COL+UNIV+2012+SPRING+ELEN+E4815+PROBLEM+SETS+FOR+SEMESTER-16+JAN+2012-3

COL+UNIV+2012+SPRING+ELEN+E4815+PROBLEM+SETS+FOR+SEMESTER-16+JAN+2012-3

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RANDOM SIGNALS AND NOISE ELEN E4815 Columbia University Spring Semester 2012 Problem Sets for the Semester 16 January 2012 I. Kalet 1
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RANDOM SIGNALS AND NOISE ELEN E4815 COLUMBIA UNIVERSITY SPRING SEMESTER 2012 Problem Set # 1 Problem Set Due Date: 24 January 2012 Problems (1-5) are from the book (MC) by S. Miller and D. G. Childers, “Probability and Random Processes: With Applications to Signal Processing and Communications”, Elsevier-Academic Press, 2004. Problems # 1-3 Chapter 2-Problems 2.18, 2.19, 2.20, 2.24 and 2.27 Problems # 4 and 5 Chapter 3-Problems 3.1 and 3.10 Problem # 6 Find the E{k},E{k 2 } and the Variance of k, for a binomial probability distribution. 2
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RANDOM SIGNALS AND NOISE ELEN E4815 COLUMBIA UNIVERSITY SPRING SEMESTER 2012 Problem Set # 2 Problem Set Due Date: 31 January 2012 Problems are from the book (MC) by S. Miller and D. G. Childers, “Probability and Random Processes: With Applications to Signal Processing and Communications”, Elsevier-Academic Press, 2004. Problems 1-4 Chapter 4- Problems 4.3, 4.4, 4.15, 4.16 , 4.20 Problems 5-6 Chapter 5- Problems 5.16, 5.21 3
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RANDOM SIGNALS AND NOISE ELEN E4815 COLUMBIA UNIVERSITY SPRING SEMESTER 2012 Problem Set # 3 Problem Set Due Date: 7 February 2012 The first six problems are taken from (MC) S. Miller and D. G. Childers, “Probability and Random Processes: With Applications to Signal Processing and Communications”, Elsevier-Academic Press, Second Edition, 2004. Problems # 1-6 (MC) Chapter 8-Problems 8.1, 8.2, 8.7, 8.8, 8.11 and 8.20 The next two problems are taken from S. Haykin, “Communications Systems”, J. Wiley and Sons, New York, Fourth Edition, 2001. Problem # 7 and 8- Chapter I, Problems 1.5 and 1.8 Problems 1.5 and 1.8, are taken from Chapter I, of Haykin’s Book (which may be found in the library-If you have any problem finding these problems please let me know). 4
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RANDOM SIGNALS AND NOISE ELEN E4815 COLUMBIA UNIVERSITY SPRING SEMESTER 2012 Problem Set # 4 Problem Set Due Date: 14 February 2012 Problem # 1 Consider the signal, x(t), shown below, which resembles the random telegraph signal. In this signal, which started at time “t” equal to - and will continue until time “t” equal to + , the voltage flips back and forth, between A volts, and 0 volts, in the following manner. The switching times, are dictated by a Poisson distribution, i.e., the probability of “k” flips in τ seconds is given by the Poisson distribution function Prob {of “k” flips in τ seconds}= [e - λτ { λτ } k ]/ k! with k=0, 1, 2, …. . x(t) A t “a flipping instant” 5
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a) Find the autocorrelation function, E {x(t+ τ ) x(t)} of the random process, x(t), and show that it is only a function of τ . b) Show that the E{x(t)} is just a constant. Therefore, this process is WSS.
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COL+UNIV+2012+SPRING+ELEN+E4815+PROBLEM+SETS+FOR+SEMESTER-16+JAN+2012-3

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