Hi Chuckk,
Chuckk Hubbard wrote:
On 6/14/06, Piotr Majdak piotr@majdak.com wrote:
And I really don't know what do you mean by "FFT, which involves distortion". You don't mean distortions as in "nonlinear distortions", don't you?
I mean that the reconstituted sound is not exactly the same as the original signal.
If:
- x is a sequence in a discrete space (i think hilbert space is OK) with
the length N,
- DFT is the discrete fourier transformation with the length of N,
- IDFT is the inverse transformation to DFT,
- we neglect all the numerical quantization and overflow for a while,
then:
IDFT(DFT(x)) = x
Even with numerical limitations it's not a problem. Try this in MATLAB: x=rand(10000,1); y=ifft(fft(x)); sqrt(sum((y-x).^2))/length(x)
I get: 1.6806e-018, which shows that the residual error of reconstruction of x is rather low even for dealing with long sequences.
And a signal with the imaginary part set to zero won't represent the same sines and cosines.
Same as what? Sorry, don't understand it...
The same as having the imaginary part set to the sums of all of the sines of the values of which the real part is the cosines.
I still can not follow your explanation, because you compare real signals with complex signals, which, of course are not the same. But...
I mean, it's not really possible for the same sum of rotating bodies to have two different sets of vertical values? The imaginary part does affect what's happening, no?
...I have the impression, that you're trying to explain komplex numbers with signals from the real world. There is no physical meaning of the set of komplex numbers! It is just a tool to perform some calculations easier (see last mail for example) and interprete the results in a "human" way (e.g. what's the sqrt(-1)?). Thus, as long as you deal with real signals, there is no imaginary part what can affect any rotation body. Introduce complex signal, there we have it :-)
br, Piotr