Probability, Random Variables, and Random Processes: Theory and Signal Processing Applications

NOTATION

Since a wide range of material is covered in this book, we present a brief overview of the notation. In many books on signal processing, the same symbol is used to denote a random process and a realization of that process, which is a deterministic waveform. In this book, we use instead the notation typical of books on probability and random processes:

• An uppercase letter denotes random variable X, random process X(t), or random sequence X[k], where t is continuous time and k is discrete time.
• Lowercase letter x is an outcome of X, x(t) is a realization (continuous waveform) of X(t), and X[k] is a realization (sequence of numbers) of X[k].

Typically, these letters are from the end of the Latin alphabet. One exception to the above notation is that uppercase K, M, and N usually denote (nonrandom) integers, as in the following sums of random variables:

(1)

If {K, M, N} turn out to be random variables in a particular problem such as in a random sum, it will be specifically mentioned.

• A bold uppercase letter denotes random vector X, random vector process X(t), or random vector sequence X[k].
• Bold lowercase letter X is a vector outcome of X, X(t) is a vector realization (vector of waveforms) of X(t), and X[k] is a vector realization (vector of sequences) of X[k].

All vectors in this book are column vectors. A row vector is obtained via transpose xT or complex conjugate transpose ...