WO2019200664A1 - 一种基于阵列天线的欠定相位恢复方法 - Google Patents
一种基于阵列天线的欠定相位恢复方法 Download PDFInfo
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- the invention relates to an underdetermined phase recovery method based on array antenna, relating to array signal processing and phase recovery intersection theory.
- phase recovery The technique of recovering the complete signal based only on the linear measurement of intensity/amplitude information of the signal, such as the Fourier transform, is commonly referred to as phase recovery.
- Phase recovery techniques are widely used in astronomy, crystallography, optical imaging, microscopy and audio signal processing. In these fields, since the intensity or amplitude of the signal is only recorded when the device acquires the signal, completely recovering the original signal without signal phase information is a challenging subject.
- phase recovery returns the original M-dimensional signal from the amplitude of the N-order linear measurement, and the mathematical model can represent
- n i is the measurement noise
- the phase recovery problem is a nonlinear non-convex problem. It is usually required that the measurement number N is greater than the dimension M of the signal to accurately recover the original signal. In theory, the number of measurements N needs to satisfy at least ⁇ (M log M) to recover the original signal with high probability.
- the phase recovery algorithms are mainly divided into two categories. One is based on the alternating optimal idea to recover the signal.
- the main representative papers include the Gerchberg-Saxton algorithm and its improved algorithm, and the other is based on the semi-determined relaxation method.
- the semi-definite programming technique is mainly used to introduce a new rank-1 variable to indirectly obtain the original signal. But this type of method will bring "matrix-lifting" problems, which will increase the amount of calculation. Recently, scholars P.
- the underdetermined phase recovery problem is often encountered, that is, the number of measurements is smaller than the dimension of the signal.
- signal recovery is usually performed in combination with certain characteristics of the signal, such as increasing sparse constraints or adding penalty terms to the objective function to recover sparse signals, depending on the sparse nature of the signal.
- certain characteristics of the signal such as increasing sparse constraints or adding penalty terms to the objective function to recover sparse signals, depending on the sparse nature of the signal.
- the invention takes the array antenna as the research background, and assumes that in the complex noise environment, the array antenna only measures the amplitude or power of the radar signal, and the phase of the incident desired signal carrier is lost.
- an array-based antenna is designed by using the MM (majorization-minimization) algorithm and the alternating directions method of mutiplier (ADMM). The underdetermined phase recovery method.
- the object of the present invention is to provide an underdetermined phase recovery method based on an array antenna, which can successfully recover the original signal in the case of carrier phase loss.
- Step 1 Establish an incident signal steering vector matrix, and design a phase recovery method model based on the array antenna;
- Step 2 Consider the sparse characteristics of the incident signal space domain and establish a model of the underdetermined phase recovery method
- Step 3 using an MM algorithm to optimize the underdetermined phase recovery model into a convex function
- Step 4 using an alternating direction multiplier algorithm to establish an objective function corresponding to the underdetermined phase recovery model and containing the incident signal parameter;
- Step 5 Using an alternate iterative update method, solving the objective function to recover the original signal.
- the incident signal steering vector matrix described in step 1 is
- N is the number of antenna elements
- M is the number of angular intervals in which an angle region ⁇ is evenly divided.
- ⁇ m is the angle at which the mth signal is incident in the angular region ⁇
- d is the spacing of the antenna elements
- ⁇ is the wavelength of the incident signal.
- step 1 The array antenna based phase recovery method model described in step 1 is
- the spatial signal sparseness characteristic of the incident signal described in step 2 means that the number of incident signals is far less than the number of angular intervals divided by the angle region ⁇ ;
- step 2 refers to the number of antenna array elements being less than the number of angular intervals divided in the angle region ⁇ , that is, N ⁇ M;
- ⁇ is a constant coefficient
- 1 is the L1 norm
- the invention introduces the phase recovery theory in the field of astronomy and image processing into the field of array signal processing, taking the array antenna as the research background, assuming that the array antenna only measures the amplitude or power of the radar signal, and adopts the maximum release estimation method to establish the initial phase recovery. Algorithm model. Then considering the spatial sparseness of the incident signal, an underdetermined phase recovery method based on the array antenna is provided. This method can successfully recover the original incident signal even when the carrier phase is lost.
- 1 is a flow chart showing the steps of the array antenna based underdetermined phase recovery method provided by the present invention
- MSE Mean Square Error
- 3a is a simulation diagram of a recovery signal when the number of iterations is 1 in the method provided by the present invention
- FIG. 3b is a simulation diagram of a recovery signal when the number of iterations is 100 in the method provided by the present invention
- FIG. 3c is a simulation diagram of a recovery signal when the number of iterations is 300 in the method provided by the present invention.
- the underdetermined phase recovery method based on the array antenna of the present invention comprises the following steps:
- Step S1 establishing an incident signal steering vector matrix, and designing a phase recovery method model based on the array antenna;
- Step S2 considering a sparse characteristic of the incident signal space domain, establishing a model of the underdetermined phase recovery method
- Step S3 using an MM algorithm to optimize the underdetermined phase recovery model into a convex function
- Step S4 using an alternating direction multiplier algorithm to establish an objective function corresponding to the underdetermined phase recovery model and containing the incident signal parameter;
- Step S5 Using an alternate iterative update method, solving the target function to recover the original signal.
- Step S1 establishing an incident signal steering vector matrix, and designing a phase recovery method model based on the array antenna;
- the maximum likelihood estimation of the original signal x can be obtained. That is, based on the phase recovery model of the array antenna, the expression is
- Step 2 Consider the sparse characteristics of the incident signal space domain and establish a model of the underdetermined phase recovery method
- the number of antenna elements is usually less than the number of intervals divided into the angle region ⁇ for the convenience of carrying and cost considerations, that is, N ⁇ M.
- the incident signal steering vector matrix A is an underdetermined matrix
- equation (1) is an underdetermined equation.
- the phase recovery model according to equation (3) cannot successfully recover the original signal.
- the underdetermined phase recovery model of the array antenna can be expressed as
- ⁇ is a constant coefficient
- 1 is the L1 norm.
- Step S3 using the MM algorithm to optimize the original non-convex objective function, and selecting a substitution function to ensure that the optimization function is a convex function;
- Equation (4) is a non-convex nonlinear problem, which will be solved by the MM algorithm.
- g_(x) is still a non-convex function. According to the MM algorithm, an alternative function needs to be sought.
- ⁇ denotes the angle operation
- ⁇ denotes the Hadamard product
- Re denotes the real part operation
- Step S4 using the ADMM algorithm idea to design an objective function based on the Lagrangian form
- ⁇ is the augmented Lagrangian multiplier and ⁇ is the penalty parameter.
- Step S5 The original iterative update method is used to restore the original signal.
- the variables x, z, and u are alternately updated.
- the values of the variables z and u at the kth time namely z k and u k
- calculate the updated value x k+1 of the vector k at the k+1th time then calculate the variables z and u at the k+1th time respectively.
- the values, z k+1 and u k+1 are then calculated as the updated value of the vector x at time k+2. And so on, until the set iteration termination condition is met, the loop is terminated. details as follows
- an iterative termination condition is set, which is terminated when the number of iterations satisfies the maximum value K or satisfies the set value of the mean square error (MSE, Mean Square Error) between x k+1 and the original signal. At this time, x k+1 is the original signal recovered.
- MSE Mean Square Error
- the number of array elements is 50, and the spacing of the antenna elements is half the wavelength of the incident signal.
- the number of incident signals is eight, and the corresponding incident angles are -60°, -40°, -20°, 0°, 10°, 40°, 60°, and 70°, and the incident signal is assumed to be Gaussian.
- Set the noise power to 1, and the signal-to-noise ratio is 30dB.
- Figure 2 shows the MSE plot between the recovered signal and the original signal for different iterations. It can be clearly seen from the figure that as the number of iterations increases, the MSE value gradually decreases. When the number of iterations is 300, the MSE value is close to 10 -4 , which proves that the algorithm can effectively recover the original signal in the absence of phase information.
- Fig. 3 shows the effect of the recovery signal distribution when the number of iterations is 1, 100 and 300 respectively.
- the original signal distribution is also shown in the figure. Since the initial value of the assumed recovery signal is a Gaussian random distribution, it can be seen from the figure that when the first iteration is completed, the recovered signal is greatly different from the original signal. When the 100th iteration is completed, the recovered signal gradually approaches the original signal. When the 300th iteration is completed, the recovery signal is basically close to the original signal, which also proves the effectiveness of the algorithm.
- the recovery process of Figure 3 corresponds to the MSE shown in Figure 2.
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Abstract
本发明提供了一种基于阵列天线的欠定相位恢复方法，其以阵列天线为研究背景，考虑天线阵元数目小于信号数目、入射期望信号相位丢失情况下，根据入射信号空域稀疏特性，并且假设在复杂噪声环境下，阵列天线只量测到了入射信号的幅值或功率，采用最大释然估计方法，建立相位恢复算法模型。然后假定入射信号在某个空间角度范围内具有空域特性，采用L1范数方法建立欠定相位恢复算法模型。再依据Majorization-Minimization算法思想，选取替代函数来确保优化函数为凸函数。最后采用交替方向乘子算法，交替更新并恢复原始入射信号。
Description
本发明为一种基于阵列天线的欠定相位恢复方法，涉及阵列信号处理和相位恢复交叉理论。
仅依据信号的线性量测强度/幅值信息，如傅里叶变换，来恢复该完整信号的技术通常称为相位恢复。相位恢复技术广泛应用于天文学、晶体学、光学成像、显微镜和音频信号处理等领域。在这些领域中，由于设备采集信号时，只记录了信号的强度或幅值，因此，在没有信号相位信息的情况下，完整恢复原始信号是一项具有挑战的课题。
从数学角度来讲，相位恢复即从N次线性量测的幅值中，恢复出原始的M维信号，数学模型可以表示
式中，a
i为已知的量测向量，n
i为量测噪声。
相位恢复问题是一个非线性非凸的问题，通常需要量测次数N大于信号的维度M，才能够准确的恢复出原始信号。在理论方面，量测次数N至少需要满足Ο(M log M)才能高概率恢复原始信号。目前相位恢复算法主要分为两类，一类是基于交替最优思想来恢复信号的，主要代表论文有Gerchberg-Saxton算法及其改进算法，另一类是基于半定松弛(semidefinite relaxation)方法，主要采用半定规划技术，引入新的秩为1的变量来间接求取原始信号。但该类方法会带来“matrix-lifting”问题，从而增加计算量。最近，学者P.Netrapalli,P.Jain和S.Sanghavi在文献(Phase retrieval using alternating minimization,IEEE Trans.Signal Process.,vol.63,no.18,pp.4814–4826,Sep.2015)中采用最速下降法，通过设定一个自适应步长，通过迭代思想来进行求解。
在实际应用中，经常会遇到欠定相位恢复问题，即量测次数小于信号的维度。在这种情况下，通常结合信号的某些特点来进行信号恢复，比如依据信号的稀疏 特性，增加稀疏约束条件或给目标函数增加惩罚项来恢复稀疏信号。文献(S.Mukherjee and C.S.Seelamantula,Fienup algorithm with sparsity constraints:Application to frequency-domain optical-coherence tomography,IEEE Transactions on Signal Processing,vol.62,no.18,pp.4659–4672,Sep.2014)中，采用凸l
1范数惩罚思想结合经典的Fienup算法来恢复一组稀疏信号，但是该方法需要知道信号的稀疏度，在实际中均不容易实现。文献(Y.Shechtman,A.Beck,and Y.Eldar,GESPAR:Efficient phase retrieval of sparse signals,IEEE Transactions on Signal Processing,vol.62,no.4,pp.928–938,Feb.2014)采用牛顿梯度迭代方法，根据粗略的稀疏度信息，通过迭代更新就可以求解。但是当稀疏度的值比较大时，该方法依然很难在实际中使用。
本发明以阵列天线为研究背景，假设在复杂噪声环境下，阵列天线只量测到了雷达信号的幅值或功率，此时入射期望信号载波相位丢失。此外，考虑天线阵元数目小于信号数目时，根据入射信号空域稀疏特性，采用MM(majorization-minimization)算法思想和交替方向乘子(ADMM，alternating directions method of mutiplier)算法，设计一种基于阵列天线的欠定相位恢复方法。
发明内容
本发明目的在于提供一种基于阵列天线的欠定相位恢复方法，能够成功实现载波相位丢失情况下原始信号的恢复。
实现本发明目的技术方案：
一种基于阵列天线的欠定相位恢复方法，其中:
步骤1：建立入射信号导向矢量矩阵，设计基于阵列天线的相位恢复方法模型；
步骤2：考虑入射信号空域稀疏特性，建立欠定相位恢复方法模型；
步骤3、利用MM算法对所述欠定相位恢复模型进行优化处理成凸函数；
步骤4、利用交替方向乘子算法，建立与欠定相位恢复模型相对应的含有入射信号参数的目标函数；
步骤5、采用交替迭代更新方法，对所述目标函数进行求解，恢复出原始信号。
具体的：
步骤1中所述的入射信号导向矢量矩阵为
上式中，N为天线阵元数目，M为某个角度区域Φ被均匀分成的角度区间个数，
其中，θ
m为角度区域Φ中第m个信号入射时的角度，d为天线阵元间隔，λ为入射信号波长。
步骤1所述的基于阵列天线的相位恢复方法模型为
上式中，
和||为分别为对向量中每一个元素取平方根和取模值操作，
为原始信号x的最大似然估计，向量x为原始入射信号，
为天线阵列所量测到的幅值信号
的第i个元素；
步骤2中所述的入射信号空域稀疏特性，指的是入射信号数目远少于角度区域Φ内所分成的角度区间个数；
步骤2中所述的欠定，指的是天线阵元数目少于角度区域Φ内所分成的角度区间个数，即N＜M；
步骤2中所述的欠定相位恢复方法模型为
上式中，β为一个常值系数，|| ||
1为L1范数。
本发明具有的有益效果：
本发明将天文学和图像处理领域的相位恢复理论引入阵列信号处理领域，以阵列天线为研究背景，假设阵列天线只量测到了雷达信号的幅值或功率，采用最大释然估计方法，建立初始相位恢复算法模型。然后考虑入射信号的空域稀疏特性，提供一种基于阵列天线的欠定相位恢复方法。该方法能够在载波相位丢失的 情况下，依然可以成功恢复原始入射信号。
图1是本发明所提供的所述基于阵列天线的欠定相位恢复方法的步骤流程图；
图2是本发明所述方法中不同迭代次数恢复信号与原始信号之间的均方误差(MSE，Mean Square Error)曲线图；
图3a是本发明提供的所述方法中迭代次数为1时恢复信号仿真图；
图3b是本发明提供的所述方法中迭代次数为100时恢复信号仿真图；
图3c是本发明提供的所述方法中迭代次数为300时恢复信号仿真图。
为使本发明的目的、技术方案及优点更加清楚、明确，以下参照附图并举实施例对本发明进一步详细说明。应当理解，此处所描述的具体实施例仅仅用于解释本发明，并不用于限定本发明。
如图1所示，本发明基于阵列天线的欠定相位恢复方法包括以下步骤：
步骤S1：建立入射信号导向矢量矩阵，设计基于阵列天线的相位恢复方法模型；
步骤S2：考虑入射信号空域稀疏特性，建立欠定相位恢复方法模型；
步骤S3、利用MM算法对所述欠定相位恢复模型进行优化处理成凸函数；
步骤S4、利用交替方向乘子算法，建立与欠定相位恢复模型相对应的含有入射信号参数的目标函数；
步骤S5、采用交替迭代更新方法，对所述目标函数进行求解，恢复出原始信号。
下面根据具体应用实施例对本发明所提供的方法做详细的说明。
步骤S1：建立入射信号导向矢量矩阵，设计基于阵列天线的相位恢复方法模型；
在本发明中，我们假设知道某角度范围Φ内有P个入射信号，但并不知道信号的入射角。我们将该角度区域Φ均匀分成M个区间，即Φ＝[θ
1，θ
m，…，θ
M]， 其中，θ
m(m＝1，２，…，M)为角度区域Φ中第m个信号入射时的角度。
考虑具有N个天线阵元的线性阵列，假设在噪声环境下，阵列天线量测到了N个幅值，表示为
则天线阵列所量测到的信号
模型表示为
式中，
和||为分别为对向量中每一个元素取平方根和取模值操作，向量x为原始入射信号，向量n为噪声，A为入射信号导向矢量矩阵，表示为
式中，
其中，d为天线阵元间隔，λ为入射信号波长。
依据最大释然估计方法，可以得到原始信号x的最大似然估计
即基于阵列天线的相位恢复模型，表达式为
步骤2：考虑入射信号空域稀疏特性，建立欠定相位恢复方法模型；
在实际情况下，比如小型雷达和无人机卫星导航接收机应用中，为了携带方便和成本考虑，天线阵元数目通常少于角度区域Φ内所分成的区间个数，即N＜M。此时，入射信号导向矢量矩阵A为欠定矩阵，式(1)即为欠定方程，根据式(3)所述的相位恢复模型无法成功恢复原始信号。
此时，假定入射信号数目也小于角度区域Φ内所分成的区间个数，即P＜M，此时入射信号具有空域稀疏特性。因此，阵列天线的欠定相位恢复模型可以表示为
上式中，β为一个常值系数，|| ||
1为L1范数。上述欠定相位恢复模型能够获得一个稀疏的解向量x。
步骤S3：用MM算法思想对原非凸目标函数进行优化处理，选取替代函数来确保优化函数为凸函数；
式(4)是一个非凸非线性问题，下面将采用MM算法思想来求解。
首先，令
然后，考虑去掉外部绝对值，存在如下情况
上式中，g_(x)仍然是非凸函数，根据MM算法思想，需要寻求一个替代函数。
在MM框架中，对于第k次迭代时的向量x
(k)，有
上式中，
定义为
∠表示取角度操作，⊙表示Hadamard乘积，Re表示取实部操作。
由上式可知，
式中，
再令
则g(x)的替代函数可以表示为
因此，通过MM算法思想，原始的相位恢复问题即可以表示为
步骤S4：采用ADMM算法思想，设计基于拉格朗日形式的目标函数；
直接求解上述(11)优化问题，比较困难，下面采用ADMM思想来进行求解。
首先，引入一个辅助向量z＝(z
1，z
2，…z
N)，并令其为
z＝Ax (12)
即z
i＝(Ax)
i，则
可以由
来表示为
式(11)中的欠定相位恢复的模型即可以写为
建立增广拉格朗日形式，即
式中，
λ为增广拉格朗日乘子，ρ为惩罚参数。
步骤S5：采用交替迭代更新方法，恢复原始信号。
在求解优化问题(15)时，采用变量x、z和u交替更新方法。首先需要假定第k时刻变量z和u的值，即z
k和u
k，计算第k+1时刻向量x的更新值x
k+1；然后再分别计算第k+1时刻变量z和u的值，即z
k+1和u
k+1，之后再计算k+2时刻向量x的更新值。以此类推，直到满足设定的迭代终止条件终止循环。具体如下
向量x的迭代更新表示为
在向量z的迭代更新时，为表示方便，首先引入中间变量q＝(q
1，q
2，…q
N)， 令
q＝Ax
k+1+u
k (17)
向量z的迭代更新表达式为
完成x
k+1和z
k+1的更新后，还需要对向量u进行更新，即
u
k+1＝u
k+Ax
k+1-z
k+1 (19)
最后，设置迭代终止条件，当迭代次数满足最大值K时或满足x
k+1与原始信号之间的均方误差(MSE，Mean Square Error)设定值时终止。此时的x
k+1即为恢复的原始信号。
为证明本发明的有效性，进行了仿真验证。
假设相控阵天线采用均匀线阵，阵元数目为50，天线阵元间隔为入射信号半波长。入射信号数目为8个，对应的入射角分别为-60°，-40°,-20°，0°，10°,40°,60°和70°，入射信号假定为高斯分布。设定噪声功率为1，信噪比均为30dB。将整个空域角度Φ均匀分为180个区间，即Φ＝[-90:1:90]。x、z和u的初值均设为高斯随机分布，惩罚参数ρ＝1，迭代步长μ＝0.5。迭代终止条件的最大次数K＝300。
图2给出了在不同迭代次数情况下，所恢复信号与原始信号之间的MSE曲线图。从该图中可以清晰看出，随着迭代次数的增加，MSE值逐渐减低。当迭代次数为300时，MSE值接近10
-4，证明了该算法在缺少相位信息的情况下，依然能够有效恢复出原始信号。
为了显示该算法恢复信号的过程，图3给出了迭代次数分别为1，100和300时的恢复信号分布效果图。为比较方便，图中也给出了原始信号分布。由于假定的恢复信号初始值为高斯随机分布，从图中可以看出，当第1次迭代完成后，恢复信号与原始信号存在很大的差异。当第100次迭代完成后，恢复信号逐渐接近原始信号。当第300次迭代完成后，恢复信号基本接近原始信号，也证明了该算法的有效性。图3恢复过程与图2中显示的MSE相互对应。
Claims (3)
- 本发明涉及一种基于阵列天线的欠定相位恢复方法，其特征在于:步骤1、建立入射信号导向矢量矩阵，设计基于阵列天线的相位恢复方法模型；步骤2、考虑入射信号空域稀疏特性，建立欠定相位恢复方法模型；步骤3、利用Majorization-Minimization算法对所述欠定相位恢复模型进行优化处理成凸函数；步骤4、利用交替方向乘子算法，建立与欠定相位恢复模型相对应的含有入射信号参数的目标函数；步骤5、采用交替迭代更新方法，对所述目标函数进行求解，恢复出原始信号。
- 根据权利要求1所述的基于阵列天线的欠定相位恢复方法，其特征在于：步骤1中所述的入射信号导向矢量矩阵为
上式中，N为天线阵元数目，M为某个角度区域Φ被均匀分成的角度区间个数，其中，θ m为角度区域Φ中第m个信号入射时的角度，d为天线阵元间隔，λ为入射信号波长；
步骤1所述的阵列天线的相位恢复方法模型为
上式中，和||为分别为对向量中每一个元素取平方根和取模值操作，
为原始信号x的最大似然估计，向量x为原始入射信号，
为天线阵列所量测到的幅值信号
的第i个元素。
- 根据权利要求2所述的基于阵列天线的欠定相位恢复方法，其特征在于：步骤2中所述的入射信号空域稀疏特性，指的是入射信号数目远少于角度区域Φ内所分成的角度区间个数；步骤2中所述的欠定，指的是天线阵元数目少于角度区域Φ内所分成的角度区间个数，即N＜M；步骤2中所述的欠定相位恢复方法模型为
上式中，β为一个常值系数，|| || 1为L1范数。
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