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Signal-to-quantization-noise ratio (SQNR orSN_qR) is widely used quality measure in analysingdigitizing schemes such aspulse-code modulation (PCM). The SQNR reflects the relationship between the maximum nominalsignal strength and thequantization error (also known as quantization noise) introduced in theanalog-to-digital conversion.

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

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

${\displaystyle \mathrm{S}\mathrm{Q}\mathrm{N}\mathrm{R}=\frac{P_{signal}}{P_{noise}}=\frac{E[x^2]}{E[{\tilde{x}}^2]}}$

The signal power is:

${\displaystyle \overline{x^2}=E[x^2]=P_{x^{\nu }}={\int }_{}^{}{x}^{2}f(x)dx}$

The quantization noise power can be expressed as:

${\displaystyle E[{\tilde{x}}^2]=\frac{x_{max}^2}{3\times 4^{\nu }}}$

Giving:

${\displaystyle \mathrm{S}\mathrm{Q}\mathrm{N}\mathrm{R}=\frac{3\times 4^{\nu }\times \overline{x^2}}{x_{max}^2}}$

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

${\displaystyle \mathrm{S}\mathrm{Q}\mathrm{N}\mathrm{R}{|}_{dB}=P_{x^{\nu }}+6.02\nu +4.77}$

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

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

• Signal to quantization noise in quantized sinusoidal - Analysis of quantization error on a sine wave

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