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.

As SQNR applies to quantized signals, the formulae for SQNR refer to discrete-time digital signals. Instead of the value- and time-continuous message signal m ( t ) {\displaystyle m(t)}, the digitized signal x ( n ) {\displaystyle x(n)} will be used. For N {\displaystyle N} quantization steps, each sample, x {\displaystyle x} requires ν = log 2 ⁡ N {\displaystyle \nu =\log _{2}N} bits. The probability distribution function (PDF) represents the distribution of values in x {\displaystyle x} and can be denoted as f ( x ) {\displaystyle f(x)}. The maximum magnitude value of any x {\displaystyle x} is denoted by x m a x {\displaystyle x_{max}}.

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

S Q N R = P s i g n a l P n o i s e = E [ x 2 ] E [ x ~ 2 ] {\displaystyle \mathrm {SQNR} ={\frac {P_{signal}}{P_{noise}}}={\frac {E[x^{2}]}{E[{\tilde {x}}^{2}]}}}

The signal power is:

x 2 ¯ = E [ x 2 ] = P x ν = ∫ x 2 f ( x ) d x {\displaystyle {\overline {x^{2}}}=E[x^{2}]=P_{x^{\nu }}=\int _{}^{}x^{2}f(x)dx}

The quantization noise power can be expressed as:

E [ x ~ 2 ] = x m a x 2 3 × 4 ν {\displaystyle E[{\tilde {x}}^{2}]={\frac {x_{max}^{2}}{3\times 4^{\nu }}}}

Giving:

S Q N R = 3 × 4 ν × x 2 ¯ x m a x 2 {\displaystyle \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:

S Q N R | d B = P x ν + 6.02 ν + 4.77 {\displaystyle \mathrm {SQNR} |_{dB}=P_{x^{\nu }}+6.02\nu +4.77}

where ν {\displaystyle \nu } is the number of bits in a quantized sample, and P x ν {\displaystyle 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 × l o g 10 ( 2 ) {\displaystyle 20\times log_{10}(2)}).

  • B. P. Lathi, Modern Digital and Analog Communication Systems (3rd edition), Oxford University Press, 1998

External links

  • - Analysis of quantization error on a sine wave