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hw02_solution

# hw02_solution - EEL 6935 Electronic Navigation Systems...

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EEL 6935 Electronic Navigation Systems Homework Solution #2 Problem 1 An exponentially correlated random process can be generated using ˙ x ( t ) = - βx ( t ) + w ( t ), where E [ w ( t )] = 0 and E [ w ( t ) w ( t + τ )] = 2 βσ 2 δ ( τ ). Then the statistics of x ( t ) are E [ x ( t )] = 0 and E [ x ( t ) x ( t + τ )] = σ 2 e - β | τ | . Use the matrix superposition integral to show that the discretized version of the above system is x k +1 = e - β Δ t x k + w k , where E [ w k ] = 0 and E [ w 2 k ] = σ 2 (1 - e - 2 β Δ t ), and Δ t = t k +1 - t k . Solution: Matrix superposition integral: x ( t k +1 ) = Φ( t k +1 - t k ) x ( t k ) + Z t k +1 t k Φ( τ ) w ( τ ) We want to put the system in this form: x k +1 = Φ(Δ t ) x k + w k Since the state vector for this system is a scalar, the state transition matrix is easy to calculate: Φ(Δ t ) = e - β Δ t . Now for the statistics of w k : E [ w k ] = E " Z Δ t 0 e - βτ w ( τ ) # = Z Δ t 0 e - βτ E [ w ( τ )] = 0 1

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E [ w 2 k ] = E ( Z Δ t 0 e - βτ ) 2 = E "( Z Δ t 0 e - βτ 1 w ( τ 1 ) 1 ) ( Z Δ t 0 e - βτ 2 w ( τ 2 ) 2 )# = E " Z Δ t 0 Z Δ t 0 e - β ( τ 1 + τ 2 ) w ( τ 1 ) w ( τ 2 ) 1 2 # = Z Δ t 0 Z Δ t 0 e - β ( τ 1 +
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