Kalman filtering used for passive synthetic aperture

In practice, beamforming-based bearing estimators typically fail to account for the impact of environmental factors on the estimating results, resulting in the so-called mismatch problem. To deal with this issue, this paper reconstructs the bearing estimation as a Kalman filter problem, which emphasizes the role of physical models in underwater acoustic signal processing. Since passive synthetic aperture (PSA) is a spatial process including the temporal evolution of a moving array, it fits a Kalman filtering structure. Kalman filter is applied directly to a lake-test dataset collected by an autonomous underwater vehicle (AUV) shell-mounted array, and the experimental findings demonstrate that the bearing estimation results’ variation is significantly decreased than the conventional beamforming (CBF).


Introduction
Passive synthetic aperture is a fundamental topic in sonar development, as the aperture is always one of the most crucial performance parameters of sonar.Williams [3] completed the first passive synthetic aperture test at sea.Most of the traditional passive synthetic aperture algorithms [4]- [5] are based on the beamformer structure, which seeks to match the data to the model.We always strive to achieve a match, but we often come across a mismatch due to our limited access to complete and accurate information about the sound field parameters.Candy et al. [6]- [8] looked into a model-based solution for underwater acoustic signal processing to deal with this issue.In recent years, Kalman filtering based on Bayesian theory has been widely used in target-bearing estimation [9]- [13] .
To deal with the mismatch problem, we need to additionally introduce priori information to improve processor performance.But how do we introduce the prior knowledge when it is in the form of a physical model?One specific implementation is to embed the physical model inside a Kalman filter, which can smoothly and self-consistently include both the procedure of propagation and the model of the signal into the signal processing framework.For the bearing estimation problem, we usually assume that the propagation model satisfies a plane-wave model, an assumption that is also implicit in the steering vectors of conventional beamforming.We would like to emphasize the dependence on the model for the bearing estimation problem and hope to exploit the intrinsic information included in the model using the Kalman filter.The following discussion focuses on the application of Kalman filtering to the plane wave-bearing estimation problem.

Time domain model
The array's spatial mobility and the sinusoidal signal's functional shape make up the two components of the model in this instance [1]- [2] .The received narrowband signal model of a moving array is shown in Figure 1. is the direction of the incoming wave.v is the speed of the motion and d is the spacing of the array elements.Assuming that the reference is the signal at the first element at the origin, the narrowband signal that the th n element received is given by: ( ) cos( ( )) , where the phase () (1) where 0 f is the source frequency and  Kalman filter contains the measurement equation as well as the state equation.When dealing with stochastic cases, the Gauss-Markov process is utilized to characterize the temporal evolution since it can incorporate statistical models of second order for both process and measurement noise.Measurement noise represents the noise contained in the received signal and the process noise represents the difference between the modeling model and the physical model.Kalman filter is the optimal choice when the measurement and the state models are both linear and the measurement and system noises are Gaussian.However, most underwater acoustic signal processing problems are nonlinear, in which case the Extended Kalman Filter (EKF) and the Unscented Kalman Filter (UKF) should be used.
The set of outputs from the array's receiver elements serves as the measurement model for the bearing estimation problem.Given the name   where a is the sound pressure amplitude and ( ) ( ) ( ) is defined, with n being the element index.
The corresponding state vector is and its corresponding state equation is: where ( ) indicates that an estimate of the situation at t is made using the data available up to time 1 t − (priori estimation), and ( ) indicates that an estimate of the situation at 1 t − is made using the data available up to time 1 t − (posteriori estimation).In addition, since the model of the bearing estimation problem is fully included in the measurement equations, in contrast to most Kalman filter configurations, the state transfer matrix A is a unitary matrix, which operates under the presumption that the state vector gradually varies over time.

Frequency domain model
The number of parameters in Equation ( 3) is three instead of two because the model here is operating in the time domain.It is possible to avoid using the sound pressure amplitude as a redundant parameter by building a model that operates in the frequency domain.However, this would require an additional phase extraction operation, which may be problematic in the case of low signal-to-noise ratios.We have to preprocess the data because the measurement equations cannot be constructed directly from the received signal.Now we consider the narrowband case operating in the frequency domain with the same received signal model as Figure 1.The complex form of the signal at the nth receiver is given by: ) There are just two parameters left for the state vector to be evaluated because we have decided to operate in the frequency domain.The Kalman filter's state equation is provided by (5) We focus on phase difference rather than phase itself, for the N elemental line array with only 1 N − measurements.Using the model mentioned above, the phase difference measurements can be written as follows:  ( ) In conclusion, the measurement equation is represented by:

Lake-test results
The experiment was carried out in Qiandao Lake, and the platform for data collection was an AUV with a six-element line array fixed outside the housing with an element spacing of 0.375m d = . The sailing speed v of the AUV was 1.5m/s .The sampling frequency s f was 52734Hz and the source frequency 0 f was 2kHz .The source was stationary and suspended in the lake test, emitting a CW signal.The AUV moves autonomously in the lake in a straight line with a uniform round trip speed and a single trip length of 100 m (Figure 2), forming a synthetic aperture.The acoustic data collected by the array was used for passive synthetic aperture processing and bearing estimation of the source.We opt to operate in the frequency domain, so we need to preprocess the received data.First of all, the collected data will be segmented.Then, the phase and frequency of the segmented signal will be extracted using the FFT to obtain the measurements required by the model.Finally, we apply the UKF to the preprocessed data to obtain the estimated values of the frequency and bearing of the sound source.
Figure 3 displays specific outcomes.Figure 3(a) displays the data's beamforming results using a conventional frequency-domain beamformer.

Conclusions
The enhancement of the performance of bearing estimation of signals can be achieved by Modelling passive synthetic aperture as a Kalman filtering problem.This explicitly accounts for the array's movement in the received signal model, evolving matched filtering to Kalman filtering.PSA naturally fits a Kalman filter structure because it is a spatial process containing the time evolution of the motion array -the time evolution echoes the iterative updating of the data, and the spatial process of the motion array corresponds to a physical model.The results also validate the effectiveness of Kalman filtering for the source frequency and bearing estimation.It can be consistently compensated in principle by measuring the deviation of the noise from the plane wave signal and the aberrations due to the motion of the array.This approach places no restrictions on the model chosen by the processor and can be easily extended to more complex models like spherical wave propagation models and situations involving non-uniformly distributed arrays.Since it doesn't significantly form a beamformer structure, it permits the estimation to be accurate without a lower limit.
elements, both of which provide the necessary bearing data.Conventional beamforming-based PSA algorithms use the spatial phase difference for bearing estimation and ignore the information contained in the Doppler.In the following, we will demonstrate how to employ Doppler's bearing information to improve bearing estimation performance by Kalman filtering.


is the phase difference extracted from the th n received signal.A second auxiliary measuring equation depending on the received frequency is also present.The Doppler relation provides it:( )

Figure 3 (
b) shows the maximum values in the bearing history plot, and Figure3(c) and (d) give the Kalman filtering results.Both source frequency and bearing estimates from Kalman filtering converge gradually to the true values during the iterative process.There is a deviation of about 0.2 Hz in the source frequency, which may have something to do with the length of the data segmentation.The variance of the bearing estimation results is significantly better than that of the conventional method, which is caused by the additional introduction of Doppler information.

Figure 3 .
Figure 3. Lake-test results: (a) CBF bearing history map; (b) CBF bearing estimation results; (c) Kalman filtering sound source frequency estimation results; (d) Kalman filter bearing estimation results.