{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

lecture-3

# lecture-3 - EE378 Statistical Signal Processing Lecture 3...

This preview shows pages 1–2. Sign up to view the full content.

EE378 Statistical Signal Processing Lecture 3 - 04/11/2007 Mean-Square Ergodicity, Wide-Sense Ergodicity, Martingales Lecturer: Tsachy Weissman Scribe: Sudeepto Chakraborty, Lei Zhao, Kamakshi S 1 Mean-Square Ergodicity Definition 1. A WSS process { X ( n ) } (with E [ X ( n )] = μ and r ( k ) = Cov ( X ( n + k ) , X ( n )) ) is 1. Mean-square ergodic in the 1 st moment if lim N →∞ E 1 2 N + 1 N n = - N X ( n ) - μ 2 = 0 , (1) 2. Mean-square ergodic in the 2 nd moment if k, lim N →∞ E 1 2 N + 1 N n = - N ( X ( n ) - μ )( X ( n - k ) - μ ) - r ( k ) 2 = 0 (2) 3. A WSS process is wide-sense ergodic if it is mean-square ergodic in the 1 st and the 2 nd moments. Example 1 { X ( n ) } is an iid process with the following distribution: X ( n ) 1 w.p. 1 / 2 - 1 w.p. 1 / 2 Verify that this process is wide-sense ergodic. Example 2 { X ( n ) } is a special case of a DC process: X ( n ) 1 n w.p. 1 / 2 - 1 n w.p. 1 / 2 E 1 2 N + 1 N n = - N X ( n ) - 0 2 = 1 0 Therefore X ( n ) is not ergodic in 1 st moment not wide-sense ergodic. In reference to Theorem 9 in lecture notes 2, this is an example of a mixture of ergodic processes (namely, a mixture of degenerate processes for which all the components are equal to the same deterministic constant). These rather extreme examples make the point that, if process “memory” is sufficiently short you will have Mean Square Ergodicity. On the other hand, if the “memory” is too long, the process will not be mean square ergodic. The rate of decay of the covariance sequence can be thought of as a measure of memory.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}