SQNR

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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 $ν = log 2 N$ 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 $x m a x$ .

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 $E[x^2]=P_{x^\nu}=\int_{}^{}x^2f(x)dx$ and will be notated $\overline{x^2}$ . The quantization noise power can be expressed $\frac{x_{max}^2}{3\times4^\nu}$

This leads to $SQNR=\frac{3\times4^\nu\overline{x^2}}{x_{max}^2}$

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 $ν$ 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 ( $20 * l o g 10 (2)$ to be exact).