SQNR
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The acronym SQNR (standing for Signal-to-Quantization Noise Ratio) is widely used in communication systems analysis, particularly in PCM (pulse code modulation) schemes.
The SQNR formula is derived from the general SNR (Signal-to-Noise Ratio) formula for the binary pulse-code modulated communication channel.
[SNR=\frac \frac]
where
Pe is the probability of received bit error
m(t) is the message signal
Since SQNR applies to quantized signals, then the formulae involved with SQNR refer to discrete-time digital signals. Thus, instead of [m(t)], we will used the digitized signal [x(n)]. For [N] quantization steps, there are [\nu=\log_2 N] bits needed for each sample, [x]. 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_].
Since SQNR, like SNR, is a ratio of signal power to some noise power, we calculate [SQNR=\frac}}=\frac^2]}] The signal power is calculated [E[x^2]=P_=\int_^x^2f(x)dx] and will be notated [\overline]. The quantization noise power can be expressed [\frac^2}]
This leads to [SQNR=\frac}^2}]
When the SQNR is desired in terms of Decibels (dB), a useful approximation to SQNR is as follows: [SQNR|_=P_+6\nu+4.8] where [\nu] is the number of bits in a quantized sample, and [P_] is the signal power calculated above. Note that for each bit added to a sample, the SQNR goes up by about 6dB.
References
- B.P.Lathi, Modern Digital and Analog Communication Systems (3rd edition), Oxford University Press, 1998
- Gimmy Chu - University Of Toronto, 2005
- Pavel Chtchetinin - University Of Toronto, 2005
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