# Signal-to-quantization-noise ratio

Signal-to-Quantization-Noise Ratio (SQNR or SNqR) is widely used quality measure in analysing digitizing schemes such as PCM (pulse code modulation) and multimedia codecs. The SQNR reflects the relationship between the maximum nominal signal strength and the quantization error (also known as quantization noise) introduced in the analog-to-digital conversion.

The SQNR formula is derived from the general SNR (Signal-to-Noise Ratio) formula for the binary pulse-code modulated communication channel:

${\displaystyle \mathrm {SNR} ={\frac {3\times 2^{2n}}{1+4P_{e}\times (2^{2n}-1)}}{\frac {m_{m}(t)^{2}}{m_{p}(t)^{2}}}}$

where

${\displaystyle P_{e}}$ is the probability of received bit error
${\displaystyle m_{p}(t)}$ is the peak message signal level
${\displaystyle m_{m}(t)}$ is the mean message signal level

As SQNR applies to quantized signals, the formulae for SQNR refer to discrete-time digital signals. Instead of ${\displaystyle m(t)}$, the digitized signal ${\displaystyle x(n)}$ will be used. For ${\displaystyle N}$ quantization steps, each sample, ${\displaystyle x}$ requires ${\displaystyle \nu =\log _{2}N}$ bits. The probability distribution function (pdf) representing the distribution of values in ${\displaystyle x}$ and can be denoted as ${\displaystyle f(x)}$. The maximum magnitude value of any ${\displaystyle x}$ is denoted by ${\displaystyle x_{max}}$.

As SQNR, like SNR, is a ratio of signal power to some noise power, it can be calculated as:

${\displaystyle \mathrm {SQNR} ={\frac {P_{signal}}{P_{noise}}}={\frac {E[x^{2}]}{E[{\tilde {x}}^{2}]}}}$

The signal power is:

${\displaystyle {\overline {x^{2}}}=E[x^{2}]=P_{x^{\nu }}=\int _{}^{}x^{2}f(x)dx}$

The quantization noise power can be expressed as:

${\displaystyle E[{\tilde {x}}^{2}]={\frac {x_{max}^{2}}{3\times 4^{\nu }}}}$

Giving:

${\displaystyle \mathrm {SQNR} ={\frac {3\times 4^{\nu }\times {\overline {x^{2}}}}{x_{max}^{2}}}}$

When the SQNR is desired in terms of decibels (dB), a useful approximation to SQNR is:

${\displaystyle \mathrm {SQNR} |_{dB}=P_{x^{\nu }}+6\nu +4.8}$

where ${\displaystyle \nu }$ is the number of bits in a quantized sample, and ${\displaystyle P_{x^{\nu }}}$ is the signal power calculated above. Note that for each bit added to a sample, the SQNR goes up by approximately 6dB (${\displaystyle 20\times log_{10}(2)}$).

## References

• B.P.Lathi , Modern Digital and Analog Communication Systems (3rd edition), Oxford University Press, 1998