Basic Knowledge Review
Fourier Series Expansion of Continuous Time Periodic Signal ( Period T )
Fourier complex coefficients 
, fundamental frequency 
Fourier transform pairs of continuous-time non-periodic signals
1. Exponential Fourier series of periodic rectangular pulse signal
% 计算并画出周期矩形脉冲信号的双边幅度谱和相位谱。
clc;clear;close all
A = 1; % 幅度
tau = 0.1; % 脉冲宽度
T = 0.5; % 周期
Omega_0 = 2*pi/T; % 基波频率 Ω0
K = 2*pi/tau/Omega_0;
k = 0 : 2*K;
F_k = A*tau/T*sinc(k*Omega_0*tau/2/pi); % 系数F_k
F_k_mag=abs(F_k); % F_k的幅度
F_k_phase = angle(F_k); % F_k的相位
k= -2*K : 2*K;
F_k_mag = [fliplr(F_k_mag(2:end)) F_k_mag]; % fliplr:左右翻转矩阵
F_k_phase = [ -fliplr(F_k_phase(2:end)) F_k_phase];
subplot(2,1,1)
stem(k*Omega_0, F_k_mag);
xlabel('k \Omega_o');
ylabel('magnitude');
grid
subplot(2,1,2)
stem(k*Omega_0, F_k_phase);
xlabel('k \Omega_o ');
ylabel('phase');
grid
Two-sided amplitude spectrum and phase spectrum of periodic rectangular pulse signal at T = 0.5
When the period parameter T increases:
Two-sided amplitude spectrum and phase spectrum of periodic rectangular pulse signal at T = 0.5
Two-sided amplitude spectrum and phase spectrum of periodic rectangular pulse signal at T = 0.5
2. The superposition of each harmonic of the periodic rectangular pulse signal
Periodic rectangular pulse signal:
Triangular Fourier series: 
, where
The first N harmonics are superimposed to obtain 
.
% 来计算画出前N次谐波叠加得到的周期矩形脉冲信号近似波形。
clc;clear;close all
t = -2 : 10^(-4) : 2;
A = 1;
tau = 1; % 脉冲宽度
T = 2; % 周期
Omega_0 = 2*pi/T; %基波频率
c0 = A*tau/T;
N = input('N=');
f_N = c0 * ones(1,length(t)); %f_N(t)
for k= 1 : 1 : N
f_N = f_N + 2*A*tau/T *sinc(k*Omega_0*tau/ 2/pi) *cos(k*Omega_0*t);
end
plot(t, f_N);
xlabel('t');
ylabel('f_N(t)');
title(['N=',num2str(N)])
axis([-2 2 -0.2 1.2]);Plot for N=25
Plot for N=50
Plot for N=1000
It can be seen from the figure that with the increase of N, the synthesized waveform is getting closer to the original rectangular pulse signal.