Striation-Based Beamforming with Two-Dimensional Filtering for Suppressing Tonal Interference

Abstract

Based on the interference spectrogram in the element–frequency domain using the data measured by the horizontal linear array, the source range can be estimated through the striation-based beamforming (SBF) method and its variants. Estimation of the striation slope is the basis for these ranging methods. But in practical scenarios, the tonal interferences and other noise make it difficult to estimate the slope. In this paper, we proposed a two-dimensional low-pass filtering method after the two-dimensional discrete Fourier transform (2D-DFT) of the element–frequency domain spectrogram. The signals can be separated from the interference and noise using this filter. For the linear frequency-modulated signal without a known waveform, we also proposed an extraction and phase compensation method based on the time–frequency spectrogram, and the acoustic data obtained can be used for source ranging. The experimental results indicate that the methods proposed are feasible.

1. Introduction

In shallow water, the range–frequency domain spectrogram of the acoustic intensity will demonstrate certain interference structures [1,2,3]. Chuprov et al. [4,5] proposed the waveguide invariant $\beta$ to describe the broadband interference pattern of the sound. It is defined as the ratio of the group and phase slowness differences. Most of the existing work assumed that $\beta$ is an approximately scalar parameter in an environment with constant water sound speed over a perfectly reflecting bottom. In shallow water, the waveguide invariant is 1 [6], while in deep water it is −3 [7]. Thus, the waveguide invariant is widely used in various fields, such as underwater communications [8,9], environmental inversion [10], source localization [11,12], and reverberation estimation in active sonar systems [13].

Passive source localization is a challenging issue in shallow waters. Classical matched-field processing (MFP) matches received array data with a dictionary of replica vectors under comprehensive a priori environmental knowledge [14,15]. It is computationally expensive and highly susceptible to environmental uncertainty when the knowledge of the ocean environment is not adequate. Other methods include the wavefront curvature method, the triangulation method, and the near-field beamforming techniques [16,17,18]. When only one array is available, these methods are only applicable to near-field sources. The source localization using the waveguide invariant includes the source range estimation as well as the potential depth determination or classification based on the depth-dependent nature of the waveguide invariant [7,19]. For source-ranging applications, usually, the value of $\beta$ is known a priori. However, under the circumstances of varying bathymetry and sound speed, etc. [20], $\beta$ is modeled as a distribution [21,22]. Then, the prior environmental information [23] or guided sources [24] is needed to calculate the value of $\beta$, which is not possible in real scenarios. Generally, the acquisition of a two-dimensional (2D) striation pattern in the time–frequency or range–frequency domain requires the relative motion between the source and the receiver and a long-time observation. All these requirements limit the application of the waveguide-invariant-based ranging methods.

The horizontal linear array (HLA), especially with a large aperture, is widely used in underwater acoustics due to its capability of the high array gain and suppressing interference sources outside the search beam. The element–frequency domain spectrogram can be obtained through a single snapshot of data and is applicable for estimating the stationary source range, whereas the range–frequency domain spectrogram requires long-time continuous observation for moving sources. Zurk et al. [25] proposed the striation-based beamforming (SBF) method that can estimate $\beta$ independent of source ranges without prior environmental information. The SBF method utilizes broadband interference versus the elements along a uniform HLA, that is, the element–frequency spectrogram of the HLA data. But, the SBF method requires the source to be located within 30° of the array broadside. To eliminate the angular constraint, Liu et al. [26] proposed the waveguide-invariant dimension striation-based beamforming (WISBF) and range-dimension striation-based beamforming (RSBF) methods. To improve the quality of the interference spectrogram, especially under multi-source scenarios, the wavenumber filtering RSBF (WF-RSBF) method was proposed [27]. This method can estimate ranges of multiple sources and suppress interference with directions different from that of the target.

The RSBF and WF-RSBF methods require that the source spectrum varies slowly. Currently, explosive sources [28] and linear frequency-modulated (LFM) signals [26] after matched filtering (MF) could satisfy the requirement. The MF of the LFM can smooth the phase spectrum of the source. However, it requires a known waveform, which is impossible for passive detection scenarios. In addition, due to limited observation time in some situations, only an incomplete LFM signal may be measured, degrading the performance of the MF. The present work proposed a method to extract the broadband spectrum of the LFM data when the frequency-changing rate of the LFM is unknown. The LFM signal is estimated by finding the maximum energies along different linear cuts in the time–frequency spectrogram. Then, the phase variation of the LFM spectrum is compensated to satisfy the requirement of the RSBF method.

Both the acoustic data preprocessed with and without the MF can be used to construct the element–frequency domain spectrogram. Then, the slope of the interference striations in the spectrogram is estimated for source ranging. However, the radiated noise of real targets, such as a ship, is characterized by a combination of line spectrums and broadband spectrums. Consequently, the element–frequency spectrogram will be contaminated by the bright energy lines of the line spectrums, resulting in an erroneous estimation of the striation slopes as zero. Then, the SBF and its variant methods would fail. Yann et al. [29] conducted a two-dimensional discrete Fourier transform (2D-DFT) on the time–frequency spectrogram of the HLA data and proposed a filtering scheme to separate the positive and negative striations spread in different regions after 2D-DFT. This method can extract the required striation in the time–frequency spectrogram, but the source azimuth should be considered when the filtering method is conducted on the element–frequency spectrogram of the HLA data. Therefore, the paper proposed a two-dimensional low-pass filtering method suitable for the HLA, which applies a mask to the 2D-DFT of the element–frequency spectrogram to exclude the interference of the line spectrum.

The rest of this article is organized as follows: Section 2 reviews the basic theories of the RSBF method. The element–frequency spectrogram of the HLA data is constructed by two methods under scenarios of known and unknown waveforms of the LFM in Section 3. Section 4 derives the two-dimensional low-pass filtering formula after 2D-DFT first, then verifies the methods proposed by comparing the ranging results based on the experimental data.

2. Review of Striation-Based Beamforming

2.1. Interference Striation Modeling

According to the normal mode theory, when the source is at the depth of $z_0$, the received pressure with frequency $\omega$ at depth $z_a$ and range $r$ can be denoted as [30]

$$\begin{array}{ll}p(r,\omega )& =\mathrm{\Omega }(\omega ){\displaystyle \sum _m{(k_{rm}(\omega )r)}^{-1/2}{\mathrm{\Psi }}_m(z_0){\mathrm{\Psi }}_m(z_a)\exp \left(j{k}_{rm}(\omega )r\right)}\\ & =\mathrm{\Omega }(\omega ){\displaystyle \sum _m{A}_m}\exp (j{k}_{rm}(\omega )r)\end{array}$$

(1)

where $\mathrm{\Omega }(\omega )$ is the source spectrum; $A_m={(k_{rm}(f)r)}^{-1/2}{\mathrm{\Psi }}_m(z_0){\mathrm{\Psi }}_m(z_a)$; ${\mathrm{\Psi }}_m$ and $k_{rm}$ are the eigenfunction and horizontal wavenumber of the mth mode, respectively.

The acoustic intensity can be given by

$$I(r,\omega )=E[pp*]={\left|\mathrm{\Omega }\right|}^2{\displaystyle \sum _m^{}{A}_m^2}+2{\left|\mathrm{\Omega }\right|}^2{\displaystyle \sum _{m,n}^{}{A}_m{A}_n\cos (\Delta k_{mn}(\omega )r)}$$

(2)

where * denotes the conjugation operation and $\Delta k_{mn}(\omega )=k_{rm}(\omega )-k_{rn}(\omega )$. When the moving source travels a certain range span, the second item on the right of Equation (2) will generate the constructive and destructive interference striations in the range–frequency spectrogram. The striation slope is described by the waveguide invariant $\beta$ [4,5].

When an HLA of a large aperture is available, similar interference striations will be observed in the array element–frequency domain. Assume the point source is at ${\mathbf{r}}_{\mathbf{0}}={\left[r_0\cos {\varphi }_0,r_0\sin {\varphi }_0,z_0\right]}^T$, where $r_0$ and ${\varphi }_0$ represent the range from the source to the center of the array and the bearing measured from the broadside of the array, respectively. The HLA of the aperture L is oriented along the y-axis, $-L/2<y<L/2$, at depth $z_a$. When the HLA is placed in shallow water, the elevation angles (or grazing angles) are very small and can be ignored [31,32]. Thus, the range from each element to the source in the far field can be approximated as [25,26]

$$\begin{array}{ll}r& ={[{(r_0\cos {\varphi }_0)}^2+{(y-r_0\sin {\varphi }_0)}^2+{(z_0-z_a)}^2]}^{1/2}\\ & \approx r_0-y\sin {\varphi }_0+y^2/(2r_0)\end{array}$$

(3)

where the quadratic term of $y$ is usually neglected under the far-field assumption [25,26].

It can be seen that any two elements in the HLA have different ranges, which will result in certain interference patterns along the array. This interference structure of the acoustic intensity is referred to as the array element–frequency domain spectrogram. In this case, even when the source is stationary, the interference striations would be observed in the array element–frequency domain spectrogram, and the source range can be estimated. The striation slope in the element–frequency spectrogram can be modeled as [25]

$$\omega ={\omega }_0-Sy,\mathrm{where}S=-\frac{d\omega }{dy}=-\frac{d\omega }{dr}\frac{dr}{dy}={\omega }_0(\beta /r_0)\sin {\varphi }_0$$

(4)

where ${\omega }_0$ and $S$ are the center frequency and the slope of the interference striations, respectively.

According to Equation (4), once the three parameters $\beta$, $S$ and ${\varphi }_0$ are provided, the source range can be calculated as $r_0={\omega }_0\beta \sin {\varphi }_0/S$. The accuracy of $\beta$ is significant for the estimation of the source range. In general, we can only obtain the ratio of the waveguide invariant to the range of the source but cannot obtain the specific value of $\beta$ or $r$ independently. This requires some prior environmental information to determine the value of $\beta$.

2.2. Striation-Based Beamforming

The RSBF method proposed by Liu et al. [26] can determine the source range directly without the prior information of waveguide invariant and the azimuth limitation. After estimating the striation slope $S$ in the element–frequency spectrogram, the RSBF extracts the pressure data along the striations and takes the compensation coefficient as

$$w_r(y,{\omega }_0,\widehat{r})=\frac{1}{L}\exp \left(-jSy\frac{\widehat{r}}{c_0}\right)\exp \left(-j{\omega }_0\frac{y\sin {\varphi }_0}{c_0}\right)$$

(5)

where $\widehat{r}$ is the scanned range.

It is worth noting that the striation-based beamforming makes an important assumption that the source spectrum varies slowly, which should satisfy $\mathrm{\Omega }(\omega +\Delta \omega )\approx \mathrm{\Omega }(\omega )$. Then the beamforming output of RSBF can be expressed as [26]

$$\begin{array}{ll}{B}_r({\omega }_0,\widehat{r})& ={\left|{\displaystyle {\int }_{-L/2}^{L/2}p(y,\omega )w_r^{*}(y,{\omega }_0,\widehat{r})dy}\right|}^2\\ & \approx {\left|p(r_0,{\omega }_0)\right|}^2{\left|\sin \mathrm{c}\left(\frac{SL(\widehat{r}-r_0)}{2c_0}\right)\right|}^2\end{array}$$

(6)

We can see that $B_r$ reaches its maximum when $\widehat{r}=r_0$ without the limitation of source azimuth. In addition, according to Equation (4), once the source range is estimated through RSBF, the value of $\beta$ can be derived as $\beta =S{r}_0/({\omega }_0\sin {\varphi }_0)$.

In a summary of the source range estimation, the first step is to construct the element–frequency spectrogram from the HLA data. Then, the striation slope $S$ is estimated from the spectrogram. Thirdly, combined with the source azimuth estimated through conventional beamforming (CBF), the SBF or RSBF is conducted to achieve the estimation of waveguide invariant or source range. Among these steps, the construction of the element–frequency spectrogram and the calculation of the striation slope are essential for the source ranging.

3. Construction of the Element–Frequency Spectrogram

The above techniques have been shown to work well for broadband sources. Unfortunately, ships of opportunity are often dominant sources in shallow waters, whose spectra are mixed by narrowband and broadband components. This section introduces the problem encountered in an experiment conducted in the South China Sea. When the LFM signal was transmitted as a broadband source, tonal interferences due to ships of opportunity were also observed. The existence of the tonal interference would significantly corrupt the element–frequency spectrogram of the HLA data and fail to estimate the striation slope $S$. Then, the performance of striation-based beamformers that assume a uniform broadband source spectrum would be degraded greatly.

An important contribution of our work is to present a universal way to suppress the tonal interference for the estimation of $S$. The SBF method and its variants require that the source spectrum varies slowly, which could be satisfied with explosive sources or LFMs after MF [26] in the existing work. During the experiment, the emitted signals from the cooperative acoustic sources that satisfy the flat-spectrum requirement of the RSBF method are the LFM signals. Therefore, the LFM signals in the experiment were used to validate the proposed slope-limited filtering method. Since the signal waveform is often unknown for passive sonars, two methods were present to construct the element–frequency spectrograms of the HLA data under two scenarios. When the waveform of the transmitted LFM is exactly known, the first one is to match filter the signal and construct the element–frequency spectrogram. Since the spectrum of the transmitted source was uncalibrated during the experiment, the waveform of the LFM is not exactly known. Thus, an extraction and phase compensation method is proposed to extract the effective segment of the LFM data with a high signal-to-noise ratio (SNR) and compensate its phase to fulfill the flat source spectrum requirement of the RSBF method. It is found in the latter sections that without the extra temporal gain of MF, the constructed element–frequency spectrogram is still able to estimate the source ranges.

3.1. Experimental Review

The experiment was conducted in an offshore area of Sanya in the South China Sea in March 2018 [33]. The water depth at the site was around 90 m. To measure the sound speed profile (SSP), a conductivity–temperature–depth (CTD) device was cast from the cooperative vessel to the full water depth at the beginning of the experiment. The measured SSP was found to have a thermocline at the depth from 30 m to 65 m. Above 30 m and below 65 m are the isovelocity layers, where the maximum and the minimum sound speeds in the water column are 1531 m/s and 1522 m/s, respectively.

During that experiment, a 128-element HLA of 6.25 m sensor spacing was placed on the ocean bottom, where only sensors from the 39th to the 128th elements worked properly. Thus, the array aperture was around 556 m. A low-frequency acoustic source was deployed to a depth of 45 m from a cooperative ship. The ship’s global positioning system (GPS) was recorded to calculate its distances and bearings relative to the array. The acoustic source repetitively transmitted a 24-s sequence consisting of LFM signals of 20 s duration swept from 45 Hz to 300 Hz and a 4-s gap. The sequence was repeated five times. The data was analyzed when the recorded range of the source to the center of the effective part of the array was about 3.24 km.

3.2. Spectrogram Construction for Known Waveform of the LFM

Assume the signal received by the nth channel of the HLA in the time domain is $p_t\left(t,n\right)$. We used the MATLAB21a program “spectrogram” to obtain the short-time Fourier transform (STFT) of $p_t\left(t,n\right)$, which presents how the frequency content of $p_t\left(t,n\right)$ changes over time. The magnitude squared of the STFT, defined as $I_{ft}\left(f,t,n\right)$, is the time–frequency spectrogram of $p_t\left(t,n\right)$, where $f=\frac{\omega }{2\pi }$. Both $f$ and $\omega$ indicate the signal frequency with different units and are interchangeably used at the convenience of this paper. The spectrogram $I_{ft}\left(f,t,128\right)$ is shown in Figure 1, where several line spectrums are evident. The SNR of the LFM is quite low due to the low source level (SL) of the acoustic source and the interference at the experimental site.

When the LFM waveform is known, the MF is applied to all channels of the HLA first, which is completed by the convolution of $p_t\left(t,n\right)$ and $s\left(-t\right)$ [34]

$$p_{t-matched}\left(t,n\right)={\displaystyle {\int }_{-\infty }^{\infty }{p}_t\left(\tau ,n\right)s\left(\tau -t\right)d\tau }$$

(7)

where $s\left(t\right)$ is the LFM waveform.

After the process of MF, the signal is coherently integrated into the time domain, resulting in the maximum SNR at the output of the filter.

The conventional beamforming (CBF) is conducted after MF in the frequency band of 100~300 Hz. Figure 2 shows the bearing time recording (BTR). It is seen that the azimuth of the LFM signal is −44.6°, and there are many interferences at other azimuth angles during the experiment.

For the five transmitted LFM sequences, Figure 1 shows that the LFM signals within the frequency band of 100~200 Hz are relatively clear. Thus, the data of each LFM sequence after MF has been extracted within a specific time window, and the Fourier transformation (FT) was conducted to construct the element–frequency spectrogram. It is worth noting that the frequency resolution should be high when performing the FT to obtain the element–frequency spectrogram. Unlike CBF, the SBF method and its variants conduct the phase compensation along the frequency axis of the striations in the element–frequency domain spectrogram. Thus, the higher the frequency resolution, the more accurate the compensation coefficient weight in Equation (5). In Figure 3, we took a 5-s time window of the MF data to conduct FT. Then, the frequency resolution was about 0.2 Hz. The interference structures for the five LFM sequences are similar, and only the third LFM was shown for analyses in the latter sections. Figure 3 illustrates that the interference structure with a low SNR can be found within the frequency band of 100~200 Hz. There are also several tonal interferences in frequencies of 80 Hz, 160 Hz, etc.

3.3. Spectrogram Construction for Unknown Waveform of the LFM

In some scenarios, the HLA may receive the LFM signals transmitted by noncooperative acoustic sources. Then, the LFM signal may be unknown. We proposed a method for constructing the element–frequency spectrogram for an unknown waveform of the LFM without using the MF. In this part, the LFM signals in the desired frequency band are estimated first based on the time–frequency spectrogram at each channel of the HLA, and the phase compensation is performed for the extracted spectrum of the signal.

The LFM signal will result in a straight line in the time–frequency spectrogram, which we refer to as the time–frequency spectral line. The line in the spectrogram is represented as

$$t=a(f-f_L)+t_0(f_L<f<f_H)$$

(8)

where $f_H$ and $f_L$ are the upper and lower frequency band limits of the selected LFM, respectively. $a$ is the unknown slope of the linear cut, and $t_0$ is the initial time.

The steps are as follows. First, for the time–frequency spectrogram at the nth channel of the HLA, choose a certain time and frequency boundary and set the scanning regions of $a$ and $t_0$. Then, integrate $I_{ft}$ along this assumed spectral line in the time–frequency spectrogram. The integrated energy can be expressed as

$$E_{ft}\left(\widehat{a},{\widehat{t}}_0,n\right)={\displaystyle {\int }_{f_L}^{f_H}{I}_{ft}}\left(f,\widehat{a}(f-f_L)+{\widehat{t}}_0,n\right)df$$

(9)

where $f_L=100\mathrm{Hz}$ and $f_H=200\mathrm{Hz}$.

Thirdly, repeat the first two steps for all the candidate $a$ and $t_0$. When the integrated intensity reaches the maximum, the estimated values of $a$ and $t_0$ match with those of the LFM, which are denoted as $\widehat{a}$ and ${\widehat{t}}_0$, respectively.

Since only the LFM signal in the band of 100~200 Hz is comparatively visible, as shown in Figure 1, data in this frequency band is extracted. In Figure 4a, we integrate the intensity $I_{ft}\left(f,t,128\right)$ along a spectral line (denoted by the dashed black line) with $\widehat{a}$ scanning from 0.06 s/Hz to 0.1 s/Hz and ${\widehat{t}}_0$ between 50 s and 70 s. The resultant $E_{ft}\left(\widehat{a},{\widehat{t}}_0,128\right)$ versus different $\widehat{a}$ and ${\widehat{t}}_0$ is shown in Figure 4b. The maximum of $E_{ft}$ occurs at $\widehat{a}$ = 0.0780 s/Hz and ${\widehat{t}}_0$ = 55.8 s, as denoted by the black cross in Figure 4b. The estimation of $\widehat{a}$ is consistent with the frequency-changing rate of the LFM (0.0784 s/Hz), as overlaid by the dashed line in Figure 4a.

Finally, the signal spectrum can be extracted for each element of the array and then stacked in a matrix to form the element–frequency spectrogram, as shown in Figure 5. In order to satisfy the requirement of slow variation in the source spectrum for the RSBF method, it is necessary to compensate for the phase of the source spectrum of the extracted signal. The spectrum of the LFM signal can be expressed as [34]

$$\mathrm{\Omega }(f)=rect(\frac{f}{KT})\exp \left(-j\pi \frac{f^2}{K}\right)$$

(10)

where $T$ is the signal duration and $K$ is the modulation rate of the LFM signal, $K=\frac{1}{\widehat{a}}$. It can be seen that the phase variation in the spectrum of the LFM comes from $\exp \left(-j\pi \frac{f^2}{K}\right)$. Thus, the extracted LFM spectrums are multiplied by $\exp \left(j\pi \frac{f^2}{K}\right)$. After the phase compensation, the corrected LFM spectrum satisfies the requirements that the source spectrum varies slowly while its interference structure remains unchanged.

To compare the extraction performance of the LFM signal with and without MF, $E_{ft}\left(0.078,{\widehat{t}}_0,128\right)$ is extracted with the estimated $\widehat{a}$, as shown by the dashed white line in Figure 4b. Figure 6 compares $E_{ft}\left(0.078,{\widehat{t}}_0,128\right)$ with $\left|p_{t-matched}\left(t,128\right)\right|$, where the amplitudes are normalized for comparison. It can be seen the MF method can have a narrower main lobe than that without the MF, indicating its superior time compression capability for time delay estimation. However, from the aspect of the output SNR, the noise levels for both methods are almost the same, although the background noise after the MF fluctuates a lot. Comparing Figure 5 with Figure 3, the element–frequency spectrograms with and without the MF exhibit very similar interference structures in the frequency band of 100~200 Hz. It validates that the LFM extraction method without the MF is feasible. What is more, the latter analyses show that the source-ranging results with and without the MF are also comparable, which is a good demonstration of the effectiveness of the LFM extraction method. However, when the spectral lines of the LFM signal are not visible in the time–frequency spectrogram, it may be difficult to extract the spectrum of the LFM signal.

4. Two-Dimensional Low-Pass Filtering in the ${\mathit{k}}_{\mathit{y}}\mathbf{-}{\mathit{k}}_{\mathit{f}}$ Domain

Estimation of the striation slope $S$ is the basis for the ranging methods using the element–frequency domain spectrogram. The common methods of extracting striation slopes include the Radon transform [35,36] and 2D-DFT [37]. The Radon transform method only accumulates the pixels in the interference spectrogram and cannot separate the signals from the interference and noise. For the 2D-DFT methods, the desired signals and the tonal interference are in different regions, which is beneficial for applying filters as discussed in this section.

4.1. Development of Two-Dimensional Low-Pass Filtering in the $k_y-k_f$ Domain

The 2D-DFT method is usually performed on the range–frequency acoustic intensity $I(r,f)$. For the element–frequency spectrogram, the acoustic intensity can be expressed as $I(y,f)$. The procedure of the 2D-DFT method to achieve the striation slope in the element–frequency spectrogram is as follows [37].

Firstly, apply the 2D-DFT to the spectrogram $I(y,f)$ to obtain $I_{2DF}(k_y,k_f)$ in the $k_y-k_f$ domain, where $k_y$ and $k_f$ are the wavenumbers in the $y$ and f axes, respectively.

Secondly, if a filtering is performed on $I_{2DF}(k_y,k_f)$, $I_{2DF-filtered}(k_y,k_f)$ will be obtained, which only contains the desired component of the target.

Thirdly, convert $I_{2DF}(k_y,k_f)$ or $I_{2DF-filtered}(k_y,k_f)$ to polar coordinates $I_{2\mathrm{DF}}(\theta ,K)$.

Lastly, integrate the energy for every hypothesized striation angle, that is $E(\theta )={\displaystyle \int I_{2\mathrm{DF}}(\theta ,K)dK}$. The angle of the striation can be estimated as $\theta +\frac{\pi }{2}$ when $E(\theta )$ reached its maximum. It is worth noting that the striation slope $S$ is linked to $\theta$ by $S=-\tan (\theta +\frac{\pi }{2})$. Once $\theta$ is estimated, $S$ is obtained and then used for further source ranging.

Ignoring the quadratic term of $y$ in Equation (3), the wavenumber in the $y$ and f axes after 2D-DFT can be expressed as [29,37]

$$k_{y\left(mn\right)}=\frac{\partial \left[\Delta k_{mn}\left(f\right)\left(r_0-y\sin {\varphi }_0\right)\right]}{\partial y}=-\Delta k_{mn}\left(f\right)\sin {\varphi }_0$$

(11)

$$k_{f\left(mn\right)}=\frac{\partial \left[\Delta k_{mn}\left(f\right)\left(r_0-y\sin {\varphi }_0\right)\right]}{\partial f}=\left(r_0-y\sin {\varphi }_0\right)\cdot \frac{\partial \Delta k_{mn}\left(f\right)}{\partial f}$$

(12)

In physical terms, $k_{f\left(mn\right)}$ is the difference of travel times between the two interfering modes $m$ and $n$, while $k_{y\left(mn\right)}$ refers to a wavenumber difference which is similar to a difference of arrival angles [7].

In shallow water waveguides, the horizontal wavenumber $k_{rm}\left(f\right)$ is bounded by $\left[\frac{2\pi f}{c_{\mathrm{bottom}}},\frac{2\pi f}{c_{\min }}\right]$, where $c_{\min }$ is the minimum sound speed in the water, and $c_{\mathrm{bottom}}$ is the sound speed in the bottom half-space. Thus, the maximum values of $k_{y\left(mn\right)}$ and $k_{f\left(mn\right)}$ are expressed respectively as [29,37]

$$k_{y,\max }=2\pi f_{\max }\sin {\varphi }_0\left(\frac{1}{c_{\min }}-\frac{1}{c_{\mathrm{bottom}}}\right)$$

(13)

$$k_{f,\max }=2\pi r_{\max }\left(\frac{1}{c_{\min }}-\frac{1}{c_{\mathrm{bottom}}}\right)$$

(14)

where $r_{\max }$ and $f_{\max }$ are the maximum source range and the maximum frequency, respectively.

As shown in Figure 3 and Figure 5, the line spectrums will cause bright horizontal lines on the element–frequency spectrograms $I(y,f)$. After conducting 2D-DFT on $I(y,f)$, the line spectrum component corresponds to the regions of $I_{2DF}(k_y,k_f)$ where the value of $k_f/k_y$ is infinite, as indicated by pink dashed line in Figure 7a. It will disturb the estimation of the striation slopes contributed by the broadband sources. Results of $I_{2\mathrm{DF}}(\theta ,K)$ and $E(\theta )$ without two-dimensional low-pass filtering are shown in Figure 7b,c. It is seen that the line spectrums have a great impact on the slope estimation, making $E(\theta )$ reach its maximum at $\theta ={90}^{\circ }$. As a result, the striation slope is wrongly estimated as 0.

Moreover, it can be found in Figure 7a that the periodic striation patterns of $I(y,f)$ due to the broadband source produce the energy distribution along a specific line. Thus, it is possible to filter out the tonal interferences by designing a two-dimensional low-pass filter bounded by the white lines, as shown in Figure 7a. The white lines L1 and L2 provide the bounds in the $k_y$ and $k_f$ axes by using Equations (13) and (14). The following is used to derive the slope of line L3. According to Reference [38], the waveguide invariant can be expressed as

$${\beta }_{mn}=-\frac{\Delta k_{mn}\left(f\right)/f}{\partial \Delta k_{mn}\left(f\right)/\partial f}$$

(15)

Combining Equations (11) and (12) with Equation (15), $k_f/k_y$ is given by

$$k_f/k_y=-r\frac{\partial \left(\Delta k_{mn}\left(f\right)\right)/\partial f}{\Delta k_{mn}(f)\sin {\varphi }_0}=\frac{r}{{\beta }_{mn}f\sin {\varphi }_0}$$

(16)

It can be seen from Equation (16) that the slope of the energies for broadband sources will be located along different lines in $I_{2DF}(k_y,k_f)$ with different azimuth angles ${\varphi }_0$ [7]. During the experiment, the value of ${\varphi }_0$ is estimated as −44.6° in Section 3.2. The value of $\beta$ may be −3 under the scenario of deep water or shallow water with an extremely thick thermocline [6,7]. But in most shallow water scenarios, $\beta$ is approximately 1. Thus, the maximum and minimum value of $k_f/k_y$ in this paper could be described as [29].

$${\left(k_f/k_y\right)}_{\max }=\frac{r_{\min }}{f_{\max }\sin {\varphi }_0}$$

(17)

$${\left(k_f/k_y\right)}_{\min }=\frac{r_{\max }}{f_{\min }\sin {\varphi }_0}$$

(18)

We need to assume the value of $r_{\max }$ when calculating the striation slope and ensure that the true range of the source is contained. Letting $r_{\min }=0$, ${\left(k_f/k_y\right)}_{\max }=0$, which is the white line along the $k_f$ axis. In fact, the signal from the source can be separated from the line spectrum only by considering the minimum value of $k_f/k_y$, as shown by the dashed line L3 in Figure 7a.

After all the boundaries of $I_{2DF}(k_y,k_f)$ for the expected source are derived, the two-dimensional low-pass filter is obtained. Inside the white lines in Figure 7a, all the energies are kept unchanged, while the energies outside of the boundaries are set as zeros.

4.2. Calculation of the Striation Slope

To obtain the desired striation slope of the broadband source in Figure 7a, the maximum and minimum sound speeds in the waveguide are taken as 1800 m/s and 1522 m/s, respectively, and the maximum range to the source is taken as $r_{\max }=5\mathrm{km}$. Figure 8 shows the results of $I_{2\mathrm{DF}}(\theta ,K)$ and $E(\theta )$ after two-dimensional low-pass filtering. It is seen from Figure 8b that the two-dimensional low-pass filter can effectively eliminate the interference of line spectrums. Then, $\theta$ is estimated as 91.9° at the maximum of $E(\theta )$ and the striation slope is 0.0332 Hz/m.

Figure 9 shows the element–frequency spectrograms with and without MF, where the black dashed lines are calculated by the estimated striation slopes. It can be seen that both the calculated slopes match the striations in the spectrograms despite the interference of line spectrums. The estimation results validate the effectiveness of the low-pass filtering method in the $k_y-k_f$ domain. When the 2D-DFT method is used in the following paper, two-dimensional low-pass filtering is adopted by default.

4.3. Comparison of Ranging Results with and without MF

Firstly, we conduct the source ranging based on the element–frequency spectrogram obtained after MF, as shown in Figure 9a. In Section 4.2, the striation slope is calculated as 0.0332 Hz/m. The ranging result of the RSBF method using the estimated $S$ in Figure 9a is shown in Figure 10, where the vertical axis in Figure 10a represents the center frequency of the interference striations. The RSBF output $B_r\left({\omega }_0,\widehat{r}\right)$ for different frequencies focuses on the same range, and the source range can be estimated through incoherent summation of $B_r\left({\omega }_0,\widehat{r}\right)$ along all frequencies, as shown in Figure 10b. Then, the estimated source range is 3.28 km, which matches well with the true source range of 3.24 km. It demonstrates the striation slope calculated after two-dimensional low-pass filtering in the $k_y-k_f$ domain is feasible.

Then, the source ranging is completed using the element–frequency spectrogram without MF in Figure 9b. Figure 11a shows the RSBF output $B_r\left({\omega }_0,\widehat{r}\right)$ along the center frequencies, while Figure 11b provides the estimated source range at 3.21 km. Compared with Figure 10a, although the beamforming outputs with MF present a narrower main lobe than those without MF, the estimated source results are close to each other. Moreover, it can be seen from Figure 6 that the output SNRs are almost the same for the two LFM signal extraction methods. Thus, it is reasonable to obtain a comparable estimation of the source range, as shown in Figure 10 and Figure 11.

To illustrate the importance of the phase compensation for the LFM signal without MF, Figure 12 shows the source-ranging results without MF and phase compensation. Compared with Figure 11a, the focus of $B_r\left({\omega }_0,\widehat{r}\right)$ shifts with frequency due to the phase variation of the source spectrum, which will result in a biased estimation of the source range. Thus, it is necessary to compensate for the phase of the LFM signal at each frequency.

5. Conclusions

In shallow waters, the received data is often contaminated by tonal interferences from ships of opportunity. The paper theoretically derives the energy distribution for the desired signal and the interference in the 2D-DFT domain of the element–frequency spectrogram of an HLA. An effective two-dimensional low-pass filter in the $k_y-k_f$ domain is proposed to exclude the tonal interference. For unknown signal waveforms, an LFM extraction method is presented to construct the element–frequency spectrogram that fulfills the flat source spectrum requirement of the SBF method and its variants. To validate the methods, experimental data in the South China Sea with a long HLA were analyzed. The experimental results demonstrate that the proposed filter can effectively eliminate the influence of line spectrums on the striation slope estimation. The spectrogram extracted without MF can provide comparable ranging results with those with MF.

It is worth noting that the SBF methods require striation patterns with constant slopes, which occur mainly in shallow water. In deep water, the interference striation patterns are very complicated, and the waveguide invariant should be modeled as a distribution. Thus, more work should be conducted when extending SBF methods in deep water. Moreover, among the existing work, the SBF method and its variants have been validated using the LFM signals, the hyperbolic frequency modulated pulse [39], and the explosive sources [40]. In the future, the study of the possibility of applying the SBF methods to other types of signals is of great importance, which meets the general requirements for passive sonars.