Pridma's Article on Fourier Transforms: Difference between revisions

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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>.
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}}
:<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 t}\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}
= \left(\frac{e^{j2\pi ft} - e^{-j2\pi f_{0}t}}{2j}\right)\frac{1}{\pi t} = \frac{sin(\pi t)}{\pi t}

Latest revision as of 20:29, 11 October 2007

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:

where is the function in the time domain that you want to transform and is the transformed function which will be in the frequency domain.


The inverse Fourier Transform is similar:

Some Properties

The Derivative of x(t)

If you take the derivative of and if you know it's transform , you can easily find the transform of the derivative:

Time Shift

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

This formula is actually fairly obvious... a time delay is really just a linear phase delay, which can be seen from the 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 .


where is the even part of and is the odd part of

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

Examples

Here are some examples of Fourier Transforms:

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, and would discard all higher frequencies.

The transfer function of said system would have to be .

To build this into a circuit, we need to find the transfer function in the time domain. We can inverse Fourier transform .

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 to ! 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!