On 25/07/2015, robert bristow-johnson <r...@audioimagination.com> wrote:
> and that's not counting noise-shaping. Since noise shaping merely changes the shape of the noise floor, that's pretty much irrelevant. Noise shaping is equivalent to applying a filter to the noise floor, for example a first order noise shaping will simply change the noise floor to a -6 dB/oct slope, like applying a first order highpass filter to white noise (like I demonstrated earlier). The overall difference will be minor (but feel free to repeat the experiment using a different noise floor, and present your results). (Also note that upsampling does not involve noise shaping at all, unless it also involves quantization.) >> Seems the conceptual misunderstanding here is that dithering does not >> mean that the error is "described as noise". > > oh dear. that's the whole point of it. > > do you understand what the difference between rectangular pdf dither and > triangular pdf dither? I could ask the same from you. > what it does to the quantization signal? the > whole purpose is to take whatever nasty non-linearity (which a staircase > function is) and turn it into something that is white (or bandlimited > white), and with it's first two moments decoupled from the value of the > input signal. Yes, "by adding noise" (using triangular, rectangular, or whatever probability distribution dither function). Sure, trivially you can "describe" noise as noise, since it is noise (what else), but it comes from _added_ noise from a noise generator, _not_ the noise from the quantization (step) function. >> Rather, dithering means >> that we deliberately *add* noise to the signal (because that has >> advantages, see the literature). > > <sigh> You cannot dither a signal if you only have a step function. That won't work. You *must* need a noise generator, hence, you're adding external noise to the signal. "<sigh>" is a really convincing technical argument, you almost convinced me. But let's see some sources: Quote from https://en.wikipedia.org/wiki/Dither: "Dither is an intentionally applied form of noise used to randomize quantization error" "[...] Lipshitz and Vanderkooy pointed out that different noise types, with different probability density functions (PDFs) behave differently when used as dither signals, and suggested optimal levels of dither signal for audio.[10][11] Gaussian noise requires a higher level for full elimination of distortion than rectangular PDF or triangular PDF noise. Triangular PDF noise has the advantage of requiring a lower level of added noise to eliminate distortion and also minimizing 'noise modulation'." "RPDF stands for "Rectangular Probability Density Function," equivalent to a roll of a dice. Any number has the same random probability of surfacing." "TPDF stands for "Triangular Probability Density Function," equivalent to a roll of two dice (the sum of two independent samples of RPDF)." "Gaussian PDF is equivalent to a roll of a large number of dice. The relationship of probabilities of results follows a bell-shaped, or Gaussian curve, typical of dither generated by analog sources such as microphone preamplifiers. If the bit depth of a recording is sufficiently great, that preamp noise will be sufficient to dither the recording." Consider the meaning of the the words: "intentionally applied form of noise", "randomize", "noise types when used as dither signals", "triangular PDF noise", "added noise", "random probability", "analog sources", "preamp noise". Let's try what happens when I dither a 16-bit constant zero signal to a 8-bit dithered signal. 16-bit constant zero signal (all samples are zero): http://morpheus.spectralhead.com/wav/zero16bit.wav Same thing dithered to 8-bit, using triangular dither (no noise shaping): http://morpheus.spectralhead.com/wav/zero8bit_dithered.wav Its spectrum looks like uniform distribution white noise: http://morpheus.spectralhead.com/img/zero8bit_dithered_spectrum.png By applying logic, consider how is it possible that the original signal was a constant zero signal (all samples zero), and the resulting dithered signal looks and sounds exactly like white noise. Some further sources to confirm: "Dithering is described in Section 5.1.2.5 as the addition of low-amplitude random noise to an audio signal as it is being quantized. The purpose of dithering is to prevent neighboring sample values from quantizing all to the same level, which can cause breaks or choppiness in the sound. Noise shaping can be performed in conjunction with dithering to raise the noise to a higher frequency where it is not noticed as much."[1] "The concept of dither is to add some random noise to the waveform in order to "break up" the statistical determineability of the stair stepped waves. We do this by literally adding noise to the signal."[2] "To dither means to add noise to our audio signal. Yes, we add noise on purpose, and it is a good thing."[3] Consider the meaning of the words: "addition of low-amplitude random noise", "add some random noise", "literally adding noise", "add noise on purpose". References: [1] "The Mathematics of Dithering and Noise Shaping " http://digitalsoundandmusic.com/5-3-7-the-mathematics-of-dithering-and-noise-shaping/ [2] "Dither Explained - An explanation and proof of the benefit of dither for the audio engineer" http://www.users.qwest.net/~volt42/cadenzarecording/DitherExplained.pdf [3] "What is dither?" http://www.earlevel.com/main/1996/10/20/what-is-dither/ To disprove these sources, you will need a stronger argument than "<sigh>". Remark: reference [3] is written by a member of this mailing list. Now let's check the source code of some dither algorithm. Dither.cpp from Audacity: https://audacity.googlecode.com/svn/audacity-src/trunk/src/Dither.cpp Here's how DITHER_NOISE is defined: // This is supposed to produce white noise and no dc #define DITHER_NOISE (rand() / (float)RAND_MAX - 0.5f) Consider that it uses C library's uniform distribution random number generator function rand() that - when normalized - generates uniform distribution white noise. Consider the comment "this is supposed to produce white noise". Let's see how rectangular dithering is defined: // Rectangle dithering, apply one-step noise inline float Dither::RectangleDither(float sample) { return sample - DITHER_NOISE; } Consider that all it does, is it merely adds "white noise" to the signal. Consider the comment "apply one-step noise". For the the source code for the functions for triangular and noise shaped dithering, see the above link. Consider that all of them are based on DITHER_NOISE (= "white noise") generated using C library's standard uniform distribution random generator function rand(). Example, at one point, triangular dither is defined as: // Generate triangular dither, +-1 LSB, flat psd float r = DITHER_NOISE + DITHER_NOISE; In other words, it's defined as sum of two white noise sources. If at this point, you still fail to realize that dithering means "adding random noise to the signal", there's really nothing else left to say, you fail at basic logic. I have shown several ways why dithering means 'adding noise to a signal'. I've shown audio examples, showing that a constant zero signal becomes noise when dithered (I've shown the very same example two different times), I've shown how certain types of noise shaping change the spectrum of the the noise floor of dithers, I've shown how some of the various PDF's look like in description and in algorithm, I've linked about 30+ external references in the topic of sigma-delta converters and dithering, and I've quoted more than a half dozen various sources. There is no point in repeating this further. If, after all this, you still fail to understand that "dithering means adding random noise to a signal", you basically 1) fail at basic logic 2) live in an alternate universe Since a ditherered constant zero signal becomes nonzero after dithering, it follows that if 0+X is nonzero, then X must be nonzero. If you have a zero entropy constant signal, and the result of a function is a noisy signal, then the only way that this can happen, is that you added noise in the function. Said otherwise, "you don't get something from nothing", unless you _add_ something - believing otherwise means you live in an alternate reality (i.e. you're delusional). _______________________________________________ music-dsp mailing list music-dsp@music.columbia.edu https://lists.columbia.edu/mailman/listinfo/music-dsp
