19CorrelationRx

# 19CorrelationRx - The Correlation Receiver Jeffrey B. Burl...

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EE 3180 Probability and Random  Signal Anaysis, ©2009 by J. B. Burl The Correlation Receiver Jeffrey B. Burl

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EE 3180 Probability and Random  Signal Anaysis, ©2009 by J. B. Burl Introduction We have talked about making decisions  based on sampled data. Here, we will extend these results to  continuous-time data with additive white  noise.
EE 3180 Probability and Random  Signal Anaysis, ©2009 by J. B. Burl Solution With Additive White  Noise Consider the example: Under H 0 :  r(t) = s 0 (t) + n(t), t 0    t   t f Under H 1 :  r(t) = s 1 (t) + n(t), t 0    t   t f where R n ( τ ) = S n δ ( τ ), n(t) is Gaussian. We can sample the data so that we can use  previous results: Under H 0 :  r k  = r(t 0  + kT) = s 0k  + n k , 1   k   N Under H 1 :  r k  = r(t 0  + kT) = s 1k  + n k , 1   k   N where n k    N(0,S n /T)

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EE 3180 Probability and Random  Signal Anaysis, ©2009 by J. B. Burl Solution With Additive White  Noise, Cont. We can form the likelihood ratio: The likelihood ratio test is then ( 29 ( 29 ( 29 ( 29 2 2 1 1 1 2 2 0 0 0 1 1 2 / 2 / 1 | 1 1 1 1 | 0 2 / 2 / 1 1 1 ( | ) 2 / ( ) ( | ) 1 2 / k k k k n n k k k k n n r s r s N N S T S T k H n k r s r s N N H S T S T k k n e e f H S T f H e e S T π - - - - = = - - - - = = Λ = = = r r r r r 1 10 00 0 01 11 1 0 ( ) ( ) ( ) H C C P C C P H η - Λ = < - r
EE 3180 Probability and Random  Signal Anaysis, ©2009 by J. B. Burl Solution With Additive White  Noise, Cont. The log-likelihood ratio test is:

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## This note was uploaded on 06/16/2010 for the course EE ee3180 taught by Professor Burl during the Spring '10 term at Michigan Technological University.

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19CorrelationRx - The Correlation Receiver Jeffrey B. Burl...

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