| %% ======================================================================
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| % 实验内容 1：方波信号的分解与合成（合集脚本）
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| % 包括：
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| % (1) 方波谐波分解（实验 1.1）
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| % (2) 方波谐波合成对比（实验 1.2）
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| % ======================================================================
|
|
|
| clc; clear; close all;
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|
|
| %% ----------------------------------------------------------------------
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| % 实验内容 1.1：方波信号的傅里叶级数分解
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| % ----------------------------------------------------------------------
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|
|
| A = 3; % 幅值
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| f = 50; % 基频 50Hz
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| w0 = 2*pi*f; % 角频率
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| T = 1/f;
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|
|
| t = 0 : 0.0001 : 2*T; % 时间：两个周期
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|
|
| harmonics_idx = [1,3,5,7,9];
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| N_terms = length(harmonics_idx);
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| harmonics_data = zeros(N_terms, length(t));
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|
|
| coeff_base = 4*A/pi;
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|
|
| % 计算谐波分量
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| for i = 1:N_terms
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| k = harmonics_idx(i);
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| harmonics_data(i,:) = (coeff_base / k) * sin(k*w0*t);
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| end
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|
|
| % ------------------------ 图 1：各谐波分量（分图显示） ------------------
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| figure('Name','方波分解 - 单独谐波','Color','w');
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| for i = 1:N_terms
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| subplot(N_terms,1,i);
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| plot(t, harmonics_data(i,:), 'LineWidth',1.5);
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| grid on;
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| ylabel(['k=',num2str(harmonics_idx(i))]);
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| if i==1
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| title('方波的前 5 项谐波分量（50Hz，幅值 3）');
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| end
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| end
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| xlabel('时间 t (s)');
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|
|
| % ------------------------ 图 2：谐波的三维 mesh 图 ----------------------
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| figure('Name','方波分解 - 3D Mesh','Color','w');
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| [X, Y] = meshgrid(t, harmonics_idx);
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| mesh(X, Y, harmonics_data);
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| view(3);
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| title('方波各次谐波的三维波形');
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| xlabel('时间 t (s)');
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| ylabel('谐波序号 k');
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| zlabel('幅值');
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| colormap jet;
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|
|
| % ------------------------ 图 3：谐波叠加 2D ------------------------------
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| figure('Name','方波分解 - 2D叠加','Color','w');
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| hold on;
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| colors = ['r','g','b','m','k'];
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| for i=1:N_terms
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| plot(t, harmonics_data(i,:), 'Color',colors(i),'LineWidth',1.3);
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| end
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| title('前五谐波在同一坐标系叠加');
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| xlabel('时间 t (s)');
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| ylabel('幅值');
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| legend("k=1","k=3","k=5","k=7","k=9");
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| grid on;
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|
|
|
|
| %% ----------------------------------------------------------------------
|
| % 实验内容 1.2：方波信号的傅里叶级数合成
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| % ----------------------------------------------------------------------
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| % =========================== 执行两种情况 ===============================
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|
|
| % x1(t): |t|<1 → 1 , |t|>1 → 0
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| % tau = 2, T = 4
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| synth_square_wave(4, 2, [5,20,100], 'x1(t): T=4, 宽度 2');
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|
|
| % x2(t): |t|<0.5 → 1 , |t|>0.5 → 0
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| % tau = 1, T = 2
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| synth_square_wave(2, 1, [5,20,100], 'x2(t): T=2, 宽度 1');
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|
|
| % =================== 定义一个用于合成的函数 ============================
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| function synth_square_wave(T, tau, N_values, fig_title)
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|
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| t_limit = 1.5 * T;
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| t = -t_limit : 0.001 : t_limit;
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| w0 = 2*pi/T;
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|
|
| % 构造理想方波
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| original = zeros(size(t));
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| for i = 1:length(t)
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| t_mod = mod(t(i) + T/2, T) - T/2;
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| if abs(t_mod) < tau/2
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| original(i) = 1;
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| end
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| end
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|
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| figure('Name', fig_title, 'Color', 'w');
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|
|
| for k = 1:length(N_values)
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| N = N_values(k);
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| f_synth = zeros(size(t));
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|
|
| % 直流项
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| a0 = tau/T;
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| f_synth = f_synth + a0;
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|
|
| % 余弦系数 an
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| for n = 1:N
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| an = 2*(tau/T)*sinc(n*tau/T);
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| f_synth = f_synth + an*cos(n*w0*t);
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| end
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|
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| subplot(length(N_values),1,k);
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| plot(t, original,'k--','LineWidth',1); hold on;
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| plot(t, f_synth,'r','LineWidth',1.5);
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| title(['谐波项数 N = ',num2str(N)]);
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| legend('理想方波','合成波形');
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| ylim([-0.2,1.2]);
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| grid on;
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| end
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| end
|