Weak fluctuating spectral line reconstruction using deep learning

The detection of weak fluctuating spectral lines emitted by underwater and surface vehicles poses a challenging problem for passive sonar system. Therefore, a spectral line reconstruction algorithm based on deep learning called the DEDAN, is proposed. The DEDAN learns the time-frequency correlation of spectral lines through end-to-end training and then reconstructs the spatial location of spectral lines. Simulation results show that the DEDAN is robust to ambient noise, and outperforms other reconstruction algorithms at a mixed signal-to-noise ratio as low as -22 dB to -26 dB. Its reconstruction performance is also verified by the measured South China Sea data.


Introduction
The field of passive sonar signal processing utilizes spectra to depict the time-frequency characteristics of underwater radiated noise, encompassing abundant single-frequency components crucial for the detection of quiet targets [1].A lofargram, which is used for low frequency analysis and recording, is commonly constructed using the short-time Fourier transform to analyze the spectral line features of passive sonar signals [2].By observing lofargram, the presence of underwater and surface vehicles can be quickly detected.
Low signal-to-noise ratio (SNR), irregular frequency fluctuations caused by channel and target motion, and strong background noise are the challenges of spectral line reconstruction.Furthermore, the term "reconstruction" implies the retrieval of potential spectral line characteristics in order to generate a lofargram that exhibits prominent spectral lines.Sophisticated image semantic features are handled to reconstruct spectral lines [3].A method based on the hidden Markov Model (HMM) is proposed to track spectral lines [4].However, the spectral lines obtained in low SNR are not reliable.Hence, new algorithms are being considered, such as deep learning.DeepLofargram was proposed to recover a single fluctuating spectral line at low SNRs under Gaussian noise.The DeepLofargram method was proposed for the recovery of a single fluctuating spectral line in low SNRs under Gaussian noise [5].
In this paper, an end-to-end model called DEDAN is proposed to reconstruct weak fluctuating spectral lines under Gaussian and weak non-Gaussian impulsive noises.First, a feature-level shrinkage module (FLS) is inserted into the encoder layer to eliminate noise-related information in lofargram.Second, spectral lines in lofargram are reconstructed by the decoder.Moreover, due to sparse spectral lines in lofargram, a special loss function is used for end-to-end training of the DEDAN.Finally, we validate DEDAN's spectral line reconstruction performance on representative simulation and actual datasets.

Signal model
Passive sonar signals can be expressed as spectral line components superimposed with noise.
where n A represents the signal amplitude.i  represents the initial phase. ()ik ft denotes the frequency of subsequent fluctuations in k t , indicating that the spectral line exhibits unpredictable variations.() k nt denotes the sampling of noise at time k t .Gaussian and weak non-Gaussian impulsive noises are present in the underwater acoustic channel, which can be described by Gaussian and α−stable distribution.According to [6], its characteristic function is as follows: where The characteristic exponent  indicates the degree of pulse characteristics in a distribution.The  -stable distribution degenerate into a Gaussian distributions when 2  = . denotes the dispersion of the distribution.The parameter  describes the skewness of a distribution.
The energies of Gaussian and non-Gaussian impulsive noise is described by the SNR and mixed signal-to-noise ratio (MSNR), which are defined as follows [7] where  denotes the signal amplitude. denotes the noise and signal variance, respectively.

DEDAN
Figure 1 shows the structure of a reconstruction neural network with encoder and decoder layers.The encoder layer composed of feature-level shrinkage modules [8] extract spectral line features on noisedominating lofargrams.The decoder layers are stacked with a series of convolution and transposed convolution layer to reconstruct lofargram with significant spectral lines.

Loss function
Due to the sparse spectral line pixels in lofargram, the loss between spectral line and noise classes is unbalanced.Hence, we adopt the class-balanced cross-entropy loss function in [5]. ,, ]

Training Phase
The loss function was optimized using mini-batch gradient descent with Adam during training.The batch size is 32.Some data augmentation methods, such as horizontal and vertical flipping of images, and grayscale maps, are used to prevent network overfitting.The network was first pre-trained with lofargrams with MSNRs of -18 to -22 dB at a learning rate of

Performance Metric
Two standard metrics, the mIoU [9] and line location accuracy (LLA) [10], are used to evaluate the reconstruction performance.
where TP , FN , and FP represent the true positives, false negatives, and false positives, respectively.denotes the Euclidean distance between the reconstructed and actual spectral lines.Following the suggestion in [10], we set 1  = .

Simulation Analysis
The performances of the proposed DEDAN are evaluated and compared with HMM, Unet [11]  As shown in Figure 2, the LLA and mIOU of the HMM vary from 0.1993 to 0.3897 and 0.4838 to 0.5431 among the five MSNRs, respectively.The mIOU and LLA of the UNet are around 0.3257 to 0.4100 and 0.5346 to 0.5679, respectively.Accordingly, DEDAN without FLS outperforms the previous two algorithms, ranging from 0.3596 to 0.4398 for LLA and 0.5432 to 0.5933 for mIOU; however, the proposed DEDAN shows outstanding results, with mIOU values from 0.3645 to 0.4650 and LLA values ranging from 0.5432 to 0.6091.Therefore, it can be illustrated that the proposed DEDAN, with impressive performance, is more robust to ambient noise, and the FLS is effective.

Experimental Data
To illustrate the DEDAN's performance, data collected during September's South China Sea experiment are utilized.The experimental geometry is shown in Figure 3，with one 12-element ocean bottom seismometer (OBS) array moored on the sea floor.The data were recorded by the ocean bottom seismometers (OBS).The sampling rate was 100 Hz.The depth of the sea is about 330m.A large number of ship radiated noise samples were collected when the experimental ship deployed and recovered OBS at ranges 0.1-60 km.We use the 2000 lofargrams provided by this experiment to finetune the pre-trained DEDAN and other deep learning algorithms.

Conclusion
A spectral line reconstruction algorithm, named DEDAN, has been studied in this paper for Gaussian/weak non-Gaussian impulsive noise.The DEDAN uses encoder-decoder pair as the core of its network architecture.FLS module is introduced in the encoder layer to suppress the noise in the lofargarm, and the decoder layer reconstructs the weak fluctuating spectral lines in lofargram.Simulation and experimental results illustrate that compared to other algorithms, the DEDAN is more robust to ambient noise, and outputs lofargram with significant spectral lines.In the future, our model will be improved by considering more underwater acoustic signals and noise distributions.
 .Ground truth label sets +  and −  correspond to spectral line and noise, respectively.The , ft p indicates the sigmoid function predicted value of the lofargram at( , )  ft.

4 10 −
, and then with lofargrams with MSNRs of -23 to -26 dB at a learning rate of 5 10 − .The training process reaches convergence in almost 250 epochs.
by our simulation.The MSNR and SNR for our simulation are both at [-19, -26] dB, with  randomly selected value of [1.9, 2].The sampling rate s f is 1000 Hz.Our main concern is low frequency within 220 Hz.Multiple fluctuating spectral lines, with harmonic relationships, are simulated in our synthetic datasets.750 Monte-Carlo simulations for each MSNR and SNR are conducted and split datasets into 85% for training and 15% for testing.Hence, a training set of 5100 lofargrams and a test set consisting of 900 lofargrams are constructed under eight different MSNRs.

Figure 3 .
Figure 3. Layout and recycling of OBS.

Figure 4
Figure 4 compares lofargrams reconstructed by HMM, UNet, DEDAN without FLS and DEDAN.HMM reconstructs spurious spectral lines, Unet and DEDAN without FLS are unable to reconstruct the complete three spectral lines, while the proposed DEDAN can reconstruct more complete spectral lines.Therefore, the proposed DEDAN is a suitable method for reconstructing weak fluctuating spectral lines under strong background noise.