Hi Ajinkya,
Thanks for your detailed response. You have brought in a new issue of the precision of each data points which I did not consider.
However, please pardon my ignorance, I do have serious concerns about your response, which unfortunately I have not understood at all.
Members, I do apologise for the level of technicality in this post.
Let's consider any mechanics problem in a spatial box of extent L. If the continuous functions f(x) defined on this box have a periodic boundary condition, the Fourier variable, in this case, the momentum becomes discrete, however there are infinite no of momenta, starting from zero to infinity.
Now think the equivalent situation here: momentum = time, and space or x = frequency. In this situation, one can understand easily that for a signal that starts at a time minus infinity and extends in discrete steps to a time plus infinity, the information can be packaged by Fourier transform to a signal in continuous freq space where it is bandlimited upto a max freq (just like the finite box in continuous space extending only upto a max of L).
All the above follows from the theory of Fourier transform.
Now think of the continuous box being a grid or lattice, that is, the space is now discretized with a lattice constant. Now you have to be a bit careful about the Fourier transform, and the delta functions will be replaced by Kronecker deltas. But all this can be done. I do this for a living day in and day out. Now you get a momentum space that is discrete and finite.
Correspondingly, a set of functions of time at discrete and finite no of points of time would be Fourier mapped to a space of freq which is bandlimited, but is also DISCRETE.
For our audio signal, that would be disastrous.
All the above follows from simple mathematical properties of Fourier transform. I have NOT assumed anything else.
So, unless you do have a signal (as a function of time) that extends from the infinite past to the infinite future (in discrete steps as in the case of the quantised signal), you are not going to get a signal in frequency space that is bandlimited and also continuous in frequency.
This is what I stated as assumption (1) of the proof.
Basically unless you are putting in an infinite amount of information, you are NOT getting out an infinite amount of information. Fourier transforms are not going to help.
I hope this clarifies my earlier post.
Thanks for your detailed response. You have brought in a new issue of the precision of each data points which I did not consider.
However, please pardon my ignorance, I do have serious concerns about your response, which unfortunately I have not understood at all.
Members, I do apologise for the level of technicality in this post.
Let's consider any mechanics problem in a spatial box of extent L. If the continuous functions f(x) defined on this box have a periodic boundary condition, the Fourier variable, in this case, the momentum becomes discrete, however there are infinite no of momenta, starting from zero to infinity.
Now think the equivalent situation here: momentum = time, and space or x = frequency. In this situation, one can understand easily that for a signal that starts at a time minus infinity and extends in discrete steps to a time plus infinity, the information can be packaged by Fourier transform to a signal in continuous freq space where it is bandlimited upto a max freq (just like the finite box in continuous space extending only upto a max of L).
All the above follows from the theory of Fourier transform.
Now think of the continuous box being a grid or lattice, that is, the space is now discretized with a lattice constant. Now you have to be a bit careful about the Fourier transform, and the delta functions will be replaced by Kronecker deltas. But all this can be done. I do this for a living day in and day out. Now you get a momentum space that is discrete and finite.
Correspondingly, a set of functions of time at discrete and finite no of points of time would be Fourier mapped to a space of freq which is bandlimited, but is also DISCRETE.
For our audio signal, that would be disastrous.
All the above follows from simple mathematical properties of Fourier transform. I have NOT assumed anything else.
So, unless you do have a signal (as a function of time) that extends from the infinite past to the infinite future (in discrete steps as in the case of the quantised signal), you are not going to get a signal in frequency space that is bandlimited and also continuous in frequency.
This is what I stated as assumption (1) of the proof.
Basically unless you are putting in an infinite amount of information, you are NOT getting out an infinite amount of information. Fourier transforms are not going to help.
I hope this clarifies my earlier post.
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sigh!! 
me i've got one more step to go. only have managed to lay hands on a red vinyl disc. sigh...
