Audio Signal Processing and Coding

3.2 DENSITY FUNCTIONS AND QUANTIZATION

In this section, we discuss the characterization of a random process in terms of its probability density function (PDF). This approach will help us derive the quantization noise equations for different quantization schemes. A random process is characterized by its PDF, which is a non-negative function, p(x), whose properties are

and

From the above equations, it is evident that the PDF area from x₁ to x₂ is the probability that the random variable X is observed in this range. Since X lies somewhere in [−∞, ∞], the total area under p(x) is one. The mean and the variance of the random variable X are defined as

Figure 3.2. (a) The Gaussian PDF and (b) The Laplacian PDF.

Note that the expectation is computed either as a weighted average (3.3) or under ergodicity assumptions as a time average (Chapter 2, Eq. 2.54). PDFs are useful in the design of optimal signal quantizers as they can be used to determine the assignment of optimal quantization levels. ...