HW 06

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Problem

Figure out why \int_{0}^{\infty} \cos(2\pi\,f\,u)\,du 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 a real odd function of time, \sgn(t)\,\!,should map to an imaginary odd function of frequency.

Proof

F[o(t)]\,\! =\int_{-\infty}^{\infty}\,o(t)\,e^{-j\,2\,\pi\,f\,t}\,dt
=\int_{-\infty}^{\infty}\,o(t)\,\left[\cos(2\,\pi\,f\,t)+j\,\sin(2\,\pi\,f\,t)\right]\,dt Euler's identity
=\int_{-\infty}^{\infty}\,o(t)\,j\,\sin(2\,\pi\,f\,t)\,dt Even function integrates out over symmetric limits
=\int_{-\infty}^{\infty}\,\left[\mbox{Im }e(t) \mbox{ and an Im }o(f)\right]\,dt
=\mbox{Im }o(f)\,\! Time integrates out
  • The odd function of time has no component (ie. 0) of frequency. Thus it is an even function in frequency.

Functions

  • Even*Even=Even
  • Odd*Odd=Even
  • Odd*Even=Odd

Incorrect Solution derived in class

F\left[\frac{\sgn (t)}{2}\right] =\int_{-\infty}^{\infty} \frac{\sgn (t)}{2} e^{-j\,2\,\pi\,f\,t}\,dt
=\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]
=\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}}
=\int_{0}^{-\infty} \frac{e^{j\,2\,\pi\,f\,u} + e^{-j\,2\,\pi\,f\,u}}{2}\,du
=\int_{0}^{-\infty} \cos(2\,\pi\,f\,u)\,du

Correct Solution

F\left[\frac{\sgn (t)}{2}\right] =\int_{-\infty}^{\infty} \frac{\sgn (t)}{2} e^{-j\,2\,\pi\,f\,t}\,dt
=\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]
=\frac{1}{2}\left[\int_{0}^{-\infty}  1\cdot e^{-j\,2\,\pi\,f\,t}\,dt+\int_{0}^{\infty} 1\cdot e^{-j\,2\,\pi\,f\,t}\,dt\right]
=\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}}
=\int_{0}^{\infty} \frac{-e^{j\,2\,\pi\,f\,u} + e^{-j\,2\,\pi\,f\,u}}{2}\,du
=\int_{0}^{\infty} -j\,\frac{e^{j\,2\,\pi\,f\,u} - e^{-j\,2\,\pi\,f\,u}}{2j}\,du
=\int_{0}^{\infty} -j\,\sin(2\,\pi\,f\,u)\,du
\ne \int_{0}^{\infty} \cos(2\,\pi\,f\,u)\,du