# can someone explain to me how Fourier Transformations work?

Discussion in 'Science' started by echelon6, Jun 10, 2012.

1. ### echelon6Member

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I tried to read everything I could find on the subject but couldn't make sense of it. Can someone explain to me how this math technique works? I want to understand this for the sake of understanding how radio interferometry works (the adding together of various low res images taken from different locations to form a higher res image)

I understand (early) uni level maths if that helps 2. ### Lucifers MentorMember

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Fourier transforms are a method of moving something from 'physical' space to a new space, where time is re-expressed as frequency. This is done through an integral that maps into the complex spectral domain. The reason it's done is that in Fourier space, there are a number of tricks you can use to simplify and solve what would otherwise be problematic problems.

As to radar inferometry - I believe it's because the measurements are actually taken in terms of the frequency of the signal coming in, so they're already in fourier space, so to work out what the physical representation is, an inverse fourier transform needs to be performed.

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4. ### echelon6Member

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I did a bit more reading on the subject and I realise there's ALOT of basics I don't understand. Anyway if anyone's willing to give it a shot at dumbing it down and explain in terms of more basic concepts, I'll be really happy to read up on more.

The specific question is basically how radio interferometry works. So basically a 2 array radio interferometer (i.e. one radio telescope looks at a part of the sky and takes a shot, and the other radio telescope is located a large distance away from the first, also looks at the same part of the sky and takes a shot, both images are combined and 'fourier transformed' to produce a very high resolution image. I want to understand that process, and also understand some other applications of this magical technique.

5. ### echelon6Member

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How can you re-express time as frequency? So for example, a sine wave describes the y position of a point on a rotating wheel. The graph is a function of y distance vs time. How can you re-express this function in terms of frequency?

6. ### noobmasteryMember

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A fourier transform expresses a function as a sum of multiple sine waves.
When you change to frequency domain, you're essentially expressing your original function as a sum of multiple sine waves of different frequencies.

For a sine wave of f=1 you'd have a fourier transform where y=1 when f=1 and y=0 elsewhere because there are no other sine wave frequencies present.

7. ### echelon6Member

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But that's what I dont understand. y varies with time, it can't always be =1 even though f=1 always

8. ### GrantMember

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If you sample your sine wave at a single point, the "frequency" that you get out of a fourier transform tells you the slope of the line at that point (whether it's going up or down sharply, or whether it's at a peak or a trough), not the absolute value.

You can't tell anything about the slope of a line from a single point on the time axis, so you need a few samples put together to figure out what the slope could be at that single point.

9. ### FoliageMember

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Simple example, you have two sine waves at different frequencies, you add them together and you get a mess, this mess in time domain looks pretty useless you can't really analyse it easily, if you convert it to fourier domain you will get two spikes at the frequency of the first sine wave and at the frequency of the second sine wave.

You can see how this becomes useful when you have hundreds or thousands of different sine waves at different amplitudes all added together, they become separated and easy to see/analyse in Fourier domain.

Does that help?

A fourier transform is at one specific point in time, eg a single slice of your waveform at t=x

Eg if at t=1 you have two sine waves with frequency 1hz and 5hz, at t=2 these frequencies could change to 10hz ad 50hz, so it only makes sense to take a fourier transform at a specific slice of time.

However obviously in the real world you cant sample a infinitesimally small chunk of time, so you have to sample at a frequency where you obtain all the available data, if you sample at speed slower than this you get aliasing errors.

Last edited: Jun 15, 2012
10. ### echelon6Member

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Cool that makes sense. So how does this technique allow radio interferometry to work?