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Unformatted text preview: EEL 6935 Electronic Navigation Systems Homework Solution #2 Problem 1 An exponentially correlated random process can be generated using x ( t ) =- x ( t ) + w ( t ), where E [ w ( t )] = 0 and E [ w ( t ) w ( t + )] = 2 2 ( ). Then the statistics of x ( t ) are E [ x ( t )] = 0 and E [ x ( t ) x ( t + )] = 2 e- | | . Use the matrix superposition integral to show that the discretized version of the above system is x k +1 = e- t x k + w k , where E [ w k ] = 0 and E [ w 2 k ] = 2 (1- e- 2 t ), and t = t k +1- t k . Solution: Matrix superposition integral: x ( t k +1 ) = ( t k +1- t k ) x ( t k ) + Z t k +1 t k ( ) w ( ) d We want to put the system in this form: x k +1 = ( t ) x k + w k Since the state vector for this system is a scalar, the state transition matrix is easy to calculate: ( t ) = e- t . Now for the statistics of w k : E [ w k ] = E " Z t e- w ( ) d # = Z t e- E [ w ( )] d = 0...
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This note was uploaded on 01/15/2012 for the course EEL 6935 taught by Professor Staff during the Fall '08 term at University of Florida.
- Fall '08