Line spectrum detection and motion parameters estimation for underwater moving target

For the detection of underwater moving target with constant speed at low signal-to-noise ratio (SNR), a track-before-detect method is proposed based on motion parameters estimation and hidden Markov model (HMM). Optimization algorithm is used for the moving target motion parameter estimation with cost function obtained by energy accumulation along line spectrum tracked with 1-D Hidden Markov model in spatially filtered LOFAR spectrum. This method is suitable for detecting weak targets with fast changes in azimuth.


Introduction
In recent years, the Track-Before-Detect (TBD) method has become a highlight research topic in the field of target detection [1] , which is mainly based on dynamic programming [2] , Hough transform [3][4] , particle filter [5][6] , etc.The essence of TBD method is the joint processing of multi-frame data to improve the performance for weak target detection.
With the increase of integration time, the change of signal parameters caused by target motion cannot be ignored.Therefore, it is necessary to estimate the state of the target at each moment in order to achieve energy focusing with accumulation of signal power spectrum along trajectory point.In reference [7], a HMM is established to extract the target azimuth trajectory in the Bearing-Time Recording (BTR) obtained for a certain bandwidth signal.A hidden Markov model is established on the omnidirectional LOFAR spectrum to estimate the frequency time series of the target signal in reference [8].In reference [9], a 2-D hidden Markov model is established on the frequency-azimuth (FRAZ) spectrum to track the target azimuth and frequency simultaneously, which requires less prior information and has better detection performance.However, it is likely to be affected by high energy interference and noise on the FRAZ spectrum.
In this paper, with three-dimensional signal space which is obtained by frequency analysis and beamforming for horizontal line array received signals, a method for weak signal detection based on trajectory tracking on two-dimensional spectrogram is used instead of tracking in three-dimensional space directly.The influence of noise and interference for trajectory tracking is reduced, especially for the noise with large energy intensity at certain point located outside the curved surface formed by the bearing-time sequence () and frequency axis in the three-dimensional signal space.

Array Signal Model
As shown in Figure1, an underwater horizontal linear array with N elements alone the x axis is used to is used to receive signals.The frequency of the target radiated harmonic signal is  0 .The origin of coordinates (0,0) is located at the center of the array.The target moves at a constant speed of  ⃗ from the initial position ( 0 ,  0 ).Assuming that the sound field is an infinitely space and the receiving array is located in the far field of the signal source, the received signal can be approximated as a plane wave.The signal received by the n-th array element can be written as [9] : (1) In equation (1),  is the signal amplitude,   is the broadband noise with power  2 ,  is the discretetime sequence number,  is the distance between the array elements,   is the sampling rate,  is the incident azimuth angle of the target signal,  0 is the initial phase of line spectrum signal and  is the sound velocity in water. .Schematic diagram of the curved surface corresponding to the azimuth sequence Perform short-time Fourier transform and beamforming on the signal received by the array, assuming that each time window contains  sampling points, the FRAZ spectra of the data for the k-th time period are obtained as follows [9]   (, ) = In equation (2),  is the possible frequency of the line-spectrum, and  is the possible azimuth of target.When the assumed values of  and  match the true received signal frequency and target azimuth at time , the power spectrum value   is the maximum.

The time-frequency spectrogram corresponding to the azimuth sequence
The dot in figure 2 shows the distribution of the trajectory points of the moving target signal in the space-time-frequency three-dimensional space.The dotted line indicates that the signal points are projected into the frequency-azimuth angle (FRAZ) spectrum, and the noise is randomly distributed in the three-dimensional space.The LOFAR spectrum and Bearing-Time Recording obtained by conventional array signal processing can be regarded as the accumulation of a certain dimension of three-dimensional space or a slice perpendicular to a certain dimension.For example, the Bearing-Time Recording can be regarded as the accumulation of three-dimensional matrix along the frequency axis, which accumulates the energy points in the full frequency range of the three-dimensional signal space not only include signals, but also a lot of noise.The LOFAR spectrum of a single beam or the Bearing-Time Recording of a single frequency point is equivalent to taking a slice of the three-dimensional matrix perpendicular to the azimuth or the frequency axis, such as the square frame in figure 2. But when the target movement causes changes in frequency and azimuth, the signal only stays for a short time on such a slice.Therefore, traditional signal processing methods are not suitable for moving targets.To solve this problem, this article uses a parameter optimization algorithm to search four parameters of the moving target, namely the initial distance  0 from the target to the array, the initial azimuth  0 of the target relative to the array ， and the velocity components   and   in the x-axis and y-axis directions for the target.Calculate the azimuth value () at each moment according to equation ( 3 (3)

Particle swarm optimization algorithm
The particle swarm optimization (PSO) algorithm is designed by simulating the predatory behavior of bird swarms.It designs a massless particle to simulate the birds in the flock, each particle has two properties: velocity and position.Each particle searches for the optimal solution in the parameter space separately, records it as the current individual extreme value, and shares it with other particles to find the current global optimal solution.All particles in the particle swarm adjust their speed and position according to the current individual extreme value and the current global optimal solution.In this paper, the parameter space consists of four target motion parameters: the initial distance  0 , the initial azimuth angle of the target relative to the array  0 , the velocity components   and   in the x and y axes for the target.The cost function is defined as the accumulation of the energy of the trajectory points correctly tracked in the LOFAR spectrum.

Hidden markov model for line-spectrum tracking
Hidden Markov model (HMM) is a dynamic Bayesian network generation model with the simplest structure, which can be used to describe a Markov process with hidden unknown parameters.The timefrequency diagram corresponding to the azimuth sequence obtained in Section 2.2 can be referred to as the spatially filtered LOFAR spectrum, which is equivalent to the ordinary LOFAR spectrum.It can show the time-varying characteristics of the line spectrum, and the change of the line spectrum frequency state can be modeled as a 1-D Markov model.
The hidden state in the model is frequency, and the elements in the observation matrix are normalized power spectrum values.Let the initial state vector obey uniform distribution.The transition probability matrix is determined by the discrete-time model satisfied by the hidden state.Since the low-frequency line spectrum of the underwater target often has a small change rate, the state transition at adjacent times approximately satisfies  +1 =   +   .  is a zero-mean Gaussian white noise satisfying (0,   ),   is determined by the stability of the line spectrum frequency.Therefore, the probability of the frequency point transferring from   to   is: After normalizing the   , the elements in the transition probability matrix A can be obtained.
Based on the HMM model parameters set above, we use the Viterbi algorithm to track the trajectory on LOFAR spectrum.Finally, the variance of the distance between two adjacent time points of the line spectrum trajectory obtained by tracking is calculated.If it is less than the set value, it is considered to be correctly tracked.

Major steps
The following steps are given to implement the algorithm: Step 1: Set the window length  0 and the step length ∆ to segment the received array signal.
Step 2：Perform Fourier transform on the time-domain signal obtained in step 1.
Step 3: According to the four parameters of the target motion (  ,  , 0 , 0 ), the azimuth angle α of the target relative to the array at each time is calculated, and the frequency domain beamforming of each signal in step 2 is performed according to the azimuth sequence to realize the spatial filtering.
Step 4: The spectrum of the multi-segment signal obtained in step 3 is spliced in chronological order to obtain the LOFAR spectrum.Step 5: The hidden Markov model is established with the frequency as the hidden state and the power spectrum value as the observation value, and the Viterbi algorithm is used to track the line spectrum in LOFAR spectrum obtained in step 4.
Step 6: Calculate the variance of the distance between the two adjacent points of the tracked trajectory point to determine whether it is correctly tracked.
Step 7: By accumulating all the energy values (power spectrum values) on the line spectrum that are correctly tracked, it is considered as the value of the cost function corresponding to the four parameters in the particle swarm optimization algorithm, and iteratively calculated until the algorithm converges.
Step 8: According to the four parameters obtained by the parameter optimization method, the trajectory of the target motion is calculated.

Simulation verification
In this paper, a continuous single-frequency signal radiated by a moving source with constant velocity   = 4.70/，  = 1.71/ is simulated, and the frequency of the transmitted signal is  0 = 303.The initial distance  0 between the source and the 27-element receiving array is 7280.1m,and the initial azimuth angle  0 is 164.05 °.The total duration of the received signal is 40 minutes, and the spectral level SNR is -23 dB after adding Gaussian white noise.
Figure 3 shows the LOFAR spectrum obtained by short-time Fourier transform after stacking 27 received signals, while Figure 4 shows the bearing time recording (BTR) with a 2Hz bandwidth.It can be observed that under the given signal-to-noise ratio conditions, the signal cannot be seen on the traditional LOFAR spectrum and BTR.Using the algorithm proposed in this paper, the PSO algorithm searches the initial distance between 6000 and 8000 m, searches the initial azimuth between 0° and 180°, and searches the y-axis component and x-axis component of the speed from 1 to 10 m / s.The frequency range of LOFAR plot is set to 301-305Hz.As shown in Table 1, the results of particle swarm optimization parameter search are   = 4.69/,   = 1.71/,  0 = 7154.01, 0 = 163.33° , which are basically consistent with the real values.Figure 5 is the time-frequency spectrum corresponding to this set of parameters, and one spectra line can be seen clearly in it.Figure 6 shows the trajectory tracked on the time-frequency spectrum.The circles are the estimated trajectory point, and the dotted lines are the real trajectory lines.For comparison, the cross shaped points in figure 4 are the result of establishing a two-dimensional hidden Markov model for tracking in the multi-frame FRAZ spectrum in the frequency range of 302Hz-304Hz.It can be seen that, in contrast, the trajectory tracked by the method proposed in this paper is basically accurate, and the performance is not good when tracking directly in the multi-frame FRAZ spectrum [9].

Conclusion
In this paper, a track-before-detect method combining particle swarm optimization algorithm and HMM for line spectrum tracking is proposed.A LOFAR spectrum is obtained with the spectrums at each moment corresponding to the target azimuth calculated with a set of assumed initial target motion parameters.Then a 1-D HMM is established to track line spectrum for each LOFAR spectrum.The energy accumulation of line spectrum is used as the value of cost function of PSO algorithm to estimate the target motion parameters.It is shown that this method is adapt to passive sonar signal detection under low SNR because of its long-time accumulation of line spectrum energy.

Figure 1 .
Figure 1.Diagram of linear array signal emitted by the target Figure 2. Schematic diagram of the curved surface corresponding to the azimuth sequence Perform short-time Fourier transform and beamforming on the signal received by the array,assuming that each time window contains  sampling points, the FRAZ spectra of the data for the k-th time period are obtained as follows[9]   (, ) = the curved surface formed by the curved line () in figure2and entire frequency axis  is obtained from the 3-D signal space.()=   0  0 +    0  0 +

Table 1 .
The comparison and relative error between the estimated values and the real values of the four parameters.