digital communication - vector approach dr. uri mahlab 1 digital communication vector space concept
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Digital communication - vector approach1Dr. Uri Mahlab
DigitalCommunicationVector Space
concept
Digital communication - vector approach2Dr. Uri Mahlab
Signal space Signal Space Inner Product Norm Orthogonality Equal Energy Signals Distance Orthonormal Basis Vector Representation Signal Space Summary
Digital communication - vector approach3Dr. Uri Mahlab
Signal Space
S(t) S=(s1,s2,…)
•Inner Product (Correlation)•Norm (Energy)•Orthogonality•Distance (Euclidean Distance)•Orthogonal Basis
Digital communication - vector approach4Dr. Uri Mahlab
Energy dttsEsT
)(0
2
ONLY CONSIDER SIGNALS, s(t)
Tt
tifts
0
0)( Tt
Digital communication - vector approach5Dr. Uri Mahlab
T
dttytxtytx0
)()()(),(
Similar to Vector Dot Product
x
yyx
cosyxyx
Inner Product - (x(t), y(t))
Digital communication - vector approach6Dr. Uri Mahlab
A
-A2A
A/2
T
Tt
t
Example
TAT
AATA
Atytx 2
4
3
2)2)((
2)
2)(()(),(
Digital communication - vector approach7Dr. Uri Mahlab
Norm - ||x(t)||
T
ExEnergydttxtxtxtx0
22)()(),()(
Extx )(
Similar to norm of vector
T
A
-A
x
xxx 2
ExT
AdttT
AtxT
2)
2cos()(
0
2
Digital communication - vector approach8Dr. Uri Mahlab
Orthogonality
0)(),( tytx T
dttytx0
0)()(
Similar to orthogonal vectors
T
A
-Ax
0yx
T
Y(t)B
y
Digital communication - vector approach9Dr. Uri Mahlab
•ORTHONORMAL FUNCTIONS
{
1)()(
0)(),(
tytx
and
tytx
TT
T
dttydttx
dttytx
0
2
0
2
0
1)()(
0)()(
T
T
X(t)
Y(t)
T/2
T/2
1
1
x
y
1)()(
0)(),(
tytx
tytx
Digital communication - vector approach10Dr. Uri Mahlab
Correlation Coefficient
EyEx
dttytx
tytx
tytx
T
0
)()(
)()(
)(),(
•In vector presentation
1 -1=1 when x(t)=ky(t) (k>0)
yx
yx cos
x
y
Digital communication - vector approach11Dr. Uri Mahlab
Example
T
TAdttytxtytx0
2
4
5)()()(),(
Now,
14.0
)87)(10(
45)(),(
2
TATA
TA
EyEx
tytx
shows the “real” correlation
t tA
-AT/2 7T/8
T
10A
X(t) Y(t)
Digital communication - vector approach12Dr. Uri Mahlab
Distance, d
ExEy2EyExd
dt)t(y)t(x)t(y)t(xd
2
T
0
222
• For equal energy signals
)1(E2d2 • =-1 (antipodal) E2d
• 3dB “better” then orthogonal signals
• =0 (orthogonal) E2d
Digital communication - vector approach13Dr. Uri Mahlab
Equal Energy Signals)1(2 Ed
E2d
E
y
x
•PSK (phase Shift Keying)
tfAty
Tt
tfAtx
0
0
2cos)(
)0(
2cos)(
•To maximize d
)()(
1
tytx (antipodal signals)
E2d
Digital communication - vector approach14Dr. Uri Mahlab
•EQUAL ENERGY SIGNALS ORTHOGONAL SIGNALS (=0)
Ed 2E
y
x
Ed 2
PSK (Orthogonal Phase Shift Keying)
tfAty
Tt
tfAtx
0
1
2cos)(
)0(
2cos)(
(Orthogonal if ...),2
3,1,
2
1)( 01 Tff
Digital communication - vector approach15Dr. Uri Mahlab
Signal Space summary• Inner Product
T
dttytxtytx0
)()()(),(
•Norm ||x(t)||
EnergydttxtxtxtxT
0
22)()(),()(
•Orthogonality
)(1)()(
0)(),(
Orthogonaltytx
if
tytx
Digital communication - vector approach16Dr. Uri Mahlab
• Corrolation Coefficient,
ExEy
dttytx
tytx
tytx
T
0
)()(
)()(
)(),(
•Distance, d
ExEy2EyExd
dt)t(y)t(x)t(y)t(xd
2
T
0
222
Digital communication - vector approach17Dr. Uri Mahlab
Modulation
Digital communication - vector approach18Dr. Uri Mahlab
Modulation Modulation
BPSK
QPSK
MPSK
QAM
Orthogonal FSK
Orthogonal MFSK
Noise
Probability of Error
Digital communication - vector approach19Dr. Uri Mahlab
)()(
)()(
so,
2cos2
)(
define We
Tt0
2cos2
)(
2cos2
)(
11
10
01
01
00
tEtx
tEtx
tfT
t
tfT
Etx
tfT
Etx
EE-)(t
Ed 2
sec
1 bit
TRbit
Binary Phase Shift Keying – (BPSK)
Digital communication - vector approach20Dr. Uri Mahlab
Binary antipodal signals vector presentationConsider the two signals:
Tt0 tf2cosT
E2)t(s)t(s c21
The equivalent low pass waveforms are:
Tt0 T
E2)t(u)t(u 21
Digital communication - vector approach21Dr. Uri Mahlab
The vector representation is – Signal constellation.
E E
Digital communication - vector approach22Dr. Uri Mahlab
The cross-correlation coefficient is:
1ss
ss)Re(
21
2112
The Euclidean distance is:
E2)Re(1E2d 2/11212
Two signals with cross-correlation coefficient of -1 are called antipodal
Digital communication - vector approach23Dr. Uri Mahlab
Multiphase signals
Consider the M-ary PSK signals:
tf2sin)1m(M
2sin
T
E2tf2cos)1m(
M
2cos
T
E2
Tt0 , M1,2,...,m )1m(M
2tf2cos
T
E2)t(s
cc
cm
The equivalent low pass waveforms are:
Tt0 , M1,2,...,m eT
E2)t(u M/)1m(2j
m
Digital communication - vector approach24Dr. Uri Mahlab
The vector representation is:
M1,2,...,m )1m(M
2sinE),1m(
M
2cosEsm
Or in complex-valued form as:
M/)1m(2jm eE2u
E
1s
4s
3s
2s
E
1s
4s
5s
3s
2s
8s
7s6s
4M 8M
Digital communication - vector approach25Dr. Uri Mahlab
Their complex-valued correlation coefficients are :
k)/M-(mj2
T
0
m*kkm
e
M1,2,...,m , M 1,2,...,k dt)t(u)t(uE2
1
and the real-valued cross-correlation coefficients are:
)km(M
2cos)Re( km
The Euclidean distance between pairs of signals is:
2/1
2/1kmkm
)km(M
2cos1E2
)Re(1E2d
Digital communication - vector approach26Dr. Uri Mahlab
The minimum distance dmin corresponds to the case which| m-k |=1
M
2cos1E2dmin
Digital communication - vector approach27Dr. Uri Mahlab
)(1 tx)(4 tx
)(3 tx )(2 tx
sE(00)
(01)(11)
(10)
sEd 2min
*
Quaternary PSK - QPSK
Digital communication - vector approach28Dr. Uri Mahlab
tf2sinT
E2)t( 02
X(t)
E
1a
2a
tf2cosT
E2)t( 01
sinT/E2a 2
T
EA
2
)tf2cos(T
E2)t(x 0
cosT/E2a1
Digital communication - vector approach29Dr. Uri Mahlab
)t(a)t(a
tf2sinsinT
E2tf2coscos
T
E2)t(x
Tt0 A0
)tf2cos(A)t(x
2211
00
0
)(1 tx)(4 tx
)(3 tx )(2 tx
sE(00)
(01)(11)
(10)
sEd 2min mind
Digital communication - vector approach30Dr. Uri Mahlab
2
E
4log
E
Mlog
EE
2/RR
sec
bits
T
1R
E2d
)4
3tf2cos(
T
E2)t(x
s
2
s
2
sb
bitsymbol
bbit
smin
04
Digital communication - vector approach31Dr. Uri Mahlab
mind
)(1 t
)(2 t
MEd
sin2
MPSK
Digital communication - vector approach32Dr. Uri Mahlab
3
E
8log
EE
8M
g.esec
bits
T
1)M(logR
s
2
sb
2bit
Digital communication - vector approach33Dr. Uri Mahlab
Consider the M-ary PAM signals
]e)t(uRe[A
tf2cosT
2A)t(s
tf2jm
cmm
c
m=1,2,….,M
Where this signal amplitude takes the discrete values (levels)
mA
M1m2Am m=1,2,….,M
The signal pulse u(t) , as defined is rectangular
U(t)= T
2Tt0
But other pulse shapes may be used to obtain a narrower signal spectrum .
Multi-amplitude Signal
Digital communication - vector approach34Dr. Uri Mahlab
Clearly , this signals are one dimensional (N=1) and , hence, are represented by the scalar components
m1m As M=1,2,….,M
The distance between any pair of signal is
|AA|)ss(d km
2
1k1mmk
2
0
M=2
M=4
)t(f1
1s
3s1s
2s
2s 4s
222
)t(f1
0
Signal-space diagram for M-ary PAM signals .
Digital communication - vector approach35Dr. Uri Mahlab
The minimum distance between a pair signals
2dmin
Digital communication - vector approach36Dr. Uri Mahlab
tf2sinT
2Atf2cos
T
2A)t(s cmscmcm
]e)t(u)jAARe[( tf2jmsmc
c
Where and are the information bearing signal amplitudes of the quadrature carriers and u(t)= .
mcA msA
T
2 Tt0
A quadrature amplitude-modulated (QAM) signal or a quadrature-amplitude-shift-keying (QASK) is represented as
Multi-Amplitude MultiPhase signalsQAM Signals
Digital communication - vector approach37Dr. Uri Mahlab
QAM signals are two dimensional signals and, hence, they are represented by the vectors
)AA(s ms,mcm The distance between a pair of signal vectors is
2kmmk |ss|d
])()[( 22ksmskcmc AAAA k,m=1,2,…,M
When the signal amplitudes take the discrete values
M,....,2,1m,M1m2 In this case the minimum distance is 2dmin
Digital communication - vector approach38Dr. Uri Mahlab
QAM (Quadrature Amplitude Modulation)
)(2 t
)(1 td
Digital communication - vector approach39Dr. Uri Mahlab
Mlog/TT
E5
2d
d2
5E
2
d3
2
d
2
1
2
d32
4
1
2
d2
4
1E
QAM16
2bitsysmbol
AVG
2AVG
2222
AVG
QAM=QASK=AM-PM
)(2 t
)(1 td
Digital communication - vector approach40Dr. Uri Mahlab
M=256M=128M=64M=32M=16M=4
+
Digital communication - vector approach41Dr. Uri Mahlab
For an M - ary QAM Square Constellation
22
S
Sn2
n
2S
d12
1ME
signal lDimentiona - One aFor
E12
6d
lbits/symbon 2M
d6
1ME
AVG
AVG
In general for large M - adding one bit requires 6dB more energy to maintain same d .
Digital communication - vector approach42Dr. Uri Mahlab
Binary orthogonal signals
Tt0 tf2sinT
E2)t(s
Tt0 tf2cosT
E2)t(s
c2
c1
Consider the two signals
Where either fc=1/T or fc>>1/T, so that
T
0
2112 dt)t(s)t(sE
1)Re(
Since Re(p12)=0, the two signals are orthogonal.
Digital communication - vector approach43Dr. Uri Mahlab
The equivalent lowpass waveforms:
Tt0 T
E2j)t(u
Tt0 T
E2)t(u
2
1
The vector presentation:
E,0s 0,Es 22
Which correspond to the signal space diagram
E
E
12d
1s
2s
Note that
E2d12
Digital communication - vector approach44Dr. Uri Mahlab
We observe that the vector representation for the equivalent lowpass signals is
]u[u
]u[u
212
111
Where
E2j0u
0jE2u
21
11
Digital communication - vector approach45Dr. Uri Mahlab
]ftm2tf2cos[T
2)t(s cm
]e)t(uRe[ tf2jm
c m=1,2,….,M Tt0
This waveform are characterized as having equal energy and cross-correlation coefficients
dte2T2
ftm2jkm
f)km(Tjef)km(T
f)km(Tsin
Let us consider the set of M FSK signals
M-ary Orthogonal Signal
Digital communication - vector approach46Dr. Uri Mahlab
r
0
T2
1
T
1
T2
3T
2
The real part of iskm
f)km(Tcosf)km(T
f)km(Tsin)Re( kmr
f)km(T2
f)km(T2sin
f
Digital communication - vector approach47Dr. Uri Mahlab
First, we observe that =0 when and . Since |m-k|=1 corresponds to adjacent frequency slots , represent the minimum frequency separation between adjacent signals for orthogonality of the M signals.
)Re( kmT2
1f
km
T2
1f
Digital communication - vector approach48Dr. Uri Mahlab
2
1s
3s
2s
2
2
Orthogonal signals for M=N=3signal space diagram
For the case in which ,the FSK signalsare equivalent to the N-dimensional vectors
1s
2s
=( ,0,0,…,0)
=(0, ,0,…,0)
Ns =(0,0,…,0, )
Where N=M. The distance between pairs of signals is
2dkmall m,k
Which is also the minimum distance.
T2/1f
Digital communication - vector approach51Dr. Uri Mahlab
Orthogonal FSK(Orthogonal Frequency Shift Keying)
,...2
3,1,
2
1)f(f
Tt0 2cos
2)(
2cos2
)(
01
11
00
T
tfT
Etx
tfT
Etx
T
tfT
tfT
tfT
tfT
010
10
02cos2
2cos2
02cos2
,2cos2
Digital communication - vector approach52Dr. Uri Mahlab
Ed 2
)(1 t
)(2 t
“0”
“1”
tfT
t
tfT
t
12
01
2cos2
)(
2cos2
)(
sec
1 bits
TRbit
Digital communication - vector approach53Dr. Uri Mahlab
ORTHOGONAL MFSK
2cos2
)(
2cos2
)(
2cos2
)(
33
22
11
tft
Etx
tft
Etx
tft
Etx
Digital communication - vector approach54Dr. Uri Mahlab
All signals are orthogonal to each other
)(1 t
)(2 t
)(3 t
Ed 2
E
E
E
sec
1)(log2
bits
TMRbit
Digital communication - vector approach55Dr. Uri Mahlab
How togeneratesignals
Digital communication - vector approach56Dr. Uri Mahlab
0 T 2T 3T 4T 5T 6T
0 T 2T 3T 4T 5T 6T
+
tfEb 02cos2
tfEb 02sin2
tf2sinT
2Atf2cos
T
2A)t(s cmscmcm
Digital communication - vector approach57Dr. Uri Mahlab
0 T 2T 3T 4T 5T 6T
0 T 2T 3T 4T 5T 6T
+
tfEb 02cos2
tfEb 02sin2
tf2sin)t(Qtf2cos)t(I)t(s ccm
)t(sm
Digital communication - vector approach58Dr. Uri Mahlab
0 T 2T 3T 4T 5T 6T
0 T 2T 3T 4T 5T 6T
+
tfEb 02cos2
tfEb 02sin2
tf2sin)t(Qtf2cos)t(I)t(s ccm
)t(sm
)t(I
)t(Q
Digital communication - vector approach59Dr. Uri Mahlab
+
tfEb 02sin2
)t(sm
)t(I
)t(Q
tfEb 02cos2
IQ Modulator
Digital communication - vector approach60Dr. Uri Mahlab
+
tfEb 02sin2
)t(sm
)t(I
)t(Q
tfEb 02cos2
IQ ModulatorPulse shaping filter
Digital communication - vector approach61Dr. Uri Mahlab
NOISE
Digital communication - vector approach62Dr. Uri Mahlab
What about Noise•White Gaussian Noise
T T
)(1 tn )(2 tn
1i
ii1 )t(a)t(n
1i
ii2 )t(a)t(n
•The coefficients are random variables !
Digital communication - vector approach63Dr. Uri Mahlab
WHITE GAUSSIAN NOISE (WGN)
Hz
Watts
2
0N)f(pn
We write
)t(n)t(n i1i
i
•All are gaussian variables•All are independent
in
in
)n(f
)...n(f)n(f,...)n,n(f)n(f
i1i
2121
Digital communication - vector approach64Dr. Uri Mahlab
•All have same probability distribution
e
2
N2
1)f(n
2
NnE
0nE
0
2i
N
n
0
i02
i
i
Digital communication - vector approach65Dr. Uri Mahlab
)t(n
1n
3n
2n
•White Gaussian Noise has energy in every dimension
0
2i
N
n
01i
i1i
e
2N
2
1)n(f)n(f
Digital communication - vector approach66Dr. Uri Mahlab
Probability of Error for Binary
SignalingThe two signal waveforms are given as
These waveforms are assumed to have equal energy E and their equivalent lowpass um(t), m=1,2 are characterized by the complex-valued correlation coefficient ρ12 .
]e)t(uRe[)t(s tf2jmm
cTt0 1,2m
Digital communication - vector approach67Dr. Uri Mahlab
The optimum demodulator forms the decision variables
Or,equivalently
And decides in favor of the signal corresponding to the larger decision variable .
T
0
*mm dt)t(u)t(rReU
*m
jm ureRe)u(
1,2m
1,2m
Digital communication - vector approach68Dr. Uri Mahlab
Lets see that the two expressions yields the same probability of error .
Suppose the signal s1(t) is transmitted in the interval 0tT . The equivalent low-pass received signal is
Substituting it into Um expression obtain
Where Nm, m=1,2, represent the noise components in the decision variables,given by
r111 NE2)NE2Re(U
r2r22 NE2)NE2Re(U
)t(z)t(ue)t(r 1j Tt0
T
0
*m
jm dt)t(u)t(ZeN
Digital communication - vector approach69Dr. Uri Mahlab
And .
The probability of error is just the probability that the decision variable U2 exceeds the decision variable u1 . But
Lets define variable V as
N1r and N2r are gaussian, so N1r-N2r is also gaussian-distributed and, hence, V is gaussian-distributed with mean value
)NRe(N mmr
)0UU(P)0UU(P)UU(P 211212
r2r1r21 NN)1(E2UUV
)1(E2)v(Em rv
Digital communication - vector approach70Dr. Uri Mahlab
And variance
Where N0 is the power spectral density of z(t) .
The probability of error is now
r0
2r2r2r1
2r1
2r2r1
2v
1EN4
NENNE2NE
NNE
r0
2
02/)mv(
v
0
1N2
Eerfc
2
1
dve2
1
dv)v(p)0V(P
2v
2v
Digital communication - vector approach71Dr. Uri Mahlab
Where erfc(x) is the complementary error function, defined as
It can be easily shown that
x
t dte2
)x(erfc2
r
0
2
2 1N2
Eerfc
2
1p
Digital communication - vector approach72Dr. Uri Mahlab
Distance, d
ExEy2EyExd
dt)t(y)t(x)t(y)t(xd
2
T
0
222
• For equal energy signals
)1(E2d2 • =-1 (antipodal) E2d
• 3dB “better” then orthogonal signals
• =0 (orthogonal) E2d
Digital communication - vector approach73Dr. Uri Mahlab
It is interesting to note that the probability of error P2 is expressed as
Where d12 is the distance of the two signals . Hence,we observe that an increase in the distance between the two signals reduces the probability of error .
0
212
2
2 N2
derfc
2
1p
Digital communication - vector approach74Dr. Uri Mahlab
0
212
2
2 N2
derfc
2
1p
r
0
2
2 1N2
Eerfc
2
1p
Digital communication - vector approach75Dr. Uri Mahlab
0
212
2
2 N2
derfc
2
1p
M=256M=128M=64M=32M=16M=4
+
2
1s
3s
2s
2
2
mindM
Ed
sin2 E E
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