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# Tele4653 l6

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### Tele4653 l6

1. 1. TELE4653 Digital Modulation & Coding Detection Theory Wei Zhang w.zhang@unsw.edu.au School of Electrical Engineering and Telecommunications The University of New South Wales
2. 2. from Digital Communications (5th Ed.) – John G. Proakis and Masoud Salehi
3. 3. MAP and ML ReceiversGoal is to design an optimal detector that minimizes the errorprobability. In other words, m = gopt (r) = arg max P [m|r] ˆ 1≤m≤M = arg max P [sm |r] (1) 1≤m≤MMAP receiver: Pm p(r|sm ) m = arg max ˆ (2) 1≤m≤M p(r)ML receiver: m = arg max p(r|sm ) ˆ (3) 1≤m≤M TELE4653 - Digital Modulation & Coding - Lecture 6. April 12, 2010. – p.2/1
4. 4. Decision RegionAny detector partitions the output space into M regions denotedby D1 , D2 , · · · , DM such that if r ∈ Dm , then m = g(r) = m, i.e., ˆthe detector makes a decision in favor of m. The region Dm ,1 ≤ m ≤ M , is called the decision region for message m.For a MAP detector we have N ′ ′ ′Dm = r ∈ R : P [m|r] > P [m |r], ∀1 ≤ m ≤ M andm = m (4) TELE4653 - Digital Modulation & Coding - Lecture 6. April 12, 2010. – p.3/1
5. 5. The Error ProbabilityWhen sm is sent, an error occurs when the received r is not inDm . M Pe = Pm P [r ∈ Dm |sm sent] / m=1 M = Pm Pe|m (5) m=1where Pe|m = p(r|sm )dr c Dm M = p(r|sm )dr (6) ′ ′ D ′ 1≤m ≤M,m =m m TELE4653 - Digital Modulation & Coding - Lecture 6. April 12, 2010. – p.4/1
6. 6. Optimum Detection in AWGNThe MAP detector for AWGN channel is given by m = arg max[Pm p(r|sm )] ˆ = arg max[Pm pn (r − sm )] N r−sm 2 1 − = arg max Pm √ e N0 πN0 r − sm 2 = arg max ln Pm − N0 N0 r − sm 2 = arg max ln Pm − 2 2 N0 1 = arg max ln Pm − Em + r · sm (7) 2 2 TELE4653 - Digital Modulation & Coding - Lecture 6. April 12, 2010. – p.5/1
7. 7. Optimum Detection in AWGNIf the signals are equiprobable, then m = arg min r − sm . ˆ (8)Nearest-neighbor detector.If the signals are equiprobable and have equal energy, m = arg max r · (sm ) ˆ (9) TELE4653 - Digital Modulation & Coding - Lecture 6. April 12, 2010. – p.6/1
8. 8. from Digital Communications (5th Ed.) – John G. Proakis and Masoud Salehi
9. 9. from Digital Communications (5th Ed.) – John G. Proakis and Masoud Salehi
10. 10. from Digital Communications (5th Ed.) – John G. Proakis and Masoud Salehi
11. 11. Error ProbabilityError Probability for Binary Antipodal Signalings1 = s(t) and s2 (t) = −s(t). The probabilities of messages 1 and2 are p and 1 − p, respectively. Assume each signal has theenergy Eb .The decision region D1 is given as N0 1 N0 1D1 = r : r Eb + ln p − Eb > −r Eb + ln(1 − p) − Eb 2 2 2 2 N0 1−p = r : r > √ ln 4 Eb p = {r : r > rth } (10)where rth = N √0 4 Eb ln 1−p . p TELE4653 - Digital Modulation & Coding - Lecture 6. April 12, 2010. – p.10/1
12. 12. Error ProbabilityError Probability for Binary Antipodal Signaling (Cont.)Pe = p p r|s = Eb dr + (1 − p) p r|s = − Eb dr D2 D1 N0 = pP N Eb , < rth 2 N0 + (1 − p)P N − Eb , > rth 2 √ √     = pQ  Eb − rth  + (1 − p)Q  Eb + rth  (11) N0 N0 2 2 2EbWhen p = 1 , we have rth = 0. Then, Pe = Q 2 N0 . TELE4653 - Digital Modulation & Coding - Lecture 6. April 12, 2010. – p.11/1
13. 13. Error ProbabilityError Probability for Equiprobable Binary SignalingSchemesSince the signals are equiprobable, the two decision regions areseparated by the perpendicular bisector of the line connecting s1and s2 .Let d12 = s2 − s1 . Therefore, the error probability is n · (s2 − s1 ) d12 Pb = P > (12) d12 2Note that n · (s2 − s1 ) is a zero-mean Gaussian r.v. with varianced2 N0 2 . Hence, 12   d2  12 Pb = Q  (13) 2N0 TELE4653 - Digital Modulation & Coding - Lecture 6. April 12, 2010. – p.12/1
14. 14. Error ProbabilityError Probability for Binary Orthogonal Signaling √The signal vector representation is s1 = ( Eb , 0) and √s2 = (0, Eb ). √It is clear that d = 2Eb and   d2  Eb Pb = Q  =Q (14) 2N0 N0 EbUsually, γb = N0 is referred to as the SNR per bit. TELE4653 - Digital Modulation & Coding - Lecture 6. April 12, 2010. – p.13/1
15. 15. from Digital Communications (5th Ed.) – John G. Proakis and Masoud Salehi
16. 16. from Digital Communications (5th Ed.) – John G. Proakis and Masoud Salehi
17. 17. from Digital Communications (5th Ed.) – John G. Proakis and Masoud Salehi
18. 18. from Digital Communications (5th Ed.) – John G. Proakis and Masoud Salehi