Re: Zero Padding in radix 2 FFT

Fred Marshall wrote:
"Stan Pawlukiewicz" <spam@xxxxxxxxxxxxxx> wrote in message news:dpgjjr$kv0$1@xxxxxxxxxxxxxxxxxxxxxx

Fred Marshall wrote:

Hmmmm.... long ago I suggested that we make it clear when we're talking about theory or practice. I don't think the notion was very well received and I think the idea was that we'd make it clear.

No such luck in this thread. We read about clear procedures / methods ("theoretical") and then somebody says: "yeah, but you can't reconstruct a [real world] waveform with it" Hmmmmm....

I wonder if we can't just use a set of (theoretical) identities to make things clear:

I never liked these so called identites. If I take two periodic functions and their periods are related by an irrational constant the sum is no longer periodic. Their resulting Fourier transform is continuous and consists of delta functions, but not at rationally related frequencies.

I find it easier to thing of everything being continuous, and use delta functions where appropriate. This makes everything linear with respect to addition of waveforms. 


Stan,

No argument there.
There are periodic functions and there are functions that aren't periodic (and there are functions that aren't periodic that are sums of periodic functions as you point out).
You mention both types. Both types are covered. So, I rather miss the point I'm afraid.

What ever wasn't linear in any of the discussion?

Anyway, if you see how I built up to the most periodic and discrete versions, I did start with the continuous and went to the more particular cases. So, I think we agree on that point.

Now, if you mean that superposition in limited domains and between them doesn't work after making those assumptions, then that's correct.

Basically if you have a truly discrete spectrum of a periodic function i.e. the sample rate and periodicity are rational multiples, and another periodic function (in continuous time) that is not a rational multiple of the sample rate, the resulting superposition can not really considered a truly discrete spectum. The DFTs are additively linear but there are frequencies in between the bins.

That is, if you have a periodic temporal waveform of frequency 1Hz that's sampled at 10Hz and transformed.
And, if you have a periodic temporal waveform of frequency (pi)Hz that's sampled at 3*piHz and transformed,
Then you can't add the resulting spectra and, inversely, you can't add the resulting samples either - because they are in different "domains" I guess you might say.
The same thing applies if you have a set of samples at 1Hz and a set of samples at 4Hz. You can't add the samples together and expect to extract the two sampled signals thereafter.
The 1Hz samples have to be interpolated to 4Hz first - in general.... all necessary filtering, bandlimiting, etc. implied.

Fred 


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