EE 451
Homework 7 Due: Friday, April 1 Read Text, Chapter 8.1-8.2.
Spring 2016
1. Consider the rate-1/2 convolutional code with encoder shown below. a. Find the truth table for the encoder encod er and draw the trellis diagram for the code. b. Determine the minimum free distance, d free , for the code. c. Assume an initial trellis state of (0,0), and determine the code word for the input bit sequence u ( n) (1,1,0,0,1,0) . Draw the corresponding path through the trellis. d. Assume an initial trellis state of (0,0), and determine the max imum likelihood decoded codeword, vˆ , and information bits, uˆ , if the received bit sequence is r (1,0,0,0,0,1,0,1,10,11,10) . Use the Viterbi Algorithm to perform the ML decoding. =
=
2. (Text, 8.1.1) A communication network has 26 nodes. Each node may fail with probability p = 0.05, independently. Find the probability that a) 2 out of 26 nodes fail; b) no more than 5 nodes no des fail; c) the number of failed nodes is between 20 and 23. 3. A (26, 13) linear binary code is used on a binary symmetric channel with bit error probability = 0.05. The codeword 0 is sent and the received 26-tuple is . Find the probability that a) the weight of r is is 2; the weight of r is is no larger than 5; the weight of r is is between 20 and 23. 4. (Text 8.1.2) A bank customer has selected her four-digit personal identification number (PIN). Assume that the four digits are selected independentl y and uniformly over the digits {0, …, 9}. Find the probabilities of the following events: the sum of the digits is a) 3; b) 7. 5. (Text 8.1-8) A network consists of ten links, 1 , 2 , ⋯ , 10 in cascade (Fig. P8.1-8 in the text). If any one of o f the links fails, the entire system fails. All links are independent and with equal probability of failure, p. a. The probability that a link does not fail is 0.98. What is the probability of failure of the network? b. The reliability of a network is the probability of no t failing. If the system reliability is required to be 0.99, what must be the failure probability of each link?
c. Repeat part a) if link 1 has probability of failure 0.03, while the other links can fail with probability 0.01. 6. (Text 8.2-1) For a certain binary non-symmetric channel, it is given that | (0|1) = 0.1 and | (1|0) = 0.2, where x is the transmitted digit and y is the received digit. Assume that (0) = 0.4 a. Determine (0) and (1). b. What is the probability that no 1s will be in the output for an input sequence of 10 digits. c. What is the probability that at least five 0s will be in the output for an input sequence of 10 digits? 7. (Text, 8.2-2) A binary symmetric channel has error probability . The probability of transmitting 1 is Q. If the receiver detects an incoming digit as 1, what is the probability that the originally transmitted digit was a) 1; b) 0? 8. (Text, 8.2-5) The pdf of a Gaussian variable, X, is given by 1 () = −(−4) /18 √ 2 Determine a) the value of C; b) ( ≥ 2) ; c) ( ≥ 1); d) ( ≥ 2).
