ECE 4601 Homework Set #1: Probability of Error in Digital Communication Systems, Assignments of Digital Communication Systems

Download ECE 4601 Homework Set #1: Probability of Error in Digital Communication Systems and more Assignments Digital Communication Systems in PDF only on Docsity! ECE 4601 HOMEOWORK SET #1 DUE: Jan. 23, 2003 Problem 1. The following problem explains how a digital communication system works. A communication system uses on-off signals to transmit symbols (0) and (+1). Symbols 0 and +1 are transmitted with equal probability (p0=p1=1/2). Assume that the transmitter sends only one symbol (either one or zero). The channel adds a normalized Gaussian random variable (noise) N to the transmitted symbol. Thus the received symbol is equal to The receiver’s task is to decide which symbol has been sent by the transmitter. The receiver observes X and compares it to a threshold that is equal to 0.5λ. The receiver decides that the transmitted symbol was +1 if X is greater than 0.5 λ, otherwise it says the transmitted symbol was 0. Let A be the event that the transmitted symbol is +1. Let also B be the event that the transmitted symbol is zero 0. a) Find the probability density function of X given that the event A is occurred. b) Find the probability density function of X given that the event B is occurred. c) If A is actually occurred, find the probability that the receiver’s decision is incorrect. In other words, find Prob{X is smaller or equal to 0.5 λ } given that A is occurred. d) If B is actually occurred, find the probability that the receiver’s decision is incorrect. In other words, find Prob{X is greater than 0.5 λ } given that B is occurred. e) Find the probability of error. In other words, you need to find the probability that either A is occurred and the receiver makes an incorrect decision OR the event B is occurred and the receiver makes a wrong decision). Hint: note that A and B are disjoint events. Problem 2. Solve the question 1.5 from the textbook (Haykin’s fourth edition book). Problem 3. Solve the question 1.12 from the textbook. Problem 4. Solve the question 1.22 from the textbook (Haykin’s fourth edition book). Problem 5. Solve the question 1.26 from the textbook.    ++ = dtransmitteis)1(if dtransmitteis)0(if N N X λ