Quantization (signal processing): Difference between revisions

Revision as of 20:22, 31 October 2021

Revision as of 11:12, 22 January 2022

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===Quantization error models===

In the typical case, the original signal is much larger than one [[least significant bit]] (LSB). When this is the case, the quantization error is not significantly correlated with the signal, and has an approximately [[uniform distribution (continuous)|uniform distribution]]. When rounding is used to quantize, the quantization error has a [[mean]] of zero and the [[root mean square]] (RMS) value is the [[standard deviation]] of this distribution, given by <math>\scriptstyle {\frac{1}{\sqrt{12}}}\mathrm{LSB}\ \approx\ 0.289\,\mathrm{LSB}</math>. When truncation is used, the error has a non-zero mean of <math>\scriptstyle {\frac{1}{2}}\mathrm{LSB}</math> and the RMS value is <math>\scriptstyle {\frac{1}{\sqrt{3}}}\mathrm{LSB}</math>. Although rounding yields less RMS error than truncation, the difference is only due to the static (DC) term of <math>\scriptstyle {\frac{1}{2}}\mathrm{LSB}</math><nowiki>. The RMS values of the AC error are exactly the same in both cases, so there is no special advantage of rounding over truncation in situations where the DC term of the error can be ignored (such as in AC coupled systems). In either case, the standard deviation, as a percentage of the full signal range, changes by a factor of 2 for each 1-bit change in the number of quantization bits. The potential signal-to-quantization-noise power ratio therefore changes by 4, or </nowiki><math>\scriptstyle 10\cdot \log_{10}(4)</math>, approximately 6&nbsp;dB per bit.

At lower amplitudes the quantization error becomes dependent on the input signal, resulting in distortion. This distortion is created after the anti-aliasing filter, and if these distortions are above 1/2 the sample rate they will alias back into the band of interest. In order to make the quantization error independent of the input signal, the signal is dithered by adding noise to the signal. This slightly reduces signal to noise ratio, but can completely eliminate the distortion.