Azimuth Estimation of Multi-LFM Signals Based on Improved Complex Acoustic Intensity Method

Abstract

The complex acoustic intensity method is one of the common methods used for the azimuth estimation of single-vector sensors. However, this method establishes a relationship between frequency and azimuth, which limits its practical applicability for multiple linear frequency modulation (LFM) signals with overlapping frequency domains. In this paper, the time–frequency distribution information of the LFM signal is combined with the complex acoustic intensity method, and more signal parameter information is used to expand the application scenario of the single-vector sensor. The proposed method first processes the time–frequency graph of the signal to obtain a stable and clear time–frequency distribution, and then obtains the acoustic intensity distribution of the signal using the time integration of the energy on the ridge of the signal to estimate the target orientation more stably. The simulation results show that the root mean square error of azimuth estimation is less than 1° when the SNR is greater than 0 dB. Furthermore, a pool experiment was carried out to verify the effectiveness of the proposed method.

1. Introduction

With the development of active detection technology, multi-component LFM signals have become widely used in sonar and radar systems [1,2,3,4,5]. The azimuth estimation of multi-component LFM signals is very important for effective countermeasures against active radar, sonar, and underwater acoustic communication systems [6]. A sound pressure array is usually used in traditional azimuth estimation techniques [7]. However, a large array with multiple elements increases the system’s overhead and is problematic in terms of the design. Additionally, problems such as array tilt and array element failure are encountered when the array is deployed in actual seawater, which increases the difficulty of estimating the underwater target azimuth. The vector sensor can synchronously measure the sound pressure and particle velocity information at a certain point in the sound field space, providing more comprehensive sound field information and broadening the dimension of signal processing [8,9,10]. Due to the receiving directivity of the vibration velocity channel, a single-vector sensor can achieve a non-fuzzy azimuth estimation of the target [11]. Vector sensors are widely used, such as in deployment problem diagnosis and weak signal-bearing measurements [12]. Therefore, azimuth estimation using a single-vector sensor has always been a concern of researchers.

In recent years, domestic and foreign scholars have conducted a significant amount of research on the azimuth estimation method of single-vector sensors. Nehorai [13] proposed the complex acoustic intensity method for azimuth estimation of single-vector sensors, analyzed the algorithm’s performance, and provided a theoretical error calculation method. Sun et al. [14] proved that sound energy flow is the maximum likelihood detection in the isotropic noise field, and the direction of sound energy flow is the maximum likelihood estimation of the direction of the sound source. Moreover, they provided the Cramér–Rao bound of this estimation method. Yao et al. [15] proposed histogram and weighted histogram azimuth estimation algorithms based on the complex acoustic intensity method, and presented the signal-processing process. They pointed out that the weighted histogram method is suitable for cases in which the sound source contains a line spectrum and wideband signal, and this signal is closer to the signal generated by ships in reality. Zhang et al. [16] proposed an improved histogram method. They pointed out that using the center-of-gravity method to obtain the energy distribution can improve the performance of the weighted histogram method. The improved histogram method can achieve higher accuracy of target azimuth estimation in the interference environment. Ashish et al. [17] proposed an azimuth estimation method based on high-order statistics, which can estimate the azimuth for multiple broadband sources. Liu et al. [18] proposed a robust vector sensor azimuth estimation method based on the subspace method; however, due to the wide natural directivity beam of the single-vector sensor, the resolution ability is poor when the target azimuth is close. Shi et al. [19] proposed a single-vector sensor azimuth estimation method based on LVD-MUSIC, which improved the resolution of the subspace class method for multiple targets but required a high amount of computation and a high SNR. Taken together, although there is a large volume of published work concerning azimuth estimation with a single-vector sensor, the subspace method is computationally intensive, and the azimuth estimation method based on acoustic intensity does not consider the time–frequency characteristics.

The LFM signal is a non-stationary signal, and time–frequency joint representation is a powerful tool for analyzing non-stationary signals [20]. Existing time–frequency estimation methods, such as short-time Fourier transform (STFT) [21] and Gabor transform [22], have fixed window sizes, which cannot meet the requirements of a high time resolution and a high frequency resolution at the same time. Wigner–Ville distribution (WVD) has good time–frequency aggregation but also has serious cross-term interference [23]. Seyed et al. [24] proposed using compressed sensing to process multi-component LFM signals, which can restrain the influence of cross-terms to a certain extent. The kernel function of Cohen class time–frequency distribution can be designed to suppress the influence of cross-terms and improve the performance of time–frequency parameter estimation. The smoothing kernels are often designed to reduce the cross-term without compromising the resolution requirement of the instantaneous frequency of the signals. Nabeel et al. [25] proposed spatial averaging and directional smoothing methods to process wideband signals with similar time–frequency distributions; however, this method needs an array as the receiver to suppress cross-terms with spatial information.

In this paper, we combine the time–frequency information of an LFM signal with the complex acoustic intensity method to improve the azimuth estimation performance of the single-vector sensor for a wideband LFM signal. Firstly, the kernel function is used to suppress the cross-term of the WVD method to obtain a clear time–frequency distribution of LFM. Secondly, the threshold segmentation algorithm is used to process the time–frequency graph to remove the Gaussian white noise distributed on the whole time–frequency graph as far as possible, and the time–frequency ridge of the LFM signal is obtained. Finally, the acoustic intensity distribution of the LFM signal with time and frequency is obtained by cross-WVD, and the energy on the ridge is integrated over time; then, the azimuth estimation of the LFM signal is obtained via the principle of complex acoustic intensity. Compared with the traditional sub-band partition method, the proposed method considers the time–frequency distribution characteristics of signals, improves the time–frequency correlation of the processing method, and can obtain better azimuth estimation accuracy.

2. Theory

2.1. The Complex Acoustic Intensity Method

In this section, the signal receiving model based on a single-vector sensor and the complex acoustic intensity method is briefly reviewed. For far-field sound source signals, the signals received by the sound pressure channel and the vibration velocity channel of the vector sensor can be approximately considered completely correlated; however, the received isotropic noises are irrelevant. In this case, taking the signals’ incidents at two different azimuths as an example, the received signals of the two-dimensional vector sensor (the vector sensor described below is two-dimensional) can be modeled as shown in Equation (1).

$$\{\begin{array}{l}P(t)=s_1(t)+s_2(t)+n_p(t)\\ V_x(t)=s_1(t)\cos {\theta }_1+s_2(t)\cos {\theta }_2+n_{vx}(t)\\ V_y(t)=s_1(t)\sin {\theta }_1+s_2(t)\sin {\theta }_2+n_{vy}(t)\end{array}$$

(1)

• $P(t)$, $V_x(t)$, $V_y(t)$: the signals received by the sound pressure channel and two orthogonal vibration velocity channels, respectively;
• $s_1(t)$, $s_2(t)$: two signals’ incidents in different azimuths, respectively;
• ${\theta }_1$, ${\theta }_2$: the incident azimuth angles of the two signals, respectively;
• $n_p(t)$, $n_{vx}(t)$, $n_{vy}(t)$: the white Gaussian noise received by each channel.

In unit time, the sound energy passing through a unit area perpendicular to the direction of energy propagation is called the sound energy flow density, expressed as ω, which can be obtained by the product of sound pressure and vibration velocity. Sound energy flow is also called acoustic intensity flow, whose mode is positively correlated with sound energy. The output of the sound energy flow detector is shown in Equation (2).

$$\overline{I(t)}=\frac{1}{T}{\displaystyle {\int }_0^T\omega (t)dt=\frac{1}{T}}{\displaystyle {\int }_0^T{\displaystyle {\int }_{\theta =0}^{360}P(t)\cdot }}V(t,\theta )d\theta dt$$

(2)

• $\omega (t)$: sound energy flow density;
• $\overline{I(t)}$: the output of the acoustic energy flow detector;
• $V(t,\theta )$: the vibration velocity vector with azimuth $\theta$;
• $T$: the integration time.

For the two-dimensional plane, $V(t,\theta )$ can be defined as in Equations (3) and (4):

$$V(t,\theta )=v(t)n$$

(3)

$$n^{\mathrm{T}}=[\cos \theta ,\sin \theta ]$$

(4)

• $v(t)$: vibration velocity waveform;
• $n$: the direction vector of the vibration velocity;
• $\theta$: direction of sound energy flow;
• $\mathrm{T}$: transposition.

According to Equation (2), at time $t$, the combined vector $\omega (t)$ is obtained by adding the density of sound energy flow in all directions in the horizontal plane. Equation (5) can be obtained by changing the integral order of Equation (2).

$$\overline{I(t)}=\frac{1}{T}{\displaystyle {\int }_0^{360}\left[{\displaystyle {\int }_{\theta =0}^{T}P(t)\cdot }V(t,\theta )dt\right]}d\theta$$

(5)

According to Equation (3), the output of the sound energy flow detector can be regarded as the time domain integration of the sound energy flow density in all directions to obtain the sound energy corresponding to each angle, and then the energy integration of all angles. That is, the output of the sound energy flow detector is the value positively correlated with the sum of the sound energy of all angles.

We can transform Equation (3) into the frequency domain to obtain Equation (6).

$$\overline{I(f)}=\frac{1}{2\pi F}{\displaystyle {\int }_{\theta =0}^{2\pi }\left[{\displaystyle {\int }_0^{F}P(f)\cdot }V(f,\theta )df\right]}d\theta$$

(6)

• $V(f,\theta )$: the Fourier transform of Equation (3).

The azimuth of the target sound source can be obtained by estimating the direction of sound energy flow. The norm of sound energy flow can be obtained from the sound energy flow received by two vertical velocity channels of the vector sensor as shown in Equation (7).

$$R(t)=\sqrt{I_x^2(t)+I_y^2(t)}$$

(7)

• $I_x(t)$: the horizontal sound energy flow, obtained by $P(t)V_x(t)$;
• $I_y(t)$: the vertical sound energy flow, obtained by $P(t)V_y(t)$.

We can transform Equation (7) into the frequency domain to obtain Equation (8).

$$R(f)=\sqrt{I_x^2(f)+I_y^2(f)}$$

(8)

The azimuth estimation of the sound source can be obtained from Equation (9).

$$\widehat{\theta (f)}=\arctan \frac{I_y}{I_x}$$

(9)

Since the vector sensor cannot directly output the sound energy flow vector, the information on the sound energy flow can be obtained by processing the received signals of the sound pressure channel and the vibration velocity channel through the cross-spectrum, as shown in Equation (10).

$$\{\begin{array}{l}{S}_{pvx}(f)=P(f)V_x^{*}(f)\\ S_{pvy}(f)=P(f)V_y^{*}(f)\end{array}$$

(10)

Since the signals received by the sound pressure channel and the vibration velocity channel have the same phase, the energy is concentrated in the real part of the cross-spectrum, and the azimuth estimation of the target sound source can be obtained by the complex acoustic intensity method shown as Equation (11).

$$\{\begin{array}{l}{I}_x(f)\approx \mathrm{Re}[S_{pvx}(f)]\\ \\ I_y(f)\approx \mathrm{Re}[S_{pvy}(f)]\\ \widehat{\theta (f)}=\arctan \frac{I_y(f)}{I_x(f)}\end{array}$$

(11)

2.2. Azimuth Estimation of Wideband LFM Signal Using the Weighted Histogram Method

According to Equation (11), the azimuth estimation accuracy of the complex acoustic intensity method is related to the energy distribution of each frequency component of the target sound source. For narrow-band single-frequency signals, the signal frequency is constant during the observation time, and a long time integral can be obtained. Therefore, better target azimuth estimation accuracy can be obtained. However, for a wideband LFM signal, the duration of each frequency component is short, and the time gain is low during the observation time, leading to a failure in accurately estimating the target azimuth using the complex acoustic intensity method. Therefore, the weighted histogram method is needed to estimate the azimuth of wideband LFM signals. The processing flow of the weighted histogram method is shown in Figure 1.

The main objective of the weighted histogram method is to find the sum of each angle’s sound energy flow norm to determine the distribution of the sound energy of each angle of the plane. In the existing algorithm, the sound energy is calculated with an equal interval angle. When the statistical interval is Δθ, 0° is taken as the starting point, and the interval is 0~Δθ, Δθ~2Δθ, …, (N − 1)Δθ~NΔθ. A total of 180/Δθ times were calculated, as shown in Equation (12).

$$\{\begin{array}{l}R(0~\Delta \theta )={\displaystyle \sum _1^{m_1}{R}_r(f)}\\ R(\Delta \theta ~2\Delta \theta )={\displaystyle \sum _1^{m_2}{R}_r(f)}\\ \hspace{1em}\hspace{1em}\hspace{1em}⋮\\ R\left[(N-1)\Delta \theta ~N\Delta \theta )\right]={\displaystyle \sum _1^{m_N}{R}_r(f)}\end{array}$$

(12)

• $R(0~\Delta \theta )$: the sum of the sound energy flow norm corresponding to all frequencies falling within the statistical interval;
• $m_1, m_2, \dots , m_N$: the number of frequencies that fall within the statistical interval;
• $R_r(f)$: the sound energy flow norm corresponding to each frequency in which the azimuth falls within the statistical interval.

According to Equation (12), the distribution of sound energy flow in each statistical interval can be obtained, and then the azimuth estimation of the wideband LFM signal can be obtained.

2.3. Improved Complex Acoustic Intensity Method for Azimuth Estimation of LFM Signal

A histogram is used to discretize the angles and estimate the target’s azimuth by obtaining the distribution of the sound energy flow from each angle. The time–frequency correlation of signals is not taken into account. In this section, a new azimuth estimation method that combines the time–frequency information of the wideband LFM signal with the complex acoustic intensity method is proposed.

2.3.1. WVD Cross-Term Suppression Method

WVD transform is an important tool for analyzing non-stationary time-varying signals because of its good time–frequency aggregation, but WVD has a quadratic time–frequency distribution, so it inevitably produces cross-terms. Since the subsequent processing needs to extract the time–frequency ridge of LFM signals, a clear time–frequency diagram is necessary. Moreover, when multiple LFM signals are received simultaneously, the influence of cross-terms is more obvious. Therefore, it is necessary to suppress the cross-terms to obtain a better effect in the subsequent processing. The time–frequency distribution of signals received by the vector sensor can be expressed as Equation (13).

$$W(t,f)={\displaystyle \int {\displaystyle \int {\displaystyle \int g(\upsilon ,\tau )}}}P(u+\frac{\tau }{2})P^{*}(u-\frac{\tau }{2})e^{-j2\pi (t\upsilon -u\upsilon -f\tau )}d\tau d\upsilon du$$

(13)

• $g(\upsilon ,\tau )$: the kernel function of WVD.

In this study, we used a compact kernel distribution (CKD) kernel given as Equation (14).

$$g(\upsilon ,\tau )=e^{2c}{e}^{\frac{c{D}^2}{{\upsilon }^2-D^2}}{e}^{\frac{c{E}^2}{{\tau }^2-E^2}}\left|\upsilon \right|<D, \left|\tau \right|<E$$

(14)

• $c$: value of 1;
• $D$: value of 0.25;
• $E$: value of 0.25.

Parameter $c$ controls the shape of the smoothing kernel, while parameters $D$ and $E$ control the extent of smoothing along the time axis and frequency axis, respectively [26]. The parameters are chosen to achieve a compromise between cross-term suppression and automatic term resolution since extensive smoothing can significantly reduce cross-terms and noise but also worsen the energy concentration of signal components along the intermediate frequency curve, while light smoothing cannot completely eliminate cross-terms.

The directional smoothing can be applied using fixed multi-directional kernels or adaptive kernels, and can further improve the time–frequency distribution’s robustness to noise and cross-terms [27]. The adaptive directional smoothing kernel is used to enhance the time–frequency distribution, shown in Equation (15).

$$\Psi (t,f)=W(t,f)\underset{t}{\ast }\underset{f}{\ast }{\gamma }_{\phi }(t,f)$$

(15)

• ${\gamma }_{\phi }(t,f)$: the directional smoothing kernel, which can be defined as Equation (16).

  $${\gamma }_{\phi }(t,f)=\frac{ab}{2\pi }\frac{d^2}{d{f}_{\phi }{}^2}{e}^{-{(a{t}_{\phi })}^2-{(b{f}_{\phi })}^2}$$

  (16)
• $a$: value of 2;
• $b$: value of 30.

Parameter $a$ controls the extent of smoothing along the major axis, whereas parameter $b$ controls the extent of smoothing along the minor axis. $t_{\phi }$ and $f_{\phi }$ can be expressed as Equation (17).

$$\{\begin{array}{l}{t}_{\phi }=t\cos (\phi )+f\sin (\phi )\\ f_{\phi }=f\cos (\phi )-t\sin (\phi )\end{array}$$

(17)

• $\phi$: the spin angle with respect to the time axis.

2.3.2. Time–Frequency Matrix Denoising Based on the Iterative Method

To extract the time–frequency ridge of the signal, it is necessary to obtain a clear time–frequency graph of the signal, and the white Gaussian noise distributed in the whole time–frequency plane can be suppressed by the iterative denoising method.

The main idea behind the iterative method, which is used to obtain the threshold for image segmentation, is to keep the mean value of the two parts, A and B, basically stable after image segmentation. In other words, with the progress of iteration, the convergence value of [mean (A) + mean (B)]/2 is taken as the segmentation threshold. To improve the convergence speed, the initial threshold is set as the intermediate value of the maximum gray value and the minimum gray value. As the average power spectrum of white Gaussian noise and signals are both constant, the threshold obtained by the iterative method can effectively separate the noise from the signal in the time–frequency domain. Assuming that the time–frequency matrix obtained by the signal after time–frequency transformation is $C(t,f)$, then the denoised time–frequency matrix can be expressed as (18).

$$C_s(t,f)=\{\begin{array}{l}C(t,f), \left|C(t,f)\right|\ge \epsilon \\ 0, \mathrm{others}\end{array}$$

(18)

Firstly, the initial threshold ${\epsilon }_1$ is obtained according to Equation (19), and then the threshold is updated by Equations (20) and (21). When it reaches ${\epsilon }_{k+1}={\epsilon }_k$, the iteration is stopped and the required threshold $\epsilon ={\epsilon }_{k+1}$ is obtained.

$${\epsilon }_1=(D+d)/2$$

(19)

$$\{\begin{array}{l}{u}_{s1}={\displaystyle \sum _{C(t,f)\in C_{s1}}{C}_{s1}/N_{s1}}\\ u_{n1}={\displaystyle \sum _{C(t,f)\in C_{n1}}{C}_{n1}/N_{n1}}\end{array}$$

(20)

$${\epsilon }_2=(u_{s1}+u_{n1})/2$$

(21)

• $D$: the maximum value of the time–frequency matrix $C(t,f)$;
• $d$: the minimum value of the time–frequency matrix $C(t,f)$;
• $C_{s1}$: the part of $C(t,f)$ greater than ${\epsilon }_1$;
• $N_{s1}$: the number of time–frequency points included in part $C_{s1}$;
• $u_{s1}$: the mean value of energy at all time–frequency points in part $C_{s1}$;
• $C_{n1}$: the part of $C(t,f)$ less than ${\epsilon }_1$;
• $N_{n1}$: the number of time–frequency points included in part $C_{n1}$;
• $u_{n1}$: the mean value of energy at all time–frequency points in part $C_{n1}$.

2.3.3. Azimuth Estimation of the Improved Complex Acoustic Intensity Method

The obtained time–frequency map is processed in the connected domain, and each connected domain is marked as $L_i(t,f)$. Then, the time–frequency ridge distribution of the LFM signal can be obtained as Equation (22).

$$F_i(t,f)=\arg \left[\underset{f}{\max } L_i(t,f)\right]$$

(22)

• $F_i(t,f)$: the ith LFM pulse ridge.

Since the signals received by each channel of the vector sensor are correlated while the noise is unrelated, the time–frequency distribution of the signal’s acoustic intensity can be obtained, as shown in Equation (23).

$$I_{P{V}_x}(t,f)={\displaystyle {\int }_{-\infty }^{+\infty }P(t+\frac{\tau }{2})}{V}_x{}^{*}(t-\frac{\tau }{2})e^{-j2\pi f\tau }d\tau$$

(23)

By combining Equations (11), (22) and (23), the azimuth estimation of the improved complex acoustic intensity method can be obtained as Equation (24).

$$\widehat{{\theta }_i(t,f)}=\arctan \frac{{\displaystyle \int \mathrm{Re}\left\{I_{P{V}_y}\right\}\otimes F_i(t,f)}dt}{{\displaystyle \int \mathrm{Re}\left\{I_{P{V}_x}\right\}\otimes F_i(t,f)}dt}$$

(24)

• $\otimes$: the Hadamard product.

Equation (24) adds the time–frequency information of the LFM signal to the complex acoustic intensity method. By integrating the time, a more stable acoustic intensity estimation can be obtained to improve the azimuth estimation performance of the complex acoustic intensity method.

3. Simulation and Analysis

The simulations were conducted on MATLAB R2016b in Windows 10. In the simulations, we assumed two LFM signals’ incidents from different azimuths: the bandwidth of $s_1$ is 4 kHz, the duration is 0.1 s, the center frequency is 8 kHz, and the incident angle is 30 degrees. The bandwidth of $s_2$ is 4 kHz, the duration is 0.1 s, the center frequency is 9 kHz, the incident angle is 50 degrees, and the noise is Gauss white noise.

To verify the extraction effect of the proposed method on time–frequency ridges, the signal receiving SNR was set as −5 dB based on the above simulation conditions. The simulation results are shown in Figure 2.

Figure 2a shows the time–frequency distribution obtained by the WVD method. It can be seen that the WVD method generates high energy at time–frequency locations where the signal does not exist. Figure 2b shows the time–frequency distribution after cross-term suppression. It can be seen that with the CKD kernel and the directional smoothing method, the time–frequency graph has better time–frequency aggregation, and the cross-term is also suppressed to a large extent. Due to the influence of noise, the time–frequency ridge of the signal cannot be effectively extracted, as seen in Figure 2b. The distribution characteristics of white Gaussian noise in the time–frequency plane are suppressed by the iterative denoising method, and the results are shown in Figure 2c. It can be seen from Figure 2c that the time–frequency distribution of the signal can be clearly obtained in the time–frequency diagram after denoising. Additionally, the time-frequency ridges are extracted correctly, as shown in Figure 2d.

Under the condition of SNR = [–5, 4] dB, the proposed method was compared with the traditional complex acoustic intensity method, 50 Monte Carlo simulations were conducted, and the results are shown in Figure 3, Figure 4 and Figure 5.

Figure 3 shows the acoustic intensity distribution of the LFM signals obtained by the cross-spectrum method, where the SNR is −5 dB. Due to the overlap of signals in the frequency domain, the acoustic intensity of different signals cannot be correctly estimated by the cross-spectrum method, and the ratio of acoustic intensity is not stable, which results in the deviation of the azimuth estimation of the target.

As can be seen in Figure 4a, with the increase in SNR, the accuracy of the target azimuth estimation using the complex acoustic intensity method is also improved, but the azimuth estimate converges to the composite azimuth of the two targets, which is caused by the overlap of signals from different azimuth in the frequency domain. Figure 4b shows the target azimuth estimation results obtained using the proposed method under different SNR conditions. It can be seen in Figure 4b that the proposed method can perform effective azimuth estimation for LFM signals. Compared with Figure 4a, the proposed method also has a better convergence effect with the improvement in SNR. This is because the proposed method considers the time–frequency characteristics of the received signal, so the signals can be distinguished from the time–frequency domain.

Figure 5 shows the distribution of RMSE obtained by the proposed method and the traditional method under different SNR conditions. With an increase in SNR, the RMSE of the two methods decreased. When the SNR was high, the traditional method still had a large RMSE because the traditional method could not estimate the azimuth of the two targets correctly. With an increase in SNR, the time–frequency map of the signal becomes clearer, and the energy distribution on the extracted time–frequency ridge becomes more stable. The azimuth estimation accuracy of the proposed method is improved.

4. Experimental Verification

To verify the effectiveness of the proposed method, an experiment was carried out in a 25 m × 15 m × 10 m anechoic pool. When the signal frequency was greater than 2 kHz, the sound absorption coefficient of the anechoic pool was greater than 0.99. The experimental hardware system mainly included two parts: a transmitting system and receiving system. The transmitting part mainly included a control module, signal generator, power amplifier, and transmitting transducer. The function of the control module was to trigger the signal generator so that different signal generators could send signals synchronously. The receiving part mainly consisted of a vector sensor (made by Harbin Engineering University, Harbin, China), a pre-amplification and filtering system (made by Harbin Engineering University, Harbin, China), and a multi-analyzer system B&K PULSE 3560D (Brüel & Kjær in Denmark). The signal sampling frequency was set at 65,535 Hz. The experimental receiving system is shown in Figure 6. The photos of the experiment and the anechoic pool are shown in Figure 7.

The position relationship between the transmitting transducers (P1, P2, P3) and the receiving system (V) in the pool is shown in Figure 8. The vector sensor and sound source were both laid 5 m underwater. There were four kinds of signals transmitted in the experiment.

Signal 1: Bandwidth B = 1 kHz, center frequency f = 8 kHz, the signal duration was 100 ms, and the frequency was decreasing.

Signal 2: Bandwidth B = 2 kHz, center frequency f = 12 kHz, the signal duration was 50 ms, and the frequency was decreasing.

Signal 3: Bandwidth B = 2 kHz, center frequency f = 5 kHz, the signal duration was 50 ms, and the frequency was increasing.

Signal 4: Bandwidth B = 2 kHz, center frequency f = 7.5 kHz, the signal duration was 100 ms, and the frequency was increasing.

In the experiments, the data with a length of 1 s were processed, and the observation time was set as 100 ms; that is, the data were divided into 10 segments, and the signal-to-noise ratio of the receiver signal was changed by changing the peak-to-peak value of the signal generator output.

• Experiments using two transmitting transducers.

Experiment 1:

We selected signal 1 and signal 2 transmitted by P1 and P2, and the signals were transmitted in a cycle. The results obtained by using the proposed method are shown in Figure 9.

Experiment 2:

We selected signal 2 and signal 3 transmitted by P2 and P3, and the signals were transmitted in a cycle. The results obtained by using the proposed method are shown in Figure 10.

Experiment 3:

We selected signal 3 and signal 4 transmitted by P1 and P3, and the signals were transmitted in a cycle. The results obtained by using the proposed method are shown in Figure 11.

• Experiments using three transmitting transducers.

Experiment 4:

We selected signal 1, signal 2, and signal 3 transmitted by P1, P2, and P3, and the signals were transmitted in a cycle. The results obtained by using the proposed method are shown in Figure 12.

Experiment 5:

We selected signal 2, signal 3, and signal 4 transmitted by P1, P2, and P3, and the signals were transmitted in a cycle. The results obtained by using the proposed method are shown in Figure 13.

It can be seen from Figure 9, Figure 10, Figure 11, Figure 12 and Figure 13 that the ridge-extraction method in this paper was able to correctly extract the time–frequency ridge of the LFM signal and accurately extract the time–frequency energy of the signal, which verifies the effectiveness of the simulation results.

With the change in time, the proposed method was able to correctly obtain the azimuth estimation results of the two targets and exhibited a good azimuth estimation performance under different SNR conditions. After calculation, the SNR corresponding to different peak-to-peak values were 7 dB, 1 dB, and −5 dB, respectively, and the azimuth estimation errors were 0.4°, 0.8° and 2.7°, respectively. The stability and effectiveness of the proposed method are verified.

5. Conclusions

This paper investigates the problem of azimuth estimation of multiple LFM signals. Because LFM signals overlap in the frequency domain, the application of the complex acoustic intensity method is limited. The main contribution of this paper is to propose a simple and feasible method to combine time–frequency parameters and the complex acoustic intensity method. The proposed method can estimate the azimuth of multiple LFM signals, which are separable in the time–frequency domain. In this method, the cross-term generated between multiple LFM signals is suppressed first. Then, the white Gaussian noise distributed in the whole time–frequency plane is suppressed by the denoising method, and the clear time–frequency ridge distribution of the LFM signal is extracted. Finally, the stable acoustic intensity distribution of the LFM signals is obtained by time integration of the energy on the ridge line, and then the target azimuth is estimated. The proposed method improves the stability and accuracy of the traditional complex acoustic intensity method for azimuth estimation of overlapping signals in the frequency domain. Based on a theoretical analysis, simulation experiments were carried out to verify the performance of the proposed method. The results show that the proposed method can accurately obtain the azimuth estimation results of multi-LFM signals, and the root mean square error of the azimuth estimation is less than 1° when the SNR is greater than 0 dB. The effectiveness of the proposed method was further verified by an anechoic pool experiment, in which it exhibited good azimuth estimation performance under different SNR conditions. These results imply that a single-vector sensor can be used to estimate the azimuth of non-cooperative underwater acoustic communication signals with certain time–frequency laws, expanding the application scenario of a single-vector sensor. This approach solves the problem of sound pressure array structure design when the array receives non-cooperative signals; in addition, there is no fuzzy problem of azimuth estimation using a single-vector sensor, providing an effective azimuth estimation method for an underwater acoustic countermeasure platform with a limited sensor installation size. For this technique to work, the LFM pulses must be separated well in terms of time and frequency. Azimuth estimation for time–frequency overlap and azimuth estimation for non-cooperative signals with more complex time–frequency distribution forms will continue to be studied in future research.