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

2) Is it possible to use Wolfram Alpha 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):

\begin{align} x(t) = \begin{cases} 2 & 0 \leq t \leq \frac{T}{2} \\ -1 & \frac{T}{2} < t \leq T \end{cases} \end{align}

My Answers: \begin{align} x(t) = \begin{cases} 2 & 0 \leq t \leq \frac{1}{2} \\ -1 & \frac{1}{2} < t \leq 1 \end{cases} \end{align}

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:

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

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  • $\begingroup$ Simplify the constant sines/cosines ! $\endgroup$
    – user65203
    Commented Jul 10, 2016 at 8:46

2 Answers 2

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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
    Commented Jul 10, 2016 at 9:25
  • $\begingroup$ @joe: depends on your exact defintion of $a_0$. $\endgroup$
    – user65203
    Commented Jul 10, 2016 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
    Commented Jul 10, 2016 at 9:31
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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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