On 24-Jun-15 15:31, Sampo Syreeni wrote:
Certainly any signal with compact support isn't bandlimited. That's the simplest form of the uncertainty principle. But even if you take a strictly bandlimited "window" function with rapid falloff (a bandlimited square/flattop convolved with itself a couple of times comes to mind), you can easily fit stuff under it which behaves nice as peaches with regard to derivative conditions, but manages to have arbitrarily high frequency content.
I don't want to take bandlimited windows. I want to take a rectangular window. But then I can apply the BLEP method to the discontinuities in the function and the derivatives arising from this window. Now, do I get a bandlimited version in the result? If I apply the window to a polynomial (particularly a straight line, occuring in the sawtooth) or to a bandlimited sine and then "BLEP" the result, then I would expect to get a bandlimited signal in the result, right? Now, how about doing the same to an exponential?
Say, the simplest form of phase modulation at low index, f(x)=sin(x+sin(x)/10). When you "window" that, the result has all of the properties you're asking for, including no discontinuities in any derivative and nice enough rolloff of them in modulus. So it obeys the Paley-Wiener-Schwartz conditions, and its Fourier transform extends into an entire function. But the resulting function still contains infinitely high frequencies, and so can't be sampled losslessly at any finite rate.
Upon a first attempt I failed to estimate the growth of derivatives of an FM sine analytically. But numerical checks of the first few derivatives were suggesting that the derivatives are not rolling off fast enough. So I would expect that this function doesn't satisfy the PWS conditions (exponential growth along the imaginary axis). Furthermore, if it did satisfy the PWS conditions, this would have meant that the signal is bandlimited (according to the PWS theorem itself).
Don't get me wrong, I'm not trying to rain on your parade. I'm pretty impressed that someone is willing to take the time to go back into the fundamentals, and I too would like to see stuff such as PM properly bandwidth limited.
I don't think that the BLEP method is capable of bandlimiting an FM sine. But it might be able to bandlimit an FM sawtooth, even if FM is exponential.
But I still don't see how you're going to get there starting with derivative conditions. To me they seem incommeasurate with what you're trying to achieve.
The derivative rolloff is exactly what is defining the convergence of the sum of the BLEPs (one BLEP per derivative). More specifically, it is a sufficient condition for the convergence of the BLEPs. So, there seems to be a strong correspondence between the convergence of the BLEPs and the bandlimitedness of the "base" signal.
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