16.322 Stochastic Estimation and Control, Fall 2004 Prof vander Velde x(t) verage rate of occurrence of change points =2 points per second R2()=E[x(x+] P(even number of change points in the interval x)xo +P(odd number of change points in the interval =(-xo) ( 2 k(1)(az) xae xoe You are generating higher harmonics in this case, as at each change point the amplitude changes sign. In the previous example, changes in amplitude at change point may be far smaller S2(o)=2(2)2 Example: Binary function with an arbitrary amplitude distribution The problem considers a binary function with a more general amplitude distribution. The distribution of a is restricted to +l with equal I probability That gives the pseudo-random binary code used by GPs Page 7 of 816.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde Page 7 of 8 Average rate of occurrence of change points = λ points per second ( ) ( ) ( ) ( ) 2 0 2 0 2 2 0 0 0,2,4,... 1,3,5,... 2 0 ( ) () ( ) (even number of change points in the interval ) (odd number of change points in the interval )( ) 1 1 ! ! 1 1 ! xx k k k k k k k R E xtxt P x P x x ex e k k e x k λτ λτ λ τ τ τ τ τ λτ λτ λ τ − − = = − = + ⎡ ⎤ ⎣ ⎦ = + − = − = − ∑ ∑ ( ) 0,1,2,... 2 0 0 2 2 0 ! k k x e k x e λ τ λ τ λ τ = ∞ − = − − = = ∑ ∑ You are generating higher harmonics in this case, as at each change point the amplitude changes sign. In the previous example, changes in amplitude at change point may be far smaller. ( ) 2 0 2 2 2(2) ( ) 2 xx x S λ ω ω λ = + Example: Binary function with an arbitrary amplitude distribution The problem considers a binary function with a more general amplitude distribution. The distribution of a is restricted to ±1with equal probability. That gives the pseudo-random binary code used by GPS