- #1

diredragon

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## Homework Statement

Given the Fourier transformation pair ##f(t) \implies F(jw)## where

##f(t) = e^{-|t|}## and ##F(jw)=\frac{2}{w^2+1}## find and make a graph of the Fourier transform of the following functions:

a) ##g(t)=\frac{2}{t^2+1}##

b) ##h(t) = \frac{2}{t^2+1}\cos (w_ot)##

## Homework Equations

3. The Attempt at a Solution [/B]

We have just recently started learning about Fourier Transforms and last class we mentioned the duality principle of the Fourier transformations and i didn't quite understand what it was all about. I suppose this is what should be applied here. I made some calculations and wanted to check if they are correct.

So the principle should be:

If ##f(t) \implies F(jw)## then ##F(t) \implies 2\pi f(-w)##

a) ##g(t)=\frac{2}{t^2+1}##

##g(t)## is clearly ##F(t)## so ##F_g(jw)## should be:

##F_g(jw) = 2\pi e^{-|w|}## because of the duality.

b)##h(t) = \frac{2}{t^2+1}\cos (w_ot)##

This one is a little trickier but it boils down to ##h(t)=g(t)\cos (w_ot)## and since ##f(t)e^{-jw_ot} \implies F(j(w-w_o))## then it's clearly:

##F_h(t) = \frac{1}{2}(F_g(w-w_o)+F_g(w+w_o))##

Is this correct? Could you also refer me to a source which explains this principle more in depth?