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Digital Communication Dr. Mahlab Uri1
Chapter 4
characterization of communication
signals and systems
Digital Communication Dr. Mahlab Uri
contents
4-1 Parts 1,2- Representation of Bandpass Signals4-1-1 Representation of Bandpass Signals
4-1-2 Representation of Linear Bandpass System
4-1-3 Response of a Bandpass Systems to a Bandpass signal
4-1-4 Representation of Bandpass Stationary Stochastic Processes
4-2 Parts 3 signal space representation4-2-1 Vector Space Concept
4-2-2 Signal Space Concept
4-2-3 Orthogonal Expansions of Signals
4-3 Parts 4,5,6 Representation of digitally modulated signals4-3-1 Memoryless Modulation Methods
4-3-2 Linear Modulation with Memory
4-3-3 Nonlinear Modulation Methods with Memory
Digital Communication Dr. Mahlab Uri
Part - 1
Digital Communication Dr. Mahlab Uri
4-1 Representation of Bandpass Signals
In the next subsections we will represent the
bandpass signals and systems in terms of equivalent
lowpass waveforms and the characterization of
band pass stationary stochastic processes
Digital Communication Dr. Mahlab Uri5
Representation of 1-1-4Bandpass Signals
Signals and channels that satisfy the condition that their bandwidth is much
smaller than the carrier frequency are termed narrowband bandpass signals
and channels.
Is a real value signal that shown in figure ts (4-1-1)
Figure (4-1-1)
Digital Communication Dr. Mahlab Uri6
a signal that contains only the positive frequencies in ts
fSfufS 2 (4-1-1) fS Is the Fourier transform of ts
fu Is the unit step function
The Equivalent time-domain expression is:
FSFfuF
dtefSts ftj
11
2
2
(4-1-2)
ts the analytic signal or pre-envelope of ts
Digital Communication Dr. Mahlab Uri7
t
jtfuF
21
tst
jts
tst
jtts
(4-1-3)
(4-1-4)
Digital Communication Dr. Mahlab Uri8
tst
jts
tst
jtts
d
t
sts
tts
11
th ts ts
tt
th ,1
Such a filter is called a Hilbert transformer
(4-1-5)
(4-1-4)
(4-1-6)
Digital Communication Dr. Mahlab Uri9
The frequency response of a Hilbert transformer
0
00
0
11 2
2
fj
f
fj
dtet
dtethfH
ft
ft
(4-1-7)
This filter is basically a phase shifter for all frequencies f
Digital Communication Dr. Mahlab Uri10
cl ffSfS
tfj
tfj
l
c
c
etsjts
etsts
2
2
1F
(4-1-8)
(4-1-9)
tfjl cetstsjts2
(4-1-10)
frequency translation
equivalent lowpass representation
Digital Communication Dr. Mahlab Uri11
tjytxtsl
tftytftxts cc 2sin2cos
tftytftxts cc 2cos2sin
The signal is a complex-value and may be expressed as: tsl
(4-1-11)
(4-1-12)
(4-1-13)
ty tx and are called the quadrature componentsof the bass band signal ts
Digital Communication Dr. Mahlab Uri12
t
cfj
etl
st
cfj
etjytxts 2
Re2
Re
tsl is usually called the complex envelope of the Real signal ts , and it basically the equivalentlowpass signal.
(4-1-14)
II- Another representation is:
Digital Communication Dr. Mahlab Uri13
tjl etats
tytxta 22
tytx
t 1tan
Finally a third possible representation is:
Where:
(4-1-15)
(4-1-16)
(4-1-17)
Digital Communication Dr. Mahlab Uri14
is called the envelop of ta ts t is called the phase of ts
ttftaeta
etsts
c
ttfj
tfj
l
c
c
2cos
Re
Re
2
2
Then:
(4-1-18)
Digital Communication Dr. Mahlab Uri15
Therefore:
tfjl cetsts 2Re
tftytftxts cc 2sin2cos
ttftats c 2cos
Are equivalent representation of bandpass signals
(4-1-12)
(4-1-14)
(4-1-18)
Digital Communication Dr. Mahlab Uri16
dtedtets
dtetsfS
ftjtfj
l
ftj
c
22
2
Re
The Fourier transform of ts is:
Using of the identity
2
1Re
(4-1-19)
(4-1-20)
Digital Communication Dr. Mahlab Uri17
clcl
ftjtfj
l
tfj
l
ffSffS
dteetsetsfS cc
2
1
2
1 222
The basic relationship between the spectrum of
the real bandpass signal and the spectrum of
the equivalent lowpass signal
fS fSl
(4-1-21)
Digital Communication Dr. Mahlab Uri18
dtets
dtts
tfj
lc
22
2
Re
The energy in the signal ts is defined as:
Using the identity we obtain:
dtttfts
dttsE
cl
l
24cos2
1
2
1
2
2
(4-1-22)
(4-1-23)
2
1Re
Digital Communication Dr. Mahlab Uri19
ta 2 varies slowly relative
the energy in bandpass signal expressed in terms of
the equivalent lowpass signal is:
dttsE l
2
2
1
Where tsl is just the envelop ta of ts
(4-1-24)
E - The energy in bandpass signal
ttftats c 2cos
Digital Communication Dr. Mahlab Uri20
Representation of Linear 2-1-4Bandpass System
In this section we will described a filter or
a system by its impulse response or by its
frequency response.
th
Digital Communication Dr. Mahlab Uri21
A linear filter or system may be described either
by its impulse response th or by its frequencyresponse fH
Since th is real
fHfH (4-1-25)
Digital Communication Dr. Mahlab Uri22
We define cl ffH as:
00
0
f
ffHffH cl
Then:
0
00
ffH
fffH cl
(4-1-27)
(4-1-26)
Digital Communication Dr. Mahlab Uri23
Using we have:
clcl ffHffHfH
(4-1-28)
tfjl
tfj
l
tfj
l
c
cc
eth
ethethth
2
22
Re2
(4-1-29)
is the inverse Fourier transform and its a complex value of thl fH l
fHfH
1F frequency translation
Digital Communication Dr. Mahlab Uri24
Response of a Bandpass 3-1-4Systems to a Bandpass signal
In this section we demonstrate that output of a bandpass
system to a bandpass input signal is simply obtained
from the equivalent lowpass input signal and the
equivalent lowpass impulse response of the system.
Digital Communication Dr. Mahlab Uri25
th
ts tr
tsl thl
ts Is a narrowband bandpass signal, with an equivalent lowpass signal
?trl
tsl
Is the impulse response of a narrowband bandpass system, with an
equivalent lowpass impulse response
th
thl
Digital Communication Dr. Mahlab Uri26
The output of the bandpass system is also a bandpass
signal, and it can be expressed as:
tfjl cetrtr 2Re(4-1-30)
And is related to the input signal and the impulse
response by the convolution integral:
dthstr
(4-1-31)
Digital Communication Dr. Mahlab Uri27
in the frequency domain expressed by: tr
fHfSfR (4-1-32)
Substituting from: fS
clcl
ftjtfj
l
tfj
l
ffSffS
dteetsetsfS cc
2
1
2
1 222
and from: fH clcl ffHffHfH
for
for
Digital Communication Dr. Mahlab Uri
We obtain the result:
clcl
clcl
ffHffH
ffSffSfR
2
1
(4-1-33)
0 cl ffS
0* clcl ffHffS
and
ts narrowband bandpass signal th impulse response of a narrowband system
0* clcl ffHffS
0 cl ffH for 0f
Digital Communication Dr. Mahlab Uri
Therefore, simplifies to :
clcl
clcl
clcl
ffRffR
ffHffS
ffHffSfR
2
1
2
1
(4-1-34)
clcl
clcl
ffHffH
ffSffSfR
2
1
Digital Communication Dr. Mahlab Uri30
The output spectrum of the equivalent lowpass
system exited by the equivalent lowpass signal:
fHfSfR lll (4-1-35)
The time domain relation is given by the
convolution integral:
dthstr lll
(4-1-36)
Digital Communication Dr. Mahlab Uri31
NOTE 1:The combination of
with gives the relationship between the
bandpass output signal and the equivalent lowpass time
functions and tsl thl
NOTE 2:This simple relationship allows as to ignore any linear
frequency translations encountered in the modulation of signal for
purposes of matching its spectral content to frequency allocation of a
particular channel.
Thus, for mathematical convenience, we shall deal
only with transmission of equivalent lowpass
signals through equivalent lowpass channels.
summary
dthstr lll
tfjl cetrtr 2Re tr
Digital Communication Dr. Mahlab Uri
Part - 2
Digital Communication Dr. Mahlab Uri33
Representation of Bandpass 4-1-4Stationary Stochastic Processes
In this section we extend the representation to sample
function of a bandpass stationary stochastic process.
Digital Communication Dr. Mahlab Uri34
tn Is a sample function definitions of wide-sense stationary stochastic process with zero mean and
power spectral density fnn
The stochastic process Is said to be a narrow
bandpass process if the width of the spectral
density is much smaller than the carrier frequency
tn
cf
definitions
fnn
Digital Communication Dr. Mahlab Uri
under this conditions can be represented
by 3 equivalent forms:
tn
ttftatn c 2cos 1
tfj cetztn 2Re 3
tftytftxtn cc 2sin2cos 2
(4-1-37)
(4-1-38)
(4-1-39)
ta The envelope of the real valued signal.
t The phase of the real valued signal. tx ty tn
The complex envelope of tn tz
The quadrature components of .
Digital Communication Dr. Mahlab Uri36
yyxx
If tn is zero mean then tx tyand must also havezero mean values. In addition the stationary of tnImplies that the autocorrelation and cross-correlation
function of tx tyand satisfy the followingproperties:
yxxy
(4-1-40)
(4-1-41)
Digital Communication Dr. Mahlab Uri37
The autocorrelation function nn of tn
tftytftxtftytftxEtntnE cccc 2sin2cos2sin2cos
(4-1-42)
tftf
tftf
tftf
tftf
ccyx
ccxy
ccyy
ccxx
2sin2cos
2cos2sin
2sin2sin
2cos2cos
tftytftxtn cc 2sin2cos
Digital Communication Dr. Mahlab Uri38
Use of the trigonometric identities:
(4-1-43)
BABABA
BABABA
BABABA
sinsin2
1cossin
coscos2
1sinsin
coscos2
1coscos
Digital Communication Dr. Mahlab Uri39
In (4-1-42) yields the result:
tf
fy
tf
f
tntnE
cxyyx
cxyyx
cyyxx
cyyxx
22cos2
1
2sin2
1
22cos2
1
2cos2
1
(4-1-44)
Digital Communication Dr. Mahlab Uri40
Must be independent of t .
cyxcxxnn ff 2sin2cos (4-1-45)
the right-hand side of:
yyxx yxxy
tn Is stationary.
yesno
?
tf
fy
tf
f
tntnE
cxyyx
cxyyx
cyyxx
cyyxx
22cos2
1
2sin2
1
22cos2
1
2cos2
1
Digital Communication Dr. Mahlab Uri41
The autocorrelation function of the equivalent
lowpass process:
tjytxtz (4-1-46)
Is defined as:
tztzEzz2
1
(4-1-47)
Digital Communication Dr. Mahlab Uri42
yxxyyyxxzz jj 2
1
Substituting into
we obtain:
(4-1-48)
If symmetry properties given in and in
are used in
we obtain:
yxxxzz j(4-1-49)
tjytxtz tztzEzz2
1
yxxy yxxyyyxxzz jj 21
yyxx
Digital Communication Dr. Mahlab Uri43
Finally, we incorporate the result given by
into
cfjzznn e 2Re(4-1-50)
Thus, the autocorrelation function nn of thebandpass stochastic process is uniquely determined
from the autocorrelation function zz of the equivalent lowpass process tz and the carrier frequency
cf .
yxxxzz j cyxcxxnn ff 2sin2cos
and we have:
Digital Communication Dr. Mahlab Uri44
czzczz
fjfj
zznn
ffff
deef c
2
1
Re 22
The power density spectrum of the stochastic process tn
(4-1-51)
fzz is the power density spectrum of the equivalent
lowpass process tz
Since the autocorrelation function of tz satisfies thethe property *zzzz , it follows that
fzz Is a real-valued function of frequency.
Digital Communication Dr. Mahlab Uri45
Properties of the Quadrature Components
The cross-correlation function of the quadrature
components of tn
Furthermore, any cross-correlation function
satisfies the condition:
xyyx(4-1-52)
yxxy
Digital Communication Dr. Mahlab Uri46
xyxy(4-1-53)
is an odd function of xy . 00 xy
tx tyand uncorrelated (for 0 only). If 0xy for all than zz Is real and the power spectral density fzz satisfies the condition
ff zzzz (4-1-54)And vice versa. That is, fzz Is symmetric about 0f
Digital Communication Dr. Mahlab Uri47
222
222
1,
yx
eyxp
in the special case in which the stationary stochastic
process tn is Gaussian the quadrature components tx tyand are jointly Gaussian.
Moreover for 0 They are statistically independent,
hence, their joint probability density function is:
(4-1-55)
The variance is defined as: 0002 nnyyxx
Digital Communication Dr. Mahlab Uri48
Representation of White Noise
NOTE: White noise is a stochastic process that is defined to have a flat (constant) power spectral density over the entire frequency range. This
type of noise cannot be expressed in terms of quadrature components, as
result of its wideband character.
The power spectral density of bandpass white noise
resulting from passing the white noise process through
a spectrally flat (ideal) bandpass filter
Figure 4-1-3
Digital Communication Dr. Mahlab Uri49
Bandpass white noise can be represented by :
tz
Bf
BfN
fzz
2
10
2
10
(4-1-56)
ttftatn c 2cos 1
tfj cetztn 2Re 3
tftytftxtn cc 2sin2cos 2
The equivalent lowpass noise has a power
spectral density:
Digital Communication Dr. Mahlab Uri50
BNzz
sin0
The autocorrelation function is:
(4-1-57)
The limiting form of zz As B approaches infinityIs
:
0Nzz (4-1-58)
Digital Communication Dr. Mahlab Uri51
yyxxzz
The power spectral density for white noise and
bandpass white noise is symmetric about 0fso 0xy for all Therefore:
(4-1-59)
That is, the quadrature components tx tyand are uncorrelated for all time shifts are the
autocorrelation functions of tz tx tyand , are allequal.
Digital Communication Dr. Mahlab Uri
Part - 3
Digital Communication Dr. Mahlab Uri53
4-2 signal space representation
In the next subsection we demonstrate that
signals are similar to vectors and we will
develop a vector representation for signal
waveform
Digital Communication Dr. Mahlab Uri54
Vector Space Concept1 -2-4
A vector with n-dimension
can be represent as a linear combination
of unit vectors or basis vectors
nvvv ......21
n
i
iievv1
Where a unit vector has length of unity and
Is the projection of vector v on ivie
(4-2-1)
Digital Communication Dr. Mahlab Uri55
The inner product of two vectors is defined as:
n
i
iievvv1
2121
The vectors are orthogonal if 021 vv
The norm of a vector (length) v is:
m
i
jvvvv1
2/1 2
For all : ji i1 mj
(4-2-2)
(4-2-4)
(4-2-3)
v
Digital Communication Dr. Mahlab Uri56
NOTE: A set of m vectors are defined as orthonormal if
the vectors are orthogonal and each one of them
are unit norm
NOTE: A set of m vectors are said to be linearly independent
if none of the vectors can be represented as a linear
combination of the remaining vectors
Digital Communication Dr. Mahlab Uri57
The triangle inequality of Two n-dimensional vectors
in the same direction:
2121 vvvv
2121 vvvv
If then we can say that the two
n-dimensional vectors are also satisfied the
Cauchy Schwartz inequality:
21 avv
(4-2-5)
(4-2-6)
Digital Communication Dr. Mahlab Uri58
The norm square (length) of the vectors:
212121 2222
vvvvvv
021 vvAnd if they orthogonal ( ) then:
2222121 vvvv
(4-2-7)
(4-2-8)
Digital Communication Dr. Mahlab Uri59
Let us take vectors:
nvvvv 1 12111 .....
1
1
1
v
vu
2v
Normalizing
subtracting the projection of onto
nvvvv 222212 .....
1u
11222 uuvvu
(4-2-11)
(4-2-12)
1v
nvvvv 232313 .....
Digital Communication Dr. Mahlab Uri60
Normalizing the vector to unit length:
2
2
2
u
uu
22311333 uuvuuvvu
3vSubtracting projection on and 2u1u
Normalized it and we will get the next
orthonormal vector :
3
3
3
u
uu
(4-2-13)
(4-2-14)
(4-2-15)
Digital Communication Dr. Mahlab Uri61
Signal Space Concept2-2-4
developing a parallel treatment to two generally
complex value signals and on some
interval of
tx2 tx1
ba,
Digital Communication Dr. Mahlab Uri62
The inner product of the complex signals is:
dttxtxtxtxb
a
2121 ,
If the inner product of the complex signals is
zero then the signals are orthogonal.
The norm of the signals is:
2/1
2
dttxtx
b
a
(4-2-16)
(4-2-17)
Digital Communication Dr. Mahlab Uri63
NOTE: As in the vector representation A set of m signals
are orthonormal if they are orthogonal and their
norms are all unity.
NOTE: A set of m signals is linearly independent, if no
signal can be represent as the linear combination of
the remaining signals.
Digital Communication Dr. Mahlab Uri64
The signals satisfy the triangle inequality:
txtxtxtx 2121
and the Cauchy-Schwartz inequality when
2/1
2
2/1
2
2 211
dttxdttxdttxtx
b
a
b
a
b
a
taxtx 12
(4-2-18)
(4-2-19)
Digital Communication Dr. Mahlab Uri65
Orthogonal Expansions of 3-2-4Signals
In this section we will develop a vector representation for
signal waveform and the equalities between a signal
waveform and its vector representation
Digital Communication Dr. Mahlab Uri66
deterministic, real-valued signal with finite energy: ts
dttss2
a set of orthonormal functions Nntfn ,......2,1,
nm
nmdttftf mn
1
0
(4-2-20)
(4-2-21)
Digital Communication Dr. Mahlab Uri67
The approximate signal by weighted linear
combination of:
ts
k
k
kk tfsts1
the coefficients ks
The error approximation:
Kksk 1,
tstste
(4-2-22)
(4-2-23)
Digital Communication Dr. Mahlab Uri68
we will like to select an to minimize the energy
of the approximate error
kse
dttfstsdttsts
k
k
kke
1
2
Two option to find the optimum of
1.differeniaing the series for each coefficient and setting the
firs derivative to zero
2.Multiply with a orthonormal function and base of the
mean-square-error criteria saying that the minimum will
obtain when the result will be zero .
ks
(4-2-24)
tfn
e
Digital Communication Dr. Mahlab Uri69
01
dttftfsts n
k
k
kk
Kn ,.....,2,1
dttftss nn
k
k
ks sdttsdttste1
22
min
(4-2-25)
(4-2-26)
(4-2-27)
Digital Communication Dr. Mahlab Uri70
if 0min
dttssk
k
tv
2
1
2
k
k
kk tfsts1
(4-2-28)
(4-2-29)
Digital Communication Dr. Mahlab Uri
Appendix
Signal space Signal Space
Inner Product
Norm
Orthogonality
Equal Energy Signals
Distance
Orthonormal Basis
Vector Representation
Signal Space Summary
71
Digital Communication Dr. Mahlab Uri
Energy dttsEsT
)(0
2
ONLY CONSIDER SIGNALS, s(t)
Tt
tifts
00)( T
t
72
Digital Communication Dr. Mahlab Uri
T
dttytxtytx0
)()()(),(
Similar to Vector Dot Product
x
yyx
cosyxyx
(x(t), y(t))-Inner Product
73
Digital Communication Dr. Mahlab Uri
A
-A2A
A/2
T
Tt
t
Example
TAT
AATA
Atytx 2
4
3
2)2)((
2)
2)(()(),(
74
Digital Communication Dr. Mahlab Uri
||x(t)||-Norm
T
ExEnergydttxtxtxtx0
22 )()(),()(
Extx )(
Similar to norm of vector
T
A
-A
x
xxx 2
ExT
AdttT
AtxT
2
)2
cos()(0
2
75
Digital Communication Dr. Mahlab Uri
Orthogonality
0)(),( tytx T
dttytx0
0)()(
Similar to orthogonal vectors
T
A
-Ax
0 yx
T
Y(t)B
y
76
Digital Communication Dr. Mahlab Uri
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
77
Digital Communication Dr. Mahlab Uri
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
78
Digital Communication Dr. Mahlab Uri
Example
T
TAdttytxtytx0
2
4
5)()()(),(
Now,
14.0
)8
7)(10(
45
)(),(2
TATA
TA
EyEx
tytx
shows the real correlation
t tA
-AT/2 7T/8
T
10A
X(t) Y(t)
79
Digital Communication Dr. Mahlab Uri
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
80
Digital Communication Dr. Mahlab Uri
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
81
Digital Communication Dr. Mahlab Uri
EQUAL ENERGY SIGNALS
ORTHOGONAL SIGNALS (=0)
Ed 2
E
y
x
Ed 2FSK (Frequency Shift Keying)
tfAty
Tt
tfAtx
0
1
2cos)(
)0(
2cos)(
(Orthogonal if ...),2
3,1,
2
1)(
01 Tff
82
Digital Communication Dr. Mahlab Uri
Signal Space summary Inner Product
T
dttytxtytx0
)()()(),(
Norm ||x(t)||
EnergydttxtytxtxT
0
22 )()(),()(
Orthogonality
)(1)()(
0)(),(
Orthogonaltytx
if
tytx
83
Digital Communication Dr. Mahlab Uri
Corrolation Coefficient,
ExEy
dttytx
tytx
tytx
T
0
)()(
)()(
)(),(
Distance, d
ExEy2EyExd
dt)t(y)t(x)t(y)t(xd
2
T
0
222
84
Digital Communication Dr. Mahlab Uri
Orthonormal Basis
Suppose we try to approximate x(t) by writing,
Xa(t) where
Suppose we have a function x(t) and we are
given a set of orthonormal functions, Nii
t,...,2,1
)(
N
iiia
txtatx1
)()()(
Question:
What are the best ai that we can select
such that Xa(t) is close to X(t) ?
85
Digital Communication Dr. Mahlab Uri
We want to minimize the distance between x(t)
and xa(t) .
N
iiia
tatxtxtxd1
)()()()(Distance
The best ai are
...
)()()(),(
2
11
22
min
0
N
N
ii
T
iii
aaExd
dtttxttxa
opt
opt
86
Digital Communication Dr. Mahlab Uri
In general is an othonormal basis in
if any function in can be written as
,...2,1
)(ii
t
T
iii
iii
dtttxttxa
where
Tttatx
0
1
)()()(),(
0)()(
2L
Sometimes called COMPLETE ORTHONORMAL SET (COS)
OTHONORMAL BASIS
2L
87
Digital Communication Dr. Mahlab Uri
A Set of Orthogonal functions - Nii t ,...2,1)(
iallfor
dttt
jiif
dttttt
T
i
T
jiji
i
0
22
0
1)()(
0)()()(),(
May be simply written as
ji
ji
if
iftt
ijji
1
0{)(),(
Is the Kronecker Deltaij
88
Digital Communication Dr. Mahlab Uri
EXAMPLE
Fourier Series
tT
22sin
T
2)t(
tT
22cos
T
2)t(
tT
2sin
T
2)t(
tT
2cos
T
2)t(
T1)t(
5
4
3
2
1
T
0
T
0
tdtT
2sin
T
2)t(x3a
tdtT
2cos
T
2)t(x2a
89
Digital Communication Dr. Mahlab Uri
. encecorrespond one-to-one a have
,...)a,a(a vector,
theand (t)function x The*
)t(x a
a)t(x
)t(),t(xa
)t(a)t(x
21
ii
1i
i
Space RepresentationSignal
90
Digital Communication Dr. Mahlab Uri
a1
a2
a3
a
The vector )3,2,1(a aaa
Represents one and only one function
(using a given basis) ,...2,1 ),( iti
3
1
)()(i
iitatx
Vector Representation
91
Digital Communication Dr. Mahlab Uri
)(1
t
)(3
t
)(2
t
X(t)
...)t(a )t(a )t(a x(t)
:as drepresente becan x(t) thusbasis theIs -
)t( and )t(),t( :use we
3a and 2a,1a,a using of Instead
332211
i
321
Vector as a Signal
92
Digital Communication Dr. Mahlab Uri
Ex)t(xa
Therefore,
a...aaaa
,...)a,a,a(a
a vector at thelook weifBut
a)t(x
)t(a,)t(aEx)t(x
)t(a)t(x
1i
2
i
2
3
2
2
2
1
2
321
1i
2
i
2
1j
jj
1i
ii
2
1i
ii
X(t) -
Norm (or Energy) In Signal Space
93
Digital Communication Dr. Mahlab Uri
. .
)(1
t
)(3
t
)(2
t
X(t)
21a
53a
32a
Ex
JoulesEx
Ex
Extx
38
38
532)( 222
94
Digital Communication Dr. Mahlab Uri
The Distance d, between x(t) and y(t) is
1
22
2
a
1
22
2
1
2
2
1 1
22
:is vectorsebetween th d distance the
b and a vectorsat thelook weifBut
)(
)()(
)()(
iiiba
b
iii
iiii
i iiiii
babad
bad
tbad
tbta
tytxd
Distance, d, In Signal Space
95
Digital Communication Dr. Mahlab Uri
The Distance between the functions equals the distance between the
vectors
X(t)
Y(t)
51a
61b
22b
43a
03b 82 ad
Y(t)=(2,6,0)
x(t)=(8,5,4)
53
)04()28()65()()( 222
d
tytxd
96
Digital Communication Dr. Mahlab Uri
bay(t)-x(t)Distance
ax(t)Energy
,...)a,a(a
a)t(x
Tt0
)t(),t(xa
)t(a)t(x
22
21
i
1i
ii
Signal Space Summary
97
Digital Communication Dr. Mahlab Uri98
Gram-Schmidt Procedure
NOTE: This procedure construct a set of orthonormal
vectors from a set of n-dimensional vectors by
normalize zing its length
98
Digital Communication Dr. Mahlab Uri
Part - 4
Digital Communication Dr. Mahlab Uri
4-3 Representation of digitally modulated signals
In the next subsections we will describe.
memoryless modulation methods.
Linear modulation with memory.
Nonlinear modulation methods with memory.
100
Digital Communication Dr. Mahlab Uri101
ASK
FSK
PSK
DSB
Digital Communication Dr. Mahlab Uri
Memoryless Modulation 1-3-4Methods
Pulse-amplitude-modulated (PAM) signals
2( ) Re ( ) ( ) 2cj f tm m m cS t A g t e A g t Cos f t
1,2,....., , 0 t T 4.3-1m M
The denote the set of M possible amplitudes
corresponding to possible k-bit blocks of symbols.
, 1 m MmA 2kM
2d - distance between adjacent signals amplitudes.
(2 1 ) , m=1,2,...,M 4.3-2mA m M d
(4-3-1)
(4-3-2)
102
Digital Communication Dr. Mahlab Uri
R/k - The symbol rate for the PAM signals
bT 1/ R bit interval
/ bT k R kT symbol interval
The M-PAM signals have energies:
2 2 2 20 0
1 1( ) ( ) 4.3-3
2 2
T T
m m m m gS t dt A g t dt A
g - the energy in the pulse g(t).
Digital PAM is also called amplitude-shift keying (ASK).
(4-3-3)
103
Digital Communication Dr. Mahlab Uri
The one-dimensional (N=1) signal:
The signal waveform
f(t) - the unit-energy
( ) ( ) 4.3-4m mS t S f t
2
( ) ( ) 2 4.3-5cg
f t g t Cos f t
1
, m=1,2,....,M 4.3-62
m m gS A
(4-3-4)
(4-3-5)
(4-3-6)
104
Digital Communication Dr. Mahlab Uri
The Euclidian distance between any pair of signal points:
The minimum Euclidean dist:
2(e)
mn
1d 2 4.3-7
2m n g m n gS S A A d m n
(e)mind 2 4.3-8gd
(4-3-7)
(4-3-8)
105
Digital Communication Dr. Mahlab Uri
Signal space diagram for digital PAM signals
0 1
00 01 11 10
000 001 011 010 110 111 101 100
(a) M=2
(b) M=4
(c) M=8
106
Digital Communication Dr. Mahlab Uri
Phase-modulated signalsThe M signals waveforms are represented as:
g(t) - the signal pulse shape
- the M possible phases of the carrier.
22 ( 1) /( ) Re ( ) , m=1,2,...,M, 0 t Tcj f tj m MmS t g t e e
2
( ) ( ) 2 ( 1) 4.3-11
2 2( ) ( 1) 2 ( ) ( 1) 2
m c
c c
S t g t Cos f t mM
g t Cos m Cos f t g t Sin m Sin f tM M
m
Phase-modulated signals is usually called phase-shift keying (PSK).
107
Digital Communication Dr. Mahlab Uri
The energy of the signal waveforms:
Represented as a linear combination of two orthonormal
signals:
The two-dimensional vectors are:
2 20 0
1 1( ) ( ) 4.3-12
2 2
T T
m gS t dt g t dt
1 1 2 2( ) ( ) ( ) 4.3-13m m mS t S f t S f t
1 22 2
( ) 2 ( ) 2 4.3-14,15c cg g
f t Cos f t f t Sin f t
2 2
( 1) ( 1) , m=1,2,...,M 4.3-162 2
g g
mS Cos m Sin mM M
(4-3-12)
(4-3-13)
(4-3-14,15)
(4-3-16)
108
Digital Communication Dr. Mahlab Uri
The Euclidean distance between signal point:
The minimum Euclidean distance: (|m-n|=1)
The preferred mapping or assignment of k information bits to the possible
phases is Gray encoding, so that the most likely errors caused by noise will result in
a single bit error in the k-bit symbol.
2kM
1/2
(e)
mn
2d 1 ( ) 4.3-17m n gS S Cos m n
M
(e)min2
d 1 4.3-18g CosM
(4-3-17)
(4-3-18)
109
Digital Communication Dr. Mahlab Uri
Signal space diagrams for PSK
0 1 11
01
00
10
011001
000
100
101
111
110
010
M=2
(BPSK)
M=4
(QPSK)
M=8
(Octal PSK)
110
Digital Communication Dr. Mahlab Uri
Quadrature Amplitude Modulation
quadrature PAM or QAM
The waveforms:
the information-bearing signal amplitudes
the signal pulse.
Alternative expression waveforms may be:
2( ) Re ( ) , m=1,2,...,M, 0 t Tcj f tm mc msS t A jA g t e
( ) ( ) 2 ( ) 2 4.3-19m mc c ms cS t A g t Cos f t A g t Sin f t
msA
( )g t
mcA
2( ) Re ( ) ( ) (2 ) 4.3-20cj f tj mm m m c mS t V e g t e V g t Cos f t
2 2 1 tan /m mc ms m ms mcV A A A A
(4-3-19)
(4-3-20)
111
Digital Communication Dr. Mahlab Uri
Representation as a linear combination of two
orthonormal signal waveforms:
1 1 2 2( ) ( ) ( ) 4.3-21m m mS t S f t S f t
1 22 2
( ) ( ) 2 ; ( ) ( ) 2 4.3-22c cg g
f t g t Cos f t f t g t Cos f t
1 21 1
4.3-232 2
m m m mc g ms gS S S A A
(4-3-21)
(4-3-22)
(4-3-23)
112
Digital Communication Dr. Mahlab Uri
The Euclidean distance between any pair of signal
vectors:
In the special case signal amplitudes take the set of
discrete values {(2m-1-M)d,m=1,2,..,M}
the Euclidean distance reach the minimum
which the same results as for PAM
2 2( ) 1 4.3-24
2
e
mn m n g mc nc ms nsd S S A A A A
( ) 2 4.3-25emin gd d
(4-3-24)
(4-3-25)
113
Digital Communication Dr. Mahlab Uri
M=4
M=8
M=16
M=32
M=64
Several signal space diagrams for rectangular QAM.
114
Digital Communication Dr. Mahlab Uri
FSK - orthogonal multi-
dimensional signalsM equal-energy orthogonal signal waveforms:
The equivalent low-pass signal waveform:
2( ) Re ( ) , m=1,2,...,M, 0 t Tcj f tm lmS t S t e
2
S ( ) 2 2 4.3-26m ct Cos f t m ftT
22( ) , m=1,2,...,M, 0 t T (4.3-27)c
j m f t
lmS t eT
(4-3-26)
(4-3-27)
115
Digital Communication Dr. Mahlab Uri
The cross-correlation coefficients:
The real part of :
2 ( ) ( )
km0
2 / (4.3-28)
2
T j m k ft j T m k fSin T m k fT e dt eT m k f
km
km kmRe( )
2 4.3-29
2
Sin T m k fCos T m k f
T m k f
Sin T m k f
T m k f
kmRe( ) 0 when 1/ 2 and m kf T
(4-3-28)
(4-3-29)
116
Digital Communication Dr. Mahlab Uri
For the case in which the M FSK signals are equivalent to the N-dimensional vectors:
The distance between pairs of signals:
1/ 2f T
1
1
1
s 0 0 .... 0 0
s 0 0 .... 0 0
s 0 0 0 .... 0 4.3-30
( ) 2ekmd
(4-3-30)
(4-3-31)
117
Digital Communication Dr. Mahlab Uri118
How to
generate
signals
Digital Communication Dr. Mahlab Uri119
0 T 2T 3T 4T 5T 6T
0 T 2T 3T 4T 5T 6T
+
tfEb 0
2cos2
tfEb 0
2sin2
tf2sinT
2Atf2cos
T
2A)t(s cmscmcm
Digital Communication Dr. Mahlab Uri120
0 T 2T 3T 4T 5T 6T
0 T 2T 3T 4T 5T 6T
+
tfEb 0
2cos2
tfEb 0
2sin2
tf2sin)t(Qtf2cos)t(I)t(s ccm
)t(sm
Digital Communication Dr. Mahlab Uri121
0 T 2T 3T 4T 5T 6T
0 T 2T 3T 4T 5T 6T
+
tfEb 0
2cos2
tfEb 0
2sin2
tf2sin)t(Qtf2cos)t(I)t(s ccm
)t(sm
)t(I
)t(Q
Digital Communication Dr. Mahlab Uri122
+
tfEb 0
2sin2
)t(sm
)t(I
)t(Q
tfEb 0
2cos2
IQ Modulator
Digital Communication Dr. Mahlab Uri123
+
tfEb 0
2sin2
)t(sm
)t(I
)t(Q
tfEb 0
2cos2
IQ ModulatorPulse shaping filter
Digital Communication Dr. Mahlab Uri
Part - 5
Digital Communication Dr. Mahlab Uri
Nonlinear Modulation 3-3-4Methods with Memory
In this section we consider a class of digital modulation
methods in witch the phase of the signal is constructed to
be continuous
125
Digital Communication Dr. Mahlab Uri
n
n nTtgItd )((4-3-50)
PAM signal
In sequence of amplitudes obtained by mapping k-bit blocksof binary digits.
g(t) rectangular pulse of amplitude 1/2T and duration T
seconds.
d(t) frequency modulate the carrier.
126
Digital Communication Dr. Mahlab Uri
Equivalent lowpass waveform
t
d ddTfjT
tv 0)(4exp2
(4-3-51)
- peak frequency deviation.
- initial phase of the carrier.df
0
127
Digital Communication Dr. Mahlab Uri
time varying phase of the carrier.
Carrier modulated signal
0);(2cos2
IttfT
tS c(4-3-52)
);( It
128
Digital Communication Dr. Mahlab Uri
time varying phase of the carrier
(4-3-53)
t
d ddTfIt )(4;
dnTgITft
n
nd
)(4
The integral of d(t) is continuous. Hence we have continuousphase signal.
The phase of the carrier in the interval is determined by integrating
TntnT )1(
It;
129
Digital Communication Dr. Mahlab Uri
(4-3-54)
ndn
k
kd InTtfITfIt )(22;1
)(2 nTtqhInn
Integrating
- accumulation (memory) of all symbols up totime (n-1)T.
h - modulation index.n
It;
130
Digital Communication Dr. Mahlab Uri
(4-3-57)
Tt
TtT
tt
tq
,2
1
0,2
0,0
)(
h, ,and q(t) are defined as:n
Tfh d2 (4-3-55)
1n
k
kn Ih (4-3-56)
131
Digital Communication Dr. Mahlab Uri
Continuous-Phase Modulation (CPM)
When expressed in the form of CPFSK becomes a special case of a general class of CPM
signals in which the carrier phase is:
(4-3-58)
n
k
kk kTtqhIIt ),(2; TntnT )1(
- sequence of M-ary information symbols.
sequence of modulation indices.
- some normalized waveform shape.
kI
kh
)(tq
It;
132
Digital Communication Dr. Mahlab Uri
Waveform q(t) as integral of g(t) pulse.
t
dgtq0
)()( (4-3-59)
a
b
FIGURE 4-3-16 Pulse shapes for full response CPM (a,b)
ttg
2cos1
2
1
1
0
tg
2
1
0
tq
2
1
0
tq
2
1
0
133
Digital Communication Dr. Mahlab Uri
c
d
ttg
cos1
4
1
tg tq
tq
2
1
2
1
2
1
4
1
0
0 0
0
FIGURE 4-3-16 Pulse shapes for partial response CPM (c,d).
dc, CPM. response partial Tfor t 0 tg ba, M.CP response full Tfor t0 tg
134
GMSK pulses
Digital Communication Dr. Mahlab Uri
FIGURE 4-3-17 Phase trajectory for binary CPFSK.
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
2 3 4 5
0
h
h2
h3
h4
h5
h
h2
h3
h4
h5
CPFSK with binary symbols .1nI Set of phase trajectories beginning at time t=0.
Phase trajectory for binary CPFSK
135
Digital Communication Dr. Mahlab Uri
Phase Tree diagram for CPFSK.
d d d d
FIGURE 4-3-18 Phase trajectory for quaternary CPFSK.
3
1
1
3
3
1
1
3
3
1
1
3
3
1
1
3
3
1
1
3
3
1
1
3
3
1
3
1
1
3
3
1
1
3
3
1
1
3
3
1
1
3
1
33
1
1
3
3
1
1
3
3
1
1
3
3
1
1
3
3
1
1
3
3
1
1
3
0
h
h2
h3
h4
h5
h
h2
h3
h4
h5
h6
h6
h6
136
Digital Communication Dr. Mahlab Uri
FIGURE 4-3-19 Phase trajectories for binary CPFSK (dashed) and binary partial response CPM
based on raised cosine pulse of length 3T (solid).
1
1
1 1
1 11 1 1
1
1
1
1
1
11
1 1
1 1t Ii,
Phase trajectory generated by the sequence:
(1,-1 , -1,-1 , 1,1 , -1,1)
137
Digital Communication Dr. Mahlab Uri
Phase trellis or phase cylinder with binary modulation
FIGURE 4-3-20 Phase cylinder for binary CPM with h=1/2 and a raised cosine
pulse of length 3T.[From sundberg (1986) ,(C) 1986 IEEE.]
138