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Problem 10.9. Use Eqs. (10.61), (10.64), and (10.66) to show that N1 and N2 are uncorrelated and therefore independent Gaussian random variables. Compute the variance of N1-N2. Solution The correlation of N1 and N2 is ( ) ( ) ( ) [ ] ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 2 1 20 0 1 20 0 0 1 2 0 1 20 2 ( ) ( ) cos 2 cos 2 2 ( ) ( ) cos 2 cos 2 2 cos 2 cos 2 2 cos 2 cos 2 0 T T T T T N N w s w t f t f s dsdt w s w t f t f s dsdt N t s f t f s ds dt N f t f t dt π π π π δ π π π π ⎡ ⎤= ⎢ ⎥⎣ ⎦ = = − = = ∫ ∫ ∫ ∫ ∫ ∫ ∫ E E E where the last line follows from Eq.(10.61). Since N1 and N2 are uncorrelated ( ) ( ) [ ] ( ) ( ) ( ) 2 2 1 2 1 1 2 2 2 2 1 2 2N N N N N N N N ⎡ ⎤ ⎡ ⎤ ⎡− = + +⎣ ⎦ ⎣ ⎦ ⎣ ⎡ ⎤ ⎡ ⎤= +⎣ ⎦ ⎣ ⎦ E E E E E E 2 ⎤⎦ The variance of the N1 term is ( ) ( ) ( ) [ ] ( ) ( ) ( ) ( ) ( ) ( ) 1 1 1 10 0 1 10 0 0 1 1 2 0 00 2 ( ) ( ) cos 2 cos 2 2 ( ) ( ) cos 2 cos 2 2 cos 2 cos 2 2 cos 2 T T T T T N N w s w t f t f s dsdt w s w t f t f s dsdt N t s f t f s ds dt N f t dt π π π π δ π π π ⎡ ⎤= ⎢ ⎥⎣ ⎦ = = − = ∫ ∫ ∫ ∫ ∫ ∫ ∫ E E E Using the double angle formula 2cos2θ = 1+2cosθ, we have ( ) ( )2 01 0 0 1 cos 4 2 2 TNN f N T π⎡ ⎤ = +⎣ ⎦ = ∫E t dt The derivation of the variance of N2 is similar and the combined variance is N0T. Excerpts from this work may be reproduced by instructors for distribution on a not-for-profit basis for testing or instructional purposes only to students enrolled in courses for which the textbook has been adopted. Any other reproduction or translation of this work beyond that permitted by Sections 107 or 108 of the 1976 United States Copyright Act without the permission of the copyright owner is unlawful.
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