Quantization

With the minimum amplitude $A_\mathit{FS-}$ maximum ampitude $A_\mathit{FS+}$ and the number of quantization steps:

$$ \Delta_a = \frac{A_\mathit{FS-} - A_\mathit{FS+}}{N_{\mathit{steps}}} $$

$$ N_{\mathit{steps}} = 2^{N_{\mathit{bits}}} $$

The plot below shows the quantization curve for a 3-bit quantizer - with 8 steps.

Typical bit depths in the history of digital music:

• 8 bit has a very distinc distortion:
  • Fairlight CMI
  • LinnDrum
• 12 bit is still recognizable:
  • E-mu SP-1200
• 16 bit has ben the standard for a long time and only causes artefacts in specific situations:
  • CD
• 24 bit makes it almost unnecessary to worry about bit depth:
  • today's recording standard in most audio interfaces
• 32 bit ....
  • improved recording standard
  • representation in DAW's

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Quantization Error¶

The quantization error $e$ is calculated as the difference between original and quantized signal:

$$ e = x-x* $$

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Probability Density of the Quantization Error¶

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Power of the Quantization Error¶

$$\begin{eqnarray} W &=& \int_{-\Delta / 2}^{\Delta / 2} e^2 p(e) d e \\ &=& \frac{1}{\Delta} \int_{-\Delta / 2}^{\Delta / 2} q^2 d e \\ &=& \frac{1}{\Delta} \left[\frac{1}{3} q^3 \right]_{-\Delta / 2}^{\Delta / 2} \\ &=& \frac{1}{3 \Delta} \left(\frac{\Delta^3}{8} + \frac{\Delta^3}{8} \right) \\ &=& \frac{\Delta^2}{12} \end{eqnarray}$$