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:

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

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

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

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

$\mathrm {SQNR} ={\frac {P_{signal)){P_{noise))}={\frac {E[x^{2}]}{E[{\tilde {x))^{2}]))$ The signal power is:

${\overline {x^{2))}=E[x^{2}]=P_{x^{\nu ))=\int _{}^{}x^{2}f(x)dx$ The quantization noise power can be expressed as:

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

$\mathrm {SQNR} ={\frac {3\times 4^{\nu }\times {\overline {x^{2)))){x_{max}^{2)))$ When the SQNR is desired in terms of decibels (dB), a useful approximation to SQNR is:

$\mathrm {SQNR} |_{dB}=P_{x^{\nu ))+6.02\nu +4.77$ where $\nu$ is the number of bits in a quantized sample, and $P_{x^{\nu ))$ is the signal power calculated above. Note that for each bit added to a sample, the SQNR goes up by approximately 6dB ($20\times log_{10}(2)$ ).

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