N_{0} }}\text{Re} \left\{ {\sum\limits_{p = 0}^{{N_{\text{p}} - 1}} {\left| {\sigma^{\prime } } \right|^{2} {\mathbf{s}}_{{{\text{E}},p}}^{\text{H}} \left[ {\frac{{{\text{d}}{\mathbf{a}}_{\text{tr}}^{*} \left( \theta \right)}}{{{\text{d}}\theta }} \odot {\mathbf{a}}_{\text{tr}} \left( \theta \right)} \right]} } \right\} $$

(B.7b)

$$ J_{21} = - {\text{E}}\left\{ {\frac{{\partial^{2} L\left( {\tau ,\theta } \right)}}{\partial \theta \partial \tau }} \right\} = - {\text{E}}\left\{ {\frac{{\partial^{2} L\left( {\tau ,\theta } \right)}}{\partial \tau \partial \theta }} \right\} = J_{12} $$

(B.7c)

$$ J_{22} = - {\text{E}}\left\{ {\frac{{\partial^{2} L\left( {\tau ,\theta } \right)}}{{\partial \theta^{2} }}} \right\} = \frac{2}{{N_{0} }}\sum\limits_{p = 0}^{{N_{\text{p}} - 1}} {\left| {\sigma^{\prime } } \right|^{2} } {\mathbf{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\varepsilon } }}_{p}^{\text{H}} \left[ {\frac{{{\text{d}}{\mathbf{a}}_{\text{tr}}^{*} \left( \theta \right)}}{{{\text{d}}\theta }} \odot \frac{{{\text{d}}{\mathbf{a}}_{\text{tr}} \left( \theta \right)}}{{{\text{d}}\theta }}} \right] $$

(B.7d)

The elements in the diagonal of ${\mathbf{J}}^{ - 1} \left( {\tau ,\theta } \right)$ are the CRBs of the corresponding $\tau$ and $\theta$, i.e.,

$$ {\text{CRB}}( \tau){ = }\frac{{J_{22} }}{{J_{11} J_{22} - J_{12} J_{21} }} $$

(B.8a)

$$ {\text{CRB}}\left( \theta \right){ = }\frac{{J_{11} }}{{J_{11} J_{22} - J_{12} J_{21} }} $$

(B.8b)