1. TELE3113 Analogue and Digital
Communications –
Detection Theory
Wei Zhang
w.zhang@unsw.edu.au
School of Electrical Engineering and Telecommunications
The University of New South Wales
6 Oct. 2009 TELE3113 1

2. Digital Signal Detection
At the receiving end of the digital communication system:
AWGN
n(t) Sampled
at t=kTs
1 if y(kTs)>λ
si(t) Receive Decision
r(t) filter y(t) y(kTs) device
0 if y(kTs)<λ
Polar NRZ Signaling
 s (t ) = + A 0≤t ≤T for 1 Threshold
si (t ) =  1 λ
s2 (t ) = − A 0≤t ≤T for 0
Noise power spectral density:
Sn(ω)=η/2
6 Oct. 2009 TELE3113 2

3. Digital Signal Detection
Suppose there are M possible signal symbols: {si} for i=1,…,M
r
We can represent these symbols in vector form si
r r
Similarly the noise vectors, n and the received signal vectors, r
Thus r r r
ri = s i + n
ϕ2
r
r n
n r
s1
r r
s2 r1
r r
n s3
r ϕ1
s4
r
n
ϕ3
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4. Digital Signal Detection
In each time interval, the signal detector makes a decision based on
r
the observation of the vector r so that the probability of correct
decision is maximized.
Consider a decision rule based on the posterior probabilities
r r
P(signal si was transmitted | received vector r ) for i = 1,2 ,K ,M
r r
= P( si | r )
The decision is based on selecting the signal corresponding to the
maximum set of posterior probabilities.
r r r
r r r f (r | si ) P( si )
Choose s i to maximize: P( si | r ) = r r
f (r ) where f(r ) =
M
r r r
∑ f(r | s m ) P( s m )
m =1
r r
where P ( s i ) is probability of si being transmitted and
r
f(r ) is the pdf function of
the received signal vector r .
This kind of decision is called maximum a posteriori probability (MAP) criterion
6 Oct. 2009 TELE3113 4

5. Digital Signal Detection
MAP criterion
r r r
r r f (r | si ) P( si ) r M
r r r
P( si | r ) = r where f(r ) = ∑ f(r | s m ) P( s m )
f (r ) m =1
r r
where f(r | si ) is called the likelihood function.
r
If the M symbols are equally probable; i.e. P ( si ) = 1 / M for all i, the decision
r r
rule based on finding the signal thatrmaximizes P( si | r ) is equivalent to
r
finding the signal that maximizes f(r | si ).
r r
The decision based on the maximum of f(r | si ) over M signal symbols
is called the maximum-likelihood (ML) criterion
6 Oct. 2009 TELE3113 5

6. Digital Signal Detection
r r r
Recall: r = si + n
For AWGN, the noise {nk} components are uncorrelated Gaussian
variables which are statistically independent
E[nk ] = 0 (zero mean) , E[rk ] = E[sik + nk ] = sik
η η
Variance σ n = E[n 2 ] − (E[n]) = → σ r2 = σ n =
2 2 2
2 2
Thus {rk} are statistically independent Gaussian variables
r r N
f(r | si ) = Π f(rk | sik ) where N is number of base vectors
k =1 1 2 2
/( 2σ n )
and f(rk | sik ) = e −( rk − sik )
2π σ n
1 − ( rk − sik ) 2 / η
= e
πη
Take natural logarithm on both sides, gives
r r −N 1 N
ln(πη ) − ∑ (rk − sik )
2
ln f(r | si ) =
6 Oct. 2009 2
TELE3113 η k =1 6

7. Digital Signal Detection
r r −N 1 N
ln(πη ) − ∑ (rk − sik )
2
ln f(r | si ) =
2 η k =1 r r r
With ln(•) is a monotonic function, the maximum of f(r | si ) over s i is
r
equivalent to finding the signal s i that minimizes the Euclidean distance:
N
r r
D(r , si ) = ∑ (rk − sik ) 2
k =1
r r
So, the ML decision criterion (maximize f(r | si ) over M signal symbols, i.e.
r
i=1,…M) reduces to finding the signal s i that is the closest in distance to
r
the received signal vector r .
Example: 3 signal symbols.
Note the decision regions formed by
the perpendicular bisectors of any
two signal symbols.
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8. Digital Signal Detection
r
Detection error will occur when the received signal vector r falls into the
decision region of other signal symbols. This is due to the presence of
strong random noise.
Consider there are two signal symbols s1 and s2 , which are spaced d apart.
The decision boundary is their perpendicular bisector.
r r r r
As, r = s i + n , the uncertainty of the received signal vector r is
r r r
mainly contributed by the random noise n (= (r − s i ) ), which is
Gaussian-distributed, around the signal symbol.
decision boundary
noise
distribution
s1 d s2
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9. Digital Signal Detection
Assume s1 is sent,
at the receiver, the probability of rr 1 −( r r
r − s1 )2 /η
With f(r |s1 ) = e
detection error is: πη
r r rr rr r rr 1 −( r
P ( s 2 is detected s1 is sent ) f(r |s1 ) = f((r -s1 )|s1 ) = f(n|s1 ) = e
n )2 /η
r r r r r πη
= P ( r − s1 > r − s 2 s1 )
∞
rr
= ∫ f (n |s1 ) dn noise
decision boundary
d/ 2
distribution
∞
1
∫
2
/η
= e −n dn
d /2 πη
∞
1 2n
∫
2
Using Q ( x) = e−y /2
dy and let y =
2π x η
∞
s1 d s2
r r 1 − y2 / 2
P ( s2 is detected s1 is sent ) = ∫ 2π
e dy
d / 2η
 d 
= Q 
 2η 
 
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10. Digital Signal Detection
r
For a signal symbol set: {si } for i = 1,...M
Detection error probability is
M
r r
Pe = ∑ P[erroneous detection|si sent ]P[ si ]
i =1
r r
M M  s k − si  r
≤ ∑∑ Q   P[ si ]
i =1 i ≠ k  2η
 

k =1
r
If all signal symbols are equally probable, i.e. P[ si ] = 1 / M
M
r r
Pe = ∑ P[erroneous detection|si sent ]P[ si ]
i =1
r r
1 M M s k − si 
≤ ∑∑ Q  2η
M i =1 i ≠ k 

 

k =1
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11. Digital Signal Detection
Calculation of error probabilities:
(
(a) Antipodal signaling: s1 = + E ,0 ; s2 = − E ,0) ( )
Pe = P ( s 2 is detected | s1 ) P ( s1 ) + P ( s1 is detected | s 2 ) P ( s 2 )
2 E   
≤Q   P ( s1 ) + Q 2 E  P ( s 2 )
 2η   2η 
    s2 s1
 2E 
= Q  [P ( s1 ) + P ( s 2 )]
 η  − E + E
 
 2E 
= Q
 η 
 signal symbol energy=E
 
Example: for NRZ signaling which takes amplitude either +A or 0. For bit
interval Tb, the energy per bit Eb=A2Tb.
 A2T   
Pe = Q  = Q Eb 
 η   η 
6 Oct. 2009  TELE3113 
  11

12. Digital Signal Detection
(
(b) Orthogonal signaling: s1 = + E ,0 ; s2 = 0,+ E ) ( )
Pe = P( s 2 is detected | s1 ) P( s1 ) + P( s1 is detected | s 2 ) P ( s 2 ) + E s2
 2E   
≤ Q  P( s1 ) + Q 2 E  P( s 2 ) s1
 2η   2η 
   
+ E
 E   E
= Q
 
[P(s1 ) + P( s 2 )] = Q 
 η
 η  
(c) Square signaling:
s1 = + ( ) ( ) (
E ,− E ; s2 = + E ,+ E ; s3 = − E ,+ E ; s4 = − E ,− E) ( )
4
Pe = ∑ P ( si is not detected | si ) P ( si ) s3 + E s2
i =1
 2 E
     

4
≤ ∑ P ( si )Q  + Q 2 2 E  + Q 2 E
  2η   2η   2η 
i =1       
 − E + E
 2E   E s4 − E s1

= 2Q  + Q 2 
  η
6 Oct. 2009  η    TELE3113 12

13. Digital Signal Detection
Integrate-and-Dump detector
r(t)=si(t)+n(t)  s (t ) = + A 0≤t ≤T for 1
si (t ) =  1
s2 (t ) = − A 0≤t ≤T for 0
t 0 +T
 a1 (t ) + no for 1
Output of the integrator: z (t ) = ∫ [si (t ) + n(t )]dt = 
t 0 +T
t0 a2 (t ) + no for 0
where a1 = ∫ Adt = AT
t0
t 0 +T
a2 = ∫ (− A)dt = − AT
t0
t 0 +T
no = ∫ n(t )dt
t0
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14. Digital Signal Detection
no is a zero-mean Gaussian random variable.
t0 +T
  t 0 +T

E{no } = E  ∫ n(t )dt  = ∫ E{n(t )}dt = 0
 t0
  t0

  t 0 +T  
2

{ }
σ no = Var{no } = E no = E  ∫ n(t )dt  
2 2 
  t0

 
 
t o +T t 0 +T
= ∫ ∫ E{n(t )n(ε )}dtdε
t0 t0
t o +T t 0 +T
η
= ∫ ∫ δ (t − ε )dtdε
t0 t0
2
t 0 +T
η ηT
= ∫
t0
2
dε =
2
1 1
pdf of no: f n (α ) =
−α 2 /( 2σ no )
2 2
e = e −α /(ηT )
o
2π σ no πηT
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15. Digital Signal Detection
 s1 (t ) = + A 0≤t ≤T for 1
As si (t ) = 
s 0 (t ) = − A 0≤t ≤T for 0 s0 s1
We choose the decision threshold to be 0. 0
− AT + AT
Two cases of detection error:
(a) +A is transmitted but (AT+no)<0 no<-AT
(b) -A is transmitted but (-AT+no)>0 no>+AT
Error probability:
Pe = P (no < − AT | A) P ( A) + P (no > AT | A) P (− A)
− AT 2 ∞ 2
e −α /(ηT )
e −α /(ηT )
= P ( A)
−∞
∫ πηT
dα + P (− A) ∫
AT πηT
dα
∞ 2
e −α /(ηT )
 2 A2T 
dα [P( A) + P (− A)]
∞ 2
e −u / 2
= ∫ πηT
Thus, Pe = Q

 Q Q(x ) = ∫ du
AT
 η   x 2π
∞ 2
e −u / 2 2α  2 Eb 
= ∫
T
du Qu =
2π ηT = Q
 η 
 Q Eb = ∫ A2 dt
2 A2T η
  0
6 Oct. 2009 TELE3113 15