Underwater Tone Detection with Robust Coherently-Averaged Power Processor

Abstract

The detection of tonal signals with unknown frequencies is an important area of study in underwater signal processing. A common approach to address this issue is to use the Discrete Fourier Transform (DFT) for observations. When a tone does not lie precisely at the discrete DFT frequency point, its energy will leak to adjacent frequency point. This phenomenon is known as scalloping loss or Picket Fence Effect (PFE). PFE leads to the degradation of detection performance based on DFT. This paper studies the problem of robust detection in the case of PFE. A coherently-averaged power processor utilizing the information of adjacent frequency bins is designed. The results of simulations and experiments show that the proposed method is robust against PFE, and is highly suitable for tone detection in practical circumstances.

1. Introduction

Over the past several decades, the detection of tonal (sinusoidal) signals embedded in noise has continuously received attention in many fields, such as sonar, radar, communication, seismology, and ocean engineering. Noise radiated from vessels consists of a mixture of broadband noise and tonal noise, mainly caused by mechanical vibration and propeller propulsion [1]. The tonal component contains the characteristics of the ship and is of great importance for target detection and recognition [2]. And marine experiments have proved that low-frequency tonal signals can propagate over a long distance and maintain remarkable phase stability [3]. Therefore, tone detection has received considerable attention in acoustic signal processing.

The problem of detecting tonal signals with unknown parameters is usually expressed as a problem of composite hypothesis testing [4,5]. There are several solutions to this problem by utilizing the statistical characteristics of DFT in different segments. A common and extensively used method is to reduce the variability in signals and noise and increase the Signal-to-Noise Ratio (SNR) by averaging process, which is referred to as the Average Power Processor (AVGPR) [6,7]. Another widely-used method is the Generalized Likelihood Ratio Test (GLRT) [8] for composite hypothesis test, which performance close to the optimal bound, especially under low probability of false alarm [9,10]. However, only the amplitude information of the tonal signals is used in these two methods. Consequently, the detection performance tends to decay due to the loss of phase information. In order to take full advantage of the phase information, a phase estimation method has been proposed and the performance of the GLRT has been greatly improved by compensating the phase difference among segments, which is referred to as the Coherent Generalized Likelihood Ratio Test (CGLRT) [11]. The AVGPR can also be enhanced by introducing phase compensation to coherently average segments, and is named as the Coherently-Averaged Power Spectral Estimate (CAPSE) [12].

However, the ship radiated noise is also gradually reduced with the development of propeller control and mechanical noise suppression technology [13,14]. Therefore, the aforementioned detectors need to be further improved, which is also the purpose of this study. Theoretically, if the tonal signal is stable enough within the observation time, the processing gain of DFT increases with the increase of DFT points [15]. However, the tonal signal usually jitters or drifts in the actual situation because of the change in machine working conditions [16], mechanical fault [17], Doppler effect [18], and so on. Therefore, the length of DFT is limited in order to increase the robustness to signal instability. Due to the discrete nature of Discrete Fourier Transform (DFT), when a tone does not align with the center of the DFT frequency bin or lie precisely on the discrete DFT frequency point, then the signal energy will leak to the adjacent frequency point. It will degrade the performances of the above detectors and is known as scalloping loss or Picket Fence Effect (PFE) [19]. PFE deserves special attention in DFT-based detectors because their performance can be improved by solving it.

In the present study, a frequency bin joint coherent AVGPR that is robust to the performance degradation caused by PFE is developed. It utilizes the information of adjacent DFT frequency bins to gather the leaked energy of signal. Simulation is carried out to analysis the performance of the detectors in the presence and absence of the PFE. A sea experiment is conducted to compare the reliability of the methods.

The remainder of this paper is organized as follows: Section 2 describes the models and presents the detection problem. Next, the proposed detector is detailed in Section 3. Simulations are presented in Section 4, for the purpose of comparing the performances of the examined detectors. The experimental results are detailed in Section 5. Finally, the conclusions reached in this study are given in Section 6.

2. Problem Statement

2.1. Model

In general, the detection problem can be considered as a choice between two hypotheses $H_0$ and $H_1$, where $H_0$ represents only noise and $H_1$ represents the presence of signal. This can be expressed as [4]:

$$\begin{array}{l}{H}_0:x\left(n\right)=g\left(n\right)\hfill \\ H_1:x\left(n\right)=s\left(n\right)+g\left(n\right)\hfill \end{array}$$

(1)

where $x\left(n\right)$ is observation; $s\left(n\right)=A\exp \left[j\left({\omega }_{0}n+\phi \right)\right]$ is the tonal signal to be detected with unknown amplitude $A$, unknown initial phase $\phi$ and unknown normalized angular frequency ${\omega }_0$, ${\omega }_0=2\pi f_0/f_s$; $f_0$ is the tone frequency; $f_s$ is the sampling rate; and $g\left(n\right)$ is an additive White Gaussian Noise (WGN) with zero mean and unknown variance ${\sigma }_g^2$.

In order to increase the robustness to signal instability while taking advantage of the DFT processing gain, it is usually used to segment the observation, take the DFT of the segments, and average the DFT results [6]. In this study, the observation sequence $x\left(n\right)$ is divided into $L$ segments of length $N$ with sliding length $M$ as follows:

$$\begin{array}{ll}{x}_l\left(n\right)& =x\left(l\cdot M+n\right)\\ & =A\exp \left[j\left({\omega }_{0}n+{\omega }_{0}lM+\phi \right)\right]+g_l\left(n\right)\\ & n=0, \dots ,N-1; l=0,\dots ,L-1.\end{array}$$

(2)

Figure 1a illustrates the segmentation of observation sequence. The segments can be overlapped if $M<N$ and contiguous if $M=N$. Assume that the total number of observation data $x\left(n\right)$ is $L_x$. The relationship among $L$, $N$, $M$ and $L_x$ is shown as follows, where $\lfloor \cdot \rfloor$ is the rounding down notation:

$$L=\lfloor \frac{L_x-N}{M}\rfloor +1.$$

(3)

The DFT of a segment can only observe the spectrum on a limited number of $N$ frequency points. When the signal frequency is aligned with the center of a DFT frequency bin or at the discrete DFT point, the full amplitude in that bin is obtained. Otherwise, when the signal frequency is between two discrete DFT frequency points (such as $k-1$ and $k$ in Figure 1b), the energy of the signal leaks to the adjacent frequency bins, causing a reduced amplitude split between the two bins, as illustrated in Figure 1b, where the digital frequency of the ${\omega }_0$ is $k_0^{*}$, ${\omega }_0=\frac{2\pi k_0^{*}}{N}$ for $N$-point DFT, $k_0^{*}=k_0+\Delta$, $k_0$ is the integer closest to $k_0^{*}$ and $|\Delta |\le 1/2$. The $N$-point DFT of $x_l\left(n\right)$ is expressed as follows:

$$\begin{array}{ll}{X}_l\left(k\right)& ={\displaystyle {\displaystyle \sum }_{n=0}^{N-1}}{x}_l\left(n\right)\exp \left(-j\frac{2\pi kn}{N}\right)\\ & =S_l\left(k\right)+G_l\left(k\right)\\ & k=0,\dots ,N-1\end{array}$$

(4)

where

$$\begin{array}{ll}{S}_l\left(k\right)& ={\displaystyle {\displaystyle \sum }_{n=0}^{N-1}}{s}_l\left(n\right)\exp \left(-j\frac{2\pi kn}{N}\right)\\ & =A\exp \left[j\right(\frac{2\pi M{k}_0^{*}}{N}l+\phi \left)\right]{\displaystyle {\displaystyle \sum }_{n=0}^{N-1}}\exp (-j2\pi \frac{k_0^{*}-k}{N}n)\\ & =A\frac{\sin \left[\pi \right(k_0^{*}-k\left)\right]}{\sin \left[\pi \right(k_0^{*}-k)/N]}\exp \left[j\right(\frac{2\pi M{k}_0^{*}}{N}l+\phi \left)\right]\exp \left[j\pi \right(k_0^{*}-k\left)\frac{N-1}{N}\right],\end{array}$$

(5)

$$G_l\left(k\right)={{\displaystyle \sum }}_{n=0}^{N-1}{g}_l\left(n\right)\exp \left(-j\frac{2\pi kn}{N}\right).$$

(6)

A variant of Dirichlet kernel is used in the derivation of $S_l\left(k\right)$. $G_l\left(k\right)$ also has a complex Gaussian distribution with $G_l\left(k\right)~CN\left(0,N{\sigma }_g^2\right)$, since it is the linear combination of $g_l\left(n\right)$. When $k=k_0$, then $S_l\left(k_0\right)$ is obtained as follows:

$$S_l\left(k_0\right)=A\frac{\sin \left(\pi \Delta \right)}{\sin \left(\pi \Delta /N\right)}\exp \left[j\left(\frac{2\pi M{k}_0^{*}}{N}l+\phi \right)\right]\exp \left(j\pi \Delta \frac{N-1}{N}\right).$$

(7)

2.2. Existing Detectors and Problem

The AVGPR [7] is a conventional averaging processor. It generally stabilizes signals more than the noise as the number of averaged segments increases, because the signals are usually more coherent than the noise. However, its shortcoming is that it only utilizes the amplitude information of the DFT. It can be expressed as follows:

$$T_{AVGPR}=\frac{1}{L}{\displaystyle \sum }_{l=0}^{L-1}{\left|X_l\right|}^2.$$

(8)

In order to use the phase information to achieve more processing gain, the phase difference $\exp \left(j\frac{2\pi M{k}_0^{*}}{N}l\right)$ between $X_0\left(k\right)$ and $X_l\left(k\right)$ $\left(l=0,\dots ,L-1\right)$ must be eliminated. This can be estimated by the $R$-point DFT (padding zeros if necessary) of the complex sequence $X_l\left(k\right)$, as follows:

$$W_k\left(v\right)={\displaystyle \sum }_{l=0}^R{X}_l\left(k\right)\exp \left(-j\frac{2\pi vl}{R}\right),\hspace{1em}v=0,\dots ,R-1.$$

(9)

The estimated phase difference $\delta \left(k\right)$ can be calculated by the following:

$$\delta \left(k\right)=\frac{\arg \max \left|W_k\left(v\right)\right|}{R}.$$

(10)

When $k=k_0$, then $\delta \left(k_0\right)=\frac{k_0^{*}M}{R}$. Multiplying $X_l\left(k\right)$ by $\exp \left[-j2\pi \delta \left(k\right)l\right]$ to compensate the phase difference, the coherent sequence $Y_l\left(k\right)$ is obtained as follows:

$$Y_l\left(k\right)=X_l\left(k\right)\exp \left[-j2\pi \delta \left(k\right)l\right].$$

(11)

The CAPSE [12] can be considered as the windowed coherent AVGPR that utilizes phase information, and can be expressed as follows:

$$T_{CAPSE}=\frac{1}{L}{\left|{\displaystyle \sum }_{l=0}^{L-1}{Y}_l\right|}^2.$$

(12)

The GLRT replaces the unknown parameters by their Maximum Likelihood Estimations (MLEs). Although there the GLRT is not optimal, in practice, it appears to work quite well [4]. The CGLRT [11] is a combination of phase compensation and the GLRT, and is given by the following:

$$T_{CGLRT}=-L\ln \frac{{\widehat{v}}_1}{{\widehat{v}}_0},$$

(13)

$$\widehat{u}=\frac{1}{L}{\displaystyle \sum }_{l=0}^{L-1}{Y}_l\left(k\right),$$

(14)

$${\widehat{v}}_0=\frac{1}{L}{\displaystyle \sum }_{l=0}^{L-1}{\left|Y_l\left(k\right)\right|}^2,$$

(15)

$${\widehat{v}}_1=\frac{1}{L}{\displaystyle \sum }_{l=0}^{L-1}{\left|Y_l\left(k\right)-\widehat{u}\right|}^2.$$

(16)

It is worth noting that the test statistics of AVGPR, CAPSE and CGLRT are all constructed with a single frequency bin sequence $X_l\left(k\right)$. When $\Delta \ne 0$, then the signal amplitude of $S_l\left(k_0\right)$ in Equation $\left(7\right)$ attenuates from $AN$ to $A\frac{\sin \left(\pi \Delta \right)}{\sin \left(\pi \Delta /N\right)}$, which is caused by PFE. The attenuated amplitude can cause a loss of processing gain up to $3.9 \mathrm{dB}$ when $|\Delta |=0.5$ and $N\ge 10$ [20]. Therefore, a tonal detector that is robust against PFE is of great importance in practice, particularly in low SNR circumstances.

3. Proposed Method

This section presents a Frequency bin Joint Coherently-Averaged Power Processor (denoted as FJ_CAVGPR) by jointly utilizing the information of adjacent DFT frequency bins. As shown in Figure 1b, when $\Delta \ne 0$, then the energy of $k_0^{*}$ will leak to adjacent discrete frequency bin $k-1$ and $k$. The FJ_CAVGPR increases robustness by collecting the signal energy leaked to adjacent frequency bins.

When $k_0^{*}$ is closer to $k-1$, $\Delta =k_0^{*}-\left(k-1\right)$, $\Delta \in \left[0, 0.5\right)$, as shown in Figure 1b. When $k_0^{*}$ is closer to $k$, $\Delta =k_0^{*}-k$, $\Delta \in \left[-0.5, 0\right)$. For the case $\Delta \in \left[0, 0.5\right)$, the DFT of observation at frequency bin $k-1$ and $k$ can be expressed as follows:

$$\begin{array}{ll}{X}_l\left(k-1\right)& =A\frac{\sin \left(\pi \Delta \right)}{\sin \left(\pi \Delta /N\right)}\exp \left(j\phi \right)\exp \left(j\pi \Delta \frac{N-1}{N}\right)\\ & \times \exp \left(j\frac{2\pi M{k}_0^{*}}{N}l\right)+{\displaystyle {\displaystyle \sum }_{n=0}^{N-1}}{g}_l\left(n\right)\exp \left(-j2\pi \frac{k-1}{N}n\right),\end{array}$$

(17)

$$\begin{array}{ll}{X}_l\left(k\right)& =A\frac{\sin \left[\pi \left(\Delta -1\right)\right]}{\sin \left[\pi \left(\Delta -1\right)/N\right]}\exp \left(j\phi \right)\exp \left[j\pi \left(\Delta -1\right)\frac{N-1}{N}\right]\\ & \times \exp \left(j\frac{2\pi M{k}_0^{*}}{N}l\right)+{\displaystyle {\displaystyle \sum }_{n=0}^{N-1}}{g}_l\left(n\right)\exp \left(-j2\pi \frac{k}{N}n\right).\end{array}$$

(18)

For the case $\Delta \in \left[-0.5, 0\right)$, the DFT of observation at frequency bin $k-1$ and $k$ can be expressed as follows:

$$\begin{array}{ll}{X}_l\left(k-1\right)& =A\frac{\sin \left[\pi \left(\Delta +1\right)\right]}{\sin \left[\pi \left(\Delta +1\right)/N\right]}\exp \left(j\phi \right)\exp \left[j\pi \left(\Delta +1\right)\frac{N-1}{N}\right]\\ & \times \exp \left(j\frac{2\pi M{k}_0^{*}}{N}l\right)+{\displaystyle {\displaystyle \sum }_{n=0}^{N-1}}{g}_l\left(n\right)\exp \left(-j2\pi \frac{k-1}{N}n\right),\end{array}$$

(19)

$$\begin{array}{ll}{X}_l\left(k\right)& =A\frac{\sin \left(\pi \Delta \right)}{\sin \left(\pi \Delta /N\right)}\exp \left(j\phi \right)\exp \left(j\pi \Delta \frac{N-1}{N}\right)\\ & \times \exp \left(j\frac{2\pi M{k}_0^{*}}{N}l\right)+{\displaystyle {\displaystyle \sum }_{n=0}^{N-1}}{g}_l\left(n\right)\exp \left(-j2\pi \frac{k}{N}n\right).\end{array}$$

(20)

In the cases of both Equations $\left(17\right),\left(18\right)$ and $\left(19\right),\left(20\right)$, the phase differences of sequences $X_l\left(k-1\right)$ and $X_l\left(k\right)$ are both $\exp \left(j\frac{2\pi M{k}_0^{*}}{N}l\right)$. Therefore, $\delta \left(k\right)$ can be used to compensate for the phase difference in $X_l\left(k-1\right)$ and $X_l\left(k\right)$ simultaneously to obtain a coherent sequence $Z_l\left(k-1\right)$ and $Z_l\left(k\right)$, as follows:

$$\begin{array}{cc}\hfill Z_l\left(k-1\right)& =X_l\left(k-1\right)\exp \left[-j2\pi \mathsf{\delta }\left(k\right)l\right],\hfill \\ \hfill Z_l\left(k\right)& =X_l\left(k\right)\exp \left[-j2\pi \mathsf{\delta }\left(k\right)l\right].\hfill \end{array}$$

(21)

Then, the expression of FJ_CAVGPR can be written by combining $Z_l\left(k-1\right)$ and $Z_l\left(k\right)$ as follows:

$$T_{FJ\_CAVGPR }\left(k\right)=\frac{1}{L}\left[{\left|{\displaystyle \sum }_{l=0}^{L-1}{Z}_l\left(k-1\right)\right|}^2+{\left|{\displaystyle \sum }_{l=0}^{L-1}{Z}_l\left(k\right)\right|}^2\right].$$

(22)

Note that $Z_l\left(-1\right)=0$. The schematic of the FJ_CAVGPR is detailed in Figure 2.

4. Simulations

In this section, the detection performance of the proposed FJ_CAVGPR is compared with those of the AVGPR, CGLRT and CAPSE via simulation.

The detection results of four tonal signals buried in WGN are presented in Figure 3. The segment number is $L=32$, the length of each segment is $N=500$ and $M=N$, which means there is no overlapping between segments. The sampling rate is $f_s=1000 \mathrm{Hz}$, the frequencies of the four tones are $100.5\mathrm{Hz}$, $200\mathrm{Hz}$, $301\mathrm{Hz}$ and $399.5\mathrm{Hz}$, respectively corresponding to $\Delta =0.25$, $\Delta =0$, $\Delta =-0.5$ and $\Delta =-0.25$. The SNR of each frequency is $-25 \mathrm{dB}$. In this study, the SNR is defined as follows:

$$SNR=10\lg \left(\frac{A^2}{2{\sigma }_g^2}\right).$$

(23)

As can be seen in Figure 3a the AVGPR fails to distinguish the first tone, and contains several false alarms. The CGLRT and CAPSE in Figure 3b,c have poor detection performance on the tone with $\Delta =-0.5$. Meanwhile, the FJ_CAVGPR can successfully detect the tonal signals, as shown in Figure 3d. It is noticed that the detected frequencies in Figure 3b,c are at $100 \mathrm{Hz}$, $200 \mathrm{Hz}$, $300 \mathrm{Hz}$, and $400 \mathrm{Hz}$. However, those in Figure 3d are at $102 \mathrm{Hz}$, $200 \mathrm{Hz}$, $302 \mathrm{Hz}$, and $400 \mathrm{Hz}$. This is because more energy enters the next frequency bin due to the presence of noise. In fact, the amplitude at $100 \mathrm{Hz}$ and $300\mathrm{Hz}$ in Figure 3d is also obviously high, which makes the peak wider and easier to detect.

Next, the probability of detection ($P_D$) versus SNR curves that indicates the minimal input SNR required to reach the given $P_D$ for a fixed probability of false alarm ($P_{FA}$) is used to evaluate the detection performance of the four detectors. $P_D$ versus SNR curves for $|\Delta |=0$ and $|\Delta |=0.5$ at $P_{FA}={10}^{-2}$ is obtained from $\mathrm{10,000}$ Monte Carlo trials, as shown in Figure 4. The noise is additive WGN, the sampling rate is $f_s=1000 \mathrm{Hz}$, $L=32$, $N=500$, and $M=N$. As can be seen that from $|\Delta |=0$ to $|\Delta |=0.5$, the required SNR of AVGPR, CGLRT and CAPSE to achieve a given $P_D$ (e.g., $P_D=0.9$) is increased by about $3.9 \mathrm{dB}$. And FJ_CAVGPR only improves by about $1.5\mathrm{dB}$. In other words, FJ_CAVGPR has a processing gain of $2.4 \mathrm{dB}$ in this circumstance. On the other hand, it is found that when $|\Delta |=0$, the FJ_CAVGPR is $0.5 \mathrm{dB}$ worse than CAPSE. Because in this case, the signal in the adjacent frequency bin is noise rather than useful information. It appears as higher background noise in the time-frequency diagram or spectrum diagram as shown in Figure 3c,d. Nevertheless, in practice, we found that this $0.5 \mathrm{dB}$ loss had little impact to the result. Therefore, it is believed that this loss is worthwhile compared with the processing gain in practical applications, the proposed FJ_CAVGPR shows good robustness against PFE.

5. Experiment

A sea experiment was conducted $45 \mathrm{km}$ south of Sanya, Hainan Province, in January 2019. The water depth of the experimental area was $90\mathrm{m}$, the sound speed in the water body was basically the same, about $1532\mathrm{m}/\mathrm{s}$, and the sea state was level II. In the experiment, an acoustic transducer was transported by an unanchored boat, and transmitted wideband noise containing $3.5 \mathrm{kHz}$ and $4 \mathrm{kHz}$ tonal signals. The acoustic signal was received by a hydrophone, which was at a depth of 40 m and a distance of about 3 km from the transducer. The sampling frequency of receiving system was $10 \mathrm{kHz}$. The hydrophone used in the experiment was customized, with a received voltage response of $-171\mathrm{dB}\pm 1\mathrm{dB}$ re $1\mathrm{V}/\mathsf{\mu }\mathrm{Pa}$ from $2\mathrm{kHz}$ to $5\mathrm{kHz}$. In order to compare the methods under low SNR condition, we added tonal signals of $2 \mathrm{kHz}$, $2.5 \mathrm{kHz}$, $3 \mathrm{kHz}$. Among them, the tone of $2.5 \mathrm{kHz}$ and $4 \mathrm{kHz}$ were stronger, and the others were relatively weak.

The Low-Frequency Analysis and Recording (LOFAR) is used to evaluate the performance of the proposed and reference methods. The LOFAR gram can be regarded as the time-frequency diagram of time on the vertical axis, and the tonal signal appears in the form of line spectrum in the diagram. The results can be intuitively compared from the LOFAR gram. The more obvious and continuous the spectral lines, the better the processing performance. Then, by taking a segment number of $L=16$, $N=2048$, $M=1024$ with $50\%$ overlapping, the processing results are shown in Figure 5. For a more intuitive comparison, we show the amplitude and test statistics rather than the binary test results, where the amplitude of AVGPR, CAPSE, and FJ_CAVGPR are normalized and expressed in $\mathrm{dB}$.

As can be seen in Figure 5a, the AVGPR cannot detect the tone of $2\mathrm{kHz}$, while the tone of $3.5\mathrm{kHz}$ is difficult to distinguish due to the strong interference nearby. The CGLRT and CAPSE respectively shown in Figure 5b,c are superior to AVGPR, but the tones at $2\mathrm{kHz}$, $3\mathrm{kHz}$ and $3.5\mathrm{kHz}$ cannot be detected continuously. In contrast, the above tones can be distinguished more clearly and continuously by FJ_CAVGPR, as shown in Figure 5d. This is because the proposed method utilizes the energy of adjacent frequency points to generate wider and brighter spectral lines on the graph. Therefore, the detector proposed in this study is also considered to be more reliable with real data when compared with the other examined methods.

6. Conclusions

PFE or scalloping loss will degrade the performance of DFT-based tonal detectors. In this study, we focus on the robust detection of tonal signals in the presence of PFE, and propose an improved coherently-average power processor. The proposed detector coherently utilizes the information from adjacent discrete frequency bins to collect the energy that leaked into them. Both the simulation and experimental results reveal that the proposed detector achieves better robustness against PFE and more reliable with real data when compared with the AVGPR, CGLRT and CAPSE. When PFE exists, the proposed method has a processing gain of $2.4 \mathrm{dB}$ compared with other methods.

In future, the sparsity of the tonal signal in frequency domain should be used, and the adaptive phase compensation method could be developed.