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Unformatted text preview: EE 7615 Solutions to Problem Set # 3 4.3 We need that R1 be suﬃcient statistic for decisions about S ∈ {s1 , s2 } =
√
√
{ E, − E } based on (R1 , R2 ). For this we need
pR2 R1 ,S (r2 r1 , s) = pR2 R1 (r2 r1 ) ∀s ∈ {s1 , s2 }, r2 , r1
The event that S = s and R1 = r1 is the same as the event that S = s
and N1 = r1 − s. Therefore we need
pN2 N1 ,S (r2 −r1 r1 −s, s) = p1 pR2 R1 ,S (r2 r1 , s1 )+p2 pR2 R1 ,S (r2 r1 , s2 ) ∀s, r2 , r1
where p1 = P (S = s1 ) and p2 = P (S = s2 ). Now the signal S is
independent of the noise components. Therefore we can write
pN2 N1 (r2 −r1 r1 −s) = p1 pN2 N1 (r2 −r1 r1 −s1 )+p2 pN2 N1 (r2 −r1 r1 −s2 ) ∀s, r2 , r1
If we set s = s1 in the above we get
pN2 N1 (r2 − r1 r1 − s1 ) = pN2 N1 (r2 − r1 r1 − s2 )
This is satisﬁed if N2 is independent of N1 .
A counterexample is if N2 = −N1 . 1 ...
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This note was uploaded on 03/05/2012 for the course EE 7615 taught by Professor Naragipour during the Fall '11 term at LSU.
 Fall '11
 Naragipour

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