Reconstructing a function from its sine Fourier series

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pat666
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Homework Statement


Hey,
This is a question from a sample exam we are given for our engineering maths exam. Firstly given that the Fourier series contains only "sin(x)" doesn't this mean that it is an "odd" function? Can the period be calculated from the given function easily?

Secondly is it possible to draw the "deleted diagram" from the given Fourier series (we haven't learn't this but it should be shouldn't it)? If it is can someone please draw it and then help me do question iii.

Thankyou

PS, if anyone is willing to draw it or get matlab/mathematica to, can you post it and then I will attempt the question and repost.

Homework Equations





The Attempt at a Solution

 
Attachments
  • Exam question page with text: Below is a diagram depicting a portion of a function r(t) with a maximum value of 1 and a minimum value of -1. Diagram Deleted. Parts (i) odd or even, (ii) period, (iii) show the Fourier series -4/pi(sin t + sin 3t/3 + sin 5t/5 + ...) = -4/pi(sin nt / n), for n = 1, 3, 5...
    sample exam.png
    100.4 KB · Views: 617

Discussion

pat666 said:

Homework Statement


Hey,
This is a question from a sample exam we are given for our engineering maths exam. Firstly given that the Fourier series contains only "sin(x)" doesn't this mean that it is an "odd" function? Can the period be calculated from the given function easily?

Secondly is it possible to draw the "deleted diagram" from the given Fourier series (we haven't learn't this but it should be shouldn't it)? If it is can someone please draw it and then help me do question iii.

Thankyou

PS, if anyone is willing to draw it or get matlab/mathematica to, can you post it and then I will attempt the question and repost.

Homework Equations


The Attempt at a Solution

If your Fourier expansion has only sine terms, it is indeed an odd function, so the An coefficient is 0. I suppose by "max and min of 1 and -1" they mean [tex]t\in[-1,1][/tex]

r(t) could be the unit step function scaled to -4, when 0<t<1. That would give you the Bn coefficient you've got, but it could've been something else too I think.

Edit: Noticed there's no -4 as a 0th term in the series, so I'm probably wrong. Looks like you need a function that integrated from -1 to 1 gives you zero, but integrated while multiplied by sin(n*t)/pi gives you -4/pi*n
 
is this a reverse square wave?
the series looks similar to this?
 
Attachments
  • Typeset math: odd square wave f(x) equals 1 for 0 < x < l and -1 for -l < x < 0, with its sine Fourier series sum over odd n of (4/(pi n)) sin(n pi x / l) equal to (4/pi)(sin(pi x/l) + sin(3 pi x/l)/3 + sin(5 pi x/l)/5 + ...).
    Screen shot 2011-05-30 at 11.26.50 PM.png
    7.8 KB · Views: 596

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