Post on 16-May-2018
TSKS01 Digital Communication
Lecture 4
Digital Modulation � Detection
Mikael Olofsson
Department of EE (ISY)
Div. of Communication Systems
2013-09-24 TSKS01 Digital Communication - Lecture 4 2
A One-way Telecommunication System
Channel
Source
encoder
Source
decoder
Source
Destination
Channel
encoderModulator
Channel
decoder
De-
modulator
Source
coding
Channel
coding
Packing
Unpacking
Error control
Error correction
Digital to analog
Analog to digital
Medium
Digital
modulation
2013-09-24 TSKS01 Digital Communication - Lecture 4 3
Situation
ChannelSender ReceiverSource Destination
Noise
ChannelVector
sender
Vector
detectorSource DestinationModulator
De-
modulator
Noise
Vector channelVector
sender
Vector
detectorSource Destination
Vector noise
2013-09-24 TSKS01 Digital Communication - Lecture 4 4
Last Time � Signals as Vectors
Signals:
Tt <≤0
( ) 1,,1,0, −= Mitsi �
ON basis:
( ) ( )���
���
≠
===� ji
jidttt ji
T
ji,0
,1,
0
δφφ
( ) 1,,1,0, −= Njtj �φ
Linear combinations:
( ) ( )�−
=
=1
0
,
N
j
jjii tsts φ
Vectors:
�
���
�
=
−1,
0,
Ni
i
i
s
s
s �
Inner products (correlation):
( ) ( ) ( ) ,,0
dttbtaba
T
�= �−
=
=1
0
N
j
jjbaba �
ON basis � ( ) baba �=,
Norms: ( )aaa ,2
=
aaa �=2
Euclidean distance:
( ) 222 , bababad −=−=
Angles: ( ) ( )ba
ba
ba
ba
⋅=
⋅=
,cos
�α
Equal!
2013-09-24 TSKS01 Digital Communication - Lecture 4 5
Last Time � Impact of White Gaussian Noise
Received: ( ) ( ) ( )tWtstX i +=
( ) ( ) ( )dtttWWW
T
jjj �==0
, φφ
Statistical properties:Noise: ( ) ( ) ( )tWtWtWN
j
jj′+=�
−
=
1
0
φ
( ) ( )tWtXN
j
jj′+=�
−
=
1
0
φ
( ) ( ) ( )dtttXXX
T
jjj �==0
, φφ jji Ws += ,
Vectors: WsX i +=
�
���
�
=
−1
0
NX
X
X �
�
���
�
=
−1
0
NW
W
W �
Wj Gaussian with mean 0.
Xj Gaussian with mean si,j.
Wj and Wk independent for j ≠ k.
Xj and Xk independent for j ≠ k.
Wj and W�(t) independent.
( )22
020 NNfR
jWW =�= σ
2013-09-24 TSKS01 Digital Communication - Lecture 4 6
AWGN � Additive White Gaussian Noise
Orthogonal noise components
are statistically independent.
X1 & X2 independent
Y1 & Y2 independent
PSD = Variance
Y2
Y1
X1
X2
σXi= σYi
= RW ( f ) = N0/22 2
2013-09-24 TSKS01 Digital Communication - Lecture 4 7
Demodulation
ChannelModulatorDe-
modulator
Noise
Projections:
Do this!
Signals are
projected on
basis functionsNoise is
projected on
basis functions
2013-09-24 TSKS01 Digital Communication - Lecture 4 8
Correlation Receiver
2013-09-24 TSKS01 Digital Communication - Lecture 4 9
Matched Filters
A filter with impulse response hj(t) = φj(T � t) is matched to φj(t).
Example: Usage (φj�hj)(t):
T 2T
Maximum at t = T.
Orthogonal signal φi(t):
Zero at t = T.
T 2T
T
φj(t)hj(t)
Then (φi�hj)(t):
(φi(t),φj(t)) = 0
2013-09-24 TSKS01 Digital Communication - Lecture 4 10
Demodulation Using Matched Filters
ChannelModulatorDe-
modulator
Noise
Do this!
Filter: Matched to
Output:
Sample at t = T : And this!
2013-09-24 TSKS01 Digital Communication - Lecture 4 11
Matched Filter Receiver
2013-09-24 TSKS01 Digital Communication - Lecture 4 12
The Vector Detector
Knowledge that can be used:
All entities are realizations of stochastic variables.
Use statistical descriptions of those stochastic variables.
The task of the vector detector:
Observe and output according to a well chosen decision rule,
designed to minimize the error probability.
x a�
Vector channelVector
sender
Vector
detectorSource Destination
Vector noise
x a�isia
2013-09-24 TSKS01 Digital Communication - Lecture 4 13
1 3
3
1
�1
φ0
φ1
0s
1s
2s
Vector Detection
?
Received vector: x
x
Goal: Minimize the error probability
{ }xXAA =≠ |�Pr
Estimated
symbol
Sent
symbol
Received
vector
Observed
realization
Equivalent decision rule:
Set if
is maximized for k = i.
{ }xXaA k == |Priaa =�
2013-09-24 TSKS01 Digital Communication - Lecture 4 14
Independent of ak.
"4"2
02
46
"20
24
680
0.01
0.02
0.03
Detection
Bayes rule: { }( )
( ){ }k
X
kAX
k aAxf
axfxXaA ==== Pr
||Pr
|
New decision rule:
Set if
is maximized for k = i.
iaa =�
( ) { }kkAXaAaxf =⋅Pr|
|
2013-09-24 TSKS01 Digital Communication - Lecture 4 15
Maximum Likelihood (ML) Detection
Assumption: { } { }1,,1,01
Pr −∈∀== MkM
aA k �
ML decision rule: Set if is maximized for k = i. iaa =� ( )kAX
axf ||
( ) ( )∏−
=
=1
0
||||
N
j
kjAXkAXaxfaxf
j
Independent variables:
( ) 02
,
1
0 0
1 NsxN
j
jkjeN
−−−
=
∏ ⋅=π
( )( )�
⋅=
−
=
−−−
1
0
2,
0
1
2
0
N
j
jkj sxNN
eNπ
Natural logarithm (strictly increasing function):
( )( ) ( ) ( )� −−−=−
=
1
0
2
,
0
0|
1ln
2|ln
N
jjkjkAX
sxN
NN
axf π ( ) ( )ksxdN
NN
,1
ln2
2
0
0 −−= π
Constant
Equivalent ML rule: Set if is minimized for k = i. iaa =� ( )ksxd ,
2013-09-24 TSKS01 Digital Communication - Lecture 4 16
1 3
3
1
�1
φ0
φ1
2s
0s
1s
ML Decision Regions
?
B2
B1
B0
x
Interprete as the nearest signal.x
!
Result:
Decision regions consisting
of all points closest to a
signal point.
Notation:
Bi is the decision region of
the signal vector si.
Thus also of the signal si (t)
and of the message ai.
The borders are orthogonal to
straight lines between signals:
In 2 dimensions: Lines.
In 3 dimensions: Planes.
In more dim: Hyperplanes.
Borders cut the lines mid-way.
2013-09-24 TSKS01 Digital Communication - Lecture 4 17
Error Probability
Symbol error probability:
{ } { } { }�−
=
=≠⋅==≠=1
0
e |�PrPr�PrM
i
iii aAaAaAAAP
{ } { }�−
=
=∉⋅==1
0
|PrPrM
i
iii aABXaA
ML detection again { } { }
��
�−∈∀== 1,,1,0
1Pr Mi
MaA i � :
{ }�−
=
=∉=1
0
e |Pr1 M
i
ii aABXM
P ( ) N
M
i
iAXdxdxaxf
M�� 1
1
0|
|1�� �
−
=
=
iBx ∉
This is generally hard to calculate.
2013-09-24 TSKS01 Digital Communication - Lecture 4 18
A Special Case � Two signals in One Dim
( )10 , ssd
B1B0
fX|A(x|a0) fX|A(x|a1)
{ } { }0100 |Pr|Pr aABXaABX =∈==∉
s0 s1
φ0
���
���
=+
>= 0
0,10,0
0 |2
Pr aAss
X
���
���
=−+
>= 00,0
0,10,0
0 |2
Pr aAsss
W���
��� −
>=2
Pr0,00,1
0
ssW
( )���
���
>=2
,Pr 10
0
ssdW
( )
�
��
�=
2
2,
0
10
N
ssdQ
( )
�
��
�=
0
10
2
,
N
ssdQ
Similarily for .{ }11 |Pr aABX =∉
This works in all
directions.
B1B0
s0
s1( )�QP =� e
2
10 ss +
2013-09-24 TSKS01 Digital Communication - Lecture 4 19
Back to M Signals in N Dimensions
1 3
3
1
�1
φ0
φ1
2s
0s
1s
B2
B1
B0
Interprete as the nearest signal.x
We had:
{ }�−
=
=∉=1
0
e |Pr1 M
i
ii aABXM
P
{ }��−
= ≠
=∈=1
0
|Pr1 M
i ij
ij aABXM
Hard to calculate!!
Could we take this to
the more simple case
in one dimension?
2013-09-24 TSKS01 Digital Communication - Lecture 4 20
The Union Bound
1 3
3
1
�1
φ0
φ1
2s
0s
1s
B2
B1
B0
Interprete as the nearest signal.x
Define overestimated regions:
( ) ( ){ }ijji sxdsxdxB ,,:, <=
B0,1B0,2
An upper bound based on
overestimating the decision
regions. We had:
{ }��−
= ≠
=∈=1
0
e |Pr1 M
i ij
ij aABXM
P
{ }��−
= ≠
=∈≤1
0
,e |Pr1 M
i ij
iji aABXM
P
Overestimated error probability:
��−
= ≠
�
��
�≤
1
0 0
,
e2
1 M
i ij
ji
N
dQ
MP
Back to the one-dim case:
( )jiji ssdd ,, =Distances:
2013-09-24 TSKS01 Digital Communication - Lecture 4 21
The Nearest Neighbour Approximation
1 3
3
1
�1
φ0
φ1
2s
0s
1s
Interprete as the nearest signal.x
��−
= ≠
�
��
�≤
1
0 0
,
e2
1 M
i ij
ji
N
dQ
MP
We had the union bound:
0,11,0 dd =
1,22,1 dd =
0,22,0 dd =
� �−
= =
�
��
�≈
1
0 : 0
mine
min,2
1 M
i ddj jiN
dQ
MP
Nearest neighbour approximation:
Dominated by the smallest distance.
jiji
dd ,min min≠
=
min,:# ddjn jii ==
�−
=
�
��
�=
1
0 0
min
2
1 M
i
iN
dQn
M
mind=
2013-09-24 TSKS01 Digital Communication - Lecture 4 22
Comments
Alternative upper bound:
As the union bound, but only consider pairs of points whose
decision regions share a common border.
At high SNR Eavg/N0:
Both the union bound on � and the nearest neighbour of �
the error probability are close to the real error probability.
Alternative approximation:
As the nearest neighbour approximation, but consider the two or
three smallest distances.
Very simple approximation:
�
��
�≈
0
mine
2N
dQP
Very simple upper bound:
( )
�
��
�⋅−≤
0
mine
21
N
dQMP
2013-09-24 TSKS01 Digital Communication - Lecture 4 23
Maximum á-Posteriori (MAP) Detection 1(2)
B1B0
s0 s1
φ0
fX|A(x|a0)Pr{A=a0}�
fX|A(x|a1)Pr{A=a1}�
Back to this rule (MAP):
Set if is maximized for k = i. iaa =� ( ) { }kkAXaAaxf =⋅Pr|
|
( ) { }( )kk aANsxd =⋅− Prln, 0
2
Equivalent reasoning leads to this rule:
Set if is minimized for k = i. iaa =�
2013-09-24 TSKS01 Digital Communication - Lecture 4 24
Maximum á-Posteriori (MAP) Detection 2(2)
The borders are still:
Straight lines (hyperplanes)
Orthogonal to straight lines
between signal points.
1 3
3
1
�1
φ0
φ1
2s
0s
1s
B2
B1
B0
But:
Borders cut the lines mid-way
only if the two probabilities are
equal.
Borders depend on N0.
Note:
A signal point may in extreme
cases not even be in its own
detection region.
2013-09-24 TSKS01 Digital Communication - Lecture 4 25
Special Case: Orthogonal Decision Borders
2s
0s1s
B2
B1B0
3sB3
1,0d
2,1d 2,0d
UB: 0,21,20,1e qqqP ++≤1,20,1e qqP +≤Alternative bound:
NN: 1,2e qP ≈ 1,20,1e qqP +≈Alternative approx.:
Exact: 1,20,11,20,1e qqqqP −+=
(orthogonal noise components are independent)
�
��
�=
0
,
ji,2N
dQq
jiDefine notation:
2
2,1
2
1,0
2
2,0 ddd +=Pythagoras:
1,2e qP >Lower bound:
2013-09-24 TSKS01 Digital Communication - Lecture 4 26
0θ
α
Cost of Non-Optimal Detection
Detected signal points: ( ) ( ) ( ) ( )ααθφθφθ coscos,, 000000 ⋅±=⋅⋅⋅±=⋅±=± aaas
d′Signals: )()( 00 tats φ⋅=
)()( 01 tats φ⋅−=
Wrong basis function in receiver:
)(0 tθ
What is the cost of this non-ideal
situation?
0φ0s
1s
ad 2=
( )αcos⋅=′ ddEffective distance:
�
��
� ′=
0
e2N
dQPResulting error probability:
Signals are
projected on
basis functions
2013-09-24 TSKS01 Digital Communication - Lecture 4 27
Example of Non-Optimal Detection
( ) ( ) 2
0
e 1025.124.252
−⋅≈≈=
�
��
N
dQPAt α = π /4:
Ideal detection: ( ) ( ) 4
0
e 109.716.3102
−⋅≈≈=
�
��
N
dQP
d′0θ
0φ0s
1s
ad 2=
α
SNR: 50
2
=N
a
( )αcos⋅=′ dd
Distances: ad 2=
Signal points: a±
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