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In the slide shown below by red arrow we have quantization noise up to fs/2, but why?

From Nyquist fs=fb/2 for good sampling but why is there noise here noise up to fs/2?

What is the logic ?

enter image description here

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    \$\begingroup\$ What is fb in fs = fb/2 ? Please add a link to the source of the image you have posted. \$\endgroup\$
    – AJN
    Sep 19 '20 at 16:06
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Quantization is a non-linear process and in general produces an output waveform with lots of harmonics which in digital domain would alias back to frequency region between \$[-\frac{f_s}{2},\frac{f_s}{2}]\$. Thus the spectrum of quantization "error" could cover the entire Nyquist band. Note that quantization error is a deterministic signal which can be calculated from the input signal and the quantizer characteristics and hence does not behave as noise at all.
For convenience, we can model the quantization error as a white noise. For this assumption to hold, we need to ensure that the input signal is "well-behaved". These assumptions include:

  1. The input signal should be able to cover the entire range of the quantizer.
  2. The input signal changes sufficiently between sample to sample so that its position within the quantization interval is random.
  3. The input signal frequency should not be harmonically related to the sampling frequency.

Essentially, these assumptions make sure that the quantization error gets a different value from sample to sample and can be considered as random (uncorrelated from sample to sample) and hence treated as 'white noise' with flat power spectral density (PSD). If these assumptions are not satisfied quantization error power will not be flat.

This can be easily simulated in MATLAB. For instance, consider an input sinusoid of frequency \$f = \frac{19}{4096}f_s\$ sampled at sampling frequency \$f_s = 1MHz\$. The quantizer rounds the sampled signal to the nearest 2 decimal position (\$\Delta = 0.01\$). Then, the quantization error can be seen to be quite random, as shown below: enter image description here

enter image description here

enter image description here

The PSD (one-sided) of the quantization error from white noise assumption comes to be \$P = 10*log10(2*\frac{\Delta^2}{12}) = 10*log10(2*\frac{0.01^2}{12}) = -107dB/Hz\$ is shown as orange line above. We see that the white noise assumption is holding quite well.

Consider another case where signal frequency, \$f = f_s/8 = 125kHz\$. Now the white noise assumption is not valid and the noise power is concentrated at \$\frac{3f_s}{4},\frac{f_s}{2}\$. This is expected since quantization error is now a periodic signal. All this is happening as assumption number 3 is not satisfied:

enter image description here enter image description here enter image description here Just to clarify, the PSD plots is showing the PSD of the output signal after quantization and the biggest tones in both the plots is the input signal.

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  • \$\begingroup\$ Yess its an FFT property if we do FFT on Fs then we have bins 0 to fs/2. am i correct? \$\endgroup\$
    – rocko445
    Sep 19 '20 at 18:48
  • \$\begingroup\$ Yes you are right \$\endgroup\$
    – sarthak
    Sep 19 '20 at 18:49
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The image mentions that they have assumed that the quantisation noise is white. Hence the same value at all frequencies. Since the noise is present in the sampled signal, all frequencies mean 0 to fs/2. For a discrete time signal, the spectrum outside the range 0 to fs/2 are copies of the values inside 0 to fs/2 range.

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  • \$\begingroup\$ yes we make a copy of the signal and noise +fs to the right -fs to the left,how does it make the noise be till fs/2 why if we seample till fs the noise is till fs/2. Its supposed to be till fs/2 \$\endgroup\$
    – rocko445
    Sep 19 '20 at 16:40
  • \$\begingroup\$ The values from fs/2 to fs is also a copy (flipped) of the values from 0 to fs. You can also imagine it as being -fs/2 to +fs/2 (total is still fs). The values in the negative frequency range are copies of the ones on the positive side. \$\endgroup\$
    – AJN
    Sep 19 '20 at 17:16
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The key word in "assumed white" is "assumed".

The model of quantization as signal + (quantization) noise is a nice way to deal with quantization under certain circumstances. In particular, it works when you either have a signal that's got a lot of spectral content, or that otherwise can be counted on to "scramble" the error due to quantization. The assumption of whiteness is especially safe if you're using an ADC that has a lot of intrinsic noise -- this will often contribute enough noise that the ADC by itself will whiten the quantization noise.

On the other hand, if you have a quiet ADC with a small input signal, quantization error can be anything but white. In particular, in a control system you can get an oscillation around a quantization point -- if you suspect that such a thing may happen, your best bet is to assume that all the quantization noise will be concentrated at the worst possible frequency.

And just to finish: in a sigma-delta converter, the quantization noise is intentionally shaped so that it is "blue" -- meaning so the noise is concentrated at the high frequencies rather than low. Then the output of the actual quantization element is low-pass filtered to yield an (often astonishingly) higher-precision, lower bandwidth result.

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Bernard Widrow (who along wih Ted Hoff developed the various adaptive algorithms used in digital echo cancellation, etc) has a fine book out, on Quantization.

In the early chapters he discusses THREE different types of quantization noise.

The 3rd type will produce TONES. These tones result from either

  • triangular inputs, with constant slewrate; as the input signal crosses the code boundaries, the quantization errors are also (very fast) triangular errors, at a constant period

or

  • very slowly changing input sigals that have SLOW Slewrate, and for small intervals of time will produce repetitive triangular quantization errors
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