ECE 614 – Principles of Digital Communications Homework 3 Assigned on: 02/16/2016 Due by: 03/03/2016 3.1 Find the projection ŝ t of st e t ut onto the span of e 3t u t , e 4t u t , (a) Sketch ŝ t and st on the same graph. (b) Calculate the energy of the error signal, Ee sˆt st dt 2 3.2 Consider the 4-PSK signal set described by st 2 Re ag t e j 2f ct , where g t ut ut 1 , and a 1, j,1, j. The receiver is expecting the carrier frequency to be 2400Hz. However, suppose the transmitter carrier frequency f c is slightly larger than 2400Hz, namely, f c 2400 f , where f 0 represents the frequency error in Hz. The receiver is unaware of the frequency error and it minimizes distance under the assumption that f 0 . Suppose there is no channel noise. Find the smallest frequency error f in Hz that causes the decision to be incorrect. 3.3 A Binary Symmetric Channel (BSC) is defined as follows. Both the input bi and the output bo of the channel take binary values bi , bo 1,1. Given bi , with probability p, we have bo bi , and with probability 1-p, we have bo bi . Here p is called the “transition probability” whose value may be different for different BSCs. Consider a cascade of L BSC’s each with the transition probability, where the output of each BSC is connected to the input of the next. a) Show that the resulting overall channel is a BSC. b) Find the error probability of the overall channel as a function of L. c) What happens as L ? 3.4 Suppose a customer “Alice” comes to a post office to mail a package. She waits at the end of a queue of 8 people. There are 4 service people, currently all available to help customers. Each service person can help one customer at a time. The time t needed for one service person to finish the service of one customer follows the exponential distribution with mean 1 (second), i.e., f t exp t ut . Assume each customer leaves the post office immediately after finishing his/her task. Find the probability that Alice is the last customer leaving the post office.