Solving a Fourier Series for a Saw-Tooth Wave

[rusty009]

Hey, I am trying to solve this question:

obtain a Fourier series for a saw-tooth wave, a periodic signal, with period T, defined such that

x(t)=At -T/2<= t >= T/2

where A has a value of 1 at the maximum value of x(t)

i) obtain the Fourier series for this periodic signal in form

x(t)= $$\frac{A0}{2}$$+$$\sum$$[An*cos(2*pi*j*n*f0*t) + Bn*sin(2*pi*j*n*f0*t)]

where the limits of the $$\sum$$ are infinity and n=1

then,

ii) obtain the series in the form

x(t)= $$\sum$$Cn* exp(2*pi*j*n*f0*t)

where the limits of the $$\sum$$ are infinity to n=- infinity

Thanks in advance !

 

[mheslep]

Hey, I am trying to solve this question:

obtain a Fourier series for a saw-tooth wave, a periodic signal, with period T, defined such that

x(t)=At -T/2<= t >= T/2

where A has a value of 1 at the maximum value of x(t)

i) obtain the Fourier series for this periodic signal in form

x(t)= $$\frac{A0}{2}$$+$$\sum$$[An*cos(2*pi*j*n*f0*t) + Bn*sin(2*pi*j*n*f0*t)]

where the limits of the $$\sum$$ are infinity and n=1

then,

ii) obtain the series in the form

x(t)= $$\sum$$Cn* exp(2*pi*j*n*f0*t)

where the limits of the $$\sum$$ are infinity to n=- infinity

Thanks in advance !

A0=0 from inspection the average of the ramp signal is zero.
An=0 since the signal is odd, no cosine terms allowed.
Bn left as an exercise to the reader

 

[rusty009]

Hey, I don't really understand. Firsty, what is A ? And could explain into detail the way you found Ao and An, thanks a lot !

 

[Redbelly98]

A is the slope of x(t), according to the definition you gave us.

Ao, An, and Bn can be found from working out the integrals given in the definition of Fourier Series. People who are experienced with Fourier Series learn to recognize situations where these are zero. Don't worry about that if it's not clear to you, just work out those integrals to find Ao, An, and Bn.