Signal-to-quantization-noise ratio explained

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:

SNR = (3 x 2 ^{2n})/(1+4 P _{e} x ( 2 ^{2n} -1 )) (m ^{2} _{m(t)})/(m ^{2} _{p(t)})

where:

P _{e} is the probability of received bit errorm _{p(t)} is the peak message signal levelm _{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:

SQNR = (P _{signal})/(P _{noise}) = (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]} = (x ^{2} _{max})/(3 x 4 ^{\nu}) Giving:SQNR = (3 x 4 ^{\nu x} \overline{x ^{2})/() }

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

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 x log _{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