Signal-to-quantization-noise ratio

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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:

is the probability of received bit error
is the peak message signal level
is the mean message signal level

As SQNR applies to quantized signals, the formulae for SQNR refer to discrete-time digital signals. Instead of , the digitized signal will be used. For quantization steps, each sample, requires bits. The probability distribution function (pdf) representing the distribution of values in and can be denoted as . The maximum magnitude value of any is denoted by .

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 is the number of bits in a quantized sample, and is the signal power calculated above. Note that for each bit added to a sample, the SQNR goes up by approximately 6dB ().

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