Re: [music-dsp] Nyquist–Shannon sampling theorem
As I remember it, the sampling theorem says that the sampling rate used to sample a signal must be at least twice the highest frequency being sampled in order to get a faithful reproduction when the samples are turned back into a (continuous) output signal. In practice, because it is necessary to band limit most signals to prevent aliasing artifacts, the sampling rate usually needs to be about 2.2 times the highest frequency being sampled, since it is impossible in practice to create low pass filters that are extremely steep, steep enough to allow a sampling rate of only 2 times the highest frequency involved to prevent aliasing. Take the standard 8000 Hz sampling rate for telephone toll quality voice used with mu-law and a-law voice codecs for long distance digital transmission. The specified upper frequency is about 3400 Hz, as I recall. 8000 Hz is more than 2 times 3400 Hz, it's close to 2.2 times. I think you can have frequencies changing amplitude and jumping in and out subject to the constraints given above. Obviously, telephone voice signal will have different frequencies at different times depending on the speaker and the words being spoken. Music has different frequencies and amplitudes throughout a particular performance. I'm not sure what you mean by discontinuous. When people speak on the telephone, there are often periods of silence between periods of speech signal. Signals that have sharp rising edges like the unit step function and the infinite impulse (Dirac Delta) function will obviously be band limited and their shape and frequency content changed by the anti-aliasing low pass filter used before sampling takes take place. I don't think you get an exact reconstruction of the signal, but with proper filtering after being converted by a digital to analog converter, you can get a signal that sounds (or looks like, on an oscilloscope) a lot like the original signal, such that it is recognizable and intelligible. Music is band limited to 20 KHz and sampled at at least 44,100 Hz for recording on CDs. What is your application? From: Doug Houghton doug_hough...@sympatico.ca To: A discussion list for music-related DSP music-dsp@music.columbia.edu Sent: Wednesday, March 26, 2014 10:42 PM Subject: [music-dsp] Nyquist–Shannon sampling theorem I can't seem to get to the bottom of this with the usual internet pages. Is the test signal, while possibly containing any number of wave compenents at various frequencies, required to be continous ansd uniform? By this I mean you can't have frequencies jumping in and out, changing in amplitude etc... I'm guessing this somehow scratches at the surface of what I've read about no signal being properly band limited unless it's infinit. I fail to see how a readable proof is possible to explain exact reconstruction of any real recording sound, whether it's music or crickets chirping. I sort of see maybe how an infinit signal could solve some of these issues, meaning any amplitude/frequency complexities over infinity may simply resolve to something that can be bandlimited and described as a frequency of a steady signal, something like that. Curouis, I am starting to suspect there is a lot of typical misconceptions about what the math really proves, I can't read the equations I'm turning to this list. -- 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
Re: [music-dsp] Nyquist–Shannon sampling theorem
The application is music. I understand the basics, my question is in the constraints that might be imposed on the signal or functon as referenced by the theory. Is it understood to be repeating? for lack of a better term, essentually just a mash of frequencies that bever change from start to finish. I'm thinking the math must consider it this way, or rather the difference is abstracted since the signal is assumed to be band limited, which means infinit, which means you can create any random signal by inject the required freuencies at the reuired amplitides and phase from start to finish, even a 20k 2ms blip in the middle of endless silence. Is that making any sense? I'm struggling with the fine points. I bet this is obvious if you understand the math in the proof. -- 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
Re: [music-dsp] Nyquistâ?Shannon sampling theorem
consider this from a wiki page A bandlimited signal can be fully reconstructed from its samples, provided that the sampling rate exceeds twice the maximum frequency in the bandlimited signal. This minimum sampling frequency is called the Nyquist rate. This result, usually attributed to Nyquist and Shannon, is known as the Nyquist-Shannon sampling theorem. An example of a simple deterministic bandlimited signal is a sinusoid of the form . If this signal is sampled at a rate so that we have the samples , for all integers , we can recover completely from these samples. Similarly, sums of sinusoids with different frequencies and phases are also bandlimited to the highest of their frequencies. The example may imply that the bandlimited signal to satisfy the theory is at it's most a complex sum of various sinusoids at different frequencies phases, amplitudes. aa9a7b3fc744c653a5629d4b3d6ae5fd.pngfb409984dea7c4b5f093208b3174ac4c.png269e6a3cdee35a7eec719c55abbf640a.png7b8b965ad4bca0e41ab51de7b31363a1.pnge34fd49d79f3869d9033f958be91021e.png-- 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
Re: [music-dsp] Nyquist–Shannon sampling theorem
sorry about all the attachments, didn't see that coming. -- 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
Re: [music-dsp] Nyquist–Shannon sampling theorem
Hi Doug, I think you’re overthinking this… There is the frequency-sensitive requirement that you can’t properly sample a signal that has frequencies higher than half the sample rate. For music, that’s not a problem, since our ears have a significant band limitation anyway. So, if we have a musical signal with lots of discontinuities, resulting in strong frequencies to, say, 100 kHz, and we make a copy with everything stripped off above 20 kHz with a lowpass filter, the two waveforms will not look alike. But they will sound alike to us. Now sample that at 44.1 kHz, 24-bit. Then push that out to a D/A converter back to a third analog waveform. It *will* look just like the second waveform, and it will sound like both it and the original waveform. By 20k 2ms blip”, I assume you mean a 2 ms step that has been band limited to 20 kHz. Sure, no problem. On Mar 26, 2014, at 9:23 PM, Doug Houghton doug_hough...@sympatico.ca wrote: The application is music. I understand the basics, my question is in the constraints that might be imposed on the signal or functon as referenced by the theory. Is it understood to be repeating? for lack of a better term, essentually just a mash of frequencies that bever change from start to finish. I'm thinking the math must consider it this way, or rather the difference is abstracted since the signal is assumed to be band limited, which means infinit, which means you can create any random signal by inject the required freuencies at the reuired amplitides and phase from start to finish, even a 20k 2ms blip in the middle of endless silence. Is that making any sense? I'm struggling with the fine points. I bet this is obvious if you understand the math in the proof. -- -- 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
Re: [music-dsp] Nyquist–Shannon sampling theorem
There is the frequency-sensitive requirement that you can’t properly sample a signal that has frequencies higher than half the sample rate. For music, that’s not a problem, since our ears have a significant band limitation anyway. This is intuitive. I think perhaps what I'm asking has more to do directly with the fourier series than sample theory. It's my understanding that the fourier theory says any signal can be created by summing various frequencies at various phases and amplitudes. So this would answer my question then that it's not really a stipulation of the function persay, since any signal at all can be described this way. -- 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
Re: [music-dsp] Nyquist–Shannon sampling theorem
It's my understanding that the fourier theory says any signal can be created by summing various frequencies at various phases and amplitudes. OK, now recall that the Fourier series describes a subset of “any signal” with a subset of “various frequencies”. It’s more like one cycle of any waveform can be created by summing sine waves of multiples of that cycle at various phases and amplitudes (a little awkward, but trying to modify your words). Fourier figured that out by observing the way heat traveled around an iron ring (hence the focus on cycles)–he wasn’t really into the recording scene back then ;-) (It’s true that Fourier techniques can be used to create more arbitrary signals, but that somewhat in the manner that movies are made from many still pictures.) So, it seems that you’re trying to match the theory of a very specific, limited portion of a signal, with one that doesn’t have those limitations. On Mar 26, 2014, at 9:46 PM, Doug Houghton doug_hough...@sympatico.ca wrote: There is the frequency-sensitive requirement that you can’t properly sample a signal that has frequencies higher than half the sample rate. For music, that’s not a problem, since our ears have a significant band limitation anyway. This is intuitive. I think perhaps what I'm asking has more to do directly with the fourier series than sample theory. It's my understanding that the fourier theory says any signal can be created by summing various frequencies at various phases and amplitudes. So this would answer my question then that it's not really a stipulation of the function persay, since any signal at all can be described this way. -- -- 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
Re: [music-dsp] Nyquist–Shannon sampling theorem
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
Re: [music-dsp] Nyquist–Shannon sampling theorem
I'm guessing this somehow scratches at the surface of what I've read about no signal being properly band limited unless it's infinit. You're talking about Sinc filtering (ideal low pass filter), which is essentially an IIR filter that needs infinite past and future samples. In practice, a very steep filter is used to attenuate the signal above the Nyquist frequency to almost nothing. A Lanczos (windowed Sinc) filter will be close to ideal. For synthesis of non-sinusoidal test waveforms, a BLIT (Band-Limited Impulse Train) oscillator will give you a perfectly band limited signal. Thor smime.p7s Description: S/MIME Cryptographic Signature -- 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
Re: [music-dsp] Nyquist–Shannon sampling theorem
On 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. -- 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… So, let’s back off. The sampling theorem says that you can recreate any signal as long as you sample at a rate of more than twice the highest frequency component. Now, how do you feel that conflicts with Fourier theory, specifically? -- 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
Re: [music-dsp] Dither video and articles
On 2014-03-26, Nigel Redmon wrote: Maybe this would be interesting to some list members? A basic and intuitive explanation of audio dither: https://www.youtube.com/watch?v=zWpWIQw7HWU Since it's been quiet and dither was mentioned... Is anybody interested in the development of subtractive dither? I have a broad idea in my mind, and a little bit of code (for once!) as well. Unfortunately nothing too easily adaptable though... Willing to copy and explain all of it, though. :) The video will be followed by a second part, in the coming weeks, that covers details like when, and when not to use dither and noise shaping. I’ll be putting up some additional test files in an article on ear level.com in the next day or so. In any case, thank you kindly. Dithering and noise shaping, both in theory and in practice is *still* something far too few people grasp for real. -- Sampo Syreeni, aka decoy - de...@iki.fi, http://decoy.iki.fi/front +358-40-3255353, 025E D175 ABE5 027C 9494 EEB0 E090 8BA9 0509 85C2-- 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
