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I have a signal composed of the summation of a set of sine waves of different frequencies. The amplitude of these sub-signals can change so many times a second.

I have been told that, if I want to retain the ability to distinguish each of the frequencies, the time-frequency uncertainty principal means there will be a limiting relationship between the duration of the time window between amplitude changes and the smallest interval between frequencies.

I found a website which seems to deal with the problem, but as a non-mathematician, I'm not sure how to utilise the formulas it shows. To be honest, I'm not even sure if it's relevant.

My question then: What is the relationship between the time interval and the minimum frequency interval?

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Roughly speaking, the relationship is that one is proportional to the inverse of the other. For a more precise answer, you might need to explicate what you mean by "the ability to distinguish each of the frequencies". – joriki Dec 12 '11 at 16:37
Are you familiar with the Nyquist rate? – dls Dec 12 '11 at 16:54
Asking this question on the signal processing stack exchange dsp.SE instead might be worthwhile. – Dilip Sarwate Dec 12 '11 at 17:57
If you search for the sampling-theorem instead of the time-frequency uncertainty principle you find something more rigorous (a theorem) instead of just a principle. – Fabian Dec 12 '11 at 18:09
Some related material is here on dsp.SE – Dilip Sarwate Dec 12 '11 at 21:29

for a energy signal i.e physically realizable, exist a uncertainty principle between duration of signal and her bandwidth. If Bq is her quadratic band and Dq is her quadratic duration then BqDq >= (8*pi)^-1 . Therefore a signal can't have both duration and band zero. But if duration tend to zero then band tend to infinity and vice versa. To notice that the particular constant (8*pi)^-1 to derive from particular systems of mesure that is used.

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