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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 TwoDimensional 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 pairwise 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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