Firstly, the L2,0 norm is approximated selleckchem by an arctan function to construct an approximate ��X�� 2,0 norm minimization problem. Then this problem is solved by a quasi-Newton method to estimate DOA.3.1. Basic Idea of the Proposed MethodLet ��i = ��X(i,:)��2, i = 1, 2, Inhibitors,Modulators,Libraries , M, we have:��X��2,0=��i=1M(��X(i,:)��2)0=���Ρ�0(8)where �� = (��1, , ��M). Considering the following arctan function:f��(s)=2��arc?tan(s22��2)(9)where �� is a positive parameter and s is a variable parameter. Then f��(s) has the following property:lim�ġ�0f��(s)={1s��00s=0(10)LetF��(��)=��i=1Nf��(��i)From Equation (10), we have:lim�ġ�0F��(��)=���Ρ�0(11)From Equations (8) and (10), then:lim�ġ�0F��(X)=��X��2,0So DOA can be obtained by solving the approximate ��X��2,0 norm minimization:{minF��(X)s.t.Y=��X+N(12)where �� is a small positive constant.
Because noise is unknown, we synchronously hope to Inhibitors,Modulators,Libraries minimize F��(X) and ����X?Y��F2. Then Equation (12) is converted into a multiple objective optimization:{minXF��(X)minX����X?Y��F2(13)Using the linear weighting method, Equation (13) can be written as:minx L��,��(X)=F��(X)+�ˡ���X?Y��F2(14)where Inhibitors,Modulators,Libraries Inhibitors,Modulators,Libraries ��?��F denotes the Frobenious norm and the parameter �� will be discussed in Section 3.2. When the parameter �� is very small, the objective function F��(X) is highly unsmoothed and contains a lot of local minimization, so its global minimization is not easy. On the other hand, if the parameter �� is larger, the objective function F��(X) is smoother and contains less local minimization.
In order to obtain Carfilzomib global minimization, we select a decreasing sequence for ��, denoted as:��=[��1,��2,?,��J], ��j+1<��jwhere ��1 is a relatively large value, and ��J is a small value. For �� = ��J?1, the solution of Equation (14) is denoted as x��j�C1, and x��j�C1 is used as the initial value for �� = ��j, thus we hope the proposed algorithm can escape from getting trapped into a local minimum and reach the global minimization for a small value �� = ��J.For some fixed value �� = ��j, minimization problem Equation (14) is solved by the quasi-Newton method in this paper. One of the most successful quasi-Newton methods is the BFGS algorithm, which is second order convergent and has good numerical stability. Therefore, we use the BFGS algorithm to solve Equation (14) for some fixed value �� = ��j.
The conjugate gradient for a matrix variable is defined as:?L��,��(X)?X*=12(?L��,��(X)?XR+i?L��,��(X)?XI)where XR and Xl denote the real part and the imaginary part respectively, X*denotes the conjugate of X. Then the conjugate
Since Global Navigation Satellite System (GNSS) sensors selleck chemicals llc (e.g., Global Positioning System or GPS of the US [1,2], Galileo of Europe, GLONASS of Russia, and Compass of China) are widely used in many fields, including navigation, surveillance and precise timing, their vulnerability to radio frequency interference (RFI) is drawing significant attention. GPS RFI [3�C5] is especially problematic for GPS-based safety-of-life services such as aviation.