Homework Three

October 5th, 2009, class notes (as interpreted by Nick Christman)

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The topic covered in class on October 5th was about how to deal with signals that are not periodic.

Given the following fourier series, what if the signal is not periodic?

$x(t)=x(t+T)=\sum _{n=-\infty }^{\infty }\alpha _{n}e^{\frac {j2\pi nt}{T}}$ where $\alpha _{n}={\frac {1}{T}}\int _{-{\frac {T}{2}}}^{\frac {T}{2}}x(t')e^{-{\frac {j2\pi nt'}{T}}}\,dt'$

To investigate this potential disaster, let's look at what happens as the period increases (i.e. not periodic). Essentially, as $T\rightarrow \infty$ we can say the following:

${\frac {1}{T}}$	$\rightarrow$	$\,df$
${\frac {n}{T}}$	$\rightarrow$	$\,f$
$\sum _{n=-\infty }^{\infty }{\frac {1}{T}}$	$\rightarrow$	$\int _{-\infty }^{\infty }({\mbox{ }})\,df$

With this, we get the following:

$x(t)=\lim _{T\to \infty }\left[\sum _{n=-\infty }^{\infty }\left({\frac {1}{T}}\int _{-{\frac {T}{2}}}^{\frac {T}{2}}x(t')e^{-{\frac {j2\pi nt'}{T}}}\,dt'\right)e^{\frac {j2\pi nt}{T}}\right]\equiv \int _{-\infty }^{\infty }\left(\int _{-\infty }^{\infty }\,x(t')e^{-j2\pi ft'}\right)e^{j2\pi tf}\,df$

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