Of michigan fall 2012 r october 3 2012 40 93 lecture

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Unformatted text preview: p2 (r0 )π2 ∆ if we have a decision rule so that r0 ∈ R2 . In order to make the largest contribution to Γ we should have a decision rule such that r ∈ Ri if pi (r )πi > pj (r )πj for all j = i . EECS 455 (Univ. of Michigan) Fall 2012 October 3, 2012 39 / 93 Lecture Notes 7 Objective p1 (r )π1 p0 (r )π0 p2 (r )π2 r0 r0 + ∆ EECS 455 (Univ. of Michigan) Fall 2012 r October 3, 2012 40 / 93 Lecture Notes 7 Objective The decision rule that minimizes the average error probability is the decision rule that maximizes M −1 pi (r )πi dr . i =0 Ri p1 (r )π1 p0 (r )π0 p2 (r )π2 r R2 EECS 455 (Univ. of Michigan) R0 R0 R1 Fall 2012 R2 R0 October 3, 2012 41 / 93 Lecture Notes 7 Example of Two-Dimensional Densities 0.08 0.06 0.04 0.02 3 0 3 2 2 1 1 0 0 −1 −1 −2 −2 −3 EECS 455 (Univ. of Michigan) −3 Fall 2012 October 3, 2012 42 / 93 Lecture Notes 7 Optimal Receiver The decision rule that minimizes average probability of error assigns r to Ri if pi (r )πi = max pj (r )πj . 0≤ j ≤ M − 1 To understand this consider the case of M = 2 where R0 ∪ R1 is the enitre observation region. Then E [Pe ] = 1 − R0 p0 (r )π0 dr − p1 (r )π1 dr . R1 If for a particular r , p0 (r )π0 > p1 (r )π1 then the error probability will be smaller if that value of r is included in the decision region for R0 rather than the decision region for R1 . EECS 455 (Univ. of Michigan) Fall 2012 October 3, 2012 43 / 93 Lecture Notes 7 Alternate Forms of the Optimal Receiver Let p (r ) be an arbitrary density function that is nonzero everywhere pi (r ) is nonzero then an equivalent decision rule is to assign r to Ri if pj (r ) pi (r ) πi = max πj . p (r ) 0≤j ≤M −1 p (r ) Thus for M hypotheses the decision rule that minimizes average error probability is to choose i so that pi (r )πi > pj (r )πj , ∀ j = i . Let Λi ,j = pi (r ) pj (r ) where i = 0, 1, . . . , M − 1, j = 0, 1, . . . , M − 1. Then the optimal decision rule is: πj for all j = i . Choose i if Λi ,j > πi EECS 455 (Univ. of Michigan) Fall 2012 October 3, 2012 44 / 93 Lecture Notes 7 Alternate Forms of the Optimal Receiver 1 We will usually assume πi = M ∀i . (If not we should do source encoding to reduce the entropy (rate)). For this case the optimal decision rule is Choose i if Λi ,j > 1 ∀ j = i . Note that the optimum receiver does a pair-wise comparison between two potential signals (for every pair). So if we know the optimum receiver for any two signals we can find the optimal receiver for M signals. EECS 455 (Univ. of Michigan) Fall 2012 October 3, 2012 45 / 93 Lecture Notes 7 Example 2: Additive White Gaussian Noise Consider three signals in additive white Gaussian noise. For additive white Gaussian noise K (s, t ) = N0 δ(t − s). Let {ϕi (t )}∞ 0 be any i= 2 complete orthonormal set on [0, T ]. Consider the case of 3 signals. Find the decision rule to minimize average error probability. First expand the noise using orthonormal set of functions and random variables. ∞ ni ϕi (t ) n(t ) = i =0 where E [ni ] = 0 and...
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