# Of michigan fall 2012 1 t 2 1 0 s3 s2 s0 s4 1 s1 s5

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Unformatted text preview: pT (t ) These are orthogonal if fc T ≫ 1. sm = e(j 2πm/8) , m = 0, 1, ..., 7 Data bits Signal sm 000 s0 √ (1, 0) √ 001 s1 ( 2/2, 2/2) 011 s2 ) √ (0, 1√ 010 s3 (− 2/2, 2/2) 110 s4 ) √ (−1, 0√ 111 s5 (− 2/2, − 2/2) 101 s6 √ (0, −1) √ 100 s7 ( 2/2, − 2/2) EECS 455 (Univ. of Michigan) Fall 2012 ϕ1 (t ) 2 1 0 s3 s2 s0 s4 -1 s1 s5 s6 ϕ0 (t ) s7 -2 -2 -1 0 1 2 October 3, 2012 33 / 93 Lecture Notes 7 Optimal Receiver Note that we can recover completely r (t ) if we know the coefﬁcients rm , m = 0, 1, .... So the optimal decision based on observing r0 , r1 , ... is also the optimal decision based on observing r (t ). Given signal si (t ) is transmitted we can determine the probability density of rm as follows. First, rm is Gaussian since it is the result of integrating Gaussian noise. Second the mean of rm , conditioned on signal si (t ) transmitted is si ,m and the variance is N0 /2. EECS 455 (Univ. of Michigan) Fall 2012 October 3, 2012 34 / 93 Lecture Notes 7 Optimal Receiver So the probability density of rm conditioned on signal si (t ) transmitted (event Hi ) is pi (rm ) = frm |Hi (rm ) = √ 2π 1 N0 /2 exp{− (rm − si ,m )2 } 2(N0 /2) Next note that rm is independent of rn for m = n. Thus k fr0 ,r1 ,...,rk |Hi (x0 , x1 , x2 , ..., xk ) = frm |Hi (xm ) m =0 k = pi (xm ) m =0 EECS 455 (Univ. of Michigan) Fall 2012 October 3, 2012 35 / 93 Lecture Notes 7 M -ary Detection Problem Consider the problem of deciding which of M hypotheses is true based on observing a random variable (vector) r . The performance criteria we consider is the average error probability. That is, the probability of deciding anything except hypothesis Hj when hypothesis Hj is true. The underlying model is that there is a conditional probability density (mass) function of the observation r given each hypothesis Hj . P {r ∈ Rm |Hi } = pi (r )dr Rm There are disjoint decision regions R0 , R1 , ..., RM −1 . When r ∈ Rm the receiver decides Hm . EECS 455 (Univ. of Michigan) Fall 2012 October 3, 2012 36 / 93 Lecture Notes 7 Decision Regions R3 R1 R5 R2 R0 EECS 455 (Univ. of Michigan) R4 R7 R6 Fall 2012 October 3, 2012 37 / 93 Lecture Notes 7 Objective Our goal is to ﬁnd the decision regions R0 , R1 , ..., RM −1 that minimize the error probability. M −1 M −1 P e , i πi E [Pe ] = = i =0 i =0 P {don’t decide Hi |Hi }πi M −1 = i =0 [1 − P {decide Hi |Hi true}] πi M −1 M −1 = i =0 πi − pi (r )πi dr i =0 Ri M −1 = 1− EECS 455 (Univ. of Michigan) pi (r )πi dr . i =0 Fall 2012 Ri October 3, 2012 38 / 93 Lecture Notes 7 Objective The decision rule that minimizes the average error probability is the decision rule that maximizes M −1 ∞ M −1 Γ= pi (r )πi dr = i =0 Ri −∞ i =0 pi (r )πi I (r ∈ Ri )dr . Consider a small region for r ∈ A = (r0 , r0 + ∆) where pi (r ) is nearly constant. If r ∈ A then the contribution to Γ is either p0 (r0 )π0 ∆ if we have a decision rule so that r0 ∈ R0 , or the contribution to Γ is p1 (r0 )π1 ∆ if we have a decision rule so that r0 ∈ R1 , or the contribution to Γ is...
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## This note was uploaded on 02/12/2014 for the course EECS 455 taught by Professor Stark during the Fall '08 term at University of Michigan.

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