This document discusses optimal detection theory for digital modulation and coding. It contains the following key points: 1) The goal of detection is to minimize error probability by choosing the optimal decision rule that maximizes the probability of the received signal given each possible transmitted signal. 2) The maximum a posteriori (MAP) and maximum likelihood (ML) receivers are introduced as optimal detectors. 3) For binary antipodal signaling in additive white Gaussian noise, the MAP detector reduces to choosing the signal closest to the received signal. 4) Expressions are provided for the error probability of binary signaling schemes in AWGN, including the well-known Q-function expression for binary antipodal signaling.

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. from Digital Communications (5th Ed.) – John G. Proakis and Masoud Salehi

3. MAP and ML Receivers
Goal is to design an optimal detector that minimizes the error
probability. In other words,
m = gopt (r) = arg max P [m|r]
ˆ
1≤m≤M
= arg max P [sm |r] (1)
1≤m≤M
MAP 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. Decision Region
Any detector partitions the output space into M regions denoted
by D1 , D2 , · · · , DM such that if r ∈ Dm , then m = g(r) = m, i.e.,
ˆ
the 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. The Error Probability
When sm is sent, an error occurs when the received r is not in
Dm .
M
Pe = Pm P [r ∈ Dm |sm sent]
/
m=1
M
= Pm Pe|m (5)
m=1
where 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. Optimum Detection in AWGN
The 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. Optimum Detection in AWGN
If 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. from Digital Communications (5th Ed.) – John G. Proakis and Masoud Salehi

9. from Digital Communications (5th Ed.) – John G. Proakis and Masoud Salehi

10. from Digital Communications (5th Ed.) – John G. Proakis and Masoud Salehi

11. Error Probability
Error Probability for Binary Antipodal Signaling
s1 = s(t) and s2 (t) = −s(t). The probabilities of messages 1 and
2 are p and 1 − p, respectively. Assume each signal has the
energy Eb .
The decision region D1 is given as
N0 1 N0 1
D1 = 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. Error Probability
Error 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
2Eb
When p = 1 , we have rth = 0. Then, Pe = Q
2 N0 .
TELE4653 - Digital Modulation & Coding - Lecture 6. April 12, 2010. – p.11/1

13. Error Probability
Error Probability for Equiprobable Binary Signaling
Schemes
Since the signals are equiprobable, the two decision regions are
separated by the perpendicular bisector of the line connecting s1
and s2 .
Let d12 = s2 − s1 . Therefore, the error probability is
n · (s2 − s1 ) d12
Pb = P > (12)
d12 2
Note that n · (s2 − s1 ) is a zero-mean Gaussian r.v. with variance
d2 N0
2 . Hence,
12
 
d2 
12
Pb = Q  (13)
2N0
TELE4653 - Digital Modulation & Coding - Lecture 6. April 12, 2010. – p.12/1

14. Error Probability
Error 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
Eb
Usually, γb = N0 is referred to as the SNR per bit.
TELE4653 - Digital Modulation & Coding - Lecture 6. April 12, 2010. – p.13/1

15. from Digital Communications (5th Ed.) – John G. Proakis and Masoud Salehi

16. from Digital Communications (5th Ed.) – John G. Proakis and Masoud Salehi

17. from Digital Communications (5th Ed.) – John G. Proakis and Masoud Salehi

18. from Digital Communications (5th Ed.) – John G. Proakis and Masoud Salehi