Compressive sensing of complex-valued data using Gaussian entropy

In this paper, we propose an effective compressive sensing algorithm based on Gaussian entropy for complex-data. Compared with the traditional mean squared error (MSE) method, we consider the full second order statistics information of Gaussian noise in the new algorithm, including relevant information and conjugate information, which makes the recovered signal closer to the original input signal. Simulation results of the synthesized 1D signal and 2D signal show that the proposed algorithm has better performance than the MSE method.


Introduction
Compressive sensing is an effective theory of signal processing.For sparse signal, the sampling rate is much lower than the Nyquist sampling rate.Compressive sensing is based on the signal sparsity principle, it projects the high-dimensional signal onto the low-dimensional space to realize the signal compression sampling, and finally the signal is reconstructed by the optimized methods [1][2][3][4][5][6][7][8][9].
Nowadays, the research on real-valued signals in the field of compressive sensing has made great progress.However, there are complex-valued signal (20Wang et al., 2018) in many aspects, such as terahertz (THz) imaging [10], synthetic aperture radar (SAR) [11], sonar (SAS) [12], magnetic resonance imaging (MRI) [13], aeronautical navigation and so on.Therefore, it is necessary to study the algorithm that reconstruct the compressively sensed signals for complex domain.Recently [14], proposed a minimized lp-norm recovery algorithm.For complex data, the algorithm can smooth the amplitude and phase data.It was shown by [15] that an algorithm for reconstructing complex terahertz data based on BFGS [16].presented a strategy of reconstructing complex data by separating the real and imaginary parts.These documents use the mean squared error (MSE) method as a cost function that measures the performance of algorithm.However, the Gaussian noise signal includes the circular signal and non-circular signal for the compressive sensing of complex domains.Most traditional signal processing algorithms assume that the signal has second-order circularity.These algorithms use the MSE criteria and only include the correlation function information of the Gaussian noise signal, which is optimal for the circular Gaussian noise signal [17][18] proposed the method by using Gaussian entropy, it took into account the full second-order statistical information of the error signal.It improved the performance of the linear and widely linear filter.In this paper, we propose a new and effective method to reconstruct the complex-valued signal based on Gaussian entropy.We consider the correlation function information and the conjugate correlation function information of the non-circular signal to achieve better reconstructed performance.
This paper is organized as follows.In Section 2, we present the basic theory of compressive sensing.In Section 3, the traditional method of MSE Complex-valued Compressive Sensing (MCCS) and the new method of Gaussian Entropy Complex-valued Compressive Sensing (ECCS) are described.The numerical simulation results for the synthesized 1D data and 2D data are described and analyzed in Section 4. Conclusions are given in Section 5.

Basics of compressive sensing
 be a signal vector, when it can use an orthonormal basis matrix , where 0 || || x means the numbers of nonzero elements in x .Using a measurement matrix to linearly transform , where  is not related to  , where M<<N.Thus, measurement vector can be taken as: where   A ,  represents the possible noise.Since M << N, the equation can not be directly solved, on the premise that the signal  can be compressed, solving the problem of ill-conditioned equations is converted to be a minimum 0 l -norm problem: so the sparse coefficient vector x can be obtained.However, the solution to minimum 0 l -norm is a Non-deterministic Polynomial (NP) hard problem, the above solution can be replaced by the minimum 1 l -norm problem [2] : l -norm minimization is a convex optimization problem, the expression (3) can be transformed into a linear programming problem.Such algorithms include gradient projection (GP) method [16], basic pursuit (BP) method [19], least angle regression (LARS) method [20] and so on.

Method
In this section, we study two algorithms based on MCCS and ECCS for reconstructing the compressively sensed complex data.In complex domain, we consider the presence of noise, the problem of 1 l -norm minimization is defined as:

MCCS algorithm
First, we use the classical MSE method to get the optimal solution of equation (7).For the MSE algorithm, the objective function can be written as: . In this paper, we solve the errors of M observation samples separately, where ) is a row vector, so we can get a set of cost function: Derive the above equation, so we have where E denotes expectation, the superscript H denotes Hermitian transpose, and Here, note that is not true in the complex domain.In order to make the signal recovery error smaller, we use the following formula to update x : where 0 ,    .

ECCS algorithm
The use of MSE method is limited to only the circular error signal, this method can get the optimal solution.However, the use of entropy method can solve many problems, even when the error signal is non-circular.In this section, the ECCS algorithm base on entropy criteria can exploit the full of secondorder statics of the error signal.Here, we introduce the entropy.
For complex-valued random variable , the probability density function (pdf) of z is defined as Where let R z and I z denoted by the real and imaginary parts of the random variable z , 1   i , and E represents expectation.
In this paper, we assume that a complex random variable z satisfies Gaussian distribution and has zero mean, so the pdf of z is written as: } 2 Therefore, we obtain the entropy of z: , z is circular.Otherwise, z is non-circular.Thus the  is used to measure the degree of second-order non-circularity, where |  is larger, the degree of non-circular is greater.
Through the above description, we can get the another objective function of compressive sensing: From the above cost function, we can easily derive the gradient of ( 16) with respect to x : According to the Newton variant updates, we can make the reconstructed signal closer to the original signal: where we can put it in the step size .Thus, the equation ( 18) can be written as:

Simulation results
In this section, we apply the synthetic 1D data and 2D data to compare the two algorithms of MCCS and ECCS.We can use the signal-to-noise ratio (SNR) to deduce that the performance of the compression sensing algorithm, the SNR is calculated as follows: Where f represents the original signal, f ˆrepresents the recovered signal.).We also show the simulation results of 64 64  2D synthesized data [18], which are mainly composed of amplitude and phase, with the middle part of the amplitude being 0.5, the other part being 1, the middle part of the phase being π/3, and the other value of phase is π/6.As shown in Figure 5 and Figure 6, we can see that the data is the original data (Figure 5 and Figure 6, left), the data recovered from the MCCS method (Figure 5 and Figure 6, middle) and the ECCS method on the right.The top of Figure 5 and Figure 6 represent the real part and the bottom represent the imaginary part.As with 1D data, the simulation results of 2D data also denote that the performance of ECCS is superior to that of MCCS when | |  is not zero.(The SNR of MCCS is 36.67 dB, the SNR of ECCS is 41.59 dB).
In the case of different input SNR, where the input SNR varies from 0.03 dB to 19.89 dB, we compared the performance of the two methods for 1D data.As shown in Table 1, we can conclude that the ECCS algorithm performance is better than the MCCS algorithm under different input SNR.
Table 1.The two methods using different input SNR.

Conclusions
In this paper, we have proposed a new cost function based on Gaussian entropy for compressed sensing of complex-valued data.We have shown and compared the simulation results for 1D and 2D synthetic data.It has been demonstrated that using the Gaussian entropy make the ECCS method have a better reconstructed performance when the | |  is not zero.We also conclude that the ECCS algorithm performance is better than the MCCS algorithm under different input SNR.Therefore, the ECCS algorithm that we proposed for compressed sensing of complex-valued data can be applied to SAR, THz, MRI and Aeronautical Navigation etc.

Firstly, the.
simulation results of different values of | |  for 1D data is shown in Figure1and Figure2, where the blue line is the original signal, which is a synthesized cosine signal Number of iterations is fixed at 128000.As shown in Figure1and Figure2, the green line consisting of plus is the signal recovered by the MCCS method, the red line consisting of points is the signal recovered by the ECCS method.