(New page: ---- <br>==Answer== <math> a_n = \frac{1}{T}\int_{0}^{T}x(t)e^{-j\frac{2\pi}{T}n t} dt = \int_{0}^{1}x(t)e^{-j2\pi n t}</math> Substitute <math>a=\pi, \;\; b=-j2\pi n...) |
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<math> a_n = \frac{1}{T}\int_{0}^{T}x(t)e^{-j\frac{2\pi}{T}n t} dt = \int_{0}^{1}x(t)e^{-j2\pi n t}</math> | <math> a_n = \frac{1}{T}\int_{0}^{T}x(t)e^{-j\frac{2\pi}{T}n t} dt = \int_{0}^{1}x(t)e^{-j2\pi n t}</math> | ||
Revision as of 09:35, 28 September 2010
==Answer==
<math> a_n = \frac{1}{T}\int_{0}^{T}x(t)e^{-j\frac{2\pi}{T}n t} dt = \int_{0}^{1}x(t)e^{-j2\pi n t}</math>
Substitute <math>a=\pi, \;\; b=-j2\pi n</math>
<math>
\begin{align}
a_n &= \int_{0}^{1}\text{sin}(at)e^{bt}dt = \left[\frac{1}{b}\text{sin}(at)e^{bt}\right]^{1}_{0} - \int_{0}^{1}frac{a\text{cos}(at)}{b}e^{bt}dt \\
&= \left(\frac{e^b}{b}\text{sin}(a)\right) - \frac{a}{b}\left(\left[\frac{\text{cos}(at)}{b}e^{bt}\right]^{1}_{0} - \int_{0}^{1}\frac{-a\text{sin}(at)}{b}e^{bt}dt\right) \\
&= \left(\frac{e^b}{b}\text{sin}(a)\right) - \frac{a}{b} \left( \left( \frac{\text{cos}(a)}{b}e^{b}-\frac{1}{b}\right) + \frac{a}{b}\int_{0}^{1}\text{sin}(at)e^{bt}dt \right) \\
&= \frac{e^b}{a}\text{sin}(a)-\frac{a\text{cos}(a)}{b^2}e^b + \frac{a}{b^2} - \frac{a^2}{b^2}\int_{0}^{1}\text{sin}(at)e^{bt}dt \\
\end{align}
</math>
As you can see <math>\int_{0}^{1}\text{sin}(at)e^{bt}dt</math> term repeats, therefore it needs to be subtracted to both sides.
Then, <math>a_n</math> can be calculated.
<math>
\frac{e^{-j2\pi n}}{-j2\pi n}\text{sin}\pi + \frac{\pi \text{cos}\pi}{4 \pi^2 n^2}e^{-j2\pi n} - \frac{\pi}{4 \pi^2 n^2} = \left(1-\frac{\pi^2}{4\pi^2 n^2}\right) \int_{0}^{1}\text{sin}(\pi t) e^{-j2\pi n t}dt = \left(1-\frac{\pi^2}{4\pi^2 n^2}\right) a_n
</math>
From this equation, <math>\frac{e^{-j2\pi n t}}{-j2\pi n}\text{sin}\pi=0</math> because of <math>\text{sin}\pi</math>, and <math>\frac{\pi \text{cos}\pi}{4 \pi^2 n^2}e^{-j2\pi n} = \frac{-\pi}{4\pi^2 n^2}</math>
There are two terms of <math>-\frac{\pi}{4\pi^2 n^2}</math>.
<math>a_n=\frac{-\frac{2\pi}{4\pi^2 n^2}}{1-\frac{\pi^2}{4\pi^2 n^2}} = -\frac{2\pi}{4\pi^2 n^2 - \pi^2} = \frac{2}{\pi(1-4n^2)}</math>
