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 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^{{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 $m(t)$ , we will use the digitized signal $x(n)$ . For $N$ quantization steps, each sample, $x$ requires $\nu =\log _{2}N$ 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 $x_{{max}}$ .

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^{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 {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\nu +4.8

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 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

Noise (physics and telecommunications)
General	Acoustic quieting Distortion Noise cancellation Noise control Noise measurement Noise power Noise reduction Noise temperature Phase distortion
Noise in...	Audio Buildings Electronics Environment Government regulation Human health Images Radio Rooms Ships Sound masking Transportation Video
Class of noise	Additive white Gaussian noise (AWGN) Atmospheric noise Background noise Brownian noise Burst noise Cosmic noise Flicker noise Gaussian noise Grey noise Jitter Johnson–Nyquist noise (thermal noise) Pink noise Quantization error (or q. noise) Shot noise White noise Coherent noise Value noise Gradient noise Worley noise
Engineering terms	Channel noise level Circuit noise level Effective input noise temperature Equivalent noise resistance Equivalent pulse code modulation noise Impulse noise (audio) Noise figure Noise floor Noise shaping Noise spectral density Noise, vibration, and harshness (NVH) Phase noise Pseudorandom noise Statistical noise
Ratios	Carrier-to-noise ratio (C/N) Carrier-to-receiver noise density (C/kT) dBrnC E_b/N₀ (energy per bit to noise density) E_s/N₀ (energy per symbol to noise density) Modulation error ratio (MER) Signal, noise and distortion (SINAD) Signal-to-interference ratio (S/I) Signal-to-noise ratio (S/N, SNR) Signal-to-noise ratio (imaging) Signal to noise plus interference (SNIR) Signal-to-quantization-noise ratio (SQNR) Contrast-to-noise ratio (CNR)
Related topics	List of noise topics Acoustics Colors of noise Interference (communication) Noise generator Radio noise source Spectrum analyzer Thermal radiation

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