EECE542: Digital Communications Theory - Homework 7 Solutions, Assignments of Theories of Communication

Download EECE542: Digital Communications Theory - Homework 7 Solutions and more Assignments Theories of Communication in PDF only on Docsity! EECE542: Digital Communications Theory Prof. M. Hayat 1 ECE542, Fall 2004 Homework 7 Solutions Problem 1 i. Union Bound: { }∑ ≠ Ρ≤Ρ i ie sentijannounce 0 , But { } ( ) ( ) ∫ ∞ +       − ==Ρ ji ij i Qdyyfsentijannounce µµ σ µµ 2 1 2 (Assume ij µµ ≥ ) If ij µµ < , { } ( ) ( ) ∫ + ∞−       − ==Ρ ji ji i Qdyyfsentijannounce µµ σ µµ2 1 2            +     +     ≤Ρ∴ σσσ 2 62 2 44 2 16 4 1 QQQe Note: For example,       σ2 26Q comes from six ji combinations for which 1=− ji . These are ( ) ( ) ( ) ( ) ( ) ( )2,3,1,2,3,4,4,3,3,2,2,1 . For the specified, values of iµ and 2σ we have: ( ) ( ) ( ) 0342.06 2 142 2 3 ≈++≤Ρ QQQe The union bound is fairly tight in this example. ii. Bhattachanya Bound: [ ]∑ ∫ ≠ ∞ ∞− ≤Ρ ji jiie dyyfyf 2 1 , )()( Use the fact that π=∫ ∞ ∞− − dte t 2 and some algebraic manipulations to find: [ ] ( ) 2 2 82 1 )()( σ µµ ji edyyfyf ji − −∞ ∞− =∫ 2.0246 4 1 222 2 92 2 1 ≈         ++≤Ρ∴ −−− σσσ eeee This is larger than the union bound because ( )ij |Ρ is now being upper bounded where in the union bound case it was calculated exactly. EECE542: Digital Communications Theory Prof. M. Hayat 2 iii. Gallager Bound: From class notes: [ ] ( ) ( )                   Ε=         =                           = ≥         ≤Ρ         + + + ≠ + −+ − ∞ ∞− ≠ + + + −− + + − ∞ ∞− ≠ +−−+ − − ∞ ∞− ≠ ++ ∑ ∫ ∑ ∫ ∑ ∫ ∑ ρ ρσ µ ρρσ µ ρσ µρσ µ ρ ρσ µ ρσ µ σρσ µρσ µ ρ σ µρ σ µ ρ ρρ σπ σπ σπσπ ρ )1()1()1(2 )1(2 )1()1(22)1( )1(2 1 1 2 1 1 2 1 1 1 1 , 222 2 2 2 22 2 2 2 2 2 2 2 2 2 2 2 2 2 1 2 1 0,)()( jiij i iji i ii Y ij ij yyy ij yy ij jiie eee dyeeee dyee dyyfyf where Y is a zero-mean Gaussian r.v with variance 2σ . Now [ ] NY ee 2 2 σαα =Ε for any ℜ∈α So, ρ µ ρ µ ρσρσ µρσ µ σπ                  Ε≤ ∑ ≠       + + + + −+ − ij Y ie j ij i eeP )1()1(2 )1(2 , 22 2 2 2 2 By Jensen’s Inequality (assuming 10 ≤≤ ρ ) Hence, ∑ ∑ ∑∑ ≠ + +      − ++ +      − ++       + + ≠ + −+ − ≠       + + ≠ + −+ − =         =                 Ε≤ ij ij ij Y ij ie jiji j ij i j ij i e eee eeeP ρρσ µµ ρρσ µ ρρρρσ µ ρ µ ρ µ ρσρσ µρσ µ ρ µ ρ µ ρσρσ µρσ µ σπ σπ 222 2 2 2 2 22 2 2 2 22 2 2 2 )1( 1 )1( 1 )1(2 1 )1( 1 )1(2 )1(2 1 )1(2 )1(2 )1()1(2 )1(2 , 2 2 See MATLAB pot to see the dependence of the bound on ρ . Note that the minimum is at 1=ρ . ∴The Gallager bound is equivalent to the Bhattachanya bound in this example. %Gallager- bound plot