This section shows how to derive that the noise power of quantization error is , where is the quantization step size.
Thus, the probability that 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 a probability distribution, but technically it is a probability density function, and to obtain probabilities, we have to integrate over one or more intervals, as above. We use probability distributions for variables which take on discrete values (such as dice), and we use probabilitydensities for variables which take on continuous values (such as round-off errors).
In our case, the mean is zero because we are assuming the use ofrounding (as opposed to truncation, etc.).
Since the quantization-noise signal is modeled as a series of independent, identically distributed (iid) random variables, we canestimate the mean by averaging the signal over time. Such an estimate is called a sample mean.
Thus, the mean is the first moment of the pdf. The second moment is simply the expected value of the random variable squared, i.e., .
“Central” just means that the moment is evaluated after subtracting out the mean, 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 the mean square. Such an estimate is called the sample variance. For sampled physical processes, the sample variance is proportional to the average power in the signal. Finally, the square root of the sample variance (the rms level) is sometimes called the standard deviation of the signal, but this term is only precise when the random variable has a Gaussian pdf.