EE 322 Probabilistic Methods for Electrical Engineers Homework 6 Assigned: 11/10. Due: 11/17 Problem 1. A digital communication system transmits a symbol S every T second. The symbol S is equal to either A or A, representing a digital bit 0 or 1, respectively. The signals A and A are used with equal probability 1/2. The signal is corrupted by noise N , 2 which is well modeled by a Gaussian random variable with zero mean and variance N . The noise is independent of the signal. The received signal R can therefore be written as R=S+N (1) Based on R, we would like to decided whether an A or A has been transmitted. The decision rule to estimate S is simply S = A sign(R) (2) where S denotes the estimate of S, and sign is the sign function. (a) Fix A = 1. 2 2 (b) Vary N from 0.05 to 1, and for each N do the following: (a) Generate 106 random binary A symbols as S. 2 (b) Generate 106 Gaussian noise samples of variance N . (c) Add the signals and noises correspondingly, to obtain 106 received signals. (d) Apply the decision rule to the received symbols. Compare the decisions with transmitted signals. Count the number of errors. And calculate the bit error rate (BER). 2 (c) Plot the error rate as a function of signal-to-noise ratio (SNR) defined as A2 /N , in log-log scale. Also plot the theoretical values for the BER (see note 15, page 11). Problem 2. Let X and N be two independent random variables. Both are Gaussian distributed. The mean and variance of X are 10 and 2, respectively. The mean and variance of N are 0 and 1, respectively. Let Y = X + N . Given that Y = 11, what is the probability that X is less than 9. Problem 3. Determine the distribution of Y = X 2 if X is Laplacian distributed: fX (x) = |x| e . 2 Problem 4. Let X be a continuous random variable that it uniformly distributed between 0 and 1. When X is given, Y is uniformly distributed between X and 2X. (a) Find the variance of Y . 1 of 2 (b) Find the correlation coefficient between X and Y . (c) Use MATLAB to generate a plot of 1000 realizations of X and Y . Plot each realization as a point labeled '+' on the two-dimensional plane. You need to include both the MATLAB code and the plot. Problem 5. Let X be a random variable such that M (s) = a + be2s + ce4s , E[X] = 3, Var(X) = 2. Find a, b, and c, and the PMF of X. Problem 6. Let X1 , X2 , . . . , X50 be fifty i.i.d. (independent and identically distributed) random variables. They all have the common PDF fXi (x) = ex u(x) P where u(x) is the step function. Define Y = 50 i=1 Xi . (3) (a) Find the mean and variance of Y (b) Find the MGF of Y . (c) Find the exact distribution of Y (d) Find the approximate distribution of Y using Central Limit Theorem (that is, assuming Y is Gaussian with the same mean and variance.) (e) Compare the two distributions by plotting them in the same figure. END OF ASSIGNMENT 2 of 2