> On Mar 26, 2014, at 10:07 PM, Doug Houghton <[email protected]> wrote: > > so is there a requirement for the signal to be periodic? or can any series of numbers be cnsidered periodic if it is bandlimited, or infinit? Periodic is the best word I can come up with. > > -- > > Well, no--you can decompose any portion of waveform that you want...I'm not sure at this point if you're talking about the discrete Fourier Transform or continuous, but I assume discrete in this context...but it's not that generally useful to, say, do a single transform of an entire song. Sorry, I'm not sure where you're going here... >
Actually, yes there IS a requirement that it be periodic. Fourier theorem says that any periodic sequence can be represented as a sum of sinusoids. Sampling theory says that any band-limited _perriodic_ signal can be properly sampled at the Nyquist rate. The trick is that any finite-duration signal can be thought of as one period of a periodic signal. This is part of the reason you get infinite repetitions in the frequency domain after you sample. ...sort of. As for frequencies jumping in and out, I think you were on the right track when you said that it's a Fourier theorem thing. Imagine you had a signal with one sinusoid that slowly fades in and out for the duration of the signal. Imagine that the the envelope of this sinusoid is the first half of a sinusoid. The envelope can be described as a sinusoid whose period is twice the signal duration. If you were to simply take these two stationary sinusoids (the envelope and the audible tone) and multiply them you end up with a spectrum that contains their sum and difference tones. In that way it can already be thought of as a tone that (slowly) pops in and out, but which is represented as a sum of two stationary sinusoids. If you wanted to have the tone come in and out more quickly you could add the first harmonic of a square wave (or several) to the envelope. For each additional harmonic you add to the envelope you get an additional two sinusoids in spectrum of the whole signal. You can keep adding harmonics up to the Nyquist frequency. This means that your frequencies can pop in and out very quickly, but only as fast as your sampling rate allows. Rinse and repeat for additional e.g. harmonics of your audible tone, additional tones, etc. Note that you can construct any envelope you could imagine, including apparently asymmetrical, or ones which are approximately zero for most of its duration pops in and out for only a small portion of it. That all comes from Fourier theorem. The important part is that each of these envelopes would be periodic in the duration of your sampled signal, as far as the sampling theorem is concerned. I hope that helps a bit -Stefan -- dupswapdrop -- the music-dsp mailing list and website: subscription info, FAQ, source code archive, list archive, book reviews, dsp links http://music.columbia.edu/cmc/music-dsp http://music.columbia.edu/mailman/listinfo/music-dsp
