对多普勒信息不敏感的低旁瓣波形设计方法与流程

技术特征：

1.对多普勒信息不敏感的低旁瓣波形设计方法，其特征在于，具体按照以下步骤实施：

步骤1：构建“时延-多普勒”范围内低自相关旁瓣波形设计的复数数学模型，对该数学模型的系数矩阵进行分割得到矩阵的实系数矩阵与虚系数矩阵并将非正定的实系数矩阵与虚系数矩阵转换为正定矩阵；

步骤2：使用ADMM方法对目标函数进行变换求解引入变量；

步骤3：用分段最小值法求解不等式约束问题；

步骤4：使用MM方法得到满足约束条件的探测波形x；

步骤5：重复步骤3、4直至x收敛。

2.根据权利要求1所述的对多普勒信息不敏感的低旁瓣波形设计方法，其特征在于，所述步骤1具体为：

同时考虑时延K_k与多普勒L_l∈{L_d,...,L_u}因素时，自相关函数写为：

r_lk＝x^HA_lkx (1)

其中，k＝1,...,K,0<K_d<K_u<N，l＝1,...,L,L_d<L_u，lk＝1,...,LK,LK＝L×K，

x为长度为N、模为1的信号序列，

$M_{lk\mathrm{mod}L}=M_k=\left[\begin{array}{cc}{O}_{k\&times;(N-k)}& I_{k\&times;k}\\ I_{(N-k)\&times;(N-k)}& O_{(N-k)\&times;k}\end{array}\right]$

$d_l={\&lsqb;1,e^{{\mathrm{jw}}_l},\mathrm{...},e^{j(N-1)w_l}\&rsqb;}^T$

算子diag(·)是将向量扩展为对角矩阵，使得D_l(n,n)＝d_l(n)；

那么，满足要求的波形设计的数学描述为：

$\begin{array}{c}\underset{x}{\min }\mathrm{\&epsiv;}\\ \begin{array}{cc}s.t.& |r_{lk}{|}^2=|x^H{A}_{lk}x{|}^2\&le;\mathrm{\&epsiv;},lk=1,\mathrm{...},LK\\ & \left|x_n\right|=1,n=1,\mathrm{...},N\end{array}\end{array},---\left(2\right)$

对系数矩阵A_lk进行分割得到：

$r_{lk}=x^H\left(\frac{{\mathrm{RE}}_{lk}+{\mathrm{jIM}}_{lk}}{2}\right)x---\left(3\right)$

其中，

1)

2)x^HRE_lkx与x^HIM_lkx均为实数；

那么，自相关函数r_lk与RE_lk、IM_lk满足关系：

4|r_lk|²＝(x^HRE_lkx)²+(x^HIM_lkx)² (4)

对系数矩阵RE_lk进行特征分解，得到特征根{λ₁,...,λ_N},记MR_lk＝min{λ₁,...,λ_N},其中min{·}表示求·中的最小元素；

若MR_lk≤0,则：

若MR_lk>0,则：

其中E_N为N阶单位矩阵，即：

正定矩阵分解为：

其中，

同理，系数矩阵IM_lk对应的正定矩阵分解为：

其中，

系数矩阵IM_lk的特征根的最小值,记为MI_lk；

那么式(1)表示为：

$r_{lk}=x^H\left(\frac{{\mathrm{MR}}_{lk}+W_{lk}^H{W}_{lk}}{2}\right)x+{\mathrm{jx}}^H\left(\frac{{\mathrm{MI}}_{lk}+V_{lk}^H{V}_{lk}}{2}\right)x---\left(7\right)$

根据信号序列x的模为1即x^Hx＝1得：

$r_{lk}=\frac{N\&times;{\mathrm{MR}}_{lk}+x^H{W}_{lk}^H{W}_{lk}x}{2}+j\frac{N\&times;{\mathrm{MI}}_{lk}+x^H{V}_{lk}^H{V}_{lk}x}{2}---\left(8\right)$

式(8)可以写成另外一种实数形式：

$r_{lk}=\frac{N\&times;{\mathrm{MR}}_{lk}+{\stackrel{\&OverBar;}{x}}^T{\stackrel{\&OverBar;}{W}}_{lk}^T{\stackrel{\&OverBar;}{W}}_{lk}\stackrel{\&OverBar;}{x}}{2}+j\frac{N\&times;{\mathrm{MI}}_{lk}+{\stackrel{\&OverBar;}{x}}^T{\stackrel{\&OverBar;}{V}}_{lk}^T{\stackrel{\&OverBar;}{V}}_{lk}\stackrel{\&OverBar;}{x}}{2}---\left(9\right)$

其中，

Re(·)与Im(·)分别表示·的实部与虚部；

则是实对称矩阵，即：

由式(9)，式(4)表示为：

$4*|r_{lk}{|}^2={(N*{\mathrm{MR}}_{lk}+{\stackrel{\&OverBar;}{x}}^T{\stackrel{\&OverBar;}{W}}_{lk}^T{\stackrel{\&OverBar;}{W}}_{lk}\stackrel{\&OverBar;}{x})}^2+{(N*{\mathrm{MI}}_{lk}+{\stackrel{\&OverBar;}{x}}^T{\stackrel{\&OverBar;}{V}}_{lk}^T{\stackrel{\&OverBar;}{V}}_{lk}\stackrel{\&OverBar;}{x})}^2---\left(10\right)$

令替代变量F_lk与G_lk分别为则式(10)表示为：

$4|r_{lk}{|}^2=F_{lk}^2+G_{lk}^2---\left(11\right)$

则式(2)表示为：

$\begin{array}{c}\underset{x}{\min }\mathrm{\&epsiv;}\\ \begin{array}{cc}s.t.& F_{lk}^2+G_{lk}^2\&le;4\mathrm{\&epsiv;},\\ & F_{lk}=N\&times;{\mathrm{MR}}_{lk}+{\stackrel{\&OverBar;}{x}}^T{\stackrel{\&OverBar;}{W}}_{lk}^T{\stackrel{\&OverBar;}{W}}_{lk}\stackrel{\&OverBar;}{x},\\ & G_{lk}=N\&times;{\mathrm{MI}}_{lk}+{\stackrel{\&OverBar;}{x}}^T{\stackrel{\&OverBar;}{V}}_{lk}^T{\stackrel{\&OverBar;}{V}}_{lk}\stackrel{\&OverBar;}{x},lk=1,\mathrm{...},LK,\\ & \left|x_n\right|=1,n=1,\mathrm{...},N\end{array}\end{array}---\left(12\right).$

3.根据权利要求2所述的对多普勒信息不敏感的低旁瓣波形设计方法，其特征在于，所述步骤2具体为：

引入拉格朗日乘子α_lk,β_lk与ADMM乘子ρ>0，式(12)表示为：

$\begin{array}{c}\underset{\stackrel{\&OverBar;}{x},\mathrm{\&epsiv;},\mathrm{\&alpha;},\mathrm{\&beta;},F,G}{\min }L=\mathrm{\&epsiv;}+{\&Sigma;}_{lk=1}^{LK}{\mathrm{\&alpha;}}_{lk}\&lsqb;F_{lk}-(N*{\mathrm{MR}}_{lk}+{\stackrel{\&OverBar;}{x}}^T{\stackrel{\&OverBar;}{W}}_{lk}^T{\stackrel{\&OverBar;}{W}}_{lk}\stackrel{\&OverBar;}{x})\&rsqb;\\ +{\&Sigma;}_{lk=1}^{LK}{\mathrm{\&beta;}}_{lk}\&lsqb;G_{lk}-(N*{\mathrm{MI}}_{lk}+{\stackrel{\&OverBar;}{x}}^T{\stackrel{\&OverBar;}{V}}_{lk}^T{\stackrel{\&OverBar;}{V}}_{lk}\stackrel{\&OverBar;}{x})\&rsqb;\\ +\frac{\mathrm{\&rho;}}{2}{\&Sigma;}_{lk=1}^{LK}{\&lsqb;F_{lk}-(N*{\mathrm{MR}}_{lk}+{\stackrel{\&OverBar;}{x}}^T{\stackrel{\&OverBar;}{W}}_{lk}^T{\stackrel{\&OverBar;}{W}}_{lk}\stackrel{\&OverBar;}{x})\&rsqb;}^2\\ +\frac{\mathrm{\&rho;}}{2}{\&Sigma;}_{lk=1}^{LK}{\&lsqb;G_{lk}-(N*{\mathrm{MI}}_{lk}+{\stackrel{\&OverBar;}{x}}^T{\stackrel{\&OverBar;}{V}}_{lk}^T{\stackrel{\&OverBar;}{V}}_{lk}\stackrel{\&OverBar;}{x})\&rsqb;}^2\\ \begin{array}{cc}s.t.& F_{lk}^2+G_{lk}^2\&le;4\mathrm{\&epsiv;},lk=1,\mathrm{...},LK,\\ & {\stackrel{\&OverBar;}{x}}_n^2+{\stackrel{\&OverBar;}{x}}_{n+N}^2=1,n=1,\mathrm{...},N\end{array}\end{array}---\left(13\right).$

4.根据权利要求3所述的对多普勒信息不敏感的低旁瓣波形设计方法，其特征在于，所述步骤3使用分段最小值法求解不等式约束问题，从而更新ε(t+1),F_lk(t+1),G_lk(t+1),α_lk(t+1),β_lk(t+1)，具体为：

①更新F_lk(t+1),G_lk(t+1)

对于给定t次的α_lk(t),β_lk(t)求解ε(t+1),F_lk(t+1),G_lk(t+1)的问题，式(13)表示为：

$\begin{array}{c}\underset{\mathrm{\&epsiv;},F,G}{\min }L=\mathrm{\&epsiv;}+{\&Sigma;}_{lk=1}^{LK}{\mathrm{\&alpha;}}_{lk}\left(t\right)\&lsqb;F_{lk}-(N*{\mathrm{MR}}_{lk}+{\stackrel{\&OverBar;}{x}}^T(t\left){\stackrel{\&OverBar;}{W}}_{lk}^T{\stackrel{\&OverBar;}{W}}_{lk}\stackrel{\&OverBar;}{x}\right(t\left)\right)\&rsqb;\\ {\&Sigma;}_{lk=1}^{LK}{\mathrm{\&beta;}}_{lk}\left(t\right)\&lsqb;G_{lk}-(N*{\mathrm{MI}}_{lk}+{\stackrel{\&OverBar;}{x}}^T(t\left){\stackrel{\&OverBar;}{V}}_{lk}^T{\stackrel{\&OverBar;}{V}}_{lk}\stackrel{\&OverBar;}{x}\right(t\left)\right)\&rsqb;\\ +\frac{\mathrm{\&rho;}}{2}{\&Sigma;}_{lk=1}^{LK}{\&lsqb;F_{lk}-(N*{\mathrm{MR}}_{lk}+{\stackrel{\&OverBar;}{x}}^T(t\left){\stackrel{\&OverBar;}{W}}_{lk}^T{\stackrel{\&OverBar;}{W}}_{lk}\stackrel{\&OverBar;}{x}\right(t\left)\right)\&rsqb;}^2\\ +\frac{\mathrm{\&rho;}}{2}{\&Sigma;}_{lk=1}^{LK}{\&lsqb;G_{lk}-(N*{\mathrm{MI}}_{lk}+{\stackrel{\&OverBar;}{x}}^T(t\left){\stackrel{\&OverBar;}{V}}_{lk}^T{\stackrel{\&OverBar;}{V}}_{lk}\stackrel{\&OverBar;}{x}\right(t\left)\right)\&rsqb;}^2\\ \begin{array}{cc}s.t.& F_{lk}^2+G_{lk}^2\&le;4\mathrm{\&epsiv;},lk=1,\mathrm{...},LK\end{array}\end{array}---\left(14\right)$

记

$\begin{array}{c}{\tilde{F}}_{lk}(t+1)=(N*{\mathrm{MR}}_{lk}+{\stackrel{\&OverBar;}{x}}^T(t\left){\stackrel{\&OverBar;}{W}}_{lk}^T{\stackrel{\&OverBar;}{W}}_{lk}\stackrel{\&OverBar;}{x}\right(t\left)\right)-\frac{{\mathrm{\&alpha;}}_{lk}\left(t\right)}{\mathrm{\&rho;}}\\ {\tilde{G}}_{lk}(t+1)=(N*{\mathrm{MI}}_{lk}+{\stackrel{\&OverBar;}{x}}^T(t\left){\stackrel{\&OverBar;}{V}}_{lk}^T{\stackrel{\&OverBar;}{V}}_{lk}\stackrel{\&OverBar;}{x}\right(t\left)\right)-\frac{{\mathrm{\&beta;}}_{lk}\left(t\right)}{\mathrm{\&rho;}}\end{array}---\left(15\right)$

式(14)简化为：

$\begin{array}{c}\underset{\mathrm{\&epsiv;},F,G}{\min }L=\mathrm{\&epsiv;}+\frac{\mathrm{\&rho;}}{2}\underset{lk=1}{\overset{LK}{\&Sigma;}}\&lsqb;{(F_{lk}-{\tilde{F}}_{lk}(t+1\left)\right)}^2+{(G_{lk}-{\tilde{G}}_{lk}(t+1\left)\right)}^2\&rsqb;\\ \begin{array}{cc}s.t.& F_{lk}^2+G_{lk}^2\&le;4\mathrm{\&epsiv;},lk=1,\mathrm{...},LK\end{array}\end{array}---\left(16\right)$

由式(14)的约束条件可知，变量F_lk,G_lk的取值是受变量ε约束的，而对于给定的ε，式(14)的F_lk(t+1),G_lk(t+1)为：

$\begin{array}{c}{F}_{lk}(t+1)=f_{lk}{\tilde{F}}_{lk}(t+1)\\ G_{lk}(t+1)=g_{lk}{\tilde{G}}_{lk}(t+1)\end{array}---\left(17\right)$

其中，

$f_{lk}=g_{lk}=\left\{\begin{array}{cc}1,& {\tilde{F}}_{lk}^2+{\tilde{G}}_{lk}^2\&le;4\mathrm{\&epsiv;}\\ {\left(\frac{4\mathrm{\&epsiv;}}{{\tilde{F}}_{lk}^2+{\tilde{G}}_{lk}^2}\right)}^{1/2},& {\tilde{F}}_{lk}^2+{\tilde{G}}_{lk}^2\&gt;4\mathrm{\&epsiv;}\end{array}\right.$

记为S_lk，则：

$S_{lk}=\left\{\begin{array}{cc}1,& {\tilde{F}}_{lk}{(t+1)}^2+{\tilde{G}}_{lk}{(t+1)}^2\&gt;4\mathrm{\&epsiv;},\\ 0,& {\tilde{F}}_{lk}{(t+1)}^2+G_{lk}{(t+1)}^2\&le;4\mathrm{\&epsiv;},\end{array}\right.,lk=1,\mathrm{...},LK---\left(18\right)$

②更新ε(t+1)

给定t次的α_lk(t),β_lk(t)求解ε(t+1)：

将式(17)、(18)代入式(16)得：

$\begin{array}{c}\underset{\mathrm{\&epsiv;}}{\min }L=\mathrm{\&epsiv;}+\frac{\mathrm{\&rho;}}{2}\underset{lk=1}{\overset{LK}{\&Sigma;}}{S}_{lk}{\left(F_{lk}\right(t+1)-{\tilde{F}}_{lk}(t+1\left)\right)}^2+\frac{\mathrm{\&rho;}}{2}\underset{lk=1}{\overset{LK}{\&Sigma;}}{S}_{lk}{\left(G_{lk}\right(t+1)-{\tilde{G}}_{lk}(t+1\left)\right)}^2\\ =\mathrm{\&epsiv;}+\frac{\mathrm{\&rho;}}{2}\underset{lk=1}{\overset{LK}{\&Sigma;}}{S}_{lk}{(1-f_{lk})}^2{\tilde{F}}_{lk}{(t+1)}^2+\frac{\mathrm{\&rho;}}{2}\underset{lk=1}{\overset{LK}{\&Sigma;}}{S}_{lk}{(1-g_{lk})}^2{\tilde{G}}_{lk}{(t+1)}^2\\ =\mathrm{\&epsiv;}+\frac{\mathrm{\&rho;}}{2}\underset{lk=1}{\overset{LK}{\&Sigma;}}{S}_{lk}{(1-f_{lk})}^2({\tilde{F}}_{lk}{\left(t+1\right)}^2+{\tilde{G}}_{lk}{\left(t+1\right)}^2)\end{array}---\left(19\right)$

忽略掉常数项，式(19)简化为：

$\underset{\mathrm{\&epsiv;}}{\min }L=A\mathrm{\&epsiv;}+B\sqrt{\mathrm{\&epsiv;}}---\left(20\right)$

其中，

由于式(18)的分段情况，式(20)是不连续的，需要分段讨论其极小值点，分段讨论式(20)时，分段的分割点取值范围为其中，ε_max是代入原问题得到的ε，ε₀是大于0且小于ε_max与的实数；

在区间[ε_d,ε_u]上，ρ>0且S_lk≥0，所以A,B>0，故式(20)取极小值时的ε为：

$\mathrm{\&epsiv;}=\left\{\begin{array}{cc}{\mathrm{\&epsiv;}}_u,& {\left(\frac{-B}{2A}\right)}^2\&gt;{\mathrm{\&epsiv;}}_u\\ {\left(\frac{-B}{2A}\right)}^2,& {\left(\frac{-B}{2A}\right)}^2\&Element;\&lsqb;{\mathrm{\&epsiv;}}_d,{\mathrm{\&epsiv;}}_u\&rsqb;\\ {\mathrm{\&epsiv;}}_d,& {\left(\frac{-B}{2A}\right)}^2\&lt;{\mathrm{\&epsiv;}}_d\end{array}\right.---\left(21\right)$

将式(21)代入式(20)得到L在各段的极小值{L₁,...,L_m}，选取{L₁,...,L_m}中最小值对应的ε为ε(t+1)；

③更新拉格朗日乘子α_lk(t+1),β_lk(t+1)

$\begin{array}{c}{\mathrm{\&alpha;}}_{lk}(t+1)={\mathrm{\&alpha;}}_{lk}\left(t\right)+\mathrm{\&rho;}\left(F_{lk}\right(t+1)-{\tilde{F}}_{lk}(t+1\left)\right)\\ {\mathrm{\&beta;}}_{lk}(t+1)={\mathrm{\&beta;}}_{lk}\left(t\right)+\mathrm{\&rho;}\left(G_{lk}\right(t+1)-{\tilde{G}}_{lk}(t+1\left)\right)\end{array}---\left(22\right).$

5.根据权利要求4所述的对多普勒信息不敏感的低旁瓣波形设计方法，其特征在于，所述步骤4具体为：

步骤4.1：推导目标函数

同理于式(14)，求解时，式(13)写为：

$\begin{array}{c}\underset{\stackrel{\&OverBar;}{x}}{\min }L=\mathrm{\&epsiv;}(t+1)+{\&Sigma;}_{lk=1}^{LK}{\mathrm{\&alpha;}}_{lk}(t+1)\&lsqb;F_{lk}(t+1)-(N*{\mathrm{MR}}_{lk}+{\stackrel{\&OverBar;}{x}}^T{\stackrel{\&OverBar;}{W}}_{lk}^T{\stackrel{\&OverBar;}{W}}_{lk}\stackrel{\&OverBar;}{x})\&rsqb;\\ +{\&Sigma;}_{lk=1}^{LK}{\mathrm{\&beta;}}_{lk}(t+1)\&lsqb;G_{lk}(t+1)-(N*{\mathrm{MI}}_{lk}+{\stackrel{\&OverBar;}{x}}^T{\stackrel{\&OverBar;}{V}}_{lk}^T{\stackrel{\&OverBar;}{V}}_{lk}\stackrel{\&OverBar;}{x})\&rsqb;\\ +\frac{\mathrm{\&rho;}}{2}{\&Sigma;}_{lk=1}^{LK}{\&lsqb;F_{lk}(t+1)-(N*{\mathrm{MR}}_{lk}+{\stackrel{\&OverBar;}{x}}^T{\stackrel{\&OverBar;}{W}}_{lk}^T{\stackrel{\&OverBar;}{W}}_{lk}\stackrel{\&OverBar;}{x})\&rsqb;}^2\\ +\frac{\mathrm{\&rho;}}{2}{\&Sigma;}_{lk=1}^{LK}{\&lsqb;G_{lk}(t+1)-(N*{\mathrm{MI}}_{lk}+{\stackrel{\&OverBar;}{x}}^T{V}_{lk}^T{\stackrel{\&OverBar;}{V}}_{lk}\stackrel{\&OverBar;}{x})\&rsqb;}^2\\ \begin{array}{cc}s.t.& {\stackrel{\&OverBar;}{x}}_n^2+{\stackrel{\&OverBar;}{x}}_{n+N}^2=1,n=1,\mathrm{...},N\end{array}\end{array}---\left(23\right)$

将式(23)中的常数项去掉后，得到关于的目标函数：

$\begin{array}{c}\underset{\stackrel{\&OverBar;}{x}}{\min }L=\frac{\mathrm{\&rho;}}{2}{\&Sigma;}_{lk=1}^{LK}{\stackrel{\&OverBar;}{W}}_{lk}{\stackrel{\&OverBar;}{x}}^4+{\&Sigma;}_{lk=1}^{LK}\left(\right(\mathrm{\&rho;}(N*{\mathrm{MR}}_{lk}-F_{lk})-{\mathrm{\&alpha;}}_{lk}){\stackrel{\&OverBar;}{W}}_{lk}{\stackrel{\&OverBar;}{x}}^2\\ +\frac{\mathrm{\&rho;}}{2}{\&Sigma;}_{lk=1}^{LK}{\stackrel{\&OverBar;}{V}}_{lk}{\stackrel{\&OverBar;}{x}}^2+\left(\mathrm{\&rho;}\right(N*{\mathrm{MI}}_{lk}-G_{lk})-{\mathrm{\&beta;}}_{lk}\left){\stackrel{\&OverBar;}{V}}_{lk}{\stackrel{\&OverBar;}{x}}^2\right)\\ \begin{array}{cc}s.t.& {\stackrel{\&OverBar;}{x}}_n^2+{\stackrel{\&OverBar;}{x}}_{n+N}^2=1,n=1,\mathrm{...},N\end{array}\end{array}---\left(24\right)$

步骤4.2：求解目标函数(24)的优化函数

式(24)右端第一项有：

$\begin{array}{c}\frac{\mathrm{\&rho;}}{2}{\&Sigma;}_{lk=1}^{LK}{\stackrel{\&OverBar;}{W}}_{lk}{\stackrel{\&OverBar;}{W}}_{lk}{\stackrel{\&OverBar;}{x}}^1=\frac{\mathrm{\&rho;}}{2}{\&Sigma;}_{lk=1}^{LK}\left(Tr{\left({\tilde{W}}_{lk}X\right)}^2\right)\\ =\frac{\mathrm{\&rho;}}{2}{\&Sigma;}_{lk=1}^{LK}{\left(vec\right({\tilde{W}}_{lk}\left)vec\right(X\left)\right)}^2\\ =\frac{\mathrm{\&rho;}}{2}{\&Sigma;}_{lk=1}^{LK}\left(vec{\left(X\right)}^{T}vec{\left({\tilde{W}}_{lk}\right)}^{T}vec\right({\tilde{W}}_{lk}\left)vec\right(X\left)\right)\end{array}---\left(25\right)$

其中，

因为，二次型函数f(x)＝x^HAx，若A,B是n×n的自共轭矩阵且A≤B，那么

$f\left(x\right)\&le;x^{H}Bx+2\mathrm{Re}\left\{x^H(A-B)x_0\right\}+x_0^H(B-A)x_0,\&ForAll;x_0\&Element;C^n---\left(27\right)$

根据式(27)

$\begin{array}{c}\frac{\mathrm{\&rho;}}{2}{\&Sigma;}_{lk=1}^{LK}vec{\left(X\right)}^T{\widetilde{\stackrel{~}{W}}}_{lk}vec\left(X\right)\\ \&le;\frac{\mathrm{\&rho;}}{2}{\&Sigma;}_{lk=1}^{LK}\left(2vec{\left(X\right)}^T\right({\widetilde{\stackrel{~}{W}}}_{lk}-{\mathrm{\&lambda;}}_{lk}^W{E}_{2N})vec\left(X\right(t))+{\mathrm{\&lambda;}}_{lk}^{W}vec{\left(X\right)}^{T}vec\left(X\right)\\ +vec{\left(X\right(t\left)\right)}^T{\mathrm{\&lambda;}}_{lk}^W{E}_{2N}-{\widetilde{\stackrel{~}{W}}}_{lk}vec\left(X\right(t)\left)\right)\end{array}---\left(28\right)$

其中，是的最大特征根，

因为所以式(28)的第二项是常数将式(28)的常数项用C_W1表示，即：

$C_{W1}=\frac{\mathrm{\&rho;}}{2}{\mathrm{\&Sigma;}}_{lk=1}^{LK}({\mathrm{\&lambda;}}_{lk}^W{N}^2+vec{\left(X\left(t\right)\right)}^T({\mathrm{\&lambda;}}_{lk}^W{E}_{2N}{\widetilde{\stackrel{~}{W}}}_{lk}\left)vec\right(X\left(t\right)\left)\right)$

则式(28)写为：

$\frac{\mathrm{\&rho;}}{2}{\mathrm{\&Sigma;}}_{k=1}^{LK}vec{\left(X\right)}^T{\widetilde{\stackrel{~}{W}}}_{lk}vec\left(X\right)\&le;C_{W1}+\frac{\mathrm{\&rho;}}{2}{\mathrm{\&Sigma;}}_{lk=1}^{LK}2vec{\left(X\right)}^T({\widetilde{\stackrel{~}{W}}}_{lk}-{\mathrm{\&lambda;}}_{lk}^W{E}_{2N})vec\left(X\right(t\left)\right)---\left(29\right)$

又

$\begin{array}{c}\frac{\mathrm{\&rho;}}{2}{\&Sigma;}_{lk=1}^{LK}2vec\left(X^T\right)({\widetilde{\stackrel{~}{W}}}_{lk}-{\mathrm{\&lambda;}}_{lk}^W{E}_{2N})vec\left(X\right(t\left)\right)\\ =\mathrm{\&rho;}{\&Sigma;}_{lk=1}^{LK}\left(Tr\right(X\left(t\right){\tilde{W}}_{lk}\left)Tr\right({\tilde{W}}_{lk}X)-{\mathrm{\&lambda;}}_{lk}^{W}Tr(X\left(t\right)\left)Tr\right(X\left)\right)\\ =\mathrm{\&rho;}{\&Sigma;}_{lk=1}^{LK}\left(Tr\right(X\left(t\right){\tilde{W}}_{lk}){\stackrel{\&OverBar;}{x}}^T{\tilde{W}}_{lk}\stackrel{\&OverBar;}{x}-{\mathrm{\&lambda;}}_{lk}^{W}Tr(X\left(t\right)\left){\stackrel{\&OverBar;}{x}}^T\stackrel{\&OverBar;}{x}\right)\\ =\mathrm{\&rho;}{\stackrel{\&OverBar;}{x}}^T{\&Sigma;}_{lk=1}^{LK}\left(Tr\right(X\left(t\right){\tilde{W}}_{lk}\left){\tilde{W}}_{lk}\right){\tilde{W}}_{lk}-{\mathrm{\&lambda;}}_{lk}^{W}X\left(t\right))\stackrel{\&OverBar;}{x}\\ =\mathrm{\&rho;}{\stackrel{\&OverBar;}{x}}^T{M}_1\stackrel{\&OverBar;}{x}\end{array}---\left(30\right)$

其中，

根据式(27)，若是M₁的最大特征根，则：

$\begin{array}{c}{\stackrel{\&OverBar;}{x}}^T{M}_1\stackrel{\&OverBar;}{x}\&le;{\mathrm{\&lambda;}}^{M_1}{\stackrel{\&OverBar;}{x}}^T\stackrel{\&OverBar;}{x}+2{\stackrel{\&OverBar;}{x}}^T(M_1-{\mathrm{\&lambda;}}^{M_1}{E}_{2N})\stackrel{\&OverBar;}{x}\left(t\right)+\stackrel{\&OverBar;}{x}{\left(t\right)}^T({\mathrm{\&lambda;}}^{M_1}-M_1)\stackrel{\&OverBar;}{x}\left(t\right)\\ ={\mathrm{\&lambda;}}^{M_1}N+\stackrel{\&OverBar;}{x}{\left(t\right)}^T({\mathrm{\&lambda;}}^{M_1}-M_1)\stackrel{\&OverBar;}{x}\left(t\right)+2{\stackrel{\&OverBar;}{x}}^T(M_1-{\mathrm{\&lambda;}}^{M_1}{E}_{2N})\stackrel{\&OverBar;}{x}\left(t\right)\end{array}---\left(31\right)$

令

${\mathrm{MM}}_1\left(x\right)=\mathrm{\&rho;}({\mathrm{\&lambda;}}^{M_1}N+\stackrel{\&OverBar;}{x}{\left(t\right)}^T({\mathrm{\&lambda;}}^{M_1}-M_1\left)\stackrel{\&OverBar;}{x}\right(t)+2{\stackrel{\&OverBar;}{x}}^T(M_1-{\mathrm{\&lambda;}}^{M_1}{E}_{2N}\left)\stackrel{\&OverBar;}{x}\right(t\left)\right)---\left(32\right)$

则

$\frac{\mathrm{\&rho;}}{2}{\mathrm{\&Sigma;}}_{lk=1}^{LK}\left|\right|{\stackrel{\&OverBar;}{W}}_{lk}\stackrel{\&OverBar;}{x}|{|}^4\&le;{\mathrm{MM}}_1\left(\stackrel{\&OverBar;}{x}\right)---\left(33\right)$

同理，式(24)右端第二项

$\frac{\mathrm{\&rho;}}{2}{\mathrm{\&Sigma;}}_{lk=1}^{LK}\left|\right|{\stackrel{\&OverBar;}{V}}_{lk}\stackrel{\&OverBar;}{x}|{|}^4\&le;{\mathrm{MM}}_2\left(\stackrel{\&OverBar;}{x}\right)---\left(34\right)$

其中，是M₁的最大特征根，是的最大特征根；

式(24)右端第三项的优化函数的推导过程同理于式(31)，优化函数为：

${\mathrm{MM}}_3\left(x\right)=({\mathrm{\&lambda;}}^{M_3}N+\stackrel{\&OverBar;}{x}{\left(t\right)}^T({\mathrm{\&lambda;}}^{M_3}-M_3\left)\stackrel{\&OverBar;}{x}\right(t)+2{\stackrel{\&OverBar;}{x}}^T(M_3-{\mathrm{\&lambda;}}^{M_3}{E}_{2N}\left)\stackrel{\&OverBar;}{x}\right(t\left)\right)---\left(35\right)$

其中，所以，式(24)的优化函数为：

$\begin{array}{c}\underset{\stackrel{\&OverBar;}{x}}{\min }{\mathrm{MM}}_1\left(\stackrel{\&OverBar;}{x}\right)+{\mathrm{MM}}_2\left(\stackrel{\&OverBar;}{x}\right)+{\mathrm{MM}}_3\left(\stackrel{\&OverBar;}{x}\right)\\ \begin{array}{cc}s.t.& {\stackrel{\&OverBar;}{x}}_n^2+{\stackrel{\&OverBar;}{x}}_{n+N}^2=1,n=1,\mathrm{...},N\end{array}\end{array}---\left(36\right)$

步骤4.3：更新

式(36)的拉格朗日函数为：

$L={\mathrm{MM}}_1\left(\stackrel{\&OverBar;}{x}\right)+{\mathrm{MM}}_2\left(\stackrel{\&OverBar;}{x}\right)+{\mathrm{MM}}_3\left(\stackrel{\&OverBar;}{x}\right)+{\mathrm{\&Sigma;}}_{n=1}^N{\mathrm{\&mu;}}_n({\stackrel{\&OverBar;}{x}}_n^2+{\stackrel{\&OverBar;}{x}}_{n+N}^2-1)---\left(37\right)$

对拉格朗日函数(37)关于求偏导

$\frac{\&part;L}{\&part;{\stackrel{\&OverBar;}{x}}_n}=\left\{\begin{array}{cc}\frac{\&part;\left({\mathrm{MM}}_1\right(\stackrel{\&OverBar;}{x})+{\mathrm{MM}}_2\left(\stackrel{\&OverBar;}{x}\right)+{\mathrm{MM}}_3\left(\stackrel{\&OverBar;}{x}\right))}{\&part;{\stackrel{\&OverBar;}{x}}_n}+2{\mathrm{\&mu;}}_n{\stackrel{\&OverBar;}{x}}_n,& n=1,\mathrm{...},N\\ \frac{\&part;\left({\mathrm{MM}}_1\right(\stackrel{\&OverBar;}{x})+{\mathrm{MM}}_2\left(\stackrel{\&OverBar;}{x}\right)+{\mathrm{MM}}_3\left(\stackrel{\&OverBar;}{x}\right))}{\&part;{\stackrel{\&OverBar;}{x}}_n}+2{\mathrm{\&mu;}}_{n-N}{\stackrel{\&OverBar;}{x}}_n,& n=N+1,\mathrm{...},2N\end{array}\right.---\left(38\right)$

因为是关于的一次函数，所以式(37)对的偏导是常数，记为d_n，令则：

${\stackrel{\&OverBar;}{x}}_n=\left\{\begin{array}{cc}-\frac{d_n}{2{\mathrm{\&mu;}}_n},& n=1,\mathrm{...},N\\ -\frac{d_n}{2{\mathrm{\&mu;}}_{n-N}},& n=N+1,\mathrm{...},2N\end{array}\right.---\left(39\right)$

将式(39)代入约束条件得

${\mathrm{\&mu;}}_n=\&PlusMinus;\frac{\sqrt{d_n^2+d_{n+N}^2}}{2}---\left(40\right)$

将式(40)代入式(36)目标函数，μ_n取值为因此

${\stackrel{\&OverBar;}{x}}_n(t+1)=\left\{\begin{array}{cc}-\frac{d_n}{\sqrt{d_n^2+d_{n+N}^2}},& n=1,\mathrm{...},N\\ -\frac{d_n}{\sqrt{d_n^2+d_{n-N}^2}},& n=N+1,\mathrm{...},2N\end{array}\right.---\left(41\right)$

将式(41)计算得到的代入

$x_n={\stackrel{\&OverBar;}{x}}_n+j{\stackrel{\&OverBar;}{x}}_{N+n}---\left(42\right)$

中，得到本轮迭代的x。