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&lt;div&gt;{{no footnotes|date=September 2011}}&lt;br /&gt;
&lt;br /&gt;
&#039;&#039;&#039;Signal-to-Quantization-Noise Ratio&#039;&#039;&#039; (&#039;&#039;&#039;SQNR&#039;&#039;&#039; or &#039;&#039;&#039;SN&amp;lt;sub&amp;gt;q&amp;lt;/sub&amp;gt;R&#039;&#039;&#039;) is widely used quality measure in analysing [[digitizing]] schemes such as PCM ([[Pulse-code modulation|pulse code modulation]]) and [[multimedia codec]]s. 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]].&lt;br /&gt;
&lt;br /&gt;
The SQNR formula is derived from the general SNR ([[signal-to-noise ratio|Signal-to-Noise Ratio]]) formula for the binary pulse-code [[Modulation|modulated]] [[Channel (communications)|communication channel]]:&lt;br /&gt;
&lt;br /&gt;
:&amp;lt;math&amp;gt;\mathrm{SNR}=\frac{3 \times 2^{2n}}{1+4P_e \times (2^{2n} - 1)} \frac{m_m(t)^2}{m_p(t)^2}&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
where&lt;br /&gt;
&lt;br /&gt;
:&amp;lt;math&amp;gt;P_e&amp;lt;/math&amp;gt; is the probability of received bit error&lt;br /&gt;
:&amp;lt;math&amp;gt;m_p(t)&amp;lt;/math&amp;gt; is the peak message signal level&lt;br /&gt;
:&amp;lt;math&amp;gt;m_m(t)&amp;lt;/math&amp;gt; is the mean message signal level&lt;br /&gt;
&lt;br /&gt;
As SQNR applies to quantized signals, the formulae for SQNR refer to discrete-time digital signals. Instead of &amp;lt;math&amp;gt;m(t)&amp;lt;/math&amp;gt;, we will use the digitized signal &amp;lt;math&amp;gt;x(n)&amp;lt;/math&amp;gt;. For &amp;lt;math&amp;gt;N&amp;lt;/math&amp;gt; quantization steps, each sample, &amp;lt;math&amp;gt;x&amp;lt;/math&amp;gt; requires &amp;lt;math&amp;gt;\nu=\log_2 N&amp;lt;/math&amp;gt; bits. The [[probability distribution function]] (pdf) representing the distribution of values in &amp;lt;math&amp;gt;x&amp;lt;/math&amp;gt; and can be denoted as &amp;lt;math&amp;gt;f(x)&amp;lt;/math&amp;gt;. The maximum magnitude value of any &amp;lt;math&amp;gt;x&amp;lt;/math&amp;gt; is denoted by &amp;lt;math&amp;gt;x_{max}&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
As SQNR, like SNR, is a ratio of signal power to some noise power, it can be calculated as:&lt;br /&gt;
:&amp;lt;math&amp;gt;\mathrm{SQNR} = \frac{P_{signal}}{P_{noise}} = \frac{E[x^2]}{E[\tilde{x}^2]}&amp;lt;/math&amp;gt;&lt;br /&gt;
The signal power is:&lt;br /&gt;
:&amp;lt;math&amp;gt;\overline{x^2} = E[x^2] = P_{x^\nu}=\int_{}^{}x^2f(x)dx&amp;lt;/math&amp;gt; &lt;br /&gt;
The quantization noise power can be expressed as:&lt;br /&gt;
:&amp;lt;math&amp;gt;E[\tilde{x}^2] = \frac{x_{max}^2}{3\times4^\nu}&amp;lt;/math&amp;gt;&lt;br /&gt;
Giving:&lt;br /&gt;
:&amp;lt;math&amp;gt;\mathrm{SQNR} = \frac{3 \times 4^\nu\times \overline{x^2}}{x_{max}^2}&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
When the SQNR is desired in terms of [[Decibels|Decibels (dB)]], a useful approximation to SQNR is:&lt;br /&gt;
:&amp;lt;math&amp;gt;\mathrm{SQNR}|_{dB}=P_{x^\nu}+6\nu+4.8&amp;lt;/math&amp;gt;&lt;br /&gt;
where &amp;lt;math&amp;gt;\nu&amp;lt;/math&amp;gt; is the number of bits in a quantized sample, and &amp;lt;math&amp;gt;P_{x^\nu}&amp;lt;/math&amp;gt; is the signal power calculated above. Note that for each bit added to a sample, the SQNR goes up by approximately 6dB (&amp;lt;math&amp;gt;20\times log_{10}(2)&amp;lt;/math&amp;gt;).&lt;br /&gt;
&lt;br /&gt;
== References ==&lt;br /&gt;
* B.P.Lathi, Modern Digital and Analog Communication Systems (3rd edition), Oxford University Press, 1998&lt;br /&gt;
&lt;br /&gt;
==External links==&lt;br /&gt;
* [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&lt;br /&gt;
&lt;br /&gt;
{{Noise}}&lt;br /&gt;
&lt;br /&gt;
[[Category:Digital audio]]&lt;br /&gt;
[[Category:Electronics]]&lt;br /&gt;
[[Category:Engineering ratios]]&lt;br /&gt;
[[Category:Noise]]&lt;/div&gt;</summary>
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