Why do we need Fourier Transform?

[hanson]

Hi all.
I am revisiting Fourier transform now and am wondering why we need Fourier transform?
I mean, what's so special of representing a function in another way (in terms of sine waves)?

Actually, I am now working on a problem. I was just told that someone worked out something in Fourier space and hence the results are steady state results. I have difficulty in understanding this.

Why results in Fourier space implies steady state results? Can someone help me out please?

 

[chroot]

The Fourier transform occurs naturally all throughout physics.

If you want to understand how an optical system will distort an image, you need to use Fourier analysis. If you want to understand how different musical instruments create their different sounds, you need Fourier analysis. If you want to understand how a quantum-mechanical system will evolve with time, you need to use Fourier analysis.

The bottom line is that many physical interactions depend on frequency, and many real-life "signals" involve complex waveforms. A complex waveform is nothing more than a superposition of simple ones, as shown by Fourier. If you want to understand how a physical system will respond to a complex stimulus, you can break that stimulus down into a superposition of simple stimuli, consider each simple stimulus in isolation, then combine the results. It's an incredibly powerful and widely applicable technique.

- Warren

 

[hanson]

The Fourier transform occurs naturally all throughout physics.

If you want to understand how an optical system will distort an image, you need to use Fourier analysis. If you want to understand how different musical instruments create their different sounds, you need Fourier analysis. If you want to understand how a quantum-mechanical system will evolve with time, you need to use Fourier analysis.

The bottom line is that many physical interactions depend on frequency, and many real-life "signals" involve complex waveforms. A complex waveform is nothing more than a superposition of simple ones, as shown by Fourier. If you want to understand how a physical system will respond to a complex stimulus, you can break that stimulus down into a superposition of simple stimuli, consider each simple stimulus in isolation, then combine the results. It's an incredibly powerful and widely applicable technique.

- Warren

I see. I think I get this. But what do you think about the steady state thing?

 

[xmavidis]

Fourier space is the reciprocal of a given space, let's say A.
A quantity in space A (i.e. the wave-vector) has a reciprocal in Fourier space (here is the wavelength).

 

[jostpuur]

Hanson, do you know what is diagonalization of a finite dimensional matrix, and what applications this technique has with systems of differential equations? If you don't know, then I'll encourage you to find out about it sooner or later, and if you do know, then one can notice that the Fourier transform can be thought of as being the diagonalization of the derivative operator. Fourier transformations are useful for PDEs in the same sense, as the diagonalization technique is useful for systems of DEs.

I'm not fully sure what steady state results are now, but I might guess that they have something to do with some solution being an eigenstate of some operator.

 

[bhrsingh]

The Fourier transform occurs naturally all throughout physics.

If you want to understand how an optical system will distort an image, you need to use Fourier analysis. If you want to understand how different musical instruments create their different sounds, you need Fourier analysis. If you want to understand how a quantum-mechanical system will evolve with time, you need to use Fourier analysis.

The bottom line is that many physical interactions depend on frequency, and many real-life "signals" involve complex waveforms. A complex waveform is nothing more than a superposition of simple ones, as shown by Fourier. If you want to understand how a physical system will respond to a complex stimulus, you can break that stimulus down into a superposition of simple stimuli, consider each simple stimulus in isolation, then combine the results. It's an incredibly powerful and widely applicable technique.

- Warren

Warren Thanks for such a simple description of Fourier trick

 

[paulfr]

As an electronics engineer I use the Fourier Transform all the time.
It essentially gives us information about a signal or system element from a frequency point of view rather than the time domain point of view.

As for the steady state issue you asked about, the FT is showing the energy in the signal at the different frequencies. We call it the Spectral content. For this we need to "integrate" the energy at the different frequencies and the signal needs to be steady to get meaningful data. If the signal which is periodic [repeating the same pattern over and over] changes so does the spectral content and the measurement would be invalid.

For example a pure musical tone is a single frequency and its FT would show a spike at that f and nothing elsewhere.[ In practice actually you would see the third and fifth harmonics too as lower signal levels.]

There is actually a standard instrument in all analog and RF electronic laboratories call a Spectrum Analyzer that displays the FT on a display similar to the way an oscilloscope displays the time domain signal.

To obtain information about transient response [non steady state] one uses the Laplace Transform.

Hope I did not overload you here.

 

[Jaynte]

It is used in many applications such as digital filters, jpeg compression etc.

For example if you have sound where the noice which you want to eliminate is
at the very high frequencies you can use Fourier analysis to be able to design the filter
you want to surpress those high frequencies.

The same for JPEG compression, you transfer the image into frequency domain and
use a filter to cancel the high frequencies out, which our eyes doesn't see anyway, and
then you can encode the image to make it much smaller, because you don't need to save those high frequencies.