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.
The SQNR formula is derived from the general signal-to-noise ratio (SNR) formula:
where:
As SQNR applies to quantized signals, the formulae for SQNR refer to discrete-time digital signals. Instead of 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 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:
The signal power is:
The quantization noise power can be expressed as:
Giving:
When the SQNR is desired in terms of decibels (dB), a useful approximation to SQNR is:
where ν ν --> {\displaystyle \nu } is the number of bits in a quantized sample, and P x ν 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)} ).
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