  ECE 614 – Principles of Digital Communications Homework 3

```ECE 614 – Principles of Digital Communications
Homework 3
Assigned on: 02/16/2016
Due by: 03/03/2016
3.1
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
Find the projection ŝ t  of st   e t ut  onto the span of e 3t u t , e 4t u t  ,
(a) Sketch ŝ t  and st  on the same graph.

(b) Calculate the energy of the error signal, Ee   sˆt   st  dt
2

3.2
Consider the 4-PSK signal set described by

st   2 Re ag t e j 2f ct

, where
g t   ut   ut  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 ut  . 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.
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