  ECE 614 – Principles of Digital Communications Homework 3

advertisement
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 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.
Download
Related flashcards
Business terms

50 Cards

Business

36 Cards

Tekken

18 Cards

Cars of China

62 Cards

Create flashcards