Underwater Transmitter Localization Based on TDOA and FDOA Considering the Unknown Time-Varying Emission Frequency

Abstract

This article considers locating a noncooperative underwater transmitter utilizing multiple receivers, such that each receiver can measure the frequency difference of arrival (FDOA) as well as the time difference of arrival (TDOA) of the transmitter’s sound. This article considers the case where the unknown emission frequency of the transmitter changes as time goes. This article addresses hybrid TDOA-FDOA localization, under the assumption that the transmitter’s maximum speed is known in advance. To the best of our knowledge, this article is unique in tackling hybrid TDOA-FDOA localization, considering the case where the unknown emission frequency changes as time goes on. Under MATLAB simulations, this article shows that the proposed hybrid localization method is comparable to the ideal case, where the time-varying emission frequency is known in advance. Furthermore, we show that the proposed localization approach outperforms the case where the emission frequency is estimated as a wrong value.

1. Introduction

In underwater environments, electromagnetic signals are easily dissipated; hence, sound is mainly analyzed for underwater transmitter localization. Underwater transmitter localization utilizing sonar measurements is important in passive transmitter tracking.

This article considers locating a noncooperative underwater transmitter utilizing multiple receivers. This article assumes that each receiver can measure the frequency difference of arrival (FDOA) as well as the time difference of arrival (TDOA) of the transmitter’s signal.

To locate a stationary transmitter, one common technique is to measure the TDOAs of the transmitter signal at multiple receivers. Each TDOA defines a hyperbola in which the transmitter can exist. The intersection of the hyperbolas gives the transmitter location estimate. When the transmitter is moving, FDOA measurements can be used in addition to TDOAs to accurately estimate the transmitter’s position and velocity [1].

This article handles the case where the emission frequency of the transmitter is not known in advance. Moreover, the emission frequency can change as time goes on [2]. This is feasible, since one considers tracking a noncooperative transmitter, which can generate a time-varying emission frequency. In practice, a transmitter cannot move with infinite speed. A transmitter’s information can be used to estimate the maximum speed of the transmitter. For instance, in the case where one tracks an underwater transmitter, one has a priori information on the maximum speed of an underwater vehicle. Thus, this article addresses hybrid TDOA-FDOA localization under the assumption that the transmitter’s maximum speed is known in advance.

Various papers [3,4,5] have tackled tracking a moving transmitter based on TDOA-only measurements of multiple sensors. Many papers [6,7,8,9,10] have handled transmitter localization utilizing hybrid TDOA-AOA measurements. Reference [6] considered tracking a transmitter, in the case where a receiver can measure the TDOA, as well as the angle of arrival (AOA) of the transmitter’s sound. Hybrid TDOA-AOA requires only two sensors to locate a transmitter [7,8]. Reference [10] addressed a TDOA-AOA hybrid location estimation utilizing only two sensors, whose bearing noise is sufficiently small. Assuming that the AOA measurement noise is sufficiently small, weighted least-squares estimation (WLSE) can be applied for TDOA-AOA localization [9].

There are many papers on transmitter localization utilizing hybrid TDOA-FDOA measurements [1,11,12,13,14,15,16]. The authors of [11] proposed an pseudolinear method for locating a moving transmitter utilizing TDOA, FDOA, and differential Doppler rate measurements. Their method was based on the pseudolinear equations and the two-step weighted least-squares estimator. Reference [1] addressed an algebraic solution for moving transmitter localization utilizing TDOA and FDOA measurements. Reference [13] addressed the transmitter localization by utilizing TDOA-FDOA measurements, by reformulating the localization problem as a weighted least-squares (WLS) problem and performing semidefinite relaxation (SDR) to obtain a convex semidefinite programming (SDP) problem. Reference [14] developed a robust SDR method for transmitter localization utilizing TDOA-FDOA measurements. Considering TDOA-FDOA measurements, [17] developed a localization method based on an improved invasive weed optimization algorithm for locating a transmitter. Reference [15] addressed a transmitter localization approach based on hybrid TDOA-FDOA, which copes with unknown propagation speed and sensor parameter errors.

As far as we know, no paper on hybrid TDOA-FDOA localization considered the case where the emission frequency is not known in advance. Moreover, no paper on hybrid TDOA-FDOA localization has considered the case where the emission frequency changes as time goes on. However, considering a noncooperative transmitter, its emission frequency may not be available in practice. Moreover, emission frequency can vary as time goes on. This inspired us to handle hybrid TDOA-FDOA localization, considering the case where the unknown emission frequency changes.

In practice, a transmitter vehicle cannot move with infinite speed. This article thus addresses hybrid TDOA-FDOA localization under the assumption that the transmitter’s maximum speed is known in advance. Considering the transmitter’s maximum speed, one derives the emission frequency range where the emission frequency can exist. By discretizing the emission frequency range, one obtains a set of feasible emission frequencies. Then, each feasible emission frequency leads to a TDOA-FDOA least-squares solution. Among all solutions, one searches for a solution that leads to the minimum residual. In this way, one can calculate the most feasible emission frequency as well as the transmitter solution.

To the best of our knowledge, this article is novel in tackling hybrid TDOA-FDOA localization, considering the case where the unknown emission frequency changes. Under MATLAB simulations, this manuscript verifies that the proposed hybrid localization method is comparable to the ideal case, where the varying emission frequency is known in advance. We show that the proposed localization approach outperforms the case where the emission frequency is estimated as a wrong value.

Section 2 states the problem handled in this work. Section 3 provides the TDOA-FDOA estimation in the case where the emission frequency is not known in advance. Section 4 discusses MATLAB simulations. Section 5 provides Conclusions.

2. Problem Formulation

This section discusses the problem handled in this article. Beforehand, one introduces several concepts. Considering a vector $\mathbf{A}$, $\mathbf{A}\left[n\right]$ indicates the n-th element in $\mathbf{A}$. Let ${\mathbf{A}}^T$ define the transpose of a vector $\mathbf{A}$. Let $\mathbf{A}[n:m]$ denote the vector ${(\mathbf{A}\left[n\right],\mathbf{A}[n+1],\dots ,\mathbf{A}\left[m\right])}^T$. For example, $\mathbf{A}[1:3]$ defines the vector composed of first 3 elements in $\mathbf{A}$. Considering a matrix $\mathbf{M}$, let $\mathbf{M}[i,i]$ define the i-th diagonal element of $\mathbf{M}$. Let $\mathbf{diag}(a,b,c,\dots )$ denote a diagonal matrix, whose diagonal elements are $a,b,c,\dots$ in this order. Let $(\mathbf{A};\mathbf{B})$ define a matrix generated by stacking a matrix $\mathbf{A}$ over another matrix $\mathbf{B}$.

Let N denote the total number of receivers. Let $s_i$ denote the i-th receiver ($i\in \{1,2,\dots ,N\}$). Let ${\mathbf{s}}_i={(x_i,y_i,z_i)}^T$ present the 3D position of the i-th receiver ($i\in \{1,2,\dots ,N\}$). Let ${\dot{\mathbf{s}}}_i={({\dot{x}}_i,{\dot{y}}_i,{\dot{z}}_i)}^T$ denote the 3D velocity of the i-th receiver ($i\in \{1,2,\dots ,N\}$). In addition, let $\mathbf{e}={(x_e,y_e,z_e)}^T$ present the transmitter’s 3D position. Let $\dot{\mathbf{e}}={({\dot{x}}_e,{\dot{y}}_e,{\dot{z}}_e)}^T$ present the transmitter’s 3D velocity.

The range between the i-th receiver and the transmitter is:

$$\begin{array}{c}\hfill r_i=\parallel \mathbf{e}-{\mathbf{s}}_i\parallel .\end{array}$$

(1)

In addition, the range rate is:

$$\begin{array}{c}\hfill {\dot{r}}_i=\frac{{(\dot{\mathbf{e}}-{\dot{\mathbf{s}}}_i)}^T(\mathbf{e}-{\mathbf{s}}_i)}{r_i}.\end{array}$$

(2)

Assume that $V_{max}$, the maximum speed of the transmitter, is known a priori. This implies that:

$$\begin{array}{c}\hfill V_{max}\ge \sqrt{{\dot{x}}_e^2+{\dot{y}}_e^2+{\dot{z}}_e^2}.\end{array}$$

(3)

In the case where one tracks an underwater transmitter, one can set $V_{max}$ as the maximum speed of an underwater vehicle.

2.1. TDOA Formulation

This subsection introduces the TDOA formulation used in this article. Let $t_i$ denote the time of arrival (TOA), when a signal is measured to arrive at $s_i$. In practice, the measurement noise exists in time arrival measurements. In other words, one utilizes:

$$\begin{array}{c}\hfill t_i=\frac{\parallel \mathbf{e}-{\mathbf{s}}_i\parallel }{C}+n_i^{to}.\end{array}$$

(4)

Here, $n_i^{to}$ indicates time measurement noise, and C indicates the signal speed. It is assumed that C is known a priori. This assumption is widely applied in TOA-based target tracking [18,19,20].

In Equation (4), $n_i^{to}$ has a Gaussian distribution with mean 0 and standard deviation ${\sigma }_{to}$. This Gaussian noise has been widely applied in TOA-based localization [21,22,23,24].

This article considers a noncooperative transmitter whose signal emission time is not known in advance. Therefore, $t_i$ in Equation (4) cannot be measured by $s_i$.

Let $t_{i,1}=t_i-t_1$ define the TDOA between $s_i$ and $s_1$. Utilizing Equation (4), one can measure:

$$\begin{array}{c}\hfill t_{i,1}=\frac{\parallel \mathbf{e}-{\mathbf{s}}_i\parallel }{C}-\frac{\parallel \mathbf{e}-{\mathbf{s}}_1\parallel }{C}+n_{i,1}^{to},\end{array}$$

(5)

where $t_{i,1}=t_i-t_1$ and $n_{i,1}^{to}=n_i^{to}-n_1^{to}$. Since $n_i^{to}$ has a Gaussian distribution with mean 0 and standard deviation ${\sigma }_{to}$, $n_{i,1}^{to}$ has a Gaussian distribution with mean 0 and standard deviation $\sqrt{2}{\sigma }_{to}$.

In Equation (5), $t_{i,1}$ defines the TDOA formulation measured by both $s_1$ and $s_i$. This time difference of arrival measurement, $t_{i,1}$, is applied in TDOA localization problems [25,26]. The TDOA measurement constructs a hyperbolic curve, indicating feasible transmitter locations [6].

Utilizing Equation (5), the transmitter range difference between $s_1$ and $s_i$ is:

$$\begin{array}{c}\hfill r_{i,1}=C\times (t_{i,1}-n_{i,1}^{to})=r_i-r_1.\end{array}$$

(6)

2.2. FDOA Formulation

This subsection introduces the FDOA formulation used in this article. The sound of the transmitter is measured by $s_i$, and $s_i$ can analyze the frequency of the sound utilizing the fast Fourier transform (FFT).

To the best of our knowledge, no paper on hybrid TDOA-FDOA localization [1,11,12,13,14,15,16] has considered the case where the emission frequency is not known in advance. As presented in [27], the frequency measurement of $s_i$ is derived as follows. Let $f^e$ denote the unknown emission frequency, which can change as time goes on. Let $f_i$ denote the frequency measurement of $s_i$, given as:

$$\begin{array}{c}\hfill f_i=f^e\times (1-\frac{{\dot{r}}_i}{C})+n_i^f,\end{array}$$

(7)

where $n_i^f$ defines the frequency measurement noise, having a Gaussian distribution with mean 0 and standard deviation ${\sigma }_f$.

Equation (7) leads to:

$$\begin{array}{c}\hfill f_{i,1}=f_i-f_1=f^e(\frac{{\dot{r}}_1-{\dot{r}}_i}{C})+n_{i,1}^f,\end{array}$$

(8)

where $f_{i,1}$ defines the FDOA formulation measured by both $s_1$ and $s_i$. In addition, $n_{i,1}^f=n_i^f-n_1^f$ has a Gaussian distribution with mean 0 and standard deviation $\sqrt{2}{\sigma }_f$.

2.3. Least-Squares Estimation

In this subsection, one assumes that $f^e$ in Equation (7) is known in advance. One addresses a least-squares estimation (LSE) to calculate the transmitter solution based on TDOA and FDOA measurements.

In practical underwater environments, the noise covariance of measurements may not be known accurately. Thus, the LSE solution ignores the noise terms in the measurements. Moreover, the LSE solution runs fast, since it does not consider the noise covariance.

We acknowledge that one can utilize the noise covariance for deriving the transmitter solution. Considering the ideal case, where the emission frequency is known in advance, the authors of [1] addressed an algebraic solution for a transmitter’s location utilizing hybrid TDOA-FDOA measurements. The algebraic solution in [1] used the noise covariance explicitly. For comparison with the proposed hybrid tracking method, we ran the algebraic solution in [1]. In MATLAB simulation section (Section 4), $algeb$ indicates a simulation result of [1]. We show that the location accuracy of the proposed estimation approach is comparable to $algeb$.

Assuming that $f^e$ in Equation (7) is known a priori, Equation (8) results in:

$$\begin{array}{c}\hfill {\dot{r}}_{i,1}={\dot{r}}_i-{\dot{r}}_1=\frac{-C\times f_{i,1}}{f^e}.\end{array}$$

(9)

Since $f_{i,1}$ is accessible, ${\dot{r}}_{i,1}$ is available, as long as $f^e$ is accessible.

Equation (6) leads to:

$$\begin{array}{c}\hfill {(r_{i,1}+r_1)}^2=r_i^2.\end{array}$$

(10)

In addition, Equation (1) leads to:

$$\begin{array}{c}\hfill r_i^2={(\mathbf{e}-{\mathbf{s}}_i)}^T\times (\mathbf{e}-{\mathbf{s}}_i).\end{array}$$

(11)

Equations (10) and (11) result in:

$$\begin{array}{c}\hfill r_{i,1}^2+{\mathbf{s}}_1^T\times {\mathbf{s}}_1-{\mathbf{s}}_i^T\times {\mathbf{s}}_i+2{({\mathbf{s}}_i-{\mathbf{s}}_1)}^T\times \mathbf{e}+2\times r_{i,1}\times r_1=0.\end{array}$$

(12)

This leads to:

$$\begin{array}{c}\hfill 0.5\times (r_{i,1}^2+{\mathbf{s}}_1^T\times {\mathbf{s}}_1-{\mathbf{s}}_i^T\times {\mathbf{s}}_i)=(-{\mathbf{s}}_i^T+{\mathbf{s}}_1^T)\times \mathbf{e}-r_{i,1}\times r_1.\end{array}$$

(13)

Let $\mathbf{S}={(\mathbf{e},r_1,\dot{\mathbf{e}},{\dot{r}}_1)}^T$ denote the transmitter state to be estimated. Let ${\mathbf{h}}_t$ be defined as:

$$\begin{array}{c}\hfill {\mathbf{h}}_t=\left(\begin{array}{c}0.5\times (r_{2,1}^2+{\mathbf{s}}_1^T\times {\mathbf{s}}_1-{\mathbf{s}}_2^T\times {\mathbf{s}}_2)\\ 0.5\times (r_{3,1}^2+{\mathbf{s}}_1^T\times {\mathbf{s}}_1-{\mathbf{s}}_3^T\times {\mathbf{s}}_3)\\ ⋮\\ 0.5\times (r_{N,1}^2+{\mathbf{s}}_1^T\times {\mathbf{s}}_1-{\mathbf{s}}_N^T\times {\mathbf{s}}_N)\end{array}\right).\end{array}$$

(14)

In addition, let ${\mathbf{G}}_t$ be defined as:

$$\begin{array}{c}\hfill {\mathbf{G}}_t=\left(\begin{array}{cccc}-{\mathbf{s}}_2^T+{\mathbf{s}}_1^T& -r_{2,1}& 0& 0\\ -{\mathbf{s}}_3^T+{\mathbf{s}}_1^T& -r_{3,1}& 0& 0\\ ⋮\\ -{\mathbf{s}}_N^T+{\mathbf{s}}_1^T& -r_{N,1}& 0& 0\end{array}\right).\end{array}$$

(15)

Then, Equation (13) leads to:

$$\begin{array}{c}\hfill {\mathbf{h}}_t={\mathbf{G}}_t\times \mathbf{S}.\end{array}$$

(16)

As one performs the time derivative of Equation (12), one obtains:

$$\begin{array}{c}\hfill r_{i,1}\times {\dot{r}}_{i,1}+{\dot{\mathbf{s}}}_1^T\times {\mathbf{s}}_1-{\dot{\mathbf{s}}}_i^T\times {\mathbf{s}}_i=\\ \hfill ({\dot{\mathbf{s}}}_1^T-{\dot{\mathbf{s}}}_i^T)\times \mathbf{e}-{\dot{r}}_{i,1}\times r_1+{({\mathbf{s}}_1-{\mathbf{s}}_i)}^T\times \dot{\mathbf{e}}-r_{i,1}\times {\dot{r}}_1.\end{array}$$

(17)

Let ${\mathbf{h}}_f$ be defined as:

$$\begin{array}{c}\hfill {\mathbf{h}}_f=\left(\begin{array}{c}{r}_{2,1}\times {\dot{r}}_{2,1}+{\dot{\mathbf{s}}}_1^T\times {\mathbf{s}}_1-{\dot{\mathbf{s}}}_2^T\times {\mathbf{s}}_2\\ r_{3,1}\times {\dot{r}}_{3,1}+{\dot{\mathbf{s}}}_1^T\times {\mathbf{s}}_1-{\dot{\mathbf{s}}}_3^T\times {\mathbf{s}}_3\\ ⋮\\ r_{N,1}\times {\dot{r}}_{N,1}+{\dot{\mathbf{s}}}_1^T\times {\mathbf{s}}_1-{\dot{\mathbf{s}}}_N^T\times {\mathbf{s}}_N\end{array}\right).\end{array}$$

(18)

In addition, let ${\mathbf{G}}_f$ be defined as:

$$\begin{array}{c}\hfill {\mathbf{G}}_f=\left(\begin{array}{cccc}({\dot{\mathbf{s}}}_1^T-{\dot{\mathbf{s}}}_2^T)& -{\dot{r}}_{2,1}& {({\mathbf{s}}_1-{\mathbf{s}}_2)}^T& -r_{2,1}\\ ({\dot{\mathbf{s}}}_1^T-{\dot{\mathbf{s}}}_3^T)& -{\dot{r}}_{3,1}& {({\mathbf{s}}_1-{\mathbf{s}}_3)}^T& -r_{3,1}\\ ⋮\\ ({\dot{\mathbf{s}}}_1^T-{\dot{\mathbf{s}}}_N^T)& -{\dot{r}}_{N,1}& {({\mathbf{s}}_1-{\mathbf{s}}_N)}^T& -r_{N,1}\end{array}\right).\end{array}$$

(19)

Then, Equation (17) leads to:

$$\begin{array}{c}\hfill {\mathbf{h}}_f={\mathbf{G}}_f\times \mathbf{S}.\end{array}$$

(20)

Utilizing Equations (20) and (16), one obtains:

$$\begin{array}{c}\hfill \mathbf{h}=\mathbf{G}\times \mathbf{S}.\end{array}$$

(21)

where $\mathbf{h}=({\mathbf{h}}_t;{\mathbf{h}}_f)$, and $\mathbf{G}=({\mathbf{G}}_t;{\mathbf{G}}_f)$.

Then, one obtains the LSE solution as:

$$\begin{array}{c}\hfill \mathbf{S}={({\mathbf{G}}^T\mathbf{G})}^{-1}{\mathbf{G}}^T\mathbf{h}.\end{array}$$

(22)

Note that in the solution $\mathbf{S}={(\mathbf{e},r_1,\dot{\mathbf{e}},{\dot{r}}_1)}^T$, $r_1$ and $\mathbf{e}$ are correlated utilizing Equation (1). In addition, ${\dot{r}}_1$ and $\dot{\mathbf{e}}$ are correlated utilizing Equation (2).

In order to avoid the case where $\mathbf{S}$ is ill-defined, we require that $\mathbf{G}$ in Equation (22) has full ranks. $\mathbf{G}$ has size $(2N-2)\times 8$. For having full ranks, we require that $2N-2\ge 8$, i.e., $N\ge 5$. This implies that we require at least five receivers.

3. Hybrid TDOA-FDOA Estimation as ${\mathit{f}}^{\mathit{e}}$ Is Not Known in Advance

This section presents how to tackle hybrid TDOA-FDOA localization, considering the case where the emission frequency $f^e$ in Equation (7) is not known in advance. Considering the transmitter’s maximum speed $V_{max}$, one derives the emission frequency range where the emission frequency can exist. By discretizing the emission frequency range, one calculates a set of feasible emission frequencies. Then, each feasible emission frequency leads to a TDOA-FDOA least-squares solution in Equation (22). Among all solutions, one searches for a solution that leads to the minimum residual. In this way, one can calculate the most feasible emission frequency as well as the transmitter solution.

One derives the range where $f^e$ in Equation (7) can exist. Utilizing Equation (7), the lower bound for $f^e$ is:

$$\begin{array}{c}\hfill f_{min}^e=mi{n}_{i\in \{1,2,\dots ,N\}}\frac{f_i\times C}{C+V_{max}}.\end{array}$$

(23)

Utilizing Equation (7), the upper bound for $f^e$ is:

$$\begin{array}{c}\hfill f_{max}^e=ma{x}_{i\in \{1,2,\dots ,N\}}\frac{f_i\times C}{C-V_{max}}.\end{array}$$

(24)

The emission frequency range is $[f_{min}^e,f_{max}^e]$. By discretizing the emission frequency range, one calculates a set of feasible emission frequencies. One calculates the set as $[f_{min}^e,f_{min}^e+\frac{\Delta }{Q},f_{min}^e+\frac{2\Delta }{Q},\dots ,f_{max}^e]$. Here, $Q+1$ denotes the number of elements in this set. In addition, $\Delta =f_{max}^e-f_{min}^e$. Let ${\widehat{f}}^e(j)$ denote the j-th element in the set. One utilizes:

$$\begin{array}{c}\hfill {\widehat{f}}^e(j)=f_{min}^e+\frac{(j-1)\times \Delta }{Q}.\end{array}$$

(25)

Utilizing the LSE approach in Section 2.3, ${\widehat{f}}^e(j)$ leads to a TDOA-FDOA solution in Equation (22). Instead of Equation (9), one utilizes:

$$\begin{array}{c}\hfill {\widehat{\dot{r}}}_{i,1}(j)=\frac{-C\times f_{i,1}}{{\widehat{f}}^e(j)}.\end{array}$$

(26)

Here, the $\widehat{a}$ operator is used to indicate that $\widehat{a}$ is an estimation of an arbitrary variable a. In Equation (26), ${\widehat{f}}^e(j)$ is used, since the true emission frequency is not known in advance.

Utilizing ${\widehat{\dot{r}}}_{i,1}(j)$ in Equation (26), Equation (22) leads to a transmitter solution:

$$\begin{array}{c}\hfill \widehat{\mathbf{S}}(j)={({\mathbf{G}}^T\mathbf{G})}^{-1}{\mathbf{G}}^T\mathbf{h}.\end{array}$$

(27)

Note that in $\mathbf{S}$, $r_1=\mathbf{S}\left[4\right]$ and $\mathbf{e}=\mathbf{S}[1:3]$ are correlated utilizing Equation (1). In addition, ${\dot{r}}_1=\mathbf{S}\left[8\right]$ and $\dot{\mathbf{e}}=\mathbf{S}[5:7]$ are correlated utilizing Equation (2). The residual associated with ${\widehat{f}}^e(j)$ is:

$$\begin{array}{c}\hfill res(j)=re{s}_1(j)+re{s}_2(j),\end{array}$$

(28)

where

$$\begin{array}{c}\hfill re{s}_1(j)=(\parallel \widehat{\mathbf{S}}(j)[1:3]-{\mathbf{s}}_1{\parallel -\widehat{\mathbf{S}}(j)\left[4\right])}^2\end{array}$$

(29)

and

$$\begin{array}{c}\hfill re{s}_2(j)=(\parallel \widehat{\mathbf{S}}(j)[5:7]-{\dot{\mathbf{s}}}_1{\parallel -\widehat{\mathbf{S}}(j)\left[8\right])}^2.\end{array}$$

(30)

Among all solutions $\widehat{\mathbf{S}}(j)$ where $j\in \{1,2,\dots ,N\}$, one searches for a solution that leads to the minimum residual. In other words, one finds $\widehat{\mathbf{S}}(j^{*})$, where

$$\begin{array}{c}\hfill j^{*}=argmi{n}_{j\in \{1,2,\dots ,N\}}res(j).\end{array}$$

(31)

Then, $\widehat{\mathbf{S}}(j^{*})$ is set as the transmitter solution, in the case where the emission frequency is not available.

4. MATLAB Simulations

MATLAB simulations are utilized to show the effectiveness of the proposed localization scheme. The scenario runs for 500 s.

In MATLAB simulations, one considers tracking an underwater transmitter utilizing sound measurements. One utilizes the sound speed as $C=1400$ m/s. In Equation (4), $n_i^{to}$ has a Gaussian distribution with mean 0 and standard deviation ${\sigma }_i^{to}=\frac{0.1}{C}$. This implies that the standard deviation for range error is 0.1 m. In Equation (7), $n_i^f$ has a Gaussian distribution, with mean 0 and standard deviation ${\sigma }^f=\frac{10}{C}$.

The sampling interval is $T=5$ seconds. As the maximum speed of a transmitter, one utilizes $V_{max}=50$ meters per second. Moreover, $Q=100$ samples are used in the feasible frequency range.

One runs $M_c=100$ Monte Carlo simulations. ${\mathbf{L}}_k^c$$(c\in \{1,2,\dots ,M_c\})$ is the transmitter’s 3D location estimate at time-step k under the c-th Monte Carlo simulation. Let ${\mathbf{L}}_k$ represent the true transmitter’s location at time-step k.

Considering the transmitter’s position, the following RMSE (in meters) is used:

$$\begin{array}{c}\hfill RMS{E}_k=\sqrt{\frac{{\sum }_{c=1}^{M_c}{\parallel {\mathbf{L}}_k^c-{\mathbf{L}}_k\parallel }^2}{M_c}}.\end{array}$$

(32)

Let ${\mathbf{V}}_k^c$$(c\in \{1,2,\dots ,M_c\})$ denote the transmitter’s 3D velocity estimate at time-step k under the c-th Monte Carlo simulation. Let ${\mathbf{V}}_k$ represent the true transmitter’s velocity at time-step k. Considering the transmitter’s velocity, the following RMSE (in m/s) is used:

$$\begin{array}{c}\hfill RMS{E}_k^V=\sqrt{\frac{{\sum }_{c=1}^{M_c}{\parallel {\mathbf{V}}_k^c-{\mathbf{V}}_k\parallel }^2}{M_c}}.\end{array}$$

(33)

As time goes on, the emission frequency $f^e$ changes, as presented in Figure 1. Note that $f^e$ is not known in advance; thus, it needs to be estimated under the proposed tracking method.

Assuming that the emission frequency is known in advance, the authors of [1] addressed an algebraic solution for a transmitter’s location, utilizing hybrid TDOA-FDOA measurements. For comparison with the proposed hybrid tracking method, one runs the algebraic solution in [1]. For convenience, let $algeb$ denote the algebraic solution in [1]. Note that $algeb$ considers the ideal case, where the emission frequency is known a priori. MATLAB simulations will show that the proposed hybrid localization method in Section 3 is comparable to the ideal case, where the emission frequency is known in advance.

4.1. Posterior Cramer–Rao Lower Bound (PCRLB)

This article utilizes the posterior Cramer–Rao lower bound (PCRLB) as the lowest estimation error of any unbiased estimator [28]. Let $\mathbf{e}={(x_e,y_e,z_e)}^T$ denote the transmitter’s 3D position. Let $\dot{\mathbf{e}}={({\dot{x}}_e,{\dot{y}}_e,{\dot{z}}_e)}^T$ define the transmitter’s 3D velocity. Let ${\mathbf{X}}_k$ denote the estimate of $(\mathbf{e};\dot{\mathbf{e}};f^e)$ at time-step k. Here, recall that $f^e$ denotes the emission frequency that needs to be estimated. Let ${\mathbf{B}}_{k|k}$ define the error covariance (uncertainty) of the state vector ${\mathbf{X}}_k$.

At every time-step k, one utilizes the extended Kalman filter (EKF) in [29] to update ${\mathbf{B}}_{k|k}$. According to [29], one obtains the recursive form of the PCRLB by merging the prediction and the update step for the error covariance of the EKF.

The transmitter moves utilizing the following motion model:

$$\begin{array}{c}\hfill {\mathbf{X}}_{k+1}={\mathbf{X}}_k+{\mathbf{n}}_k,\end{array}$$

(34)

where ${\mathbf{n}}_k$ indicates the process noise.

One ignores the effect of process noise in the PCRLB. Then, the error covariance (uncertainty) of the state vector ${\mathbf{X}}_{k+1}$ is predicted as

$$\begin{array}{c}\hfill {\mathbf{B}}_{k+1|k}={\mathbf{B}}_{k|k}.\end{array}$$

(35)

This implies that the PCRLB considers the case where there is no uncertainty in the motion of the transmitter.

Utilizing Equations (5) and (8), the hybrid TDOA-FDOA measurement equation is:

$$\begin{array}{c}\hfill M({\mathbf{X}}_k)=(t_{2,1};t_{3,1};\dots t_{N,1};f_{2,1};f_{3,1};\dots f_{N,1}).\end{array}$$

(36)

Utilizing Equations (5) and (8),

$$\begin{array}{c}\hfill \mathbf{R}=\mathbf{diag}(2{\sigma }_{to}^2,2{\sigma }_{to}^2,\dots ,2{\sigma }_f^2,2{\sigma }_f^2)\end{array}$$

(37)

indicates the error covariance in hybrid TDOA-FDOA measurements. $2{\sigma }_{to}^2$ denotes the error covariance associated with $t_{i,1}$, where $i\le N$. In addition, $2{\sigma }_f^2$ denotes the error covariance associated with $f_{i,1}$, where $i\le N$.

Utilizing Equations (35)–(37), the recursive form of the PCRLB is:

$$\begin{array}{c}\hfill {({\mathbf{B}}_{k+1|k+1})}^{-1}={({\mathbf{B}}_{k|k})}^{-1}+{\mathbf{M}}_J^T{(\mathbf{R})}^{-1}{\mathbf{M}}_J.\end{array}$$

(38)

Here, ${\mathbf{M}}_J$ is the Jacobian matrix of $M({\mathbf{X}}_k)$ in Equation (36) given by ${\mathbf{M}}_J=\frac{\partial M(\mathbf{X})}{\partial \mathbf{X}}{|}_{\mathbf{X}={\mathbf{X}}_k}$.

One utilizes the PCRLB in Equation (38) as the lowest estimation error of any unbiased estimator [28]. In Equation (38), the initial error covariance utilizes:

$$\begin{array}{c}\hfill {\mathbf{B}}_{0|0}=\mathbf{diag}({\delta }^2,{\delta }^2,{\delta }^2,V_{max}^2,V_{max}^2,V_{max}^2,{({\delta }_f)}^2).\end{array}$$

(39)

Here, $\delta >0$ and ${\delta }_f>0$ denote the initial position uncertainty and the initial frequency uncertainty respectively. In simulations, we utilize $\delta ={\delta }_f=1$.

${\mathbf{B}}_{k|k}[j,j]$ defines the j-th diagonal element in ${\mathbf{B}}_{k|k}$. $\sqrt{{\mathbf{B}}_{k|k}[j,j]}$ indicates the lower bound for the estimation error of ${\mathbf{X}}_{\mathbf{k}}\left[\mathbf{j}\right]$. Since ${\mathbf{X}}_k$ denotes the estimate of $[\mathbf{e};\dot{\mathbf{e}};f^e]$ at time-step k, the PCRLB for transmitter’s position at time-step k is set as:

$$\begin{array}{c}\hfill PCRL{B}_k=\sum _{j=1}^3\sqrt{{\mathbf{B}}_{k|k}[j,j]}.\end{array}$$

(40)

Since ${\mathbf{X}}_k$ denotes the estimate of $[\mathbf{e};\dot{\mathbf{e}};f^e]$ at time-step k, the PCRLB for transmitter’s velocity at time-step k is set as:

$$\begin{array}{c}\hfill PCRL{B}_k^V=\sum _{j=4}^6\sqrt{{\mathbf{B}}_{k|k}[j,j]}.\end{array}$$

(41)

4.2. Scenario 1

Scenario 1 is as follows. One utilizes $N=6$ receivers in total. The receivers are located at [100, 0, 100], [−100, 1500, 1], [200, 1000, 10], [100, 3000, 1], [−5100, −1000, 10], and [5100, 5000, 1] in meters, respectively. The initial transmitter’s location is $[-1000,-1000,200]$ in meters. Initially, the transmitter’s velocity is [10, 0, 0] in m/s. The transmitter performs a coordinates turn in the xy-plane with turn rate $\omega =0.5$ degree per second. The transmitter moves utilizing the following coordinate turn:

$$\begin{array}{c}\hfill {\mathbf{X}}_{k+1}={\mathbf{F}}^c{\mathbf{X}}_k,\end{array}$$

(42)

where

$$\begin{array}{c}\hfill {\mathbf{F}}^c=\left(\begin{array}{cccccc}1& 0& 0& dt\times S_w& -dt\times C_w& 0\\ 0& 1& 0& dt\times C_w& dt\times S_w& 0\\ 0& 0& 1& 0& 0& dt\\ 0& 0& 0& cos(\omega \times dt)& -sin(\omega \times dt)& 0\\ 0& 0& 0& sin(\omega \times dt)& cos(\omega \times dt)& 0\\ 0& 0& 0& 0& 0& 1\end{array}\right).\end{array}$$

(43)

Here, $S_w=\frac{sin(\omega \times dt)}{\omega \times dt}$, and $C_w=\frac{1-cos(\omega \times dt)}{\omega \times dt}$.

Figure 2 illustrates Scenario 1. In Figure 2, the transmitter’s path is depicted with red circles. Deployed receivers are marked with black diamonds.

Considering Scenario 1 in Figure 2, Figure 3 illustrates the RMSE for transmitter’s position ($RMS{E}_k$) as k varies. The emission frequency $f^e$ changes, as presented in Figure 1. In Figure 3, $unknownEmitFreq$ represents the case where the emission frequency is not known in advance. Under $unknownEmitFreq$, the proposed location scheme in Section 3 is used for deriving the transmitter solution. In Figure 3, $knownEmitFreq$ indicates the case where the emission frequency is known a priori. Under $knownEmitFreq$, the LSE solution in Section 2.3 is used for deriving the solution. In addition, $algeb$ defines the algebraic solution in [1]. In Figure 3, $wrongEmitFreq$ indicates the case where the emission frequency is set as $f_{2,1}$ in Equation (8). In this case, the LSE solution in Section 2.3 is used to derive the solution. Note that the proposed hybrid localization method in Section 3 is comparable to the ideal case, where the emission frequency is known in advance.

Regarding Scenario 1 in Figure 2, Figure 4 illustrates the RMSE for the transmitter’s velocity ($RMS{E}_k^V$ in Equation (33)) as k varies. The emission frequency $f^e$ changes, as presented in Figure 1. $wrongEmitFreq$ does not appear in this plot, since its RMSE is too large. Note that the proposed hybrid localization method in Section 3 is comparable to the ideal case, where the emission frequency is known in advance.

Considering Scenario 1,

Table 1. Computation time for all MC simulations (Scenario 1).

Alg.	Comp. Time (s)
$wrongEmitFreq$	134
$knownEmitFreq$	133
$algeb$	130
$unknownEmitFreq$	124

presents the computational time for all MC simulations, considering every algorithm in Figure 4. Note that the proposed location scheme $unknownEmitFreq$ in Section 3 runs as fast as $knownEmitFreq$.

4.3. Scenario 2

Scenario 2 is as follows. One utilizes $N=7$ receivers in total. Every receiver is located at [100, 0, 100], [−100, 1500, 1], [200, 1000, 10], [100, 3000, 1], [−5100, −1000, 10], [5100, 5000, 1], and [510, 0, 10], respectively.

The initial transmitter’s location is $[-1000,-1000,200]$ in meters. The transmitter moves with a constant velocity [10, 0, 0] in m/s.

Figure 2 illustrates Scenario 2. In Figure 2, the transmitter’s path is marked with red circles. Deployed receivers are marked with black diamonds.

Regarding Scenario 2 in Figure 5, Figure 6 illustrates the RMSE for the transmitter’s position ($RMS{E}_k$ in Equation (32)) as k changes. The emission frequency $f^e$ changes, as presented in Figure 1. In Figure 6, $unknownEmitFreq$ indicates the case where the emission frequency is not known in advance. In Figure 6, $knownEmitFreq$ represents the ideal case, where the emission frequency is known a priori. $algeb$ denotes the algebraic solution in [1]. $wrongEmitFreq$ indicates the case where the emission frequency is set as $f_{2,1}$ in Equation (8). $wrongEmitFreq$ shows worse performance than other solutions, since it utilizes a wrong emitting frequency. The worst performance of $wrongEmitFreq$ indicates that the unknown emission frequency can significantly degrade the localization performance. Figure 6 shows that the proposed hybrid localization method in Section 3 is comparable to the ideal case, where the emission frequency is known in advance.

Considering Scenario 2, Figure 7 illustrates the RMSE for the transmitter’s velocity ($RMS{E}_k^V$ in Equation (33)) as k varies. The emission frequency $f^e$ changes, as presented in Figure 1. $wrongEmitFreq$ does not appear in this plot, since its RMSE is too large.

Considering Scenario 2,

Table 2. Computation time for all MC simulations (Scenario 2).

Alg.	Comp. Time (s)
$wrongEmitFreq$	130
$knownEmitFreq$	138
$algeb$	141
$unknownEmitFreq$	161

presents the computational time for all MC simulations, considering every algorithm in Figure 7. Acknowledge that the proposed location scheme $unknownEmitFreq$ in Section 3 runs slightly slower than other algorithms. In the Conclusions, we address a parallel processing approach to improve the time efficiency of the proposed location scheme.

4.4. The Effect of Changing the Emission Frequency

We check the effect of changing the emission frequency. As time goes on, the emission frequency $f^e$ changes, as presented in Figure 8. See that one has A large variation in $f^e$ as time goes on.

Regarding Scenario 2 in Figure 5, Figure 9 illustrates the RMSE for the transmitter’s position ($RMS{E}_k$ in Equation (32)) as k changes. As plotted in Figure 8, one has a large variation in $f^e$ as time goes on. The worst performance of $wrongEmitFreq$ indicates that the unknown emission frequency can significantly degrade the localization performance. Note that the proposed hybrid localization method in Section 3 is comparable to the ideal case, where the emission frequency is known in advance.

Regarding Scenario 2, Figure 10 illustrates the RMSE for the transmitter’s velocity ($RMS{E}_k^V$ in Equation (33)) as k varies. As plotted in Figure 8, one has a large variation in $f^e$ as time goes on. $wrongEmitFreq$ does not appear in this plot, since its RMSE is too large. Note that the proposed hybrid localization method in Section 3 is comparable to the ideal case, where the emission frequency is known in advance.

One next considers the case where the emission frequency does not change as time goes on. One has $f^e=1000$ Hz at all times. Regarding Scenario 2 in Figure 5, Figure 11 illustrates the RMSE for the transmitter’s position ($RMS{E}_k$ in Equation (32)) as k changes. In Figure 11, one has $f^e=1000$ Hz at all times. The worst performance of $wrongEmitFreq$ indicates that the unknown emission frequency can significantly degrade the localization performance. Note that the proposed hybrid localization method in Section 3 is comparable to the ideal case, where the emission frequency is known in advance.

Regarding Scenario 2, Figure 12 illustrates the RMSE for the transmitter’s velocity ($RMS{E}_k^V$ in Equation (33)) as k varies. In Figure 12, one has $f^e=1000$ Hz at all times. $wrongEmitFreq$ does not appear in this plot, since its RMSE is too large.

5. Conclusions

This article tackles locating an underwater transmitter utilizing hybrid TDOA-FDOA measurements, in the case where the unknown emission frequency varies as time goes on. Utilizing MATLAB simulations, this article demonstrates the superiority of the proposed location scheme in Section 3, by comparing it with the ideal case, where the emission frequency is known a priori. Furthermore, MATLAB simulations show that the proposed localization approach outperforms the case where the emission frequency is estimated as a wrong value. The proposed location scheme can be used to locate a ground or aerial target that emits RF signals. In the future, experiments will be performed utilizing real sonar sensors, in order to demonstrate the effectiveness of the proposed hybrid localization more rigorously.

We can further improve the computational efficiency of the proposed localization methods using parallel processing. Here, parallel processing is a computing technique where multiple streams of data processing tasks co-occur through multiple central processing units (CPUs) working concurrently. Recall that in Section 3, the emission frequency range is $[f_{min}^e,f_{max}^e]$. By discretizing the emission frequency range, one calculates a set of feasible emission frequencies, given as $[f_{min}^e,f_{min}^e+\frac{\Delta }{Q},f_{min}^e+\frac{2\Delta }{Q},\dots ,f_{max}^e]$. Here, $Q+1$ denotes the number of feasible emission frequencies. In the case where we can use multiple CPUs, we can apply parallel processing to improve the time efficiency. In the case where we have M CPUs, each CPU can process $\frac{Q+1}{M}$ frequencies in a parallel manner. In this way, we can improve the time efficiency of the proposed localization methods.

In practice, receiver position and velocity errors can occur [30]. For instance, in underwater environments, sea current leads to receiver position errors. The uncertain receiver location can seriously deteriorate the localization performance. The authors of [31] addressed a Taylor series method using hybrid TDOA-FDOA to estimate both the transmitter state and receiver position simultaneously. Reference [12] considered locating multiple transmitters under hybrid TDOA-FDOA measurements, in the presence of sensor position and velocity errors. In the future, we will handle hybrid TDOA-FDOA target tracking considering both receiver position and velocity errors.