BPSK Probability of Error
BPSK Probability of Error
BPSK Probability of Error
Outline
Introduction to Digital Communications
Signal (Vector) Space Representations
Digital Modulation Schemes (M-ASK, M-PSK, M-FSK)
Performance Measures for Modulation Schemes
- Bandwidth (spectral) efficiency
- Power efficiency
A/D
Source
Encoder
Channel
Encoder
Modulator
Channel
Recovered
message
signal (analog)
D/A
Source
Decoder
Channel
Decoder
Demodulator
B
0.01 0.1
fc
B
: Fractional bandwidth
fc
ECE414 Wireless Communications, University of Waterloo, Winter 2012
Channel capacity
Bandwidth
Signal-to-noise ratio
Signal-Space Representations
Consider a modulation format where the transmitted signal waveforms belong to
M
the modulation set sm t m
1 0 t Ts .
Ts
sm,n sm t
0
n*
t dt st sm,nn t
n 1
Es
Ts
2
sm
0
t dt sm2 ,n sm
sk t sk sk ,1, sk ,2 ,..., sk , N
sl t sl sl ,1, sl ,2 ,..., sl , N
Correlation
Euclidean
Distance
n1
k , l m 1,2,...M
1
sk ( t ), sl ( t ) =
Ts
d ( sk ( t ), sl ( t )) =
2
Ts
s (t )s (t )
k
dt = sk , sl = s k sl = sk,n s*l,n
n=1
Ts
sk ( t ) - sl ( t ) dt = d ( s k , sl ) = sk,n - sl,n
2
n=1
sm t Am cos2f c t
0 t Ts
0,
otherwise
sm t sm Am Ts 2
1-dimensional
Ts
M-ASK (contd)
Examples of M-ASK
Signal Constellations
M=4
Bandpass Modulation Signal
11 10 00 01
10
m 2 m 1 M , m 1,2...M
0 t Ts
Basis Functions
2 Ts cos2f ct , 0 t Ts
2 Ts sin 2f ct , 0 t Ts
1 t
2 t
otherwise
otherwise
0,
0,
Signal-Space Representation
sm t s m A Ts 2 cos m , A Ts 2 sin m
2-dimensional
Ts
11
2 t
Es
Es
1 t
-A
Equivalent Lowpass Signal
s1 Es ,0
s 2 Es ,0
A
t
-A
12
00 s1 t A cos2f ct
s 2 0, Es
01 s2 t A cos2f ct 2
10 s3 t A cos2f ct s 3 Es ,0
s 4 0, Es
11 s4 t A cos2f ct 3 2
2 t
Signal-Space
Representation
Es
Es
Es
1 t
Es
13
0 t Ts
Am Am2 ,r Am2 ,i
m 1,2,...M
0 t Ts
m arctgAm,i Am,r
Examples of QAM
Signal Constellations
14
QAM (contd)
Baseband (Equivalent Lowpass) Representation
Basis Functions
2 Ts cos2f ct , 0 t Ts
1 t
otherwise
0,
2 Ts sin 2f ct , 0 t Ts
2 t
otherwise
0,
Signal-Space Representation
sm t s m Am,r Ts 2, Am,i Ts 2
2-dimensional
Signal Energy
Ts
15
f m mf , m 1,2...M
0 t Ts
Cross Correlation
k ,l
1 Ts j 2 k l ft
sinT k l f jT k l f
e
dt
e
Ts 0
T k l f
k ,l
sin2T k l f
2T k l f
For f 1 2T and k l
k ,l 0
16
M-FSK (contd)
Assuming frequency separation f 1 2T, the signal-space representation
for the M-FSK signals are given as N-dimensional vectors, where N=M.
2 Ts cos2 f c f m t , 0 t Ts
m t
0,
otherwise
s1 t s1 Es
0 0 ...... 0
s2 t s2 0
Es
0 ...... 0
.
.
.
sM t s M 0 0 0 ......
Es
Ts
17
Main lobe (null-to-null) bandwidth: The width of the main spectral lobe.
Fractional power-containment bandwidth: The frequency interval that
contains (1-) of the total signal power, e.g. 99.9% of the total power.
Bounded PSD bandwidth: The frequency interval where the PSD stays
above a prescribed certain threshold, e.g. sidelobes peaks 40 dB below its
maximum value
Roughly speaking, bandwidth of the modulation scheme is proportional to
the dimension number.
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Pe
Pb e Pe
log2 M
Two common mapping forms are natural mapping and Gray mapping.
In Gray mapping, the neighbour points differ in only one digit. It should be
noted that Gray mapping is not possible for every signal constellation.
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Sensitivity to interference
Robustness to impairments encountered in a wireless channel
20
M-FSK
log 2 M
bits/sec
Data rate
T
Data rate
2
BW null to null Hz
T
BW roughly
Data rate 1
hs =
= log 2 M [ bits/sec/Hz]
BW
2
hs =
log 2 M
bits/sec
T
M
Hz
2T
21
g t ak pt kTs
Ts : Signal interval
pt : Pulse shape
g f
1
2
P f a f
Ts
where
P f F pt
a f Ra n e j 2fnTs
n
1
*
Ra ( n) = E ak ak+n
22
g t ak pt kTs
Ra n E
ak ak*n
E
a
,
n 0 A2 , n 0
k
*
0, n 0
Eak E ak n , n 0
a f F Ra n Ra n e j 2fnTs A2
n
Pulse shaping
t - Ts / 2 FT
- j 2 p f (Ts
p (t ) =
P ( f ) = Tssinc ( fTs ) e
Ts
p(t)
T/2 T
P ( f ) = Tssinc ( fTs )
23
2)
P( f )
Ts
Fa ( f ) = A 2Tssinc2 ( fTs )
24
1
FT
j 2fct
G f fc G* f fc See
S
f
s t Re g t e
Tutorial 1
1
See Ch.4 of Digital Communications
g f f c *g f f c
4
by Proakis for the proof
1
1
A2Ts sinc2 f f c Ts A2Ts sinc2 f f c Ts
4
4
s f
Null-to-null bandwidth
25
g t ak jbk pt kTs
k
ak , bk A,3 A
E ak jbk 2 ,
n 0 10 A2 , n 0
n0
E ak jbk ak n jbk n , n 0
0,
10 2
10
A Ts sinc2 f f c Ts A2Ts sinc2 f f c Ts
4
4
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t
pt A sin , 0 t T
T
AT jfT
1
1
P f
e
sinc fT - sinc fT
2
Full-Cosine Pulse
pt
A
t
1
cos
0 t T
2
T
P f
AT jfT
2sinc fT sinc fT 1 sinc fT 1
e
4
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ln 2 f 2
A jfT ln 2
P f e
exp
2 B
T
2
pt
2 2
2
t T
1 4 t T
0 1
: Roll-off factor
T
,
0
2T
1
T
1 1
T
P f 1 cos f
,
2
T
2T
2T
1
0
,
f
2T
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Square
Full-cosine
Half-sinusoid
Gaussian
29
Square
BW=2/T
Half-sinusoid
BW=3/T
Full-cosine
BW=4/T
2/T
3/T
4/T
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0 1
: Roll-off factor
BW
1
T
1
2
BW
T
T
1/T
2/T
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r t
nt
r
Demodulator
Detector
sm
r t sm t nt r s m n
32
Correlation-type demodulator
Matched-filter demodulator
33
m1, 2...M
max
m1, 2...M
pr s m ps m
pr
max pr s m ps m
m1, 2...M
max pr s m
Bayes Theorem
p r : Common for all
ps m 1 M, i.e. Equally probable
messages
m1, 2...M
The conditional pdf pr s m is called the likelihood function and the decision
criterion based on the maximization of pr s m over the M signals is called the
maximum likelihood (ML) criterion.
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For an AWGN channel, the components of the noise vector n are zeromean Gaussian random variables with variance N0/2
nk2
f nk
exp 2
2
2
2 2 N0
1
nk2
1
exp
N 0
N0
k 1,2...N
k 1
k 1
1
1
2
rk sm,k
exp
k 1 N 0
N0
N 0 N
1 N
2
exp
rk sm,k
2
N 0 k 1
35
m k 1
Distance metrics
min r 2 r s m s m
m
max r s m
Correlation metrics
36
s2 t s1 t
s2 t A cos2f ct 2E T cos2f ct
Unlike other M-PSK for M>2, we can represent this special form of BPSK
signal as 1-dimensional signal. The basis function is given as
2 T cos2f ct , 0 t T
1 t
0,
otherwise
.dt
1 t
Euclidean
Distance
Decoder
37
r t 1 t dt s1 t wt 1 t dt E n
where
def T
n wt 1 t dt ~ N 0, N 0 2
0
r t 1 t dt E n
38
b0 E
b 1 E
r 0
1
Here we have two possible alternatives, therefore we can use a zero threshold
detector as an optimal detector.
Let P(e) denote the error probability
Pe P b 0, b 1 P b 1, b 0
P b 0 b 1 Pb 1 P b 1b 0 Pb 0
Bayes Theorem
Pb 0 Pb 1 1 / 2
P b 0 b 1 P b 1b 0
Pe P b 0 b 1
P b 0 b 1 Pr 0 b 1 f r b 1dr
0
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P b 0 b 1 f r b 1dr
0
2
1 r E
dr
exp
N 0 0
N 0
y2
1
exp dy
2 2 E N0
2
r E
N0 2
2E
Q
N0
1 y2 2
where Q-function is defined as Q x
dy
e
2 x
40
01 s2 t A cos2f c t 2
10 s3 t A cos2f ct 3 2
11 s4 t A cos2f ct 2
2 Ts cos2f ct , 0 t Ts
1 t
0,
otherwise
2 Ts sin 2f ct , 0 t Ts
2 t
0,
otherwise
.dt
r t
1 t
Detector
s min r sm
m1, 2,3, 4
.dt
2 t
41
Decision
regions
First, we calculate P(c), i.e. the probability of making a correct decision. Then,
probability of error is simply found as P(e)=1-P(c).
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d 2 Es
d 2
def T
nI wt 1 t dt ~ N 0, N 0 2
0
def T
nQ wt 2 t dt ~ N 0, N 0 2
d 2
Pc s1 P nI d 2, nQ d 2
P nI d 2 P nQ d 2
Es
Q
N0
Es
1 Q
N 0
P n Q
n ~ N , 2
2
Q 1 Q
43
Pc Pc s1 Pc s2 Pc s3 Pc s4
Pe 1 Pc 2Q
2Q
Es Es
Q
N 0 N 0
Es 2 Eb
2 Eb 2 Eb
Q
N 0 N 0
44
1
1 s2 t A cos 2 f c t
T
.dt
r t
1 t
Detector
s min r sm
m1, 2,3, 4
.dt
2 t
2 Ts cos2 f c 1 2T t , 0 t Ts
1 t
otherwise
0,
2 Ts cos2 f c 1 T t , 0 t Ts
2 t
otherwise
0,
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d 2 Es
Es
Es
Es
Pe Q
N0
1 t
Es nI , nQ
max r s m
m
Es
Pe s1 Pr s 2 r s1 PnQ nI Es Pn Es Q
N0
nQ , nI ~ N 0, N0 2
def
n nQ nI ~ N 0, N 0
Es
Due to symmetry, Pe Pe s1 Q
N0
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Pe s m P d l d m s m
l 1
l m
Pd l d m s m
M
l 1
l l
Ps m s l s m
M
Pe s m : Probability of making a
decision error when sm was sent
Union Bound (U-B)
P Ai P Ai
i i
Ps m s l : The probability of choosing sl
instead of the originally transmitted sm
l 1
l l
47
l m
l m
where d l ,m s l s m
N
2 0
2
d l ,m
Q
l 1 2 N 0
l m
U B M
UB B M
Q x e
x2 / 2
dl2,m
4 N0
l 1
l m
1 M
1 M M d l ,m
Pe
Pe s m
Q
M m1
M m1 l 1 2 N 0
l m
48
Pe s m Q l ,m M 1Q min
2 N0
l 1 2 N 0
l m
U B M
Minimum Euclidean
distance bound
1 M
1 M M dl ,m M 1Q d min
Pe
Pe s m
Q
2N
0
M m1
M m1 l 1 2 N 0
l m
P(e) is dominated by the minimum Euclidean distance of the signal
constellation.
49
d l ,m ~
d min
Q
N dmin,m Q
2 N0
l 1 2 N 0
l m
U B M
Pe s m
d min
1 M
1 M
~
Pe
N dmin,m Q
P e s m
M m1
M m1
2 N0
d min
N dmin Q
2 N0
N dmin
1 M
N dmin ,m
M m1
50
Es
2
d min
4 Es sin 2 4 Eb log2 M sin 2
M
M
N dmin
1, M 2
2, M 2
2
Replacing d min
and N dmin into the formula on p.50, we obtain
2 Eb
,
M 2
Q
N
0
Pe
2Q 2 Eb log M sin 2 , M 2
2
N
M
0
51
52
s1
s2
s3
s4
s5
s6
s7
s8
s9
s10
s11
s12
s13
s14
s15
d min
d min 2 A Ts 2
2
Esavg
5
Es 4 Eb
1 M
2
Esm 5 A Ts
M m1
8
Ebavg
5
53
2 neighbours
s0 , s3 , s12 , s15
4 neighbours
s0
s1
s2
s3
2 neighbours
s4
s5
s6
s7
3 neighbours
s8
s9
s10
s11
s12
s13
s14
s15
3 neighbours
s1 , s2 , s4 , s8 , s13 , s14 , s7 , s11
4 neighbours
s5 , s6 , s9 , s10
N dmin
1 M
N dmin, m 3
M m1
54
Power efficiency decreases with increasing M, but not early as fast as M-PSK.
55
f m m 2T , m 1,2...M
0 t Ts
d min 2 Es
Es
Eb log2 M
Pe M 1Q
M 1Q
N0
N0
56
Pe Q
N0
BFSK
Eb
Pe Q
N0
57
58
dk
bk
+1
-1
-1
-1
+1
-1
+1
-1
+1
-1
-1
+1
ak
+1
59
Mapper
bk 1
Differential
Encoder
a k 1
s DPSK t
A cosc t
t kT
2 cos c t
yQ t
2 sin c t
dt
yk
1 Symbol
Delay
zk
sgn Re
.
y k* 1
60
k 1T t kT
yt yI t jyQ t
ak Ae j N t
y k yt dt Ae j ak dt N t dt ATe j ak N k
T
N k ~ N 0,2 N 0T
def
def
2 2
Pe Pbk 1bk 1 PReyk yk*1 0 bk 1 P1 2 bk 1
z k sgn Re y k y k* 1 1 2
61
def
1 y k y k 1 2 2 y k y k 1 2
1 ATe j ak ak 1 / 2 N k N k 1 2
2 ATe j ak ak 1 / 2 N k N k 1 2
E1 ATe j ak ak 1 / 2
E2 ATe j ak ak 1 / 2
2
1
Var1 E 1 E1 E N k N k 1 N k N k 1 * N 0T
4
2
1
Var2 E 2 E2 E N k N k 1 N k N k 1 * N 0T
4
62
ak 1
bk
ak ak 1 2
ak ak 1 2
+1
+1
+1
+1
-1
+1
-1
-1
-1
-1
+1
-1
+1
-1
-1
1R , 1I ~ N 0, 2
Var 1 N 0T
1 1R j1I
Complex Gaussian
E2 ATe j
2
2 R ~ N AT cos , 2 2 I ~ N AT sin ,
2 2 R j2 I Complex Gaussian
Var 2 N 0T
where 2 N 0T 2
1 : Rayleigh, 2 : Rician
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1
1
f 1 2 exp 2
2
2
1
2
f 2 2 exp 2 2 2
2
2
I 0 2
Pe P1 2 P1 2 f 2 d
0
1
2
1
P 1 2
exp 2 d 1
exp u du
2 2 2
exp 2 2 2
def
1 : Rayleigh
2 : Rician
1
2 2
64
def
2
2
x x
1 x
x 2
2 exp
I
dx
2 0
2
2 0
2
def
m 2
m2 x
x 2 m 2 mx
1
I
exp 2 2 exp
dx
AT
2 0 2
2
2
2 0
A T
1
exp
2
2N0
=1
E
1
exp
2
N0
2 N 0T 2
E s 2 t dt A2T 2
T
65
10
Coherent
Differential
-1
10
-2
BER
10
-3
10
-4
10
-5
10
-6
10
6
SNR [dB]
10
12
66
g t ak p t kTs
k
hT t
hC t
hR t
Detector
wt
Actual Channel
Equivalent Channel
67
ISI terms
1, n 0
heq nTs
0, otherwise
l
H eq f constant
Ts
l
68
1
2W
l
H eq f
Ts
l
For this case, there is no choice for Heq to satisfy Nyquist criterion.
Ts
1
2W
l
H eq f
Ts
l
Ts , f W
H eq f
0, otherwise
69
Ts
1
2W
l
H eq f
Ts
l
A particular pulse shape which satisfies the above property and has been widely
used in practical applications is raised cosine. (See page 28) The Nyquist
pulse takes zero at the sampling points for adjacent signalling intervals.
X f l Ts cons.
l
70
HT f H R f
Under the ideal channel assumption , i.e. H C f 1
HT f H R f
H eq f
Ts ,
Ts
Ts
1
f
,
H RRC f H RC f cos
2Ts
2
0,
1
2Ts
1
1
f
2Ts
2Ts
f
1
2Ts
71
g t ak p t kTs
k
ak 1
Maximum instantaneous
power=(1.6)2=4.1 [dB]
Dynamic range=4.1 [dB]
Average power=1=0 [dB]
Peak power Avg. power 4.1dB
For this example, we observe large dynamic range of instantaneous power
and large peak/average ratio. These make the design of TX power amplifier
difficult.
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ak , bk 1
ak pt kTs bk pt kTs
k
k
ak pt kTs bk pt kTs Ts 2
k
k
Both terms can not pass through zero simultaneously, hence significantly
increasing the minimum instantaneous power and reducing dynamic range of the
signal. PSD and BER remain unchanged. This is known as Offset QPSK (OQPSK).
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ak jbk
e j 2 ,e j
ak jbk
e j 4 ,e j3 4
ak jbk
e j 4 , e j3 4 , for even k
e j 2 , e j , for odd k
74
75
Continuous FSK
We can get perfect temporal properties by using continuous FSK (CFSK)
g t ak pt kTs
ak 1
where
def t
qt p kTs d
h: Modulation index
76
n 1
h ak 2h
1/2Ts
k 0
t nTs
an
2Ts
3h
t
Ts
2h
h
0
-h
-2h
-3h
1/2
t
Ts
Phase Tree
+1
+1
n=0,1,..
-1
+1
-1
-1
Ts
2Ts
3Ts
77
t; a 2h ak 2h
k 0
t nTs
ha
an 2 n t n hnan
2Ts
2Ts
nTs t n 1Ts
han
t n hnan
st cos2f ct t ; a cos2 f c
2Ts
h
t n hn
st cos2 f c
2Ts
h
t n hn
an 1 st cos2 f c
2Ts
an 1
78
even k
t
,0 t 2Ts
cos
pt 2Ts
0,
otherwise
The transmission rate on the two orthogonal carriers is 1/2Ts bits/sec so that
the combined transmission rate is 1/Ts bits/sec.
79
Continuous phase is assured in MSK while 90 and 180 phase changes are
observable for OQPSK and QPSK respectively.
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81
Gaussian MSK
The spectral efficiency of MSK can be further improved by prefiltering.
g t ak pt kTs
k
Gaussian
LPF
MSK
Modulator
H f exp
B exp
ln 2
ln 2
B 2
82
T
T
2BT
ln2
BT: Normalized
3dB-Bandwidth
Phase pulse
x
1 1
qt 1
exp
2
2
x1
t T / 2
T
x2
x2
xQ x
x1
t T /2
T
83
For BT , the pulse shape takes its original unfiltered form , i.e. rectangle
pulse. GMSKMSK
The frequency pulse has a duration of 2Ts although signaling rate is 1/Ts. Such a
LPF will result in intersymbol interference which requires sequence estimation for
optimal detection.
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85