Fourier Series Question

Webmonkey

Hi There, Just messing about with some maths there.

I've a little problem.

Question: A function is defined by:

f(t)= 6 when 0<=t<0.5
f(t)= -6 when 0.5<=t<1

and f(t) = f(t+1)

Find the fourier series representation of the function.

My (obviously wrong) Solution:

the T , time in this is 1, thus omega w = 2*PI

This function is an odd function seeing that f(t) = -f(-t).
Thus Sine terms only.

So:



So we only got bn term:



Now solving the bn, i get:

48/n if n is odd
0 if n is even

This is the case, as you get a 1-1 of cosine terms when even thus calapsing whole thing to 0.

Now the fourier series is:



Since a0,an = 0 we get:



Now by plotting this answer, i get this:




Don't suppose anyone know why this is the case?

Should go from +6 to -6, not +36 -36 what ever it is. Can't read accuretly enough

Thanks.

Michael Collins

Did you forget that the period T = 2pi when you calculated Bn? I think the answer for Bn should be 24/ (n.pi)

Webmonkey

You're dead right - Just discovered this messing about here now.

Will T always be = 2Pi for bn?

Michael Collins

Ah, dammit...good question there, my last post is wrong. T of course equals 1 as per the period. Omega w = 2pi, however, and this will come out as a denominator in the integration of sin(n.w.t). You must have forgotten this part?

There's a few differnent forumlae for getting Bn for a Fourier series (they all work out the same way in the end), I was getting confused between them...

Webmonkey

Funnly enough using T as 2Pi when working out the Bn, I end up with 24/n*pi.

This plots perfectly, with the amplitude as 6. This is really confusing!

Michael Collins

Yeah, 24 / (n.pi) is the correct answer for Bn alright. But it's just a coincidence that it works out this time.

I knew what the answer should be so I could see you were out by a factor of 2pi and for some reason I thought T should be 2pi, but nope, T is always the period.

The factor of 2pi you're missing is from the integration of sin(n.w.t), when you integrate that it becomes -cos(n.w.t) / (n.w), but w = 2pi, so there you have it!

Webmonkey

Ah that makes sense now.

My lecture always said that sin(n.w.t) was -cost(n.t)/n as w is mostly always one by us.

Glad got that sorted now, plus better understanding now.

Thanks excellent stuff

Michael Collins

No bother. Yeh sometimes courses will make the equations simplier by eliminating pi etc. but I think this leads to more confusion at some point down the line (like today!).

Out of curiosity, how are you plotting the series? Using MATLAB? If so have you got an m-file for that purpose?

Webmonkey

I'm using Maple, find it very handy for checking things. Plus our course is done with maple too so we got a nice little file running for each topic with step by step.