Why are fourier transforms so powerful?
- Thread starter K.I.L.E.R
- Start date
Ok, *very roughly* in classical mechanics there is a theorem that goes something like this.
For every physical system that admits a saddle point, small departures from equilibrium (or stability) to first order admits a harmonic oscillator solution. So (intuitively) if you nudge a ball at the bottom of a parabolic well very gently, its going to want to oscillate back and forth.
Thus the natural langauge of physics is fourier analysis. Say you have some complicated system, it suffices in large part to perturb your system away from its equilibrium point, expand it out in a fourier series and identify the components to first order. All the complicated and presumably nonanalytic physics away from its saddle point, then can be analyzed in terms of simple functiosn (sins and cosines) to second, third.... infinite order away with controlable errors.
This line of reasoning goes through in quantum mechanics, and quantum field theory. Not to mention other nice properties, including the existance of the inverse fourier transform and various powerful algorithms for numerical work (eg the fft)
The mathematical reasons are so vast its really hard to do justice to it. Ultimately fourier analysis is a consequence of group theory (little known fact). However, in analysis the main thing is that given some periodic function with a set of nice properties (eg the derivatives are continous) the fourier integral is guarenteed to exist and will have various uniform convergence properties.
For every physical system that admits a saddle point, small departures from equilibrium (or stability) to first order admits a harmonic oscillator solution. So (intuitively) if you nudge a ball at the bottom of a parabolic well very gently, its going to want to oscillate back and forth.
Thus the natural langauge of physics is fourier analysis. Say you have some complicated system, it suffices in large part to perturb your system away from its equilibrium point, expand it out in a fourier series and identify the components to first order. All the complicated and presumably nonanalytic physics away from its saddle point, then can be analyzed in terms of simple functiosn (sins and cosines) to second, third.... infinite order away with controlable errors.
This line of reasoning goes through in quantum mechanics, and quantum field theory. Not to mention other nice properties, including the existance of the inverse fourier transform and various powerful algorithms for numerical work (eg the fft)
The mathematical reasons are so vast its really hard to do justice to it. Ultimately fourier analysis is a consequence of group theory (little known fact). However, in analysis the main thing is that given some periodic function with a set of nice properties (eg the derivatives are continous) the fourier integral is guarenteed to exist and will have various uniform convergence properties.
To add a bit to the above, in electronics its difinitely vital. You'll do a FT and/or Laplace to determine the stability of your system, the amount and sort of noise which you will have to filter out/compensate, the development of signal properties over different frequency ranges etc, etc...
Yeah, Fourier transforms are used all the time in physics. The basic idea is that if the system you're working with can be described with a linear differential equation (basically, the differential equation doesn't depend on the squares of the function you're attempting to solve for or any of its derivatives), and your differential equation is second-order (most are: this means they include no more than second derivatives), then you can solve the differential equation for a single sin/cosine function.
Then, once you've solved the equation for this one function, you can just add up a bunch of sin/cosine functions to get a completely general solution. In essence, this allows you to break up the problem into much smaller pieces.
Then, once you've solved the equation for this one function, you can just add up a bunch of sin/cosine functions to get a completely general solution. In essence, this allows you to break up the problem into much smaller pieces.
_xxx_ said:
Just to add a little more...
In digital communications, you're pretty much dealing with bandwidth in the channel and the signal itself. Looking at the frequency response (FT of the time-domain signal) can tell you just how much you can filter out without destroying the message one intended. You can also find out how power is spread out among the different frequencies in the power spectral density. blah blah blah, there are other things to look at too with random processes and noise.
oh the joy of being in EE...
Well, if you were willing to spend the processing power required to do the fourier transform to convert from frequency-space to real-space, then yes, it would allow for very realistic water. But the real problem is that the fourier transform is in itself a fairly expensive operation, so you'd probably just want to try to solve the differential equation iteratively instead.K.I.L.E.R said:
It depends upon what you mean by discrete. When you're talking about particles, in Fourier space, one waveform is a specific momentum. So, if you're describing particles with definite momentum, the Fourier representation is exact.
And for another example where Fourier transforms are excellent, take the early universe. The early universe can be well-described by linear theory, which makes it amenable to Fourier analysis. When one does the analysis based upon some particular theory, one obtains a specific statistical distribution of overdensities and underdensities in the universe, which can be best-measured by observing the hot/cold spots on the cosmic microwave background.
Another, more down to Earth, example comes about from looking at AC circuits: when you're dealing with alternating current driven by a sinusoidal waveform, it becomes extremely convenient to use Fourier analysis.
And for another example where Fourier transforms are excellent, take the early universe. The early universe can be well-described by linear theory, which makes it amenable to Fourier analysis. When one does the analysis based upon some particular theory, one obtains a specific statistical distribution of overdensities and underdensities in the universe, which can be best-measured by observing the hot/cold spots on the cosmic microwave background.
Another, more down to Earth, example comes about from looking at AC circuits: when you're dealing with alternating current driven by a sinusoidal waveform, it becomes extremely convenient to use Fourier analysis.
Well, I doubt they had any....FFT is pretty damned efficient as it is.Ingenu said:
But my primary concern with using FFT's with water simulation would be in dealing with disturbances. I mean, it'd be nice and easy to take a specific starting point and evolve it forward in time in Fourier space. But it could be pretty challenging to deal with adding disturbances to the water.
My experience with simulating water was a very simple kernel done in 'real' space, not frequency. It perhaps is not completely accurate, but is good enough.
http://freespace.virgin.net/hugo.elias/graphics/x_water.htm has a reasonable discussion of that (its where I cribbed my code from)
http://freespace.virgin.net/hugo.elias/graphics/x_water.htm has a reasonable discussion of that (its where I cribbed my code from)
Yeah, you won't get proper dispersion with a formula like that, though. That is the nice thing about going to Fourier space: simulating the proper dispersion relation becomes trivial. You need to do a little bit more work to get it correct in real space, and may need a finer mesh as well.
