基于Capon和信号子空间的变形算法实现波束形成附matlab代码

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本文探讨了在天线阵列信号处理中，Capon和基于信号子空间的MVDR和最小范数算法如何通过优化波束形成权重向量来提升波束形成性能，以抑制干扰和噪声。

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🔥 内容介绍

波束形成是天线阵列信号处理中的一项关键技术，它通过对来自不同方向的信号进行加权求和，来增强特定方向的信号并抑制其他方向的干扰。本文介绍了两种基于 Capon 和信号子空间的变形算法，即最小方差无失真响应 (MVDR) 算法和最小范数算法，用于实现波束形成。这些算法通过优化波束形成权重向量，来最小化波束形成输出的方差或范数，从而提高波束形成性能。

引言

波束形成在雷达、声纳、通信和医学成像等众多应用中至关重要。它可以增强目标信号，抑制干扰和噪声，提高系统性能。传统波束形成算法，如相控阵和时延求和，虽然简单易行，但对于具有复杂空间分布的信号，其性能往往受限。

Capon 算法

Capon 算法是一种自适应波束形成算法，它通过最小化波束形成输出的方差来优化波束形成权重向量。该算法的数学表达式为：

w_opt = arg min_w w^H R w s.t. w^H a_d = 1

其中：

w_opt 为最优波束形成权重向量
R 为协方差矩阵
a_d 为期望信号的方向向量

Capon 算法的优点在于其能够抑制来自干扰方向的信号，并对期望信号具有良好的空间分辨率。然而，该算法对噪声敏感，并且计算复杂度较高。

信号子空间算法

信号子空间算法是一种基于信号子空间和干扰子空间的波束形成算法。它将协方差矩阵分解为信号子空间和干扰子空间，并通过投影波束形成权重向量到信号子空间来优化权重向量。

MVDR 算法

MVDR 算法是信号子空间算法的一种，它通过最小化波束形成输出的方差来优化波束形成权重向量。该算法的数学表达式为：

w_opt = arg min_w w^H P_n w s.t. w^H a_d = 1

其中：

P_n 为干扰子空间投影矩阵

MVDR 算法对噪声和干扰具有良好的抑制能力，并且计算复杂度较低。然而，该算法对信号子空间和干扰子空间的划分敏感，并且可能出现空间泄漏现象。

最小范数算法

最小范数算法是信号子空间算法的另一种，它通过最小化波束形成权重向量的范数来优化权重向量。该算法的数学表达式为：

w_opt = arg min_w ||w||_2^2 s.t. w^H a_d = 1

最小范数算法对噪声和干扰具有良好的鲁棒性，并且计算复杂度较低。然而，该算法对期望信号的方向向量估计精度要求较高，并且可能出现波束宽度过宽的问题。

📣 部分代码

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Title:            EgregiousDataPadding.m% Description:      Script for calculating the temperature homogeneity;%                   the dimensionless Nusselt and Reynolds numbers; and the%                   required power for achieving specific flow conditions.%                   Results are saved in the file results.txt%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Acquire file path and set it as current directory % Returns the full path of the current script folder = fileparts( which(mfilename) ); % Adds all subdirectories addpath( genpath(folder) );%% Data input tic;                            % timing the script dInner = 22 * 10 ^ -3;          % inner cylinder diameter, [m]dOuter = 40 * 10 ^ -3;          % outer cylinder diameter, [m]voltRes = 29.5;                 % voltage of resistance, [V]ampRes = 0.63;                  % current of reistance, [A]position = 0.05:0.1:0.9;        % axial positions of thermocouples, [m]subsystems = 0:0.1:0.9;         % nine subsystems of control volume tempData = zeros(11, 4, 17);    % steady-state temperature measurements, [degree Celsius]                                % columns 1-9: resistance temperature along axial coordinate                                 % column 10: inlet temperature                                 % column 11: outlet temperature                                 timeData = zeros(1, 4, 17);     % time, [sec]voltFan = zeros(1, 4, 17);      % voltage of fan, [V]ampFan  = zeros(1, 4, 17);      % current of fan, [A]c = 0;YGrid', 'on');        % Caption         legend(handles(k).hAxes{i}, [handles(k).hE{1}, handles(k).hP{17},...                    handles(2).hP{1:4}], 'Data ({$\it\mu$} $\pm$ {$\it\sigma$})',...                    'axial flow', '1 inlet', '2 inlets', '3 inlets', '4 inlets',...                    'Location', 'NorthOutside', 'Orientation', 'horizontal',...                     'NumColumns', 3);                      % Axes name         if k == 1            xlabel(handles(k).hAxes{i}, 'Re');            ylabel(handles(k).hAxes{i}, '$\bar{Nu}$');        else            xlabel(handles(k).hAxes{i}, 'Flowrate [$m^3$/s]');            ylabel(handles(k).hAxes{i}, 'Power consumption [Watt]');        end                PlotDimensions(handles(k).hFig{i}, 'centimeters', [15.747, 8], 12);        ChangeInterpreter(handles(k).hFig{i}, 'latex');    endend%% Potential energy efficiency index% Potential energy efficiency potentialEn = zeros(1, 1, 17);potentialEnErr = zeros(1, 1, 17);% Potential energy equationpee = @(Q, W) Q / W;for k = 1:17   for i = 1:4       [potentialEn(1, i, k), potentialEnErr(1, i, k)] =...           UncertaintyPropagation(pee,...           [thermalEn(1, i, k) wattFan(1, i, k)],...           [thermalEnErr(1, i, k) wattFanErr(1, i, k)]);   endend swirlTh(:, 1) = potentialEn(1, 1, 1:16);swirlTh(:, 2) = potentialEnErr(1, 1, 1:16);axialTh(1, 1) = potentialEn(1, 1, 17);axialTh(1, 2) = potentialEnErr(1, 1, 17);relswirlTh = zeros(16, 2);relthermal = @(swirlTh, axialTh) (swirlTh / axialTh - 1) * 100;for i = 1:16    [relswirlTh(i, 1), relswirlTh(i, 2)] = UncertaintyPropagation(...        relthermal, [swirlTh(i, 1) axialTh(1, 1)],...        [swirlTh(i, 2) axialTh(1, 2)]);end handles(1).hFig{6} = figure('Name', 'bar chart');handles(1).hAxes{6} = axes('Parent', handles(1).hFig{6});set(handles(1).hFig{6}, 'Color', 'w');hold on;therm = [relswirlTh(1:4, 1)'; relswirlTh(5:8, 1)';...         relswirlTh(9:12, 1)'; relswirlTh(13:16, 1)'];thermerr = [relswirlTh(1:4, 2)'; relswirlTh(5:8, 2)';...         relswirlTh(9:12, 2)'; relswirlTh(13:16, 2)'];bar(handles(1).hAxes{6}, therm);ngroups = size(therm, 1);nbars = size(therm, 2);% Calculating the width of each group of barsgroupwidth = min(0.8, nbars/(nbars + 1.5));for i = 1:nbars    x = (1:ngroups) - groupwidth/2 + (2*i-1) * groupwidth / (2*nbars);    handles(1).hPe{i} = errorbar(handles(1).hAxes{6}, x,...        therm(:, i), thermerr(:, i), '.');endhold offxlabel(handles(1).hAxes{6}, 'Inlet angle [degrees]');ylabel(handles(1).hAxes{6}, 'Potential energy efficiency [$\%$]');legend(handles(1).hAxes{6}, {'1 inlet', '2 inlets', '3 inlets', '4 inlets'},...          'orientation', 'horizontal', 'Location', 'Northoutside');      colororder(colorsFit(1:4, 1:3) / 255);set(handles(1).hAxes{6}, 'XTick', 1:4, 'XTickLabel', {'45$^o$', '60$^o$', '75$^o$',...        '90$^o$'});handles(1).hAxes{6}.XAxis.TickLength = [0 0];for i = 1:nbars   set(handles(1).hPe{i}, 'Color', colorsErr(5, :) / 255); endPlotDimensions(handles(1).hFig{6}, 'centimeters', [15.747, 8], 12);ChangeInterpreter(handles(1).hFig{6}, 'latex');toc;                        % end code timer runningTime = toc;%% Save results in results.txtres = fopen('results.txt', 'w'); disp('Results printed in the file: results.txt ');fprintf(res, 'Analysis output report, written on %s\n', datetime('now'));fprintf(res, 'Elapsed time: %4.2f seconds \n \n', runningTime);fprintf(res, 'Results are printed in a way that greatly simplifies their implementation in a LaTeX table \n \n');latextbl2 = [stats(2).c(:, 1) .* 10e-10 stats(2).cerr(:, 1) .* 10e-10 stats(2).c(:, 2) stats(2).cerr(:, 2) stats(2).stat(:, 1)];latextbl = [latextbl1 latextbl2];fprintf(res, '------------------------------------------------------ \n');fprintf(res, 'Power fit curves for Nu = aRe^b and P = aQ^b; columns: a ua b ub r^2 \n \n');fprintf(res, '%5.2f & %5.2f & %5.2f & %5.2f & %5.4f && %5.2f & %5.2f & %5.2f & %5.2f & %5.4f \\\\ \n', latextbl');fprintf(res, '\n');latexnu = zeros(4, length(nussavg(1:4, 1)) * 2);latexpw = zeros(4, length(pwavg(1:4, 1)) * 2);c = 0;for i = 1:4        if i == 1    else    c = c + 4;    end       latexpw(i, 1:2:end-1) = pwavg(1+c:4+c, 1);   latexpw(i, 2:2:end) = pwavg(1+c:4+c, 2);   latexnu(i, 1:2:end-1) = nussavg(1+c:4+c, 1);   latexnu(i, 2:2:end) = nussavg(1+c:4+c, 2);endfprintf(res, '------------------------------------------------------ \n');fprintf(res, 'Average Nusselt numbers; rows: angle degrees, columns: number of inlets\n \n');fprintf(res, '& %5.2f & %5.2f && %5.2f & %5.2f && %5.2f & %5.2f && %5.2f & %5.2f \\\\ \n', latexnu');fprintf(res, '\n');% Print results in way a LaTeX-friendly wayfprintf(res, '------------------------------------------------------ \n');fprintf(res, 'Average Power values; rows: angle degrees, columns: number of inlets\n \n');fprintf(res, '& %5.2f & %5.2f && %5.2f & %5.2f && %5.2f & %5.2f && %5.2f & %5.2f \\\\ \n', latexpw');fprintf(res, '\n');latextbl = [nuss(:, 1) nusserr(:, 1) nuss(:, 2) nusserr(:, 2) nuss(:, 3) nusserr(:, 3) nuss(:, 4) nusserr(:, 4)];fprintf(res, '------------------------------------------------------ \n');fprintf(res, 'Thermal improvement index; rows: angle degrees, columns: number of inlets\n \n');fprintf(res, '& %5.2f & %5.2f && %5.2f & %5.2f && %5.2f & %5.2f && %5.2f & %5.2f \\\\ \n', latextbl');fprintf(res, '\n');latextbl = [power(:, 1) powererr(:, 1) power(:, 2) powererr(:, 2) power(:, 3) powererr(:, 3) power(:, 4) powererr(:, 4)];fprintf(res, '------------------------------------------------------ \n');fprintf(res, 'Power improvement index; rows: angle degrees, columns: number of inlets \n \n');fprintf(res, '& %5.2f & %5.2f && %5.2f & %5.2f && %5.2f & %5.2f && %5.2f & %5.2f \\\\ \n', latextbl');% Print results in way a LaTeX-friendly waylatextbl = [therm(:, 1) thermerr(:, 1) therm(:, 2) thermerr(:, 2) therm(:, 3) thermerr(:, 3) therm(:, 4) thermerr(:, 4)];fprintf(res, '------------------------------------------------------ \n');fprintf(res, 'Potential efficiency index; rows: angle degrees, columns: number of inlets \n \n');fprintf(res, '& %5.2f & %5.2f && %5.2f & %5.2f && %5.2f & %5.2f && %5.2f & %5.2f \\\\ \n', latextbl');fclose(res);