A Method Reconstruct Ballistic Target Based on Compression Sensing

In order to solve the problem of large data volume and complex processing in the processing of ballistic target echo signal, compressing sensing is applied to the processing of ballistic target echo signal. Firstly, the mathematical model of the cone-cylinder ballistic target’s precession is constructed, the signal processing when the radar radiates sparse Orthogonal Frequency Division Multiplexing-Linear Frequency (OFDM-LFM) signal is analyzed, and then match this process with compressed sensing decision branch; a new denoising recovery algorithm for signal-to-noise ratio(SNR) based on (SAMP) algorithm was proposed, and finally verify the effectiveness of the algorithm through simulation experiments.


Introduction
With the rapid development of ballistic technology, ballistic target has become a direct threat to national security. In recent years, the attack-defence confrontation of ballistic targets has become increasingly fierce, and the signal processing of ballistic targets has attracted more and more attention [1] .
In the current era of big data, the signal becomes more and more complex, and the difficulty of data processing, transmission and storage is greatly increased. The traditional Nyquist sampling will face great obstacles [2] . Compressed Sensing [3][4][5] (CS), a new data processing method, breaks through these obstacles. CS theory points out that if the signal is sparse, a sparse representation of the signal can be projected from a high-dimensional space to a low-dimensional space using an observation matrix unrelated to the sparse dictionary basis, and then the high-dimensional original signal can be reconstructed from a small amount of data by the algorithm.
The use of Orthogonal Frequency Division Multiplexing [6] (OFDM), which is widely used in the field of communications, to modulate traditional LFM signals produces a new signalling system, the OFMD-LFM signal. The OFMD-LFM signal system combines the advantages of both OFDM and LFM signals. As a multi-carrier modulation system, it is often used in MIMO radar systems in radar; it simultaneously transmits multiple mutually orthogonal chirped subcarrier frequencies, each with a large time-width bandwidth product, which can be realized simultaneously.
In order to solve the above shortcomings, the OFDM-LFM signal system is first introduced into the MIMO radar system. Then, based on the current situation of complex and variable battlefield environment and serious electromagnetic interference, a compressed sensing method for low SNR is proposed.

The model of Ballistic target
In order to adapt to the signal form, MIMO radar system is adopted in this paper, including M respectively. The geometry diagram of the precession of the cone-cylinder ballistic target of the MIMO radar is shown in Figure 1.
The distances from the canter of mass O to the m-th radar transmitting unit and the n-th radar receiving unit are Tm R and Rn R , respectively. The cone height is 1 h ,the cylinder height is 2 h , the bottom radius is r , and the half cone angle is  . The distance between the canter of mass O and '' O is a and the Z axis is the precession axis. The angle between the line of sight of the m-th radar transmitting unit and the cone rotating shaft is By constructing an observation vector , the observed signal can be obtained.
Among them, T =  is denoted as CS information operator. Reconstructing the original signal x from the observed signal y using a corresponding algorithm essentially solves a zero norm problem, It can be seen that this is a NP hard − problem because of MN  . Noise is inevitably introduced during signal observation and measurement. The basic principle of compressed sensing in the noise background is studied below.
In the noise background, you can rewrite (5) as: Among them, n is the signal noise. Rewrite (9) as: y x w n w T e  =  + =  +  + = + Where n is the measurement noise and e n w =  + is the total noise, and e can be regarded as a zero-mean Gaussian process.
When noise is mixed into the signal, it will destroy the original sparseness of the signal and turn the sparse signal into an approximately sparse signal [8] . At this time, the measurement matrix is used to observe the signal (the observation process is similar to a whitening process), and the energy of the signal and the noise are aliased, so that the useful signal and the interference noise cannot be distinguished, ultimately, the influence of noise will be magnified NM times. This phenomenon is called the noise folding [9] (NF) effect.

Sparse OFDM-LFM Signal System
Suppose that one of the MIMO radar transceiver systems transmits P-cluster sparse OFDM-LFM pulse trains, each of which contains N sub-pulses, in which the expression of the i-path sub-pulse in the sparse OFDM-LFM pulse trains of cluster m is as follows: Where k t is the fast time,  is the frequency modulation slope, q is an arbitrary natural number, and the frequency difference between the adjacent two subcarriers can be expressed as qT f = . For OFDM signals, orthogonality is satisfied between signals between the same cluster of bursts.
Where  is the electromagnetic scatter coefficient of the target and 2Rc  = is the echo delay time of the target.
Set the reference signal as Where 00 2Rc  = is the echo delay time of the reference point. Through "Dechirp" processing: After LPF filtering, the obtained result is obtained by performing FFT in the fast time domain and eliminating the RVP term: Where 12 , xx are constants. This process can also be written as follows.

Denoising algorithm based on adaptive threshold
The traditional algorithm gives a threshold  , and when the residual is less than  , the iteration is terminated. However, when noise is introduced into the radar signal, the situation becomes complicated, and the fixed threshold cannot assume the responsibility of eliminating the influence of noise.
This paper proposes an algorithm-based adaptive threshold denoising algorithm, which calculates Step8: Output: Reconstructed signal x . 32 sub-pulses are randomly selected from the 64 sub-pulses of the (1, 1) transceiver element, and the rate of reduction is 50%. First simulate with a signal-to-noise ratio of 10dB.  Fig. 2 and Fig. 3 are high resolution range images and ISAR images of the original radar echo signals, respectively; Fig. 4 is a two-dimensional ISAR image of the signal reconstructed by the OMP algorithm; Fig. 5 is reconstructed by using the denoising algorithm proposed in this paper. The ISAR image of the signal. It can be seen that in the case of high SNR, both the OMP algorithm and the denoising algorithm can reconstruct the original signal better and obtain a clearer two-dimensional ISAR image.

Simulation Analysis
The simulation is performed below with a signal-to-noise ratio of -10 db.  Fig. 9 ISAR image of the reconstructed signal by denoising algorithm Fig. 6 and Fig. 7 are high resolution range images and ISAR images of the original radar echo signals, respectively; Fig. 8 is a two-dimensional ISAR image of the signals reconstructed by the OMP algorithm; Fig. 9 is respectively using the denoising algorithm proposed in this paper. A two-dimensional ISAR image of the constructed signal. It can be seen that in the case of lower signal-to-noise ratio, the OMP algorithm has been seriously interfered by noise, and it is impossible to distinguish the useful signal and noise. The obtained two-dimensional ISAR is disorganized. However, the denoising algorithm proposed in this paper can suppress it better. Noise interference, more accurate extraction of useful signals, the two equivalent scattering points of the ballistic target can still be seen from the two-dimensional ISAR image.
Next, the normalized mean square error of the OMP algorithm and the denoising algorithm under the same rate reduction and different SNR conditions are compared. The normalized mean square error is defined as: 2 2 In the signal-to-noise ratio (-10dB, 10dB) between 2dB intervals, 100 times of Monte Carlo simulation under each SNR condition, the noise uses Gaussian white noise, the simulation results are shown below. Fig. 10 Comparison between denoising algorithm and OMP algorithm It can be seen that the reconstruction performance of OMP algorithm and the denoising algorithm proposed in this paper is ideal in the case of high signal-to-noise ratio. But when the signal-to-noise ratio is low, especially when the signal-to-noise ratio is less than 0 db, the reconstruction effect of OMP algorithm decreases sharply, and there will be many wrong sparse solutions, while the denoising algorithm can still restore useful information in the signal.

Conclusion
In this paper, the conical cylindrical ballistic target is mathematically modeled, and the compressed sensing theory of noisy signals is analyzed. Then the sparse OFDM-LFM signal system is analyzed, and the signal system is deduced in the echo processing process. The sparse dictionary base and observation matrix corresponding to the compressed sensing are obtained from the specific process. Then an algorithm-based adaptive threshold denoising algorithm is proposed for the signal folding effect in compressed sensing. Finally, the simulation results show that the algorithm has better performance. Anti-noise performance, can extract useful information in the original signal at a lower signal to noise ratio.