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I'm studying the quantization of an audio signal and in particular the SQNR (Signal to Quantization Noise Ratio).

The book on which the study says that:

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where:

  • N is the number of bits in the digital representation
  • V indicates that the signal voltage varies between -V and +V.

I don't understand how these calculations were carried out. Is SQNR not simply the "digital" version of the RMS? I know the RMS is calculated as power signal divided by power error, am I wrong?

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The signal voltage is a sinusoid signal variant from $-V$ to $+V$, thus the RMS value

$$V_{rms} = \frac{V}{\sqrt{2}} = \frac{2^N \Delta V / 2}{\sqrt{2}}$$

because $2V = 2^N\Delta V$ where $\Delta V$ is quantization step.

Quantization noise is modeled as uniform random variable in $[-\Delta V /2, +\Delta V /2]$ thus its standard deviation is $V_n = \Delta V / \sqrt{12}$.

SQRN is defined as $\mathrm{SQNR}=\frac{V_{rms}^2}{V_n^2}$

$$\sqrt{\mathrm{SQNR}}=\frac{V_{rms}}{V_n} = 2^N \sqrt{\frac{3}{2}}$$

$$\mathrm{SQNR}_{dB}=20\log_{10}\left(\frac{V_{rms}}{V_n}\right) = 20\log_{10}\left(2^N \sqrt{\frac{3}{2}}\right) = 6.02N + 1.76$$

You must have mistyped your question.

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