| Re: FFT example. Help! [message #19879] |
Tue, 02 May 2000 00:00  |
Peter Brooker
Messages: 28
Registered: July 1999
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Junior Member |
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thanks for taking the time to reply. This helps allot!!
peter brooker
Alan Barnett wrote:
> I think your confusion stems from a misunderstanding of what the FFT
> does. There are at least three related, but not identical, operations
> that are referred to as "Fourier transforms".
>
> 1) Fourier integral.
> A function f(x) that is "well behaved", _continuous_ and _not periodic_
> has a Fourier transform F(k) that is "well behaved", continuous and not
> periodic. F and f are related by the Fourier transform and Fourier
> inversion formulas:
> F(k)=int(f(x) exp(-i kx) dx)
> and
> f(x)=int(F(k) exp(i kx) dx)
> The limits of the integral are + and - infinity.
>
> 2) Fourier series.
> If the function f(x) is a periodic function of x with period L, its
> Fourier transform is zero for all k that are not integral multiples of 2
> Pi /L, and the integral diverges if k is an integral multiple of 2 Pi /
> L. F(k) can be expressed as a sum of dirac delta functions, but it is
> more convenient to simplify the notation and write f(x) as a Fourier
> series:
> f(x) = sum(An sin(n Pi x / L) + Bn cos(n Pi x / L ) )
> where the Fourier coefficients An and Bn are
> An=(2/L) int(f(x) sin(n pi x / L) dx)
> Bn=(2/L) int(f(x) cos(n pi x / L) dx)
>
> 3) Discrete Fourier transform (DFT)
> If the function F(x) is a periodic function of x with period L and is only
> defined at N equally spaced discrete points xn = n L / N, n = 0, N-1, then
> its Fourier transform is a periodic function of k = 2 Pi / L and is
> nonzero only at the discrete values km = 2 m Pi / L, m = 0, N-1. Instead
> of considering the functions F(x) and f(k), which are sums of delta
> function, it is easier to consider Fm and fn, the coefficients of the
> delta functions. These coefficients are related by the discrete fourier
> tranform:
>
> Fm = sum( fn exp(-i 2 Pi m n / N))
>
> and its inverse
>
> fn = (1/N) sum( Fm exp(i 2 Pi m n / N ) )
>
> The FFT is a fast algorithm for computing the discrete fourier transform.
>
> If you want an example using the FFT that "makes sense" (I asume that you
> mean one that you can compute analytically), you have to remember that the
> FFT computes the discrete Fourier transform, not the Fourier integral.
> The simplest example is
> fn = 1 for all n
> Fm = N for m = 0; 0 otherwise
>
> The FFT is frequently applied to the problem of estimating the Fourier
> integral of a non-periodic function whose value is only known at a
> discrete set of sampled points. One then must cope with errors introduced
> by sampling (aliasing) and truncation (Gibbs ringing).
>
> For a good overview, look in Numerical Recipes by Press et al.
>
> I hope this helps.
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