Signal-to-quantization-noise ratio

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


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

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

The signal power is:
 * $$\overline{x^2} = E[x^2] = P_{x^\nu}=\int_{}^{}x^2f(x)dx$$

The quantization noise power can be expressed as:
 * $$E[\tilde{x}^2] = \frac{x_{max}^2}{3\times4^\nu}$$

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

When the SQNR is desired in terms of decibels (dB), a useful approximation to SQNR is:
 * $$\mathrm{SQNR}|_{dB}=P_{x^\nu}+6.02\nu+4.77$$

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 ($$20\times log_{10}(2)$$).