HW 06: Difference between revisions
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==Problem== |
==Problem== |
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Figure out why <math>\int_{0}^{\infty} \cos(2\pi\,f\,u)\,du</math> seems to equal an imaginary odd function of frequency, but there is no j. |
Figure out why <math>\int_{0}^{\infty} \cos(2\pi\,f\,u)\,du</math> seems to equal an imaginary odd function of frequency, but there is no j. |
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==Background== |
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This is the incorrect solution derived in class. Cosine is incorrect, because an odd function of time, <math>\sgn(t)\,\!</math>,should map to |
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==Solution== |
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{| border="0" cellpadding="0" cellspacing="0" |
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|<math>F\left[\frac{\sgn (t)}{2}\right]</math> |
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|<math>=\int_{-\infty}^{\infty} \frac{\sgn (t)}{2} e^{-j\,2\,\pi\,f\,t}\,dt</math> |
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|<math>=\frac{1}{2}\left[\int_{-\infty}^{0} -1\cdot e^{-j\,2\,\pi\,f\,t}\,dt+\int_{0}^{\infty} 1\cdot e^{-j\,2\,\pi\,f\,t}\,dt\right]</math> |
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|<math>=\underbrace{\frac{1}{2}\int_{0}^{-\infty} e^{j\,2\,\pi\,f\,u}\,du}_{\begin{matrix}u=-t \\ du=-dt\end{matrix}}+\underbrace{\frac{1}{2}\int_{0}^{\infty} e^{-j\,2\,\pi\,f\,u}\,du}_{\begin{matrix}u=t \\ du=dt\end{matrix}}</math> |
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|<math>=\int_{0}^{-\infty} \frac{e^{j\,2\,\pi\,f\,u} + e^{-j\,2\,\pi\,f\,u}}{2}\,du</math> |
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|<math>=\int_{0}^{-\infty} \cos(2\,\pi\,f\,u)\,du</math> |
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==Solution== |
==Solution== |
Revision as of 14:00, 2 December 2008
Problem
Figure out why seems to equal an imaginary odd function of frequency, but there is no j.
Background
This is the incorrect solution derived in class. Cosine is incorrect, because an odd function of time, ,should map to