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ExamRP921110 - and quadrature component t q and draw and zz...

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Midterm Exam of Stochastic Processes Nov. 10, 2003 1. (10 points) A random process ( ) cos( ) t t ϖ = + x r ϕ where random variables r and ϕ are independent and ϕ is uniformly distributed in ( , ) π π - . Proof that ( ) t x is WSS. 2. (15 points) Let [ ] n x be a WSS random process with power spectrum ( ) j xx S e ϖ and [ ] n y be the output of an LTI system with input [ ] n x . Let the frequency response of the LTI system be ( ) j H e ϖ . Find ( ), ( ) and ( ). j j j xy yx yy S e S e S e ϖ ϖ ϖ 3. (10 points) The process ( ) t x is WSS and normal with { ( )} 0 E t = x and 2 ( ) 2 x R e τ τ - = (a) Find the probability density function (pdf) of x (4), (4) ( ) x f p . (b) Find the joint pdf of x (4) and x (10), (4), (10) ( , ) x x f p q . 4. (15 points) The process ( ) t x is WSS with ( ) 2 ( ) x R τ δ τ = . Let ( ) 2 ( ) ( ) t t t n + = y y x for all t. Find 2 1 2 { ( )}, ( , ), and ( ). yy y E t R t t S ϖ y 5. (20 points) Let ( ) t x be a narrowband signal with ( ) xx S ϖ shown below. Use Rice’s representation, ˆ ( ) ( ) ( ) t t j t = + z x x and 0 ( ) ( ) ( ) ( ) j t t t j t t e ϖ - = + = w i q z , to find the inphase component
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Unformatted text preview: and quadrature component ( ) t q , and draw ( ), ( ), ( ) and ( ) zz ww ii iq S S S jS . 6. (5 points) Give a definition of cyclostationary processes. 7. (15 points) Let f ( t ) be a transmitting signal with waveform shown below. Let the received signal be ( 10) ( ) when ( ) is transmitted ( ) ( ) when ( ) is not transmitted f t t f t t t f t-+ = v x v and ( ) t v be a zero-mean white noise with power spectrum 2 ( ) v S ϖ σ = . (a) Design a matched filter to determine whether f ( t ) is transmitted or not. Draw the impulse response of the matched filter. (b) Explain the procedure of detecting f ( t ) by using the matched filter. 8. (10 points) Show that if ( ) 0 for > S = , then ( ) (0)cos( ) for 2 R R π τ στ <...
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