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:

$\mathrm{S}\mathrm{N}\mathrm{R}=\frac{3\times 2^{2n}}{1+4P_e\times (2^{2n}-1)}\frac{m_m(t{)}^2}{m_p(t{)}^2}$

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 $\nu ={\log }_{2}N$ 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:

$\mathrm{S}\mathrm{Q}\mathrm{N}\mathrm{R}=\frac{P_{signal}}{P_{noise}}=\frac{E[x^2]}{E[{\tilde{x}}^2]}$

The signal power is:

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

The quantization noise power can be expressed as:

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

Giving:

$\mathrm{S}\mathrm{Q}\mathrm{N}\mathrm{R}=\frac{3\times 4^{\nu }\times \overline{x^2}}{x_{max}^2}$

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

$\mathrm{S}\mathrm{Q}\mathrm{N}\mathrm{R}{|}_{dB}=P_{x^{\nu }}+6.02\nu +4.77$

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

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