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Of michigan 0 t s0 s1 fall 2012 s5 2 n0 october

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Unformatted text preview: v. of Michigan) Fall 2012 October 3, 2012 82 / 93 Lecture Notes 7 Example s0 (t ) = 1ϕ0 (t ) + 0ϕ1 (t ) s1 (t ) = −1ϕ0 (t ) + 0ϕ1 (t ) √ s2 (t ) = 2ϕ0 (t ) + 3ϕ1 (t ) √ s3 (t ) = 0ϕ0 (t ) + 3ϕ1 (t ) √ s4 (t ) = −2ϕ0 (t ) + 3ϕ1 (t ) √ s5 (t ) = 2ϕ0 (t ) − 3ϕ1 (t ) √ s6 (t ) = 0ϕ0 (t ) − 3ϕ1 (t ) √ s7 (t ) = −2ϕ0 (t ) − 3ϕ1 (t ) EECS 455 (Univ. of Michigan) Fall 2012 October 3, 2012 83 / 93 Lecture Notes 7 Example ϕ1 (t ) ϕ0 (t ) EECS 455 (Univ. of Michigan) Fall 2012 October 3, 2012 84 / 93 Lecture Notes 7 Example ϕ1 (t ) R4 s4 R1 R7 s7 EECS 455 (Univ. of Michigan) R3 s3 R2 s2 R0 s1 ϕ0 (t ) s0 R6 s6 Fall 2012 R5 s5 October 3, 2012 85 / 93 Lecture Notes 7 Pair-Wise Decision Regions ϕ1 (t ) R1,0 s3 s4 R1 s7 s2 R0 s6 1 P2 (s0 → s1 ) = Q ( ) = Q ( σ EECS 455 (Univ. of Michigan) ϕ0 (t ) s0 s1 Fall 2012 s5 2 ) N0 October 3, 2012 86 / 93 Lecture Notes 7 Pair-Wise Decision Regions s R340 R3 , ϕ1 (t ) s3 R s0 0 s1 s7 EECS 455 (Univ. of Michigan) s2 s6 Fall 2012 ϕ0 (t ) s5 October 3, 2012 87 / 93 Lecture Notes 7 Pairwise Distance P2 (s0 → s3 ) = Q ( d 0, 3 2 ) = Q( ) 2σ 2σ 4 2 ) ) = Q( 2 4σ N0 √ d 0, 4 12 ) = Q( ) P2 (s0 → s4 ) = Q ( 2σ 2σ = Q( = Q( EECS 455 (Univ. of Michigan) Fall 2012 12 ) = Q( 4σ 2 6 ) N0 October 3, 2012 88 / 93 Lecture Notes 7 Pairwise Distance s0 s0 s1 s2 s3 s4 s5 s6 s7 s1 s4 s5 s6 s7 √ √ 0 2 2 2 23 √ 2 2 23 √ 2 0 2 2 2 √3 2 2 √ 23 √ 2 23 0 2 4 23 √ 27 4 2 2 2 0 2 4 4 √ √ 23 √ 23 √ 2 4 2 0 4 23 √ √ 27 2 2323 √ 27 4 0 2 4 2 2 4 4 2 0 2 √ √ 23 √ 23 2 27 4 23 4 2 0 EECS 455 (Univ. of Michigan) s2 s3 Fall 2012 October 3, 2012 89 / 93 Lecture Notes 7 Union Bounds Pe,0 = ≤ = Pe,2 = ≤ Pe,3 = ≤ Pe,1 √ √ 2 2 2 23 2 2 23 Q( ) + Q( ) + Q( ) + Q( ) + Q( ) + Q( ) + Q( ) 2σ 2σ √ 2σ 2σ 2σ 2σ 2σ 23 2 ) 5Q ( ) + 2Q ( 2σ 2σ Pe,4 = P3,5 = P3,7 √ √ 23 4 27 2 ) + 2Q ( ) + Q ( ) 2Q ( ) + 2Q ( 2σ 2σ 2σ 2σ Pe,6 √ 23 4 2 ) + 2Q ( ) 4Q ( ) + Q ( 2σ 2σ 2σ EECS 455 (Univ. of Michigan) Fall 2012 October 3, 2012 90 / 93 Lecture Notes 7 Union Bounds Pe = ≤ 1 8 7 Pe ,i i =0 √ √ 1 2 23 4 27 [26Q ( ) + 14Q ( ) + 12Q ( ) + 4Q ( )] 8 2σ 2σ 2σ 2σ EECS 455 (Univ. of Michigan) Fall 2012 October 3, 2012 91 / 93 Lecture Notes 7 Union Bounds 1.5 Eb = N0 N0 ≤ 1 26Q ( 8 4Eb ) + 14Q ( 3N0 ≤ Pe 3.25Q ( 4Eb ) + 2.75Q ( 3N0 EECS 455 (Univ. of Michigan) 12Eb ) + 12Q ( 3N0 12Eb ) + 1.5Q ( 3N0 Fall 2012 16Eb ) + 4Q ( 3N0 28Eb ) 3N0 16Eb ) + .5Q ( 3N0 28Eb ) 3N0 October 3, 2012 92 / 93 Lecture Notes 7 Union Bounds 0 10 Pe Union Bound −2 10 Simulation −4 10 −6 10 0 EECS 455 (Univ. of Michigan) 2 4 6 Fall 2012 8 10 E b /N 0 ( dB ) 12 October 3, 2012 93 / 93...
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