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## Appendix A: Round-Off Error Variance

This section shows how to derive that the noise power of quantization error is , where is the quantization step size.

Each round-off error in quantization noise is modeled as a uniform random variable between and . It therefore has the

probability density function(pdf)

Thus, theprobabilitythat a given roundoff error lies in the interval is given by

assuming of course that and lie in the allowed range . We might loosely refer to as aprobability distribution, but technically it is a probabilitydensityfunction, and to obtain probabilities, we have to integrate over one or more intervals, as above. We use probabilitydistributionsfor variables which take ondiscretevalues (such as dice), and we use probabilitydensitiesfor variables which take oncontinuousvalues (such as round-off errors).The

meanof a random variable is defined as

In our case, the mean is zero because we are assuming the use ofrounding(as opposed to truncation, etc.).The

mean of a signalis the same thing as theexpected valueof , which we write as . In general, the expected value ofanyfunction of a random variable is given by

Since the quantization-noise signal is modeled as a series of independent, identically distributed (iid) random variables, we can

estimatethe mean byaveragingthe signal over time. Such an estimate is called asample mean.Probability distributions are often be characterized by their

moments. The th moment of the pdf is defined as

Thus, the mean is thefirstmoment of the pdf. The second moment is simply the expected value of the random variable squared, i.e., .The

varianceof a random variable is defined as thesecond central momentof the pdf:

“Central” just means that the moment is evaluated after subtracting out themean, that is, looking at instead of . In the case of round-off errors, the mean is zero, so subtracting out the mean has no effect. Plugging in the constant pdf for our random variable which we assume is uniformly distributed on , we obtain the variance

Note that the variance of can be estimated by averaging over time, that is, by computing themean square. Such an estimate is called thesample variance. For sampled physical processes, the sample variance is proportional to theaverage powerin the signal. Finally, the square root of the sample variance (therms level) is sometimes called thestandard deviationof the signal, but this term is only precise when the random variable has a Gaussian pdf.EE 278 is the starting course on statistical signal processing at Stanford if you are interested in this and related topics. A good text book on the subject is [13].