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1) Please can my answers be checked, including my final Fourier series.

2) Is it possible to use Wolframalpha to check my answers? If so, how will I go about doing this?

Deduce the Fourier series for the following period waveform (the waveform is given for 1 period):

$$x(t)=\left\{ \begin{align} & 2\quad\quad\quad\quad,0<=t<=T/2 \\ & -1\quad\quad\quad, T/2<=t<=t \\ \end{align} \right.$$

My Answers:

$$x(t)=\left\{ \begin{align} & 2\quad\quad\quad\quad,0<=t<=1/2 \\ & -1\quad\quad\quad, 1/2<=t<=1 \\ \end{align} \right.$$

Calculated the 3 coefficients:

$$a_0 = 1$$ $$a_n = 0$$ $$b_n = \frac{1}{\pi.n}(2+cos\pi.n -3(-1)^2)$$

The final Fourier Series:

$$x(t)=\frac{1}{2} +\frac{1}{\pi}[(5+cos2\pi)sin(2\pi.t) + \frac{(6+cos6\pi).sin(6\pi.t)}{3} + ...] + \frac{1}{\pi}[\frac{(cos4\pi -1).sin(4\pi.t)}{2} + \frac{(cos8\pi-1).sin(8\pi.t)}{4} +...]$$

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  • $\begingroup$ Simplify the constant sines/cosines ! $\endgroup$ – Yves Daoust Jul 10 '16 at 8:46
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Except for the constant term (indeed $\frac12$), this is a so-called square waveform, the transform of which is well-know.

See http://mathworld.wolfram.com/FourierSeriesSquareWave.html or https://en.wikipedia.org/wiki/Square_wave.

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For convenience, we first deduce the mean value

$$a_0=\int_0^{1/2}2\,dt-\int_{1/2}^1dt=\frac12.$$

Then the function is odd, so there will be sine terms only, and by symmetry we can integrate on a half period

$$b_n=2\int_0^{1/2}\frac32\sin(2\pi nt)\,dt=-\left.\frac 3{2\pi n}\cos(2\pi nt)\right|_0^{1/2}=-3\frac{\cos(\pi n)-1}{2\pi n}.$$

Only the terms for odd $n$ are nonzero and

$$b_{2m+1}=\frac3{\pi(2m+1)}.$$

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  • $\begingroup$ Hi Yves, i'm a little confused. what is the formula that I will have to use for $$a_0$$ and the other coefficients? I thought $$a_0 = 1$$ $\endgroup$ – joe Jul 10 '16 at 9:25
  • $\begingroup$ @joe: depends on your exact defintion of $a_0$. $\endgroup$ – Yves Daoust Jul 10 '16 at 9:28
  • $\begingroup$ thanks Sir, so in this case what I did is totally incorrect? Or I should use another formula to for for each coefficient? $\endgroup$ – joe Jul 10 '16 at 9:31

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