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

7.9 INNOVATIONS AND MEAN-SQUARE PREDICTABILITY

In Sections 7.10 and 7.11, we describe models of a random sequence that are based on the predictability of future outcomes given past outcomes. In many engineering problems, we are interested in a random sequence that can be modeled or decomposed as follows:

(7.207)

for , where X[k] is a random signal of interest and W[k] is a “noise” process. Usually, W[k] is assumed to be an uncorrelated Gaussian sequence with zero mean, in which case it is known as additive white Gaussian noise (AWGN). In this model, we only have access to Y[k] which is sometimes called the measurement sequence, and the goal is to estimate or predict X[k]. In a communication system, X[k] is the transmitted signal, W[k] is receiver noise (caused by the transmission medium, receiver amplifiers, and so on), and Y[k] is the received signal. In subsequent chapters, we describe various techniques and algorithms for estimating X[k]. Here, we focus on rewriting Y[k] in terms of a predicable part and an unpredictable part known as the innovations, which is the “new information” in X[k] that cannot be estimated from past outcomes. Although the innovations can be defined for a continuous-time random process, we describe it only for a discrete-time random sequence. If the time-indexed ...