Re: fourier transform


Richard Mathar wrote:
In article <1161464183.886456.122350@xxxxxxxxxxxxxxxxxxxxxxxxxxx>,
rxvtcalc@xxxxxxxxx writes:
hi

fourier transform is not unique right?
The Fourier Integral transform creates only one unique result
for each function.
I can transform a periodic function with frequency f into functions
with frequency f,2f,3f,...to arbitrary acuracy right?

If "transform" relates to the Fourier Integral transform, there will
indeed generally be values at f, 2f, 3f etc in the frequency domain.
However, if the input function is periodic and contains only a single
frequency f in the time domain, the Fourier integral transform will
contain only values at f (and -f, depending how exactly you define a
periodic function with frequency f and whether you take the complex
Fourier transform, or the cosine transform etc).
If you approximate the Fourier Integral transform by the Discrete
Fourier Transform, the output will be periodic with the Nyquist
frequency set by the size of the interval in the time domain.

Ok, I was thinking something like: X(t) -> A1 cos(t)+A2 cos(2t)+A3
cos(3t) ...
decompose the first term:
A1 cos(t) = B1 sin(t)+B2 sin(2t)+B3 sin(3t) ...
decompose the first term again:
B1 sin(t) = C1 cos(t)+C2 cos(2t)+C3 cos(3t) ...

Can this work?



now I can further transform any of the decomposed functions into higher
frequency ones right?

This does not make sense in the Fourier Integral or in the Discrete
Fourier Transform. By definition, the output of a Fourier Integral
Transform has values at frequencies and other frequencies; if you chop
off components at some frequencies, they do not mystically resurface
elsewhere. You have to define what "transform" you want to use (frequency
doubling?) to achive such a result. For the Discrete Fourier Transform, all
the components at f equal the components at f+f(nyq), f+2f(nyq), f-f(nyq) etc;
so there is no information "left over" at the higher frequencies
that warants further decomposition in the Fourier sense.

now in theory, I can pick an arbitrary high frequency function k(t)
with small amplitude, and decompose any periodic function into "a tree
of" k(t)'s right?

If you undersampled the original function with a discrete Fourier
transform with too low frequency, the output contains sums of the
components at the original frequency and those mapped at these from
the higher orders. I do not think that these can be decomposed easily


thanks


--
---
Richard J Mathar
http://www.strw.leidenuniv.nl/~mathar

.

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