Hi Doug- To address some of your general questions about Fourier analysis and relationship to sampling theory:
Broadly speaking any reasonably well-behaved signal can be decomposed into a sum of sinusoids (actually complex exponentials but don't worry about that detail for now). There are several flavors of Fourier analysis corresponding to different classes of signals. I.e., there are variants for continuous-time signals and discrete-time signals, and also for periodic signals and general signals. For the case of periodic signals you use what's called the Fourier Series (there you add up harmonic components), for general signals you use the Fourier Transform (this uses both harmonic and inharmonic components). The key difference between discrete time and continuous time from a Fourier analysis perspective (either series or transform) is that continuous time signals can have arbitrarily high frequencies, but discrete-time signals can only admit a finite bandwidth (related to the sampling rate). So in all cases of Fourier analysis, we're decomposing the signal into sinusoids. As you have noted, sinusoids all extend off to +/- infinity. You are correct to note that this corresponds to steady-state analysis, when used in for example a circuit analysis context. One consequence of this is that any perfectly bandlimited signal, like in the Sampling Theorem, also has to extend to +/- infinity. The other way around is also true: any signal that only lasts a finite length of time must contain frequencies all the way up to infinity. So, strictly speaking, it is true that the conditions of the Sampling Theorem cannot ever be truly fulfilled in practice, since all practical signals are necessarily time-limited. There is *always* theoretically some level of aliasing in practice because of the time-limiting constraint. But it turns out that this doesn't present much of a difficulty in practice. To see why, consider another question you raised: if arbitrary signals can be decomposed into sums of sinusoids that all repeat off to infinity, how do we represent signals that are concentrated in a particular time, or have frequency components that turn off midway through, etc.? It's easy enough to prove that the Fourier transform is invertible - i.e., that all of the information in the signal is contained in the sinusoidal parameters - but where does all that time information go? It turns out that it gets folded into the relationship between the amplitudes and phases of the various sinusoids in a complicated way - the information is still there and recoverable, but it's not usually easy to discern from looking at the output of the Fourier transform. For a signal where a given frequency "turns on" at some particular time, the transform is only going to give you a single parameter for that frequency, corresponding to the average energy over all time. The information about it turning on and off shows up at "sidelobes" around the frequency in question. Likewise, the time signal already contains all the information about the sinusoid parameters, folded up in a way that's hard to see. Why does that help us with the Sampling Theorem? Well, it turns out that while perfectly bandlimited signals can't also be perfectly time-limited, they *can* still have they're energy concentrated in one place in time. They don't ever go to zero and stay there, but they can die off, even quite strongly. The pertinent example here is the sinc signal, which is the ideal reconstruction filter used in the Sampling Theorem. This signal has energy concentrated at time 0, and falls off as 1/n from there. So if we truncate such a signal at some reasonable length, we can capture nearly all of its energy. And then if we take the Fourier Transform of the truncated signal, we will see that it is no longer perfectly bandlimited, but the energy in the signal outside the desired bandwidth will be very low. We can get further control over this by using a window instead of simply truncating (or even fancier ideas), but you get the general idea. In engineering terms, it's possible to build reconstruction filters with reasonable delay and very good stopband rejection - 100dB and beyond, pretty much the useful range of human hearing. In practice it is not the finite-time constraint on stopband rejection that limits sampling performance, but rather other more arcane circuits and systems considerations. E On Wed, Mar 26, 2014 at 10:07 PM, Doug Houghton <doug_hough...@sympatico.ca>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. > -- > 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 > -- 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
