Introduction Definition: A random process is said to be ergodic if all e time averages of any sample function are equal to the corresponding ensemble averages(expectations) Note: if a process is ergodic, all time and ensemble averages are interchangeable. Because time average cannot be a function of time, the ergodic process must be stationary, otherwise the ensemble averages would be a function of time. But not all stationary processes are ergodic. xdc([x([x(t)=mx (x(l)=lim F2 [x(ODt (6-6b) x(O)=x/(x)dx=m2(6-6c) 5 =V<x((>=o+m<(6-7)5 Introduction • Definition: A random process is said to be ergodic if all time averages of any sample function are equal to the corresponding ensemble averages(expectations) • Note: if a process is ergodic, all time and ensemble averages are interchangeable. Because time average cannot be a function of time, the ergodic process must be stationary, otherwise the ensemble averages would be a function of time. But not all stationary processes are ergodic. = < ( ) > = σ + (6 - 7) [ ( )] = [ ] ( ) = (6 - 6c) [ ( )] (6 - 6b) 1 [ ( )] = lim [ ] [ ] (6 - 6a) 2 2 2 ∞ -∞ T/2 -T/2 ∫ ∫ rms x x x x T→→ d c x X x t m x t x f x dx m x t dt T x t x = x(t) = x(t) =m