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Unformatted text preview: 1 EE 7615 Digital Communication Solutions to Exam I Oct. 2010 Problem 1 1) Y ( t ) = Z  N w ( t ) h ( ) d = 1 T Z T N w ( t ) d 2) E [ Y ( T )] = 1 T Z T E [ N w ( t )] d = 0 var [( Y ( T )] = E [ Y 2 ( T )] = 1 T 2 Z T Z T E [ N w ( t 1 ) N w ( t 1 )] d 1 d 2 = N 2 T 2 Z T Z T ( 1 2 ) d 1 d 2 = N 2 T 2 Z T d 2 = N 2 T We could have also used the frequency domain approach but the above is easier. Problem 2 1) Using the Bayes rule we can write p M  R ( m i  r ) = p R  M ( r  m i ) p M ( m i ) p R ( r ) where p M ( m i ) = 1 / 2 and p R ( r ) = . 5 h p R  M ( r  m ) + p R  M ( r  m 1 ) i 2) Since the two messages are equally likely, we use the maximum likelihood rule. g ( r ) = m iff p R  M ( r  m ) p R  M ( r  m 1 ) Replacing the formulas for these two PDFs and simplifying we get ( r a 1 ) 2 ( r a ) 2 which results in r a + a 1 2 . 3) We have P ( E ) = . 5 P ( E  m ) + . 5 P ( E  m 1 ) . Now P ( E  m ) = P ( R > a + a 1 2  m ) = Z a + a 1 2 1 [1 + ( r a )...
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This note was uploaded on 03/05/2012 for the course EE 7615 taught by Professor Naragipour during the Fall '11 term at LSU.
 Fall '11
 Naragipour

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