Signal-to-quantization-noise ratio

Short description: Measure for analyzing digitizing schemes

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:

[math]\displaystyle{ \mathrm{SNR}=\frac{3 \times 2^{2n}}{1+4P_e \times (2^{2n} - 1)} \frac{m_m(t)^2}{m_p(t)^2} }[/math]

where:

[math]\displaystyle{ P_e }[/math] is the probability of received bit error

[math]\displaystyle{ m_p(t) }[/math] is the peak message signal level

[math]\displaystyle{ m_m(t) }[/math] is the mean message signal level

As SQNR applies to quantized signals, the formulae for SQNR refer to discrete-time digital signals. Instead of [math]\displaystyle{ m(t) }[/math], the digitized signal [math]\displaystyle{ x(n) }[/math] will be used. For [math]\displaystyle{ N }[/math] quantization steps, each sample, [math]\displaystyle{ x }[/math] requires [math]\displaystyle{ \nu=\log_2 N }[/math] bits. The probability distribution function (PDF) represents the distribution of values in [math]\displaystyle{ x }[/math] and can be denoted as [math]\displaystyle{ f(x) }[/math]. The maximum magnitude value of any [math]\displaystyle{ x }[/math] is denoted by [math]\displaystyle{ x_{max} }[/math].

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

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

The signal power is:

[math]\displaystyle{ \overline{x^2} = E[x^2] = P_{x^\nu}=\int_{}^{}x^2f(x)dx }[/math]

The quantization noise power can be expressed as:

[math]\displaystyle{ E[\tilde{x}^2] = \frac{x_{max}^2}{3\times4^\nu} }[/math]

Giving:

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

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

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

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

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