SQNR

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 $u=log\_2\; N$ 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 $SQNR=frac\{P\_\{signal\{P\_\{noise=frac\{E\; [x^2]\; \}\{E\; [\; ilde\{x\}^2]\; \}$The signal power is calculated $E\; [x^2]\; =P\_\{x^\; u\}=int\_\{\}^\{\}x^2f(x)dx$ and will be notated $overline\{x^2\}$.The quantization noise power can be expressed $frac\{x\_\{max\}^2\}\{3\; imes4^\; u\}$

This leads to $SQNR=frac\{3\; imes4^\; uoverline\{x^2\{x\_\{max\}^2\}$

When the SQNR is desired in terms of Decibels (dB), a useful approximation to SQNR is as follows:$SQNR|\_\{dB\}=P\_\{x^\; u\}+6\; u+4.8$ where $u$ is the number of bits in a quantized sample, and $P\_\{x^\; u\}$ 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).

References

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

* Dr. Gimmy Chu - University Of Toronto, 2005

* Comrade Pavel Chtchetinin - University Of Toronto, 2005

External links

* [http://www.dsplog.com/2007/03/19/signal-to-quantization-noise-in-quantized-sinusoidal/ Signal to quantization noise in quantized sinusoidal] - Analysis of quantization error on a sine wave