【 NLS 】Newton – Raphson Iteration Procedure of TOA

【 NLS 】Newton – Raphson Iteration Procedure of TOA - Based Positioning

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发布时间：2019-04-28

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上篇博文：说到了三种局部迭代算法进行TOA定位，分别为：

Newton – Raphson

Gauss – Newton

steepest descent method

这篇博文，针对 Newton – Raphson 这种迭代方法进行MATLAB编程（给出部分程序），对TOA定位进行仿真，仿真内容包括：

一次定位示意图

牛顿—拉夫森迭代方法收敛性讨论

定位误差与信噪比之间的关系

Illustration of NLS Approach for TOA - Based Positioning in a Single Trial

Consider a 2 - D geometry of L = 4 receivers with known coordinates at (0, 0), (0,10), (10, 0), and (10, 10), while the unknown source position is ( x , y ) = (2, 3).

Note that the source is located inside the square bounded by the four receivers. For presentation simplicity, the range error variance， $\sigma^2_{TOA,l}$ ，is assigned proportional to $d^2_{l}$ .

and we define the signal - to - noise ratio ( SNR ) as $d^2_l/\sigma^2_{TOA,l}$ .

如下图示，菱形表示基站位置，空心圆表示目标位置，实心圆表示估计出来的位置，发现二者几乎重合。说明可以进行定位，但有一定的误差。

2. Examine the convergence rates of the Newton – Raphson for the NLS approach in a single trial at SNR = 30 dB.

从上图看出收敛情况，该方法很快就可以收敛。貌似可以设置迭代次数为10，甚至更小，但话虽如此，毕竟这只是一次实验的收敛情况，所以迭代次数还是稍微大点为妙。取30保险。

对应的代码部分为：

clcclearclose all% == generating range measurements == %%We use x1, x2, x3 and x4 to assign the%receiver positions and X is a 4x1 matrix to store%their coordinates while the source position to be determined is%x. For 2D positioning, the dimension is L=2. The noise-free%range vector is d and its noisy version is r. The noise%component in r is a zero-mean white Gaussian vector with%variances given by sigma2. Moreover, both SNR in dB and number%of iterations in the local search schemes, denoted by dB and%iter, respectively, are set to 30.x1 = [0,0];x2 = [0,10];x3 = [10,0];x4 = [10,10];X = [x1;x2;x3;x4]'; % matrix for receiver positionsx = [2,3]'; % source position to be determinedL = size(X,2); % number of receiversd = (sqrt(sum((x*ones(1,L)-X).^2,1))).'; %noise-free rangesdB = 30; % SNR (in dB) which is defined as the mean of squared distance over noise variancesigma2 = d.^2/10^(dB/10); % sigma2--square of sigma, here we use: SNR_dB = 10log(d^2/sigma^2)r = d + randn(L,1).*sqrt(sigma2);iter = 30;x1 = x; % copy x for plot% == position estimation == %% == NLS Newton-Raphson algorithm == %x_nr = zeros(iter,2);x=[3,2]'; %initial guess valuefor i=1:iter    H=hessian_nls(X,x,r);    g=grad_nls(X,x,r);    x=x-inv(H)*g;    x_nr(i,:)=x;end%convegence analysisiter_no = 1:iter;figure(1)plot(iter_no,x_nr(:,1));xlabel('iter times');ylabel('x');figure(3)plot(iter_no,x_nr(:,2));xlabel('iter times');ylabel('y');

下面分析定位误差，也就是均方根误差，实验次数为100次，观测信噪比从-5dB 到30dB时的均方根误差rmse。

如果，所有的单位为km，则信噪比为30时的定位误差是282m，还行；信噪比为20时候的定位误差888m，看要求的标准吧。

用到的函数：

function g = grad_nls(X,x,r)% NLS gradient computation% --------------------------------% g = grad_nls(X,x,r);% g = gradient vector % X = matrix for receiver positions% x = 2D position estimate% r = TOA measurement vector%L = size(X,2); % number of receiverst1 = 0;t2 = 0;ds = sum((x*ones(1,L)-X).^2,1);ds = ds';for i=1:L    t1 = t1 + (r(i)-ds(i)^(0.5))*(x(1)-X(1,i))/ds(i)^(0.5);    t2 = t2 + (r(i)-ds(i)^(0.5))*(x(2)-X(2,i))/ds(i)^(0.5);endg=-2.*[t1; t2];

function H = hessian_nls(X,x,r)% NLS Hessian matrix computation% --------------------------------% H = hessian_nls(X,x,r);% H = Hessian matrix % X = matrix for receiver positions% x = 2D position estimate% r = TOA measurement vector%L = size(X,2); % number of receiverst1 = 0;t2 = 0;t3 =0;ds = sum((x*ones(1,L)-X).^2,1);ds = ds';for i=1:L    t1 = t1 + (x(1)-X(1,i))^2/ds(i)-(r(i)-ds(i)^(0.5))*(x(2)-X(2,i))^2/ds(i)^(1.5);    t2 = t2 + (x(2)-X(2,i))^2/ds(i)-(r(i)-ds(i)^(0.5))*(x(1)-X(1,i))^2/ds(i)^(1.5);    t3 = t3 + r(i)*(x(1)-X(1,i))*(x(2)-X(2,i))/ds(i)^(1.5);endH=2.*[t1 t3;      t3 t2];

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