Expand the following signal into its trigonometric Fourier S

Expand the following signal into its trigonometric Fourier Series. Show all working.

Derive the exponential Fourier Series of the following signal (triangle waveform):

Solution

Matlab codes to find trigonometric fourier series;

%clearing window, variables and figures
clear all;
close all;
clf;
clc;
syms t;
f=0.5+0.4*exp(-t/2);
sum=0;
for k=0:1:10
if k==0
a=(1/(2*pi))*int(f*cos(k*t),-pi,pi);
end
a=(1/pi)*int(f*cos(k*t),-pi,pi);
b=(1/pi)*int(f*sin(k*t),-pi,pi);
sum=sum+a*cos(k*t)+b*sin(k*t);
end
sum %trigonometric fourier series
Result;

(5734161139222659*pi)/18014398509481984 + (5734161139222659*exp(-pi/2)*(exp(pi) - 1))/22517998136852480 - (5734161139222659*exp(-pi/2)*cos(t)*(exp(pi) - 1))/112589990684262400 - (5734161139222659*exp(-pi/2)*sin(t)*(exp(pi) - 1))/56294995342131200 + (5734161139222659*cos(2*t)*exp(-pi/2)*(exp(pi) - 1))/382805968326492160 - (5734161139222659*cos(3*t)*exp(-pi/2)*(exp(pi) - 1))/833165931063541760 + (441089318401743*cos(4*t)*exp(-pi/2)*(exp(pi) - 1))/112589990684262400 - (5734161139222659*cos(5*t)*exp(-pi/2)*(exp(pi) - 1))/2274317811822100480 + (5734161139222659*cos(6*t)*exp(-pi/2)*(exp(pi) - 1))/3265109729843609600 - (5734161139222659*cos(7*t)*exp(-pi/2)*(exp(pi) - 1))/4436045632959938560 + (5734161139222659*cos(8*t)*exp(-pi/2)*(exp(pi) - 1))/5787125521171087360 - (441089318401743*cos(9*t)*exp(-pi/2)*(exp(pi) - 1))/562949953421312000 + (5734161139222659*cos(10*t)*exp(-pi/2)*(exp(pi) - 1))/9029717252877844480 + (5734161139222659*sin(2*t)*exp(-pi/2)*(exp(pi) - 1))/95701492081623040 - (17202483417667977*sin(3*t)*exp(-pi/2)*(exp(pi) - 1))/416582965531770880 + (441089318401743*sin(4*t)*exp(-pi/2)*(exp(pi) - 1))/14073748835532800 - (5734161139222659*sin(5*t)*exp(-pi/2)*(exp(pi) - 1))/227431781182210048 + (17202483417667977*sin(6*t)*exp(-pi/2)*(exp(pi) - 1))/816277432460902400 - (40139127974558613*sin(7*t)*exp(-pi/2)*(exp(pi) - 1))/2218022816479969280 + (5734161139222659*sin(8*t)*exp(-pi/2)*(exp(pi) - 1))/361695345073192960 - (3969803865615687*sin(9*t)*exp(-pi/2)*(exp(pi) - 1))/281474976710656000 + (5734161139222659*sin(10*t)*exp(-pi/2)*(exp(pi) - 1))/451485862643892224

Continuation of matlab codes to find exponential fourier series;

f1=0.4+0.48*t;
f2=1.6-0.48*t;
sum=0;
for k=-10:1:10
a=int(f1*exp(-1i*2*pi*k*t),0,2.5/2)+int(f2*exp(-1i*2*pi*k*t),1.25,2.5);
sum=sum+(1/2*pi)*a*exp(1i*2*pi*k*t);
end
simplify(sum, \'Steps\', 100)

Result;

((6*sin(2*pi*t))/25 - (2*sin(6*pi*t))/75 + (6*sin(10*pi*t))/625 - (6*sin(14*pi*t))/1225 + (2*sin(18*pi*t))/675 + (2*pi*sin(2*pi*t))/5 + (2*pi*sin(6*pi*t))/15 + (2*pi*sin(10*pi*t))/25 + (2*pi*sin(14*pi*t))/35 + (2*pi*sin(18*pi*t))/45 + (7*pi^2)/8 + (6*sin(2*pi*t)^2)/25 + (2*sin(6*pi*t)^2)/75 + (6*sin(10*pi*t)^2)/625 - 259/1875)/pi