Audio Signal Processing and Coding

2.9 DISCRETE-TIME RANDOM SIGNALS

In signal processing, we generally classify signals as deterministic or random. A signal is defined as deterministic if its values at any point in time can be defined precisely by a mathematical equation. For example, the signal x(n) = sin(πn/4) is deterministic. On the other hand, random signals have uncertain values and are usually described using their statistics. A discrete-time random process involves an ensemble of sequences x(n,m) where m is the index of the m-th sequence in the ensemble and n is the time index. In practice, one does not have access to all possible sample signals of a random process. Therefore, the determination of the statistical structure of a random process is often done from the observed waveform. This approach becomes valid and simplifies random signal analysis if the random process at hand is ergodic. Ergodicity implies that the statistics of a random process can be determined using time-averaging operations on a single observed signal. Ergodicity requires that the statistics of the signal are independent of the time of observation. Random signals whose statistical structure is independent of time of origin are generally called stationary. More specifically, a random process is said to be widesense stationary if its statistics, upto the second order, are independent of time. Although it is difficult to show analytically that signals with various statistical distributions are ergodic, it can be shown that a stationary ...