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From: dbd on 24 Jun 2010 19:21
On Jun 24, 3:13 pm, "Mat Hunt" <hunt_...(a)hotmail.com> wrote:
> So can anyone give me a clear and simple methodology to go from a Fourier transformed function F(k), back to the original function at a point, x say? So this would be a single number?  I would have thought the the people at matlab would have thought this one through.
>
> Mat

You persist in having continuous function expectations for discrete
finite implementations (fft, ifft). I think the people at Matlab
understand this There are symbolic and discrete implementations. With
data samples, you have picked the discrete. Abandon your symbolic
expectations. There are those who try to cure your problem with
fftshift and ifftshift, but frankly, until you understand that what
the fft and ifft functions give you are different from the infinite/
continuous theory of Fourier and how they are different, I doubt that
a review of fftshift and ifftshift syntax can have any useful meaning
to you.

Good Luck!

Dale B. Dalrymple
From: Mat Hunt on 27 Jun 2010 15:57
Okay, all I am asking is if I give a function F=F(k) which is the Fourier transform of a function f=f(x) and another point x, say then I have a number as an output which is f(x). Can the ifft function do this?

Mat
From: Greg Heath on 28 Jun 2010 02:24
On Jun 27, 3:57 pm, "Mat Hunt" <hunt_...(a)hotmail.com> wrote:
> Okay, all I am asking is if I give a function F=F(k) which is the Fourier transform of a function f=f(x) and another point x, say then I have a number as an output which is f(x). Can the ifft function do this?
>
> Mat

Yes.

However, (once again):

The fft and ifft functions assume that the waveforms are defined over

x = dx*(0:N-1); % dx = 1/Fs , Fs is the spatial sampling frequency

and

k = dk*(0:N-1); df = Fs/N

The defining summation equations in the documentation of fft and ifft
are periodic with period N.

Therefore if you start out with k data that are defined over
different intervals, e.g.,

kb = dk* [ -ceil((N-1)/2) : floor((N-1)/2 ]

you have to assume the function is periodic outside
of the given interval and find that portion that lies between
0 and dk*(N-1).

Hope this helps.

Greg
From: Mat Hunt on 28 Jun 2010 06:15
The problem is that my function isn't periodic. I am looking for a travelling wave solution which vanishes as x\rightarrow\pm\infty

Mat
From: Greg Heath on 30 Jun 2010 10:24
On Jun 28, 6:15 am, "Mat Hunt" <hunt_...(a)hotmail.com> wrote:
> The problem is that my function isn't periodic. I am looking for a travelling wave solution which vanishes as x\rightarrow\pm\infty
>
> Mat

99.9% of the waveforms that are used in DFT/FFT are
not periodic. However, when you use the DFT/FFT you
assume that, within a finite interval, the waveform
can be represented by a weighted sum of sines and
cosines of a fundamental frequency and it's harmonics.
Because of that, when that sum is evaluated outside
of the defining interval, it is periodic.

Therefore if you are using the DFT/FFT to represent
nonperiodic phenomena, you have to make that defining
interval sufficiently long.

For your physical problem it seems like there is
a system of linear partial differential equations
in x and t which you are trying to solve using dual
Fourier-Laplace transforms in space-time. I'd have
to know the complete mathematical specification of
the problem including initial and boundary conditions
before I can make any more relevant comments.

Hope this helps.

Greg
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