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Fourier Series of Even Square Wave 
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#1
Mar712, 08:32 PM

P: 778

1. The problem statement, all variables and given/known data
[itex]0.5\leq{t}\leq{1.5}, T=2[/itex] The wave is the attached picture. I need to determine the Fourier Series of the wave in the picture. I know that [tex]f(t)=a_0+{\sum}_{n=1}^{\infty}a_ncos(n\omega_0t)+{\sum}_{n=1}^{\infty}b _nsin(n\omega_0t)[/tex] where [itex]a_0=\bar{f}=0[/itex] due to being an even function. Furthermore, [itex]b_n=0[/itex] due to being an even function also. That leaves, [tex]a_n=\int_{0.5}^{0.5}cos(n\omega_0t)dt\int_{0.5}^{1.5}cos(n{\omega}_0t)dt=0\frac{1}{n\omega_0}(sin(1.5n{\omega}_0t)sin(0.5n{\omega}_0t))=\frac{1}{n{\omega}_0}(sin(0.5n{\omega}_0t)sin(1.5n{\omega}_0t))[/tex] therefore, [tex]f(t)=\frac{1}{{\omega}_0}\sum_{n=1}^{\infty}cos(n{\omega}_0t)(sin(0.5n{ \omega}_0t)sin(1.5n{\omega}_0t))[/tex] Is this suffice as an answer, or am I missing something? My textbook is lacking examples so I just would like to know if I am doing it right. Thanks! 


#2
Mar712, 10:30 PM

P: 778

I also had encountered another equation deeper in the chapter that states
[tex]f(t)=A/2+(2A/\pi)\sum_{n=1}^{\infty}\frac{sin((2n1)\omega{_0}t)}{2n1}[/tex] 


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