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:

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

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

The inverse Fourier Transform is similar:

${\mathcal {F}}^{-1}[X(f)]=x(t)=\int _{-\infty }^{\infty }X(f)e^{-j2\pi ft}\,df$

Some Properties

The Derivative of x(t)

If you take the derivative of $x(t)\!$ and if you know it's transform $X(f)\!$, you can easily find the transform of the derivative:

${\frac {dx(t)}{dt}}={\frac {d}{dt}}\int _{-\infty }^{\infty }X(f)e^{j2\pi ft}df=\int _{-\infty }^{\infty }X(f){\frac {d}{dt}}\left[e^{j2\pi ft}\right]df=\int _{-\infty }^{\infty }j2\pi fX(f)e^{j2\pi ft}df={\mathcal {F}}^{-1}\left[j2\pi fX(f)\right]$

Time Shift

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

$x(t-t_{0})=\int _{-\infty }^{\infty }X(f)e^{j2\pi ft}df=\int _{-\infty }^{\infty }e^{-j2\pi ft_{0}}X(f)e^{j2\pi ft}df={\mathcal {F}}^{-1}\left[e^{-j2\pi ft_{0}}X(f)\right]$

This formula is actually fairly obvious... a time delay is really just a linear phase delay, which can be seen from the $e^{-j2\pi ft_{0}}$ 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 $e$.

$x(t)=x_{o}(t)+x_{e}(t)\!$

$x_{e}(t)={\frac {x(t)+x(-t)}{2}}$

$x_{o}(t)={\frac {x(t)-x(-t)}{2}}$

where $x_{e}(t)\!$ is the even part of $x(t)\!$ and $x_{o}(t)\!$ is the odd part of $x(t)\!$

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

${\mathcal {F}}[x(t)]=\int _{-\infty }^{\infty }x(t)e^{-j2\pi ft}dt=\int _{-\infty }^{\infty }[x_{e}(t)+x_{o}(t)]\left(cos(2\pi ft)-jsin(2\pi ft)\right)dt$

$=\int _{-\infty }^{\infty }x_{e}(t)cos(2\pi ft)dt-j\int _{-\infty }^{\infty }x_{o}(t)sin(2\pi ft)dt$

$=X_{e}(f)+jX_{o}(f)\!$

Examples

Here are some examples of Fourier Transforms:

${\mathcal {F}}\left[cos(2\pi f_{0}t)\right]=\int _{-\infty }^{\infty }\left({\frac {e^{j2\pi f_{0}t}+e^{-j2\pi f_{0}t}}{2}}\right)e^{-j2\pi ft}dt={\frac {1}{2}}\left(\int _{-\infty }^{\infty }e^{-j2\pi (f-f_{0})t}dt+\int _{-\infty }^{\infty }e^{j2\pi (f+f_{0})t}dt\right)$

$={\frac {1}{2}}\delta \left(f-f_{0}\right)+{\frac {1}{2}}\delta \left(f+f_{0}\right)$

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, $f_{0}\!$ and would discard all higher frequencies.

The transfer function of said system would have to be $H(f)={\begin{Bmatrix}0,|f|>f_{0}\\1,|f|<f_{0}\end{Bmatrix}}$.

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

$h(t)={\mathcal {F}}^{-1}\left[H(f)\right]=\int _{-f_{0}}^{f_{0}}1e^{j2\pi ft}df=\left[{\frac {e^{j2\pi ft}}{j2\pi ft}}\right]_{f=-f_{0}}^{f_{0}}=\left({\frac {e^{j2\pi ft}-e^{-j2\pi f_{0}t}}{2j}}\right){\frac {1}{\pi t}}={\frac {sin(\pi t)}{\pi t}}$

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 $-\infty$ to $\infty$! 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!