Definition: The expected value of a continuous random variable is denoted and is defined by
where denotes the probability density function (PDF) for the random variable v.
Example: Let the random variable be uniformly distributed between and , i.e.,
Then the expected value of is computed as
Thus, the expected value of a random variable uniformly distributed between and is simply the average of and . For a stochastic process, which is simply a sequence of random variables, means the expected value of over ``all realizations'' of the random process . This is also called an ensemble average. In other words, for each ``roll of the dice,'' we obtain an entire signal , and to compute , say, we average together all of the values of obtained for all ``dice rolls.'' For a stationary random process , the random variables which make it up are identically distributed. As a result, we may normally compute expected values by averaging over time within a single realization of the random process, instead of having to average ``vertically'' at a single time instant over many realizations of the random process.C.2 Denote time averaging by
Then, for a stationary random processes, we have . That is, for stationary random signals, ensemble averages equal time averages. We are concerned only with stationary stochastic processes in this book. While the statistics of noise-like signals must be allowed to evolve over time in high quality spectral models, we may require essentially time-invariant statistics within a single frame of data in the time domain. In practice, we choose our spectrum analysis window short enough to impose this. For audio work, 20 ms is a typical choice for a frequency-independent frame length.C.3 In a multiresolution system, in which the frame length can vary across frequency bands, several periods of the band center-frequency is a reasonable choice. As discussed in §5.5.2, the minimum number of periods required under the window for resolution of spectral peaks depends on the window type used.
Stationary Stochastic Process