High resolution spectrum estimation of virtual array based on FIM

Underwater small aperture sonar spectral estimation methods are limited by physical dimensions, resulting in insufficient azimuthal resolving power. In this paper, we propose that based on the Fourier Integral Method(FIM) equivalent to a virtual array of extended aperture, after using subarray smoothing, combined with the spectral estimation method, we get the improved FIM-CBF, FIM-Capon and FIM-MUSIC, and give the applicable conditions of these three improved methods, and use the virtual aperture expansion and noise averaging suppression, to achieve the improvement of the azimuthal resolving power of the spectral estimation method under the low SNR.


Introduction
Imaging sonar is often used to measure the shape and position of underwater objects, and azimuth resolution is an important performance indicator.The azimuth resolution is affected by the array aperture.As the array aperture increases, the azimuth resolution also increases.However, in reality, due to the physical size of the platform, the aperture of the sonar array cannot be expanded indefinitely, which leads to limited azimuth resolution.In order to improve the azimuth resolution and suppress noise and interference, researchers have proposed some classical DOA estimation methods, such as CBF, high resolution method Capon[1] , MUSIC [2] ,etc.Based on these spectral estimations, some processing methods were proposed, such as the MIMO sonar virtual array method [3], which obtained a large virtual array through special array deployment, and Fourier Integral Method (FIM), Toeplitz averaging and reconstruction based on covariance matrix [4][5][6], etc.These methods can improve the azimuth resolution and noise suppression ability to a certain extent, but the improvement effect is limited by the physical aperture.
The aim of this study is to discuss the spectral estimation problem after the virtual array based on FIM.After the covariance matrix Toeplitz average of the data is equivalent to a virtual array element, it can be combined with a variety of spectral estimation methods through smoothing processing.The applicable conditions were given to further improve the azimuth resolution under lower signal-to-noise ratio and compare the resolution of various processing methods.

FIM principal process
It is assumed that there is a linear array, N receiving array elements, the array element spacing is d, the center frequency of the narrowband signal is 0 f , c is the underwater sound velocity, and the signal In (2.2), p s denoted the 1 L  dimensional signal vector of the narrowband signal radiated by the p th target, L is the number of snapshots, X is the NL  dimensional matrix, and the array manifold  is the incidence angle of the pth target relative to the normal direction of the receiving array, Z is NL  dimensional noise matrix, and each row in the noise matrix is orthogonal to each other.Two receiving array elements were selected in turn from N receiving array elements for crosscorrelation, and 2 N cross-correlation output can be obtained, namely: and 2 N cross-correlation outputs can be considered as the signals received by 2 N virtual array elements.For the ith and jth receiving array elements, the cross- correlation is: element, and there were Nm  kinds of ways to take i and j.From equations (2.3) and (2.4), it can be seen that there are 2N-1 values of m, which means there are 2N-1 positions of virtual array elements.For each position, there are multiple cross-correlation output overlaps, which can be considered as a copy of the received signal for each position with a plurality of virtual array elements.
The 2N-1 element output can be obtained by averaging the cross-correlation output of each overlapping position, that is, Toeplitz averaging.The averaged 2N-1 outputs can be regarded as received by 2N-1 virtual array elements [6][7].At this time, the received signal for each virtual receiving element is: When The value of q is 1-N, 2-N... 0,....., N-1.Average the number of 2 N to obtain 2N-1 numbers to remove the overlap get 2N-1 output.And then equivalent to 2N-1 virtual array element receiving, greatly reduces the influence of interference, obtained in low signal-to-noise ratio under the good detection effect.At this time,   p a  becomes: (2.7)

Subarray smoothing processing
Because of the cross-correlation and Toeplitz average, the 2N-1 outputs formed are cross spectral average outputs.The waveform information of the incident signal has been lost, and only the phaseshift component of the signal is left.The signals obtained are coherent.To avoid affecting subsequent spectral estimation processing, spatial smoothing is necessary.For them, the coherent processing of the smooth part of the sub array is adopted.2N-M subarrays are divided, and the number of elements of each subarray is M, then the covariance matrix after the smoothing of the subarray is obtained as follows: At this time, the smoothed covariance matrix is: The corresponding scanning vector is: If the number of subarray elements increases, the number of subarrays decreases, which can easily lead to the failure of decoherence in multi-target situations.If the number of subarray elements decreases, the number of subarrays increases, and it may achieve good decoherence results.However, the main lobe width of the power spectrum widens, while the azimuth resolution decreases and the number of targets that can be resolved decreases [8]. Figure 1 shows the FIM processing flow.

N received signals crosscorrelation
Cross-correlation output i-j the same term for the mean The 2N-1 crossspectral outputs are equivalent to virtual array elements The cross-spectrum output is decoded by Subarray smoothing

Spectral estimation
After Toeplitz averaging and subarray smoothing of the signal sampling covariance matrix of the receiving array, a signal sampling covariance matrix of the virtual array is obtained, and then spectral estimation is performed on the covariance matrix of this virtual array.

FIM-CBF
The average output power of the entire array is: In Eq. (3.1), L is the number of snapshots.When

 
wa   , it is conventional beamforming.

 
a  is the scan vector.

FIM-Capon
The core idea of Capon is to ensure that the signal in a certain direction is normally received, while signals or interference in other incident directions are suppressed.Mathematically, the output power of the array is minimized under the condition that the signal output in the required direction is guaranteed to be a constant [1].For FIM-Capon, after using the processed array sampling covariance matrix equivalent to that received by the virtual array and smoothing and decoherence its subarrays, the obtained covariance matrix is minimized when the signal output is at a fixed value.

HH d
Minimize w Rw subject to w a   (3.3) Bring (2.9) and (2.10) into (3.3) to obtain FIM-Capon, the output power of the array can be derived from the above conditions as:

FIM-MUSIC
MUSIC essentially uses the orthogonality of signal subspace and noise subspace to achieve DOA estimation [2].If the number of sources is not known, the subspaces cannot be effectively separated, and MUSIC fails.When the number of sources is known, the received sampling covariance matrix of the receiving array element is equivalent to the sampling covariance matrix received by the virtual array element after Toeplitz averaging and reconstruction.After its subarray is decohered smoothly, and the smoothed covariance matrix is decomposed to obtain the signal subspace and noise subspace.The orthogonality of the two is used to estimate the direction of arrival.
The characteristic decomposition of the covariance matrix of the received signal is: (3.5) Actually, solving for DOA is implemented as a minimum optimization search: The spectral estimation formula is: (3.7) Bring (2.12) and (2.13) into (3.7) to obtain the spectral estimation formula of FIM-MUSIC :

Numerical simulation and analysis
The sonar array is set as 96-element ULA with a center frequency of 450 kHz and a bandwidth of 20 kHz.The underwater sound velocity is set to 1500 m / s, and the 96-element ULA is arranged in a half-wavelength array.It is assumed that there are two targets, the background is Gaussian white noise, and the signal waveforms radiated by the targets are narrow-band frequency-division sine wave pulses.The sampling frequency is set to 1000 kHz, the number of snapshots L = 9600, and the superposition of signal and noise signals are subjected to band-pass filtering.The 512-order FIR band-pass filter is designed using the Chebyshev approximation principle, with the passband frequencies ranging from 440kHz to 460kHz.The upper and lower limits of the stopband cutoff frequency are 435 kHz and 465 kHz, respectively.The SNR is the ratio of the signal to the full-band noise power.The signal-to-noise ratio is the ratio of the signal to the full-band noise power， and the beam sweep range is from -90° to 90° at 0.1° intervals.
The center frequency of the target 1 signal is 450kHz and the power is 1 10 , the center frequency of the target 2 signal is 440kHz and the power is 1 10 .
When SNR = -15dB, -30dB, the band-pass filtering results are shown in Figure 2   From Figure 3 (a), (b) and (c), it can be seen that FIM has the effect of suppressing noise level.The azimuth resolution of FIM-CBF is higher than that of CBF, which breaks through the Rayleigh limit of CBF relative to physical size.From Figure 3 (a) and (b), it can be seen that the FIM-CBF method can still distinguish targets of 15 ° and 16.5 ° at low SNR compared with CBF, but its ability to improve azimuth resolution is limited by subarray division processing.
From Figure 3 (a) and (c), it can be seen that the effect of improving azimuth resolution is not obvious.Under the same SNR, two targets of 15 ° and 16 ° cannot be distinguished, but from the main lobe width, the azimuth resolution is still higher than the traditional CBF.Compared with CBF, the azimuth resolution of FIM-CBF is not significantly improved, but the anti-noise ability is significantly enhanced, which is suitable for distinguishing targets at low SNR.

Comparison of FIM-Capon Goniometric Performance
Targets 1 and 2 are at 15° and 16°, respectively, with SNR=-15dB and -30dB.After band-pass filtering, they are processed by the FIM-Capon method, as shown in Figure 4(a) and (b).Targets 1 and 2 are at 15° and 15.5° respectively, SNR=-15dB.After band-pass filtering, they are processed by the FIM-Capon method, as shown in Figure 4(c c) that FIM has the effect of suppressing noise level, and the azimuth resolution of FIM-Capon is higher than that of Capon.From Figure 4 (a), (b), it can be seen that the performance of the algorithm decreases significantly with the decrease of the SNR, and it is unable to discriminate the target effectively.It can be seen from Figure 4 (a) and (c) that FIM-Capon has higher azimuth resolution under the same SNR.
Compared with Capon, the anti-noise ability of FIM-Capon is not obviously enhanced, but it has higher azimuth resolution, which is suitable for the situation where the SNR is relatively low and requires high azimuth resolution.The target 1 and 2 are at 15 ° and 15.5 °, respectively, with SNR = -15 dB, -20 dB, -30 dB.After band-pass filtering, the FIM-MUSIC method is processed, as shown in Figure 5 (a), (b), (c).

Performance comparison of FIM-MUSIC angle measurement
As can be seen from Figure 5(a), (b), and (c), the azimuth resolution using FIM-MUSIC is higher than that of MUSIC, and the performance of the algorithm decreases as the SNR decreases.From Figure 5(c), it can still be successfully resolved at low SNR, and the FIM processing improves the noise immunity of MUSIC.

Comprehensive comparison
The target 1 and 2 are at 15 ° and 16 ° respectively, with SNR = -20 dB, as shown in Figure 6  The spectral estimation methods optimized by the FIM method have higher resolution than the traditional spectral estimation method.Among them, FIM-MUSIC has the best performance, but the number of sources needs to be known in advance.It is difficult to realise that the real data can be perfectly divided into signal subspace and noise subspace.FIM-Capon can obtain good azimuth resolution when the SNR reaches a certain condition.FIM-CBF loses a certain resolution after subarray smoothing, but it is still higher than CBF resolution.

Conclusion
A method is proposed for using Toeplitz averaging to form a virtual array and combining it with spectral estimation.This effectively suppresses noise and improves the azimuth resolution of imaging sonar without changing the physical size of the array.Under low signal-to-noise ratio conditions, FIM-CBF can improve azimuth resolution, but the improvement effect is limited, while FIM-Capon significantly improves the azimuth resolution under lower SNR conditions.If the number of sources is known, FIM-MUSIC provides the optimal azimuth resolution.


is the power of the pth incident signal.Let m i j  , for a fixed value of m, as the phase shift component of the signal received by a virtual array

Figure 2 .
Figure 2. Filter processing results under different conditions (a) SNR=-15dB, (b) SNR=-30dB After passing the sampled data from the receiving array through the band-pass filter, the CBF, Capon, MUSIC, FIM-CBF, FIM-Capon and FIM-MUSIC methods are processed respectively, and their performances are compared.4.1.Performance comparison of FIM-CBF angle measurement Targets 1 and 2 are at 15° and 16.5°, respectively, with SNR=-15dB and -30dB.After band-pass filtering, they are processed by the FIM-CBF method, as shown in Figure.3(a) and (b).Targets 1 and 2 are at 15° and 16°, respectively, with SNR=-15dB.After band-pass filtering, they are processed by the FIM-CBF method, as shown in Figure.3(c).