signal-to-quantization-noise ratio

{{Short description|Measure for analyzing digitizing schemes}}

{{no footnotes|date=September 2011}}

Signal-to-quantization-noise ratio (SQNR or SN_qR) 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:

:$\backslash mathrm\{SNR\}=\backslash frac\{3\; \backslash times\; 2^\{2n\}\}\{1+4P\_e\; \backslash times\; (2^\{2n\}\; -\; 1)\}\; \backslash frac\{m\_m(t)^2\}\{m\_p(t)^2\}$

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 $\backslash nu=\backslash log\_2\; N$ bits. The probability distribution function (PDF) represents 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:

:$\backslash mathrm\{SQNR\}\; =\; \backslash frac\{P\_\{signal\}\}\{P\_\{noise\}\}\; =\; \backslash frac\{E[x^2]\}\{E[\backslash tilde\{x\}^2]\}$

The signal power is:

:$\backslash overline\{x^2\}\; =\; E[x^2]\; =\; P\_\{x^\backslash nu\}=\backslash int\_\{\}^\{\}x^2f(x)dx$

The quantization noise power can be expressed as:

:$E[\backslash tilde\{x\}^2]\; =\; \backslash frac\{x\_\{max\}^2\}\{3\backslash times4^\backslash nu\}$

Giving:

:$\backslash mathrm\{SQNR\}\; =\; \backslash frac\{3\; \backslash times\; 4^\backslash nu\backslash times\; \backslash overline\{x^2\}\}\{x\_\{max\}^2\}$

When the SQNR is desired in terms of decibels (dB), a useful approximation to SQNR is:

:$\backslash mathrm\{SQNR\}|\_\{dB\}=P\_\{x^\backslash nu\}+6.02\backslash nu+4.77$

where $\backslash nu$ is the number of bits in a quantized sample, and $P\_\{x^\backslash nu\}$ is the signal power calculated above. Note that for each bit added to a sample, the SQNR goes up by approximately 6 dB ($20\backslash times\; log\_\{10\}(2)$).