Frequency estimation of underwater acoustic targets based on improved FRI method

The line spectrum information of a ship’s radiated noise signal is often regarded as an important feature used to distinguish between targets. In this paper, an improved FRI (finite rate of innovation) method is proposed to estimate the low-frequency line spectra of the radiated noise from underwater acoustic passive targets under the condition of finite sampling points. First, a preliminary rough estimation of the frequency is performed using the Fourier transform. Then, the multichannel signal data are iteratively fitted based on the polynomial-ratio form model. The optimal estimation results are finally determined based on the residuals between the data and the model in the fitting process. Simulation experiments prove that compared with the original FRI algorithm, the method proposed in this paper can estimate line spectral frequency more efficiently under the condition of low signal-to-noise ratio (SNR), and can achieve fast convergence with shorter operation time. The experimental data processing results prove that the method can realize high-precision extraction of line spectrum information of ship noise.


Introduction
In the field of underwater acoustic detection, the line spectrum information of ship noise is regarded as an important feature in distinguishing multi-targets.Ship noise contains line spectrum signal and continuous spectrum noise.Among them, the line spectrum signal is characterized by low frequency and high power, and its specific information is related to the actual hull structure and mechanical working characteristics of the ship, which is an important analytical factor for target detection, tracking, and identification in the field of underwater acoustics.
Underwater acoustic targets usually exhibit low-speed motion characteristics, at this time, there are small-scale frequency variations in the line spectral information, and to obtain the line spectral frequencies more accurately, it is necessary to utilize high-precision frequency estimation methods.Commonly used estimation methods include algorithms such as minimum variance distortionless response (MVDR) and multiple signal classification (MUSIC) [1,2], but these methods are not only difficult in real-time processing due to a large amount of computation but also require the use of ongrid estimation methods.Due to the presence of grid scanning errors, these algorithms are difficult to achieve accurate estimation for continuously varying frequency information even under high SNR conditions.Refined grids are often required to obtain higher accuracy, which further increases the difficulty of real-time processing.The Fourier transform, over-zero detection, and other conventional methods have the problem of low estimation accuracy [3].The FRI algorithm [4][5][6][7][8] achieves the estimation of multiple signals based on the finite rate of innovation model through the zeroed filter with the polynomial ratio method, which has gained applications in DOA and frequency estimation.
This paper proposes a high-precision line spectral frequency estimation method based on the improved FRI algorithm, which is less computationally intensive compared to the original FRI algorithm and can better avoid the local optimal estimation caused by noise under low SNR conditions, thus realizing a fast and gridless line spectral frequency estimation method for underwater acoustic targets.The effectiveness of the method is verified by simulation data analysis and experimental data processing.

Signaling model
The underwater acoustic signal from multiple channels can be represented as a superposition of the line spectrum and the continuous spectrum: where, K is the number of the line spectrum, N is the number of sampling points, L represents the number of channels, , nl r is the continuous spectrum signal, and the multi-line spectrum signal , nl s is the FRI signal from the n -th sampling point and the l -th channel..The N point sample L channel signal can be fully represented by the amplitude   and frequency    =1, , k kK .The purpose of sparse reconstruction is to estimate the parameters from the measurement samples.
For the line spectrum estimation problem, both the broadband continuous spectrum and the environmental noise are expressed as broadband noise characteristics, therefore, the continuous spectrum and the environmental noise are collectively referred to as broadband noise in the later content.The noisy signal is denoted as: where, , nl w is the noise from the n -th sampling point and the l -th channel.At this point, the sparse reconstruction problem can be expressed as:

FRI estimation method
The DFT transform , ˆnl s of the FRI signal can be expressed as the ratio structure of two polynomials −1, Kl P and Thus, the above sparse reconstruction problem is translated into a model-fitting problem with polynomial ratios: Under the linear iterative methods, the i -th iterative problem can be expressed as: Use the MSE criterion as an iterative convergence criterion: where,  2 l denotes the noise energy of each channel.Represent the numerator and denominator of the ratio polynomial as a matrix: where, , NK W and W , p and q are the corresponding coefficients.
The matrix representation of the novel iterative method can be obtained as: where I is the unit array.To ensure the uniqueness of the solution, add the linear constraint = 0 1 H i qq , and let 0 q be the randomized initial vector.The optimization problem is obtained as: The problem is solved using the Lagrange multiplier method with the objective function at the i -th iteration: Resolved as: From the zero points k z of K Q , it can be solved that the frequency and the signal amplitude of each frequency can be expressed as: The randomized initial vector 0 q is an initialized estimation of the line spectral frequency.The original FRI algorithm randomly initializes the initial vectors by performing the first loop several times to achieve the estimation.In noisy scenarios, a large amount of initialization randomization is usually required to obtain convergence results, which leads to an increase in computational time.At the same time, the method is easy to fall into local optimization, which makes the iterative estimation result deviate from the real value and generates a lot of time for invalid operations.

Improved FRI estimation methods
This method first performs amplitude averaging on the results of the Fourier transform of multichannel signals, and uses the method of peak extraction to roughly estimate the frequency.Although the estimation accuracy of Fourier transform is insufficient, its estimated value is clearly closer to the real frequency compared to the random initialization frequency, avoiding a large number of invalid initialization cycles.
The initialization frequency is expressed as: where ( ) is the peak extraction function.
At this point, the expression for initializing 0 q is: where  is any minimal value constant in the interval ( )  0,2 N .In noisy scenarios, the optimization problem often suffers from overfitting.Therefore, in this paper, we use the estimation result corresponding to the minimum residual in the iterative process as the optimal estimation result.The residual estimation criterion is denoted as: In the same way as the FRI algorithm, from the zeros k z of K Q , the signal frequency can be solved to be expressed as: In addition, due to the estimated frequency of zero point, the magnitude expression of the original FRI algorithm can not be calculated, this paper continues the derivation of the numerator denominator of the formula for the factorization and elimination of the same zero component ( ) − − 1 1 k zz, the magnitude of the expression is obtained as: To compare the estimation ability of magnitude under actual data processing conditions, equation (20) is used in the subsequent analysis of all methods.

Simulation Performance Analysis
The simulation of the received data is a multilinear spectral signal with 10 channels and 200 sample points at a sampling frequency of 2 kHz, with signal frequencies of 105 Hz and 705 Hz.Performance analysis was conducted using 1000 Monte Carlo simulations.As shown in figure 1, the original FRI algorithm and the improved FRI algorithm can estimate the signal frequency without the limitation of grid accuracy under the condition of a high SNR.In addition, the amplitude estimation errors of the comparative algorithms are shown in figure 2. However, the original FRI algorithm tends to fall into the local optimum when the number of iterations is insufficient or the SNR is low, resulting in a significant increase in the estimation error.Under the same SNR condition, to obtain more accurate estimation capability, it is necessary to repeat the initialization loop several times, resulting in longer computing time.In contrast, the improved FRI algorithm in this paper can obtain a lower estimation error than the original FRI algorithm in terms of frequency and signal amplitude above -16dB SNR.At the same time, the processing time is significantly shorter than that of the original FRI algorithm in figure 3.

Experimental data processing
The SWellEx-96 test data were processed using the improved FRI method proposed in this paper.In the test scenario, the receiving array remains stationary while the transmitting sound source is moving with the ship.The transmitting signal contains multiple line spectral structures, the processing window is 2s, and the data of 10 channels of horizontal arrays with a duration of 40 minutes are selected for processing.4 and figure 5, it can be seen that the resolution of STFT timefrequency analysis is only 0.5 Hz under the condition of the window length of 2 s.Due to the frequency shift of the line spectrum caused by the target motion, low-resolution STFT algorithm can not observe the process of frequency shift effectively.In contrast, the improved FRI algorithm proposed in this paper can observe the frequency shift process of the multi-line spectrum, and its trend is consistent with the results of the STFT time-frequency analysis, which realizes the accurate estimation of the frequency without the limitation of the grid.

Conclusion
The FRI method transforms the sparse reconstruction problem into a parameter estimation problem by establishing a sparsified parameter model, which realizes the accurate reconstruction of the signal on a continuous parameter domain, and its estimation process does not require grid scanning, so there is no off-grid error caused by the insufficient grid accuracy of algorithms.Simulation analysis proves that the improved FRI algorithm proposed in this paper has better frequency and amplitude estimation capability than the original FRI algorithm under the condition of low SNR, while the processing time is shorter.The experimental data processing proves that the algorithm proposed in this paper can be effectively applied to the estimation of line spectrum frequency of underwater acoustic targets, and has the advantage of reconstructing the signal with high accuracy even under low sampling rates.

Figure 4 . 6 Figure 5 .
Figure 4. Results of time-frequency analysis of radiated acoustic signal.