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Spectral Audio Signal Processing
    Beginning Statistical Signal Processing
       Random Variables & Stochastic Processes
          Expected Value

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Expected Value



Definition: The expected value of a continuous random variable $ v\in(-\infty,\infty)$ is denoted $ E\{v\}$ and is defined by

$\displaystyle E\{v\} \isdef \int_{-\infty}^\infty x \, f_v(x) dx $

where $ p_v(x)$ denotes the probability density function (PDF) for the random variable v.



Example: Let the random variable $ v(n)$ be uniformly distributed between $ a$ and $ b$, i.e.,

$\displaystyle p_v(x) = \left\{\begin{array}{ll} \frac{1}{b-a}, & a\leq x \leq b \\ [5pt] 0, & \hbox{otherwise}. \\ \end{array}\right. $

Then the expected value of $ v(n)$ is computed as

$\displaystyle E\{v\} = \int_a^b x \frac{1}{b-a} dx = \frac{1}{2}\frac{b^2-a^2}{b-a} = \frac{b+a}{2}. $

Thus, the expected value of a random variable uniformly distributed between $ a$ and $ b$ is simply the average of $ a$ and $ b$.

For a stochastic process, which is simply a sequence of random variables, $ E\{x(n)\}$ means the expected value of $ x(n)$ over ``all realizations'' of the random process $ x(\cdot)$. This is also called an ensemble average. In other words, for each ``roll of the dice,'' we obtain an entire signal $ x(n),\, n=0,\pm1,\pm2,\cdots$, and to compute $ E\{x(0)\}$, say, we average together all of the values of $ x(0)$ obtained for all ``dice rolls.''

For a stationary random process $ x = \{x(n),\, n=0,\pm1,\pm2,\cdots\}$, the random variables $ x(n)$ 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. Denote time averaging by

$\displaystyle {\cal E}_n\{x(n)\} \isdef \lim_{N\to\infty}\frac{1}{2N+1}\sum_{n=-N}^N x(n). $

Then, for a stationary random processes, we have $ E\{x(n)\} = {\cal E}_n\{x(n)\}$. 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.D.2 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 §1.7, the minimum number of periods required under the window for resolution of spectral peaks depends on the window type used.

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Next: Mean
written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.


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