Signal-to-quantization noise ratio

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:

:$\mathrm=\frac \frac$

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 $\nu=\log_2 N$ bits. 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 $x_$.

As SQNR, like SNR, is a ratio of signal power to some noise power, it can be calculated as:
:$\mathrm = \frac = \frac$
The signal power is:
:\overline = E ^2= P_=\int_^x^2f(x)dx
The quantization noise power can be expressed as:
:E tilde^2= \frac
Giving:
:$\mathrm = \frac$

When the SQNR is desired in terms of decibels (dB), a useful approximation to SQNR is:
:$\mathrm, _=P_+6.02\nu+4.77$
where $\nu$ is the number of bits in a quantized sample, and $P_$ is the signal power calculated above. Note that for each bit added to a sample, the SQNR goes up by approximately 6dB ($20\times log_(2)$).

References

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

External links

Signal to quantization noise in quantized sinusoidal
- Analysis of quantization error on a sine wave

{{Noise

Digital audio
Engineering ratios
Noise (electronics)