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In mathematics, the term ergodic is used to describe a dynamical system which, broadly speaking, has the same behavior averaged over time as averaged over space. -from wikipedia

From the perspective of engineering a signal processing system, does knowing whether a signal is ergodic change the plan of attack for analyzing the signals? I've always found this a fascinating concept, but I don't really know what do do with it once the determination is made. What analysis methods would be more pertinent knowing this information?

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up vote 8 down vote accepted

The concept of space as used in the statement ascribed to Wikipedia is presumably that of the ensemble of all possible signals, not space as in "Space: the final frontier...". In the absence of parallel universes in which different signals from the ensemble can be observed simultaneously, it is rare for engineers to determine that a signal is ergodic by comparing time averages to ensemble averages. More commonly, signals are assumed to be ergodic so that the values of time averages can be used as estimates of ensemble averages. For example, an engineer takes $1000$ samples of a stochastic signal and plots a histogram of values. This looks vaguely bell-shaped and so the signal is modeled as a Gaussian random process whose mean is the mean of the $1000$ samples etc. Ultimately, the proof of the pudding is in the eating. If this Gaussian model leads to predictions that differ wildly from observed results, the assumption of ergodicity might be discarded, or maybe $10,000$ samples, or $100,000$ samples might be tried (for example, predicting traffic at $5:30$ pm based on observations of traffic between $2:00$ am and $3:00$ am is rarely accurate, but observations over several weeks might lead to a better model).

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A good example of a non-ergodic system is a pool table (or similarly ray tracing in a rectangular environment). A large number of balls (or rays) see a random set of reflection angles (according to a "diffuse" distribution), however every single ball only sees the same angle of reflection again and again during its entire path. Let's say the energy abortion at every bounce is a function of angle of incidence. In an ergodic system (for a decent number of bounces) you could simply use the absorption at an "average" angle and multiply with the number of bounces. However, since this is non-ergodic, you'd get the wrong result. You will have to calculate the absorption based on the specific angle for each ball.

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