(New page: == HW4.1 == periodic signal: <math> \begin{align} x(t) = 32 + 8sin(2\pi t) + 22cos(2\pi t) + 2cos(2\pi t+\pi /2)\end{align} </math> re-writing the signal in the form of <math> \begin{alig...) |
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Revision as of 07:52, 26 September 2008
HW4.1
periodic signal: $ \begin{align} x(t) = 32 + 8sin(2\pi t) + 22cos(2\pi t) + 2cos(2\pi t+\pi /2)\end{align} $
re-writing the signal in the form of $ \begin{align} e^{jw_0t} \end{align} $ we get $ \begin{align} x(t) = 32 + \frac{8}{2j} \left ( e^{j 2 \pi t} - e^{-j 2 \pi t} \right ) + 11 \left ( e^{j 2 \pi t} + e^{-j 2 \pi t} \right ) + \left ( e^{j 2 \pi t + \pi /2 } + e^{-j 2 \pi t + \pi /2} \right ) \end{align} $
then we can convert these to the coefficients of the Fourier series
$ \begin{align} a_0 = 32 \\ a_1 = 11 - 4/j \\ a_{-1} = 11 + 4/j \\ a_2 = e^{ \pi /2} \end{align} $
