# Signal-to-quantization-noise ratio

Signal-to-quantization-noise ratio (SQNR or SNqR) is widely used quality measure in analysing digitizing schemes such as pulse-code modulation (PCM). 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 signal-to-noise ratio (SNR) formula:

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

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:

${\displaystyle \mathrm {SQNR} |_{dB}=P_{x^{\nu }}+6.02\nu +4.77}$

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