Signal-to-quantization-noise ratio

The acronym SQNR (standing for Signal-to-Quantization Noise Ratio) is widely used in communication systems analysis, particularly in PCM (pulse code modulation) schemes.

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

$SNR={\frac {(3(2^{2}*2^{n}))}{(1+4*Pe(2^{2}*2^{n})-1))}}{\frac {(mean(m(t))^{2}}{(m(t)peak)^{2})}}$

where

Pe is the probability of received bit error

m(t) is the message signal

Since SQNR applies to quantized signals, then the formulae involved with SQNR refer to discrete-time digital signals. Thus, instead of $m(t)$ , we will used the digitized signal $x(n)$ . For $N$ quantization steps, there are $\nu =\log _{2}N$ bits needed for each sample, $x$ . 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 $x_{max}$ .

Since SQNR, like SNR, is a ratio of signal power to some noise power, we calculate $SQNR={\frac {P_{signal}}{P_{noise}}}={\frac {E[x^{2}]}{E[{\tilde {x}}^{2}]}}$ The signal power is calculated $E[x^{2}]=P_{x^{\nu }}=\int _{}^{}x^{2}f(x)dx$ and will be notated ${\overline {x^{2}}}$ . The quantization noise power can be expressed ${\frac {x_{max}^{2}}{3\times 4^{\nu }}}$

This leads to $SQNR={\frac {3\times 4^{\nu }{\overline {x^{2}}}}{x_{max}^{2}}}$

When the SQNR is desired in terms of Decibels (dB), a useful approximation to SQNR is as follows: $SQNR|_{dB}=P_{x^{\nu }}+6\nu +4.8$ 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 about 6dB ( $20*log_{10}(2)$ to be exact).