>what other presumption is there? i, personally, have never seen a sequence of >samples of audio or music that was not "equidistant and linearly sampled". it's > what we call "uniform sampling".
Some of this new stuff in compressive sensing/sparse reconstruction involves non-uniform sampling. Not that I've seen it used much in practice in audio and music, and anyway it would typically be done on top of conventional sampling anyway in those contexts (and largely invisible to the audio/music side of things, at least if done right). >dunno what you're getting at, Theo. both graphics appear fully as expected to me. Yeah I'm at a loss as well. Isn't this stuff well-explained on Wikipedia, and in every book that covers spectral analysis? E On Mon, Dec 8, 2014 at 11:01 AM, robert bristow-johnson < r...@audioimagination.com> wrote: > On 12/8/14 11:52 AM, Theo Verelst wrote: > >> Hi, >> >> Having seen a lot of subjects here lately around basic subjects, I think >> it might be interesting for some to think about yet another not so highly >> spun mathematical subject directly relevant to certain DSP activities (I >> use it too). >> >> It's a common notion to take it that if we take a sequence of samples >> (presuming a equidistant and linearly sampled signal with sufficiently >> accurate digital sample representation) >> > > what other presumption is there? i, personally, have never seen a > sequence of samples of audio or music that was not "equidistant and > linearly sampled". it's what we call "uniform sampling". > > > we can apply the well known Fast Fourier Transform to them, to get a set >> of frequency+phase tuples. Of course there's a correlation between the >> magnitude of the transformed frequency components and frequencies present >> in the signal we have sampled (presume for the moment we honored the >> Niquist criterion). However, if we want to be accurate, or claim generality >> like present in the (continuous, infinite) Fourier transform, what I'm >> pointing at is that without precautions, it isn't a good idea to presume >> the FFT transformed "spectrum" is the same, or even close to the Fourier >> spectrum of the sampled signal. If sampling (and if needed reconstruction) >> is accurate, the frequencies present in the digital version of an analog >> signal should of course be exactly the same as in the the analog signals >> that was sampled. >> >> Let's look at simple examples of the errors that can take place. Here's a >> decent (simple) example, 8 harmonics of a square wave that fits exactly in >> the FFT interval, i.e. if we take an fft length (in terms of samples) of >> 256 we make sure the fundamental frequency of the square wave corresponds >> to 256 samples, too: >> >> http://www.theover.org/Musicdspexas/fft_square8.png >> >> To a certain accuracy, the measured frequency components (shown at the >> bottom of the figure) will have the same magnitude as the components summed >> together to make for the (above) waveform. >> >> Now say we take a saw wave (with all harmonics), and we take a frequency >> which in the sample domain doesn't correspond to a multiple of the FFT >> interval, we are going to get a *wrong* frequency graph: >> >> http://www.theover.org/Musicdspexas/fft_sawpl20perc.png >> >> dunno what you're getting at, Theo. both graphics appear fully as > expected to me. > > besides > > 1. Uniform sampling (and the effects thereof) > > it's about > > 2. Windowing (and the effects thereof) > > and the > > 3. Periodic extension inherent to the DFT (and the effects thereof). > > > there is aliasing involved in the line spectra regarding "the effects > thereof". > > not much else happening. > > > ?? > > -- > > r b-j r...@audioimagination.com > > "Imagination is more important than knowledge." > > > > -- > 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
