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Tonal signal detection in passive sonar systems using atomic norm minimization

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Abstract

Frequency estimation of a tonal signal in passive sonar systems is crucial to the identification of the marine object. In the conventional techniques, a basis mismatch error caused by the discretization of the frequency domain is unavoidable, resulting in a severe degradation of the object detection quality. To overcome the basis mismatch error, we propose a tonal frequency estimation technique in the continuous frequency domain. Towards this end, we formulate the frequency estimation problem as an atomic norm minimization problem. From the numerical experiments, we show that the proposed technique is effective in identifying the tonal frequency components of marine objects.

Introduction

Recently, passive sonar has received much attention as a means to detect acoustic signals radiated from marine objects (e.g., submarines, ships, and marine animals) [14]. The passive sonar system is preferred since an intentional signal transmission is unnecessary.

Roughly speaking, the signal detected by the passive sonar consists of four components: tonal signal, propeller noise, hydrodynamic noise, and ambient noise [5] (see Fig. 1). Among them, the tonal signal, the sum of sinusoidal tones occurred by a marine object [6], has received special attention as a tool to identify the marine object. Since each object can be characterized by the frequency dependent features, accurate identification of the frequency components in a tonal signal is of great importance for the target object detection.

Fig. 1
figure1

Types of signal in underwater environments

Traditionally, an approach to use the discrete Fourier transform (DFT) has been popularly employed for implementation simplicity [7]. One well-known drawback of this approach is that the frequency resolution (spacing between adjacent points in the frequency domain) is limited. In fact, since the frequency resolution of the DFT-based approach is inversely proportional to the observation interval, a small observation time might cause a basis mismatch in the frequency domain, resulting in a severe degradation of the object detection quality. To overcome the shortcoming, various spectral estimation techniques have been proposed over the years [810]. Representative techniques include root multiple signal classification (root-MUSIC) [8], estimation of signal parameters via rotational invariance techniques (ESPRIT) [9], and matrix pencil algorithm [10]. These approaches estimate the frequency components using the eigenvectors in the signal subspace. In obtaining the signal subspace, the correlation matrix constructed from the large number of sampled data is required. In addition, the number of frequency components should be known a priori or estimated in advance, to determine the number of eigenvectors used in the frequency estimation.

As an alternative approach, compressed sensing (CS)-based techniques have been employed in identifying the tonal frequency components [11, 12]. It is now well-known from the theory of CS that a high-dimensional signal can be recovered from a relatively small number of measurements as long as the desired signal is sparse (i.e., the number of nonzero elements in the signal is small). The main observation behind the CS-based techniques is that a tonal signal consists of a small number of frequency components. Motivated from this observation, the CS-based techniques construct a discretized frequency basis and then identify the basis components representing the tonal signal. While the CS-based approaches are effective in terms of the computational complexity and the accuracy of the estimated frequency components, a basis mismatch error caused by the discretization of the frequency axis is unavoidable.

The primary goal of this paper is to put forth a new approach to estimate the frequency components of the tonal signal. The key idea of the proposed approach is to formulate the tonal frequency estimation problem as an atomic norm minimization (ANM) problem [1315]. In a nutshell, ANM can be thought as an extension of the CS technique in which the sensing matrix is allowed to have an infinite number of columns (atoms). One clear benefit of using the ANM technique is that one can avoid the basis mismatch error caused by the discretization of the frequency domain, since ANM identifies the desired frequency components in a continuous domain.

From the simulations using synthetic and real experimental data, we demonstrate that the proposed detection technique is effective in identifying the tonal frequency components. In particular, when the number of frequency components is larger than 15, the proposed technique shows roughly two times larger success probability than those of the conventional techniques. By the success probability, we mean the probability of estimating all the tonal frequency components within an error of 1/N where N is the number of measurements.

We briefly summarize the notations used in this paper. Vectors are written as boldface lowercase letters (e.g., r), and matrices are denoted by boldface uppercase letters (e.g., A). For a vector \(\mathbf {r} \in \mathbb {C}^{N}\), r[n] is the nth element of r. \(\text {Toep}(\mathbf {r}) \in \mathbb {C}^{N \times N}\) is the Hermitian Toeplitz matrix with r as its first column. For a matrix \(\mathbf {A} \in \mathbb {C}^{N \times L}\), \(\mathbf {a}_{l} \in \mathbb {C}^{N}\) is the lth column of A. AT and A are the transpose and the conjugate transpose of A, respectively. AF is the Frobenius norm of A. For a square matrix \(\mathbf {M} \in \mathbb {C}^{N \times N}\), tr(M) is the trace of M.

Signal model

We consider the practical scenarios where the tonal signal dominates the received signal and is used in detecting a marine object. The signal r(t) detected by the passive sonar is expressed as [5, 6, 16]

$$ r(t) = r_{\text{tonal}}(t) + r_{\text{propeller}}(t) + r_{\text{hydro}}(t) + u(t). $$
(1)

rtonal(t) is the tonal signal generated from machinery components of a marine object, and rpropeller(t) is the propeller noise produced by the rotation of a propeller and then modulated by cavitation. rhydro(t) is the hydrodynamic noise caused by the friction between a marine object and waves, and u(t) is the ambient noise including surface noise, molecular motions, and seismic noise. Furthermore, it is well-known that the tonal signal is expressed as a linear combination of complex sinusoids, i.e.,

$$\begin{array}{*{20}l} r_{\text{tonal}}(t) &= \underset{i=1}{\overset{k}{\sum}} A_{i} \cos (2 \pi f_{i} t), \end{array} $$
(2)

where \(A_{i} \in \mathbb {C}\) is the (complex) amplitude of the ith frequency component [16]. Also, the propeller noise is given by [6]

$$\begin{array}{*{20}l} r_{\text{propeller}}(t) &= \left (1 + \underset{m=1}{\overset{p}{\sum}} B_{m} \cos \left (2\pi (m \widetilde{f}_{0}) t \right) \right) c(t), \end{array} $$
(3)

where \(\widetilde {f}_{0}\) is the undesired harmonic frequency caused by propeller rotations, \(B_{m} \in \mathbb {C}\) is the amplitude of the mth harmonic frequency component, and c(t) is the Gaussian cavitation noise. Then, the signal \(\mathbf {r}_{l} \in \mathbb {C}^{\widetilde {N}}\) sampled at the lth channel can be expressed as

$$\begin{array}{*{20}l} \mathbf{r}_{l}[n] &= \sum_{i=1}^{k}A_{il} \cos \left (2\pi \frac{f_{i}}{f_{s}} n \right) \\ &~~+ \left (1 + \sum_{m=1}^{p} B_{ml} \cos \left (2\pi \frac{m \widetilde{f}_{0}}{f_{s}} n \right) \right) \mathbf{c}_{l}[n] \\ &~~+ \mathbf{v}_{l}[n], \end{array} $$
(4)

where fs is the sampling frequency and vl is sum of the hydrodynamic noise and the ambient noise.

Proposed method

In this section, we describe the proposed technique to estimate the tonal frequency components (f1,,fk) from r1,,rL. The proposed technique consists of two main operations: (1) pre-filtering to suppress the undesired frequency components and noise and (2) estimation of the tonal frequency components from the filtered signal using a modified ANM. We first describe the pre-filtering step and then demonstrate the tonal frequency estimation using the modified ANM.

Noise suppression via pre-filtering

In the pre-filtering step, we first apply the low-pass filter (LPF) followed by the decimation to the sampled signals \(\{ \mathbf {r}_{i} \}_{i=1}^{L}\). In order to prevent the aliasing phenomenon, we set the cutoff frequency fc of the LPF to be smaller than fs/α where α is the decimation ratio. Since the desired tonal signal consists of low frequency components [6], we need to preserve the tonal component yet filter out unwanted high frequency interferences and noise. Using the low-pass filter with the cutoff frequency fc and decimation with the decimation ratio α (α>1), the output of LPF and decimation process \(\widetilde {\mathbf {r}}_{l}\) at the lth channel is given by

$$ \begin{aligned} \widetilde{\mathbf{r}}_{l}[n] = & \underset{i=1}{\overset{k}{\sum}} A_{il} \cos \left (2 \pi \frac{f_{i}}{f_{s}/\alpha} n \right) + \widetilde{\mathbf{c}}_{l}[n] + \widetilde{\mathbf{v}}_{l}[n], \\ & n = 0,\cdots,N, \end{aligned} $$
(5)

where N is \(\widetilde {N}/\alpha \), \(\widetilde {\mathbf {c}}_{l}[n]\) is the filtered propeller noise, and \(\widetilde {\mathbf {v}}_{l}[n]\) is the sum of the filtered hydrodynamic and ambient noises.

Also, we used an auto-correlation filter defined as \(R_{\mathbf {x}\mathbf {x}}(\tau) = F^{-1}[\vert \hat {X}(f) \vert ^{2}]\) where \(\hat {X}(f)\) is the Fourier transform of x, F−1 is an inverse Fourier transform function, and τ is a time-lag variable. Note that an auto-correlation filter has the property of boosting the periodic signal and suppressing the uncorrelated stochastic noise [17]. Thus, using the auto-correlation filter, desired periodic tonal signal is strengthened while propeller and ambient noise components are suppressed [18]. As a result, the pre-filtered output pl at the lth channel is given by

$$\begin{array}{*{20}l} \mathbf{p}_{l}[n] &= \underset{i=1}{\overset{k}{\sum}} A_{il} \cos \left (2 \pi \frac{f_{i}}{f_{s}/\alpha} n \right) + \mathbf{w}_{l}[n] \end{array} $$
(6)
$$\begin{array}{*{20}l} &= \mathbf{s}_{l}[n] + \mathbf{w}_{l}[n], \end{array} $$
(7)

where

$$\begin{array}{*{20}l} \mathbf{s}_{l}[n] &= \underset{i=1}{\overset{k}{\sum}} A_{il} \cos \left (2 \pi \frac{f_{i}}{f_{s} / \alpha} n \right), \end{array} $$
(8)

and wl is the output of the auto-correlation filter corresponding to the input \(\widetilde {\mathbf {c}}_{l} + \widetilde {\mathbf {v}}_{l}\). In [17, 18], it has been shown that if \(\widetilde {\mathbf {c}}_{l}\) and \(\widetilde {\mathbf {v}}_{l}\) are white Gaussian, then the power spectrum of wl is negligible.

Note that

$$\begin{array}{*{20}l} \frac{\mathbf{s}_{l}}{\sqrt{N}} &=\underset{i=1}{\overset{k}{\sum}} \frac{A_{il}}{2} \mathbf{a} \left (\frac{f_{i}}{f_{s}/\alpha} \right) + \underset{i=1}{\overset{k}{\sum}} \frac{A_{il}}{2} \mathbf{a} \left (1-\frac{f_{i}}{f_{s}/\alpha} \right), \end{array} $$
(9)

where \(\mathbf {a}(f) = [ 1, \ e^{j2\pi f}, \ \cdots, \ e^{j2 \pi f \left (N-1 \right)}]^{T} / \sqrt {N}\). Thus, \(\mathbf {S} = [ \mathbf {s}_{1} \ \cdots \ \mathbf {s}_{L} ] \in \mathbb {C}^{N \times L}\) can be expressed as

$$\begin{array}{*{20}l} \mathbf{S} &= \underset{i=1}{\overset{k}{\sum}} \left (\mathbf{a} \left (\frac{f_{i}}{ f_{s}/\alpha} \right) + \mathbf{a} \left (1 -\frac{f_{i}}{f_{s}/\alpha} \right) \right) \mathbf{x}_{i}^{*}, \end{array} $$
(10)

where \(\mathbf {x}_{i} = \frac {\sqrt {N}}{2} [ A_{i1} \ \cdots \ A_{iL} ]^{*}\).

Tonal frequency estimation via modified ANM

The primary goal after the pre-filtering is to estimate the tonal frequency components (f1,,fk) in S from the pre-filtered signal P=[p1 pL]=S+W. To this end, we first introduce an atom A(f,b), defined as

$$ \mathbf{A}\left(f, \mathbf{b} \right) = \mathbf{a} \left(f \right) \mathbf{b}^{*} \in \mathbb{C}^{N \times L}, $$
(11)

where f[0,1] and \(\mathbf {b} \in \mathbb {C}^{L}\) is an 2-normalized vector (i.e., b2=1). We also denote the set of atoms as

$$\begin{array}{*{20}l} \mathcal{A} &= \{ \mathbf{A} \left(f, \mathbf{b} \right): f \in \left[ 0, 1 \right], \mathbf{b} \in \mathbb{C}^{L}, \| \mathbf{b} \|_{2} = 1 \}. \end{array} $$
(12)

In a nutshell, the atoms in \(\mathcal {A}\) are building blocks of the desired signal S, i.e., S can be expressed as an affine combination of some atoms in \(\mathcal {A}\):

$$\begin{array}{*{20}l} \mathbf{S} &= \underset{i=1}{\overset{k}{\sum}} \| \mathbf{x}_{i} \|_{2} \mathbf{A} \left (\frac{f_{i}}{f_{s}/\alpha}, \frac{\mathbf{x}_{i}}{\| \mathbf{x}_{i} \|_{2}} \right) \\ &~~~+ \underset{i=1}{\overset{k}{\sum}} \| \mathbf{x}_{i} \|_{2} \mathbf{A} \left (1-\frac{f_{i}}{f_{s}/\alpha}, \frac{\mathbf{x}_{i}}{\| \mathbf{x}_{i} \|_{2}} \right). \end{array} $$
(13)

One can notice that while \(\mathcal {A}\) contains an infinite number of atoms, the number of atoms used in the representation (13) is very small. In this sense, we can readily say that S is sparse with respect to \(\mathcal {A}\).

Conventional ANM

In the same vein as the 0-norm minimization technique in CS, the sparse signal S (with respect to \(\mathcal {A}\)) can be reconstructed from P by solving [15, 19]

$$ \begin{aligned} &\underset{\mathbf{X}}{\min} ~~~~~~ \frac{1}{2} || \mathbf{X} - \mathbf{P} ||_{F}^{2} + \tau || \mathbf{X} ||_{\mathcal{A}, 0}, \end{aligned} $$
(14)

where τ(>0) is the pre-determined regularization parameter and \(\| \mathbf {X} \|_{\mathcal {A}, 0}\) is defined as

$$\begin{array}{*{20}l} \| \mathbf{X} \|_{\mathcal{A}, 0} &= \underset{K}{\inf} \{ K: \mathbf{X} = \underset{i=1}{\overset{K}{\sum}} c_{k} \mathbf{A}_{k}, c_{k} \ge 0, \mathbf{A}_{k} \in \mathcal{A} \}. \end{array} $$
(15)

Finding the solution to (14), however, requires a combinatorial search over all possible subspaces spanned by the atoms in \(\mathcal {A}\), so that direct (exhaustive search) approach is infeasible for most practical scenarios. To perform the reconstruction task, the ANM technique can be employed. The key idea of ANM is to replace the nonconvex term \(\| \mathbf {X} \|_{\mathcal {A}, 0}\) in (14) with its convex surrogate atomic norm \(\| \mathbf {X} \|_{\mathcal {A}}\) defined as

$$ \| \mathbf{X} \|_{\mathcal{A}} = \inf \left\{ \sum_{k} c_{k} : \mathbf{X} = \sum_{k} c_{k} \mathbf{A}_{k}, c_{k} \ge 0, \mathbf{A}_{k} \in \mathcal{A} \right \}. $$
(16)

In other words, ANM reconstructs S by solving [15]

$$ \underset{\mathbf{X}}{\min} ~~~~~~ \frac{1}{2}\Vert \mathbf{X}-\mathbf{P} \Vert^{2}_{F} + \tau \Vert \mathbf{X} \Vert_{\mathcal{A}}. $$
(17)

By using [15, Theorem 1]

$$\begin{array}{*{20}l} \| \mathbf{X} \|_{\mathcal{A}} &= \inf \left \{ \frac{1}{2} \left (\text{tr}(\mathbf{W}) + \text{tr}(\text{Toep}(\mathbf{u})) \right) \right. \\ &~~~~~~~~~: \left. \left[\begin{array}{ll} \mathbf{W} & \mathbf{X}^{*} \\ \mathbf{X} & \text{Toep}(\mathbf{u}) \end{array}\right] \succeq 0 \right \}, \end{array} $$
(18)

and thus we obtain the equivalent form of (17) as

$$ \begin{aligned} &\underset{\mathbf{X}, \mathbf{W}, \mathbf{u}}{\min} ~~~ \Vert \mathbf{X}-\mathbf{P} \Vert^{2}_{F} + \tau \left (\text{tr}(\mathbf{W}) + \text{tr}(\text{Toep}(\mathbf{u})) \right) \\ &~~\text{s.t.}~~~~ \left[\begin{array}{ll} \mathbf{W} & \mathbf{X}^{*} \\ \mathbf{X} & \text{Toep}(\mathbf{u}) \end{array}\right] \succeq 0 \end{aligned}. $$
(19)

Note that the problem (19) can be solved effectively via a well-known SDP solver (e.g., SDPT3 [19]). Note also that since the computational complexity of the SDP is \(\mathcal {O}(M^{5/2})\) [20] and the number of samples M is reduced to \(N=\widetilde {N}/\alpha \) by the decimation process, the computation complexity of the estimation step is marginal.

We now present a method to estimate the frequency components (f1,,fk) in S (see (13)). To this end, we first derive a dual problem of (17). By introducing a new variable Y=X for the problem (17), the Lagrangian L(X,Y,Z) associated with (17) is expressed as

$$\begin{array}{*{20}l} L&(\mathbf{X}, \mathbf{Y}, \mathbf{Z}) \\ &~~= \frac{1}{2} \| \mathbf{X} - \mathbf{P} \|_{F}^{2} + \tau \| \mathbf{Y} \|_{\mathcal{A}} + \text{Re}(\text{tr}(\mathbf{Z}^{*}(\mathbf{X} - \mathbf{Y}))) \\ &~~= \left (\frac{1}{2} \| \mathbf{X} - \mathbf{P} \|_{F}^{2} + \text{Re}(\text{tr}(\mathbf{Z}^{*} \mathbf{X})) \right) \\ &~~~~~~+ (\tau \| \mathbf{Y} \|_{\mathcal{A}} - \text{Re}(\text{tr}(\mathbf{Z}^{*} \mathbf{Y}))), \end{array} $$
(20)

where Z is the dual variable. One can easily show that

$$\begin{array}{*{20}l} &\underset{\mathbf{X}}{\min} \ \frac{1}{2} \| \mathbf{X} - \mathbf{P} \|_{F}^{2} + \text{Re}(\text{tr}(\mathbf{Z}^{*} \mathbf{X})) \\ &~~= \frac{1}{2}(\| \mathbf{P} \|_{F}^{2} - \| \mathbf{P} - \mathbf{Z} \|_{F}^{2}) \end{array} $$
(21)

and

$$\begin{array}{*{20}l} &\underset{\mathbf{Y}}{\min} \ \tau \| \mathbf{Y} \|_{\mathcal{A}} - \text{Re}(\text{tr}(\mathbf{Z}^{*} \mathbf{Y})) \\ &~~= \underset{\mathbf{Y}}{\min} \ \| \mathbf{Y} \|_{\mathcal{A}} \left (\tau - \text{Re} \left (\text{tr} \left (\mathbf{Z}^{*} \mathbf{Y} / \| \mathbf{Y} \|_{\mathcal{A}} \right) \right) \right) \\ &~~= \left\{\begin{array}{ll} 0 & \| \mathbf{Z} \|_{\mathcal{A}}^{*} \le \tau \\ -\infty & \| \mathbf{Z} \|_{\mathcal{A}}^{*} > \tau, \end{array}\right. \end{array} $$
(22)

where \(\| \mathbf {Z} \|_{\mathcal {A}}^{*}\) is the dual norm of the atomic norm defined as \(\| \mathbf {Z} \|_{\mathcal {A}}^{*} = \underset {\| \mathbf {X} \|_{\mathcal {A}} \le 1}{\sup } \text {Re}(\text {tr}(\mathbf {X}^{*} \mathbf {Z}))\). From (20)–(22), the dual problem of (17) is given by

$$ \begin{aligned} &\underset{\mathbf{Z}}{\max} ~~~~~~ \frac{1}{2} \left (\| \mathbf{P} \|_{F}^{2} - \| \mathbf{P} - \mathbf{Z} \|_{F}^{2} \right) \\ &~\text{s.t.} ~~~~~~ \| \mathbf{Z} \|_{\mathcal{A}}^{*} \le \tau. \end{aligned} $$
(23)

Let \(\widehat {\mathbf {Z}}\) be the solution to (23). Then, the correlation \(Q(f) = \| \widehat {\mathbf {Z}}^{*} \mathbf {a}(f) \|_{2}\) between the solution \(\widehat {\mathbf {Z}}\) and the atom a(f) is maximized when f is one of the tonal frequency components (i.e., f{f1,,fk}). Specifically, Q(f) satisfies [15]

$$\begin{array}{*{20}l} \left\{\begin{array}{ll} Q(f) = \tau, & \text{if} \ f \in \{ f_{1}, \cdots, f_{k} \} \\ Q(f) < \tau, & \text{if} \ f \notin \{ f_{1}, \cdots, f_{k} \} \end{array}\right.. \end{array} $$
(24)

Therefore, we can estimate the desired frequency components (f1, , fk) by solving Q(f)=τ.

Modified ANM

One potential limitation of the conventional ANM technique is that the performance is not so appealing when the source vectors s1,,sL are strongly correlated. For better understanding of this issue, we take the following example. If the correlation between s1 and s2 is extremely strong (i.e., \(\frac {| \langle \mathbf {s}_{1}, \mathbf {s}_{2} \rangle |}{\| \mathbf {s}_{1} \|_{2} \| \mathbf {s}_{2} \|_{2}} \approx 1\)), then s1s2 (up to a constant factorFootnote 1) and therefore the corresponding measurement vectors p1 and p2 would be also similar (i.e., p1p2). As a result, ANM cannot obtain any additional information on the frequency components from p2, which in turn implies that there is no benefit of using multiple measurement vectors p1 and p2.

In order to achieve reliable performance even when the source vectors are highly correlated, we exploit an orthonormal basis of the measurement space in estimating the frequency components. By the measurement space, we mean the subspace spanned by the measurement vectors p1,,pL. Specifically, we replace the measurement matrix P in (23) with the orthonormal basis U of span(P). Suppose the observation matrix \(\mathbf {P} \in \mathbb {R}^{N \times L}\) has full column rank (i.e., rank(P)=L), then an orthonormal basis of the measurement space can be obtained by performing an singular value decomposition (SVD) operation on P. Specifically, if P=U Σ VH where \(\mathbf {U} \in \mathbb {R}^{N \times L}\), \(\boldsymbol {\Sigma } \in \mathbb {R}^{L \times L}\), and \(\mathbf {V} \in \mathbb {R}^{L \times L}\), then U is an orthonormal basis of span(P). After obtaining the orthonormal basis U, we reformulate (23) as:

$$ \begin{aligned} &\underset{\mathbf{Z}}{\max} ~~~~~~ \frac{1}{2} \left (\| \mathbf{U} \|_{F}^{2} - \| \mathbf{U} - \mathbf{Z} \|_{F}^{2} \right) \\ &~\text{s.t.} ~~~~~~ \| \mathbf{Z} \|_{\mathcal{A}}^{*} \le \tau \end{aligned} $$
(25)

Let \(\widetilde {\mathbf {Z}}\) be the solution to (25). Then, the proposed technique estimates the desired tonal frequency components (f1, , fk) by solving

$$\begin{array}{*{20}l} Q(f) = \| \widetilde{\mathbf{Z}}^{*} \mathbf{a}(f) \|_{2} = \tau. \end{array} $$
(26)

We depict the overall block diagram of the proposed scheme in Fig. 2. To verify the performance of the modified ANM technique, we perform the numerical simulation in the scenario where the source vectors are highly correlated. In our simulation, we generate the strongly correlated source vectors s1,s2 such that |〈s1,s2〉|≈0.99, and each source vector consists of k tonal frequency components. As a metric to evaluate the performance of each technique, we employ the exact reconstruction ratio (ERR) defined as

$$\begin{array}{*{20}l} \text{ERR} &= \frac{\text{Number of successful trials}}{\text{Number of trials}}. \end{array} $$
Fig. 2
figure2

Block diagram of the passive sonar systems using proposed frequency estimation scheme

Here, each trial is defined to be successful if all the estimated frequencies \(\widehat {f}_{i}\) satisfy \(|f_{i} - \widehat {f}_{i} | < \frac {1}{2N}\) where fi is the ith target frequency component and N is the number of measurements. Note that we perform 2000 independent trials for each point of the technique. In Fig. 3, we plot the ERR performance of each technique as a function of k. Overall, the ERR performance improves with the number of measurement vectors L. One can see from Fig. 3 that the performance improvement of the modified ANM technique performs much better than that of the conventional ANM method. The key reason for this result is that p1p2 and thus the conventional ANM obtains little information on the frequency components from p2. Additionally, one can observe that the ERR performance of the modified ANM is the same as that of the conventional ANM when L=1, since the orthonormal basis of the measurement space coincides with the measurement vector (up to a constant factor) in this case. In Table 1, we summarize the critical sparsities of the proposed scheme and the conventional ANM technique. By the critical sparsity, we mean the maximum number of frequency components at which ERR is above 0.9. From the table, we can observe that our scheme outperforms the conventional ANM technique for all L. For example, when L=5, the critical sparsity of the proposed scheme is three times higher than that obtained by the conventional ANM technique.

Fig. 3
figure3

The ERR performance of the conventional and modified ANM techniques when the source vectors are highly correlated

Table 1 Critical sparsity performance comparison of modified ANM and conventional ANM

Simulation results and discussion

In this section, we study the performance of the proposed frequency estimation technique using both synthetic and real datasets.

Simulation setup

First, we test the performance of the proposed technique using synthetic data. In our simulations, we generate the tonal signal whose frequency components are uniformly chosen at random in the range of 0–500 Hz and amplitudes of frequency components are drawn i.i.d. from a Gaussian distribution \(\mathcal {N}(1, 0.1)\). The base frequency \(\widetilde {f}_{0}\) of the propeller noise is set to 10 Hz, and we assume the scenario where the propeller noise consists of three harmonic components (i.e., p=3 in (4)). The other noise component vl in (4) is generated as a white Gaussian with mean zero and variance σ2. The sampling frequency fs is set to 10 kHz. In the pre-filtering step, we set the cutoff frequency of the LPF to 500 Hz and the decimation ratio α to 10. In our simulations, the following frequency estimation techniques are considered and we perform 5000 independent trials for each point of the technique:

  1. 1)

    Root-MUSIC [8]

  2. 2)

    ESPRIT [9]

  3. 3)

    Matrix pencil [10]

  4. 4)

    Conventional ANM [19]

  5. 5)

    Modified ANM

We next verify the performance of the proposed technique using real data obtained from the shallow water evaluation cell experiment 96 (SWellEx-96). Note that the SWellEx-96 was conducted 12 km off the coast of San Diego, CA, in May 1996, and the target source signal consists of 13 frequency components (49, 64, 79, 94, 112, 130, 148, 166, 201, 235, 283, 338, and 388 Hz). In our simulations, we divide 4000 measurement vectors into 200 groups (20 measurement vectors in each group). Using each group of measurement vectors, we estimate the tonal frequency components.

Results and discussion

In Fig. 4, the ERR performance of each technique is plotted as a function of k (number of desired tonal frequency components). This simulation is performed in the scenario where the signal-to-noise ratio (SNR) is \(\text {SNR} = 10\log ((\sum \limits _{i=1}^{k}|A_{i}|^{2})/(\sum \limits _{m=1}^{p}|B_{m}|^{2} + \sigma ^{2})) = 10\) dB. As shown in Fig. 4, the ERR performance of the proposed technique is better than those of the other frequency estimation techniques in the whole range of k by a large margin.

Fig. 4
figure4

ERR performance as a function of k

In Fig. 5, we plot the mean square error (MSE) performance of conventional ANM method and modified ANM as a function of k. The MSE is defined as

$$ \text{MSE} = \frac{1}{k} \sum_{i=1}^{k} \left(\widehat{f}_{i}-f_{i} \right)^{2}. $$
Fig. 5
figure5

MSE performance as a function of k

Note that the MSE performance shows how close each of the estimated frequency components is to the original frequency components. We measured the MSE only in the case where the estimation was successful. As depicted in Fig. 5, the modified ANM performs better than the conventional ANM.

In Fig. 6, we plot the MSE performance of the proposed method and other line spectrum estimation methods as a function of SNR. As shown in Fig. 6a, ANM-based algorithms outperform conventional line spectrum methods in all SNR region. In particular, when SNR = 0 dB, the MSE of the proposed scheme is 1.5 times lower than that of the conventional ANM technique (see Fig. 6b).

Fig. 6
figure6

MSE performance as a function of SNR. a MSE performance comparison of line spectrum estimation techniques as a function of SNR. b MSE performance comparison of ANM-based scheme as a function of SNR

In our real experiment test, we say that the ith frequency component fi is successfully estimated if \(|f_{i} - \widehat {f}_{i}| < 5\). As a metric to measure the performance of the proposed technique, we exploit the success ratio rsuccess and the mean square error MSEsuccess of the successfully estimated frequency components. Let J be the index set of successfully estimated frequency components. Then, rsuccess and MSEsuccess are defined as

$$\begin{aligned} &r_{\text{success}} = \frac{|J|}{13}, \\ &\text{MSE}_{\text{success}} = \frac{1}{|J|} \underset{j \in J}{\sum} |f_{j} - \widehat{f}_{j}|^{2}. \end{aligned} $$

In Table 2, we summarize the performance of the conventional ANM-based method and the proposed technique. One can see that the success ratio rsuccess of the proposed technique is roughly 2.5 times higher than that of the conventional ANM-based technique. Furthermore, the MSEsuccess of \(\{ \widehat {f}_{j} : j \in J \}\) obtained by the proposed method is 1.2 times smaller than that acquired by the conventional method.

Table 2 Performance comparison of ANM-based algorithms

Conclusion

In this paper, we proposed an ANM-based frequency estimation technique to identify the tonal frequency components in the passive sonar system. The proposed technique consists of two major steps. In the first step, we applied the LPF, decimation, and the auto-correlation filter to the signal detected by the passive sonar, filtering out the unwanted propeller noise and high-frequency interferences. In the second step, we formulated the tonal frequency estimation problem as an ANM problem and then estimated the desired frequency components using an orthonormal basis of the measurement space. In doing so, we could achieve reliable performance even when the source vectors are strongly correlated. Finally, using both synthetic and real datasets, we demonstrated that the proposed technique is effective in estimating the tonal frequency components.

Availability of data and materials

Please contact the author for synthetic data request, and the SWellEx-96 dataset used in this work is available at http://swellex96.ucsd.edu/s5.htm.

Notes

  1. 1.

    Note that \(\frac {| \langle \mathbf {s}_{1}, \mathbf {s}_{2} \rangle |}{\| \mathbf {s}_{1} \|_{2} \| \mathbf {s}_{2} \|_{2}} \le 1\) and the equality is attained if and only if s1/s12=s2/s22.

Abbreviations

ANM:

Atomic norm minimization

CS:

Compressed sensing

DFT:

Discrete Fourier transform

ERR:

Exact reconstruction ratio

ESPRIT:

Estimation of signal parameters via rotational invariance techniques

LPF:

Low-pass filter

MSE:

Mean square error

MUSIC:

Multiple signal classification

SDP:

Semi-definite programming

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Acknowledgements

The authors would like to thank the Agency for Defense Development (ADD) in Korea for supporting this work (contract No. UD190005DD).

Funding

This work was supported by the Agency for Defense Development (ADD) in Korea under the contract No. UD190005DD.

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All the authors have participated in writing the manuscript. All authors read and approved the manuscript.

Correspondence to Wooyoung Hong.

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Keywords

  • Passive sonar
  • Tonal signal detection
  • Atomic norm minimization