View source for Pridma's Article on Fourier Transforms

=Mark's Wiki Article on Fourier Transforms=

==About the Fourier Transform==
The '''Fourier Transform''' is a way to change a function of time into a function of frequency. This comes in very handy at times as it allows you to talk about a function in the frequency domain and will make some things very obvious about a function when you see it in the frequency domain that would not be very obvious in the time domain.

==Formulas==
The Fourier Transform is as follows:

:<math>
\mathcal{F}[x(t)] = X(f) = \int_{-\infty}^{\infty} x(t) e^{-j2\pi ft}\, dt
</math>

where <math>x(t) \!</math> is the function in the time domain that you want to transform and <math>X(f) \!</math> is the transformed function which will be in the frequency domain.


The inverse Fourier Transform is similar:

:<math>
\mathcal{F}^{-1}[X(f)] = x(t)=\int_{-\infty}^{\infty} X(f) e^{-j2\pi ft}\, df
</math>

==Some Properties==

===The Derivative of x(t)===
If you take the derivative of <math>x(t) \!</math> and if you know it's transform <math>X(f) \!</math>, you can easily find the transform of the derivative:

:<math>\frac{dx(t)}{dt} = \frac{d}{dt}\int_{-\infty}^{\infty} X(f) e^{j2\pi ft} df = \int_{-\infty}^{\infty} X(f) \frac{d}{dt} \left[e^{j2\pi ft}\right] df = \int_{-\infty}^{\infty} j2\pi fX(f) e^{j2\pi ft}df = \mathcal{F}^{-1}\left[j2\pi fX(f)\right]</math>

===Time Shift===
The time shift property of the Fourier Transform is fairly straight-forward as well.

:<math>x(t-t_{0}) = \int_{-\infty}^{\infty} X(f) e^{j2\pi ft} df = \int_{-\infty}^{\infty} e^{-j2\pi ft_{0}} X(f) e^{j2\pi ft} df = \mathcal{F}^{-1}[e^{-j2\pi ft_{0}} X(f)]</math>

This formula is actually fairly obvious... a time delay is really just a linear phase delay, which can be seen from the <math>e^{-j2\pi ft_{0}}</math> part of the inverse Forier Transform shown above.


===Even and Odd Parts===
You can also split up a function to a sum of it's even and odd parts and take a Fourier Transform using sine and cosine instead of <math>e</math>.
:<math>x(t) = x_{o}(t) + x_{e}(t) \!</math>

:<math>x_{e}(t) = \frac{x(t) + x(-t)}{2}</math>

:<math>x_{o}(t) = \frac{x(t) - x(-t)}{2}</math>


where <math>x_{e}(t) \!</math> is the even part of <math>x(t) \!</math> and <math>x_{o}(t) \!</math> is the odd part of <math>x(t) \!</math>

Once you have the function split up into it's even and odd parts, you can use the following Fourier Transform formula:

:<math>\mathcal{F}[x(t)] = \int_{-\infty}^{\infty} x(t) e^{-j2\pi ft} dt = \int_{-\infty}^{\infty} [x_{e}(t) + x_{o}(t)](cos(2\pi ft) - j sin(2\pi ft)) dt</math>

::::<math>= \int_{-\infty}^{\infty} x_{e}(t) cos(2\pi ft) dt - j \int_{-\infty}^{\infty} x_{o}(t) sin(2\pi ft) dt</math>

::::<math>= X_{e}(f) + j X_{o}(f) \!</math>

==Examples==

Here are some examples of Fourier Transforms:

:<math> \mathcal{F}\left[cos(2\pi f_{0}t)\right] = \int_{-\infty}^{\infty} \left(\frac{e^{j2\pi f_{0}t} + e^{-j2\pi f_{0}t}}{2}\right) e^{-j2\pi ft} dt = \frac{1}{2}\left(\int_{-\infty}^{\infty} e^{-j2\pi(f-f_{0})t} dt + \int_{-\infty}^{\infty} e^{j2\pi(f+f_{0})t} dt\right)</math>

::::::<math> = \frac{1}{2} \delta\left(f-f_{0}\right) + \frac{1}{2} \delta\left(f+f_{0}\right)</math>

==Useful Applications==

One application of Fourier Transforms is in answering the question, "Why can't we build a brick-wall low pass filter?"

An ideal low pass filter would keep all the frequencies below a certain frequency, <math> f_{0} \!</math> and would discard all higher frequencies.

The transfer function of said system would have to be <math>H(f) = \{_{0, |f| > f_{0}}^{1, |f| < f_{0}}</math>.

To build this into a circuit, we need to find the transfer function in the time domain. We can inverse Fourier transform <math>H(f) \!</math>.

:<math>h(t) = \mathcal{F}^{-1}\left[H(f)\right] = \int_{-f_{0}}^{f_{0}} 1 e^{j2\pi ft} df = \left[\frac{e^{j2\pi ft}}{j2\pi ft}\right]_{f=f_{0}}^{f_{0}}
= \left(\frac{e^{j2\pi ft} - e^{-j2\pi f_{0}t}}{2j}\right)\frac{1}{\pi t} = \frac{sin(\pi t)}{\pi t}
</math>

If we take a closer look at the transfer function, we'd notice that if we input an impulse function, the output would have to be non-causal: the output would be a sine wave extending from <math>-\infty</math> to <math>\infty</math>! So, it is impossible to build  an exact "brick-wall" low pass filter, unless we know the input an infinite amount of time before it even happened!