Signal-to-quantization-noise ratio

Measure for analyzing digitizing schemes

This article includes a list of references, related reading, or external links, but its sources remain unclear because it lacks inline citations . Please help improve this article by introducing more precise citations. (September 2011) (Learn how and when to remove this message)

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:

\mathrm {SNR} ={\frac {3\times 2^{2n}}{1+4P_{e}\times (2^{2n}-1)}}{\frac {m_{m}(t)^{2}}{m_{p}(t)^{2}}}

{\displaystyle \mathrm {SNR} ={\frac {3\times 2^{2n}}{1+4P_{e}\times (2^{2n}-1)}}{\frac {m_{m}(t)^{2}}{m_{p}(t)^{2}}}}

where:

P_{e}

{\displaystyle P_{e}} is the probability of received bit error

m_{p}(t)

{\displaystyle m_{p}(t)} is the peak message signal level

m_{m}(t)

{\displaystyle 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 $m(t)$ {\displaystyle m(t)}, the digitized signal $x(n)$ {\displaystyle x(n)} will be used. For $N$ {\displaystyle N} quantization steps, each sample, $x$ {\displaystyle x} requires $\nu =\log _{2}N$ {\displaystyle \nu =\log _{2}N} bits. The probability distribution function (PDF) represents the distribution of values in $x$ {\displaystyle x} and can be denoted as $f(x)$ {\displaystyle f(x)}. The maximum magnitude value of any $x$ {\displaystyle x} is denoted by $x_{max}$ {\displaystyle x_{max}}.

As SQNR, like SNR, is a ratio of signal power to some noise power, it can be calculated as:

\mathrm {SQNR} ={\frac {P_{signal}}{P_{noise}}}={\frac {E[x^{2}]}{E[{\tilde {x}}^{2}]}}

{\displaystyle \mathrm {SQNR} ={\frac {P_{signal}}{P_{noise}}}={\frac {E[x^{2}]}{E[{\tilde {x}}^{2}]}}}

The signal power is:

{\overline {x^{2}}}=E[x^{2}]=P_{x^{\nu }}=\int _{}^{}x^{2}f(x)dx

{\displaystyle {\overline {x^{2}}}=E[x^{2}]=P_{x^{\nu }}=\int _{}^{}x^{2}f(x)dx}

The quantization noise power can be expressed as:

E[{\tilde {x}}^{2}]={\frac {x_{max}^{2}}{3\times 4^{\nu }}}

{\displaystyle E[{\tilde {x}}^{2}]={\frac {x_{max}^{2}}{3\times 4^{\nu }}}}

Giving:

\mathrm {SQNR} ={\frac {3\times 4^{\nu }\times {\overline {x^{2}}}}{x_{max}^{2}}}

{\displaystyle \mathrm {SQNR} ={\frac {3\times 4^{\nu }\times {\overline {x^{2}}}}{x_{max}^{2}}}}

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

\mathrm {SQNR} |_{dB}=P_{x^{\nu }}+6.02\nu +4.77

{\displaystyle \mathrm {SQNR} |_{dB}=P_{x^{\nu }}+6.02\nu +4.77}

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