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wss prb3 - EE 179 Introduction to Communications Winter...

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EE 179 Introduction to Communications Winter 2011 Homework 5 Due Thursday, Feb 10 4pm (Total: 100 pts.) 1. Random Cosine Signals [15 pts] Given a random process x ( t ) = a cos( ω c t + Θ) where ω c is a constant and a and Θ are independent RVs uniformly distributed in the ranges [ - 1 , 1] and [0 , 2 π , respectively. (a) Sketch the ensemble of this process. (b) Determine x ( t ). (c) Determine R x ( t 1 , t 2 ). (d) Is the process wide-sense stationary? (e) Is the process ergodic? (f) Find the average power P x (that is, its mean square value x 2 ( t )) of the random process x ( t ). 2. Stationarity [10 pts] Let X and Y be statistically independent Gaussian-distributed random variables with zero mean and unit variance. Define the Gaussian process Z ( t ) = X cos(2 πt ) + Y sin(2 πt ) . (a) Determine the joint probability density function of the random variables Z ( t 1 ) and Z ( t 2 ) obtained by observing Z ( t ) at times t 1 and t 2 , respectively. (b) Is the process Z ( t ) stationary? Why? 3. PSD of Random Binary Wave: [15 pts] Let x ( t ) be the infinite pulse train shown below with period T and random time shift τ , where τ is uniformly distributed between [0 , T ]. EE 179 - Introduction to Communications - Spring 2009 Homework 4: Due Thursday 5/7 at 4pm (no late HWs). 1. (20 points) Let x ( t ) be the infinite pulse train shown below with period T and random time shift τ , where τ is uniformly distributed between [0 , T ].
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  • Spring '11
  • Naragipour
  • Probability theory, Stochastic process, probability density function, Stationary process, random process

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