Definition:
The expected value of a continuous random variable
is denoted
and is defined by
(C.12) |
Example:
Let the random variable
be uniformly distributed between
and
, i.e.,
(C.13) |
(C.14) |
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
(C.15) |
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.