Signal-to-quantization-noise ratio: Information from Answers.com

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Signal-to-Quantization-Noise Ratio (SQNR or SN_qR) is widely used quality measure in analysing digitizing schemes such as PCM (pulse code modulation) and multimedia codecs. 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 SNR (Signal-to-Noise Ratio) formula for the binary pulse-code modulated communication channel:

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

where

P e

is the probability of received bit error

m (t)

is the peak message signal level

$\overline{m(t)}$ is the mean message signal level

As SQNR applies to quantized signals, the formulae for SQNR refer to discrete-time digital signals. Instead of $m (t)$ , we will use the digitized signal $x (n)$ . For $N$ quantization steps, each sample, $x$ requires $ν = log 2 N$ 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 $x m a x$ .

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:

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

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

$\mathrm{SQNR}|_{dB}=P_{x^\nu}+6\nu+4.8$

where $ν$ 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 6dB ( $20\times log_{10}(2)$ ).

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

v • d • e Noise (in physics and telecommunications)

Main articles	List of noise topics · Noise measurement · Noise temperature · Noise reduction · Distortion · Phase distortion

Noise in...	Electronics · Telecommunications · Audio · Video · Images

Class of noise	Burst noise · Jitter · Additive white Gaussian noise (AWGN) · Johnson–Nyquist noise · Cosmic noise · Gaussian noise · White noise · Grey noise · Shot noise · Flicker noise · Quantization error (or q. noise)

Engineering terms	Noise spectral density · Phase noise · Noise figure · Statistical noise · Pseudorandom noise · Channel noise level · Circuit noise level · Effective input noise temperature · Equivalent noise resistance · Equivalent pulse code modulation noise · Impulse noise (audio) · Noise floor · Noise shaping

Ratios	Carrier-to-noise ratio (C/N) · Signal-to-noise ratio (S/N, SNR) · Eb/N0 (energy per bit to noise density) · Es/N0 (energy per symbol to noise density) · Carrier-to-receiver noise density (C/kT) · dBrnC · Signal-to-quantization-noise ratio (SQNR) · Signal, noise and distortion (SINAD) · Signal-to-noise plus interference (SNIR) · Signal-to-interference ratio (S/I) · Signal to noise ratio (image processing)

See also: Displayed average noise level · Interference (communication) · Quantization error · Radio noise source · Thermal radiation

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