hw6 - SCHOOL OF ENGINEERING SCIENCE SIMON FRASER UNIVERSITY...

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SCHOOL OF ENGINEERING SCIENCE SIMON FRASER UNIVERSITY ENSC 428 – Digital Communications Spring 2008 Homework #6 due March. 26, 2008 Wednesday -Daniel Lee Assume that all random processes discussed in this homework is wide-sense stationary and zero- mean. The random process can be complex-valued. We define auto-correlation of random process X ( t ) as () ( ) () * XX EXt X t φτ τ  ≡+  . Power spectral density of X ( t ) is defined as: () () [ ] exp 2 XX XX Sf j f d φ τπ ≡− ; i.e., the Fourier transform of the auto-correlation. We can also take the Laplace transform exp XX XX ss d ττ Φ≡ 1 . Consider inputting random process X ( t ) to a linear time-invariant system with impulse response h ( t ) and denote by Y ( t ) the output random process. Express the Laplace transform exp YY YY d in terms of ( ) XX s Φ and ( ) ( ) [ ] exp Hs ht s td t . Also express output power spectral density ( ) ( ) [ ] exp 2 YY YY jf d in terms of () XX and the Fourier transform of h ( t ). Derive the relations.
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hw6 - SCHOOL OF ENGINEERING SCIENCE SIMON FRASER UNIVERSITY...

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