Fourier Transforms

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Fourier Transforms

A Fourier transform is used to transform a function from the time domain into one from the frequency domain. This can be very useful, because it allows easier minipulation of the function in many cases. The Fourier Transform is often written this way:

$X(f)=\int _{-\infty }^{\infty }x(t)e^{-j2\pi ft}\,dt$

This takes your time function, $x(t)$ and makes it into a frequency function $X(f)$. This function also has an inverse, namely, the inverse Fourier Transform, and it looks like this:

$x(t)=\int _{-\infty }^{\infty }X(f)e^{j2\pi ft}\,df$

This does exactly the opposite of the Fourier Transform. Instead of taking time and converting it to frequency, the inverse transform takes a function in frequency, $X(f)$, and converts it into a time function, $x(t)$.

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Fourier Transform Pairs

Most people don't want to do the Fourier Transform integrals all that often, but fortunately it is possible to put general Fourier Transform/Inverse Fourier Transform pairs together, and then from those pairs, perform the particular Fourier Transform you are working on.

${\mathcal {F}}[x(t)]=X(f)$
${\mathcal {F}}[dx/dt]=j2\pi fX(f)$
${\mathcal {F}}[x(t-t_{0})]=e^{-j2\pi ft_{0}}X(f)$
${\mathcal {F}}[e^{j2\pi f_{0}t}x(t)]=X(f-f_{0})$
${\mathcal {F}}[\cos(2\pi f_{0}t)x(t)]={\frac {1}{2}}X(f-f_{0})+{\frac {1}{2}}X(f+f_{0})$
${\mathcal {F}}[x(at)]={\frac {1}{\left|a\right\vert }}X\left({\frac {f}{a}}\right)$
${\mathcal {F}}[x(t)*y(t)]=X(f)Y(f)$
${\mathcal {F}}[x(t)y(t)]=X(f)*Y(f)$
${\mathcal {F}}[\sum _{n=-\infty }^{\infty }\alpha _{n}e^{\frac {j2\pi ft}{T}}]=\sum _{n=-\infty }^{\infty }\alpha _{n}\delta \left(f-{\frac {n}{t}}\right)$
The last transform is actually the transform of a Fourier Series. As you can see, when you take the Fourier Transform of a Fourier Series, you get out the frequency components, $\alpha _{n}\delta \left(f-{\frac {n}{t}}\right)$, of the function.
Another thing that you might notice is the symmetry between the forward and inverse transforms. One way to describe this symmetry is:
${\mathcal {F}}[X(t)]=x^{*}(f)$
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