# Compute SQNR (Signal to Quantization Noise Ratio)

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:

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?

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.