Say you have a signal that is 16 bits in amplitude, 2^16 = 65,536 different levels. A continuous analog signal has no steps at all (until the quantum level), but if you convert (quantize) that signal to sixteen bits, then you have 65,536 steps 1 unit (lsb) tall. Thinking of just one step going from 0 to 1, the analog signal can take on any value between 0 and 1, but when you convert it to a digital number you get 0 or 1 and nothing in between. The difference is the quantization (conversion) error. If the actual signal is 0.5, then you have an error of 0.5 between the real signal and what the converter says is the signal. For this example, the steps are all of value 1, so (0,1,2,3,4... 65,535). If the actual signal level is 3.141592654... (pi) then the error would be 0.141592654..., the difference to the nearest threshold (3). That error is (becomes) the quantization noise.
All my other derivations are longer and uglier, sorry... It took me long enough to type in this one, not used to equations in Word, then got blindsided because I couldn't copy and paste it here without losing the symbols.
I don't fully understand the situation once I start asking myself more questions about real signal amplitudes, band limiting, sample rates and the work done by converters. I think I've reached the point that if I don't do the math for the whole path there's little to be gained from more description and the usual rules of thumb, like the one that motivated this thread, which are such crutches for laymen like me. If anything you've made me want school again.
I suggest you follow the Technical Articles link in my signature and read about sampling and such. That may help understand this one.
The problem is that I had Greek symbols embedded throughout the text as well as in the equations so it was a lot to deal with. I did copy and convert all the equations to images in Word and pasted them in, but then still had Greek symbols interspersed throughout the text. I tried creating an image of just a Greek delta, for example, but never could get it to align and flow smoothly in the text. After wasting way too much time I just used Windows snip to create images of the pages I could just paste into the post. Lesson learned; I'll either avoid trying to create articles like this, or just copy and paste the sheets as images like I did this time.
I hope I won't muddy the waters further. Here we have a signal which is sampled but not quantized yet:
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Do you mean a wavelength that is equal to the Planck distance? (I am not very learned in quantum physics but it seems to me that the point where an analog signal would have steps rather than a continuous waveform would only be where the wavelength is as short as the Planck length, because any points between +/- the Planck length would not be measurable).
I suppose but for real materials there are issues above that level. I don't want to go there in this thread; physicists already know that stuff so I'll let them explain it. My quantum mechanics and materials courses were long ago. This thread is all at the macro level where we consider analog "continuous" and digital "discrete", i.e. quantized in time and amplitude. For the SNR equation time drops out and we are left with just amplitude (and thus resolution, number of bits) determining the SNR.
Thanks for this. We are lucky to have you participating in this forum.
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On the contrary, I think this is a really good way of explaining quantization error. You know what they say, a picture speaks a thousand integrals
