of 142
8/12/2019 Notes 2013a
1/142
WIRELESS COMMUNICATIONSlecture notes (part 1)
c Prof. Giorgio Taricco c
Politecnico di Torino
2013/2014
1 c Prof. Giorgio Taricco c WIRELESS COMMUNICATIONS
8/12/2019 Notes 2013a
2/142
Copyright Notice
All material in this course is the property of the Author. Copyrightand other intellectual property laws protect these materials.Reproduction or retransmission of the materials, in whole or inpart, in any manner, without the prior written consent of the
copyright holder, is a violation of copyright law. A single printedcopy of the materials available through this course may be made,solely for personal, noncommercial use. Individuals must preserveany copyright or other notices contained in or associated withthem. Users may not distribute such copies to others, whether or
not in electronic form, whether or not for a charge or otherconsideration, without prior written consent of the copyright holderof the materials.
2 c Prof. Giorgio Taricco c WIRELESS COMMUNICATIONS
8/12/2019 Notes 2013a
3/142
Basic concepts
Outline
1 Basic concepts
2 Digital modulations over the AWGN channel
3 c Prof. Giorgio Taricco c WIRELESS COMMUNICATIONS
8/12/2019 Notes 2013a
4/142
8/12/2019 Notes 2013a
5/142
Basic concepts
Reference books in Wireless Communications
S. Benedetto and E. Biglieri, Principles of Digital Transmission:With Wireless Applications. Kluwer.
A. Goldsmith, Wireless Communications. Cambridge UniversityPress.
U. Madhow, Fundamentals of Digital Communication. Cambridge
University Press.A. Molisch, Wireless Communications. Wiley.
J. Proakis and M. Salehi, Digital Communications (4th Edition).McGraw-Hill.
T. Rappaport, Wireless Communications: Principles and Practice(2nd Edition). Prentice-Hall.
D. Tse and P. Viswanath, Fundamentals of WirelessCommunication. Cambridge University Press.
5 c Prof. Giorgio Taricco c WIRELESS COMMUNICATIONS
B i
8/12/2019 Notes 2013a
6/142
Basic concepts
Reference books in Wireless Communications
S. Benedetto and E. Biglieri, Principles of Digital Transmission:With Wireless Applications. Kluwer.
A. Goldsmith, Wireless Communications. Cambridge UniversityPress.
U. Madhow, Fundamentals of Digital Communication. Cambridge
University Press.A. Molisch, Wireless Communications. Wiley.
J. Proakis and M. Salehi, Digital Communications (4th Edition).McGraw-Hill.
T. Rappaport, Wireless Communications: Principles and Practice(2nd Edition). Prentice-Hall.
D. Tse and P. Viswanath, Fundamentals of WirelessCommunication. Cambridge University Press.
5 c Prof. Giorgio Taricco c WIRELESS COMMUNICATIONS
B i t M d l f di it l i ti t
8/12/2019 Notes 2013a
7/142
Basic concepts Model of a digital communication system
Model of a digital communication system
CHANNEL MODELThe main subject of this course is the study of digitalcommunications over a transmission channel.
To this purpose, it is useful to characterize the model of adigital communication system in order to get acquainted withits different constituent parts.
TOP LEVEL CLASSIFICATION
The model can be divided into three sections, as illustrated in
the following picture:1 Theusersection2 Theinterfacesection3 Thechannelsection
6 c Prof. Giorgio Taricco c WIRELESS COMMUNICATIONS
8/12/2019 Notes 2013a
8/142
Basic concepts Model of a digital communication system
8/12/2019 Notes 2013a
9/142
Basic concepts Model of a digital communication system
Model of a digital communication system
CHANNEL MODELThe main subject of this course is the study of digitalcommunications over a transmission channel.
To this purpose, it is useful to characterize the model of adigital communication system in order to get acquainted withits different constituent parts.
TOP LEVEL CLASSIFICATION
The model can be divided into three sections, as illustrated in
the following picture:1 Theusersection2 Theinterfacesection3 Thechannelsection
6 c Prof. Giorgio Taricco c WIRELESS COMMUNICATIONS
8/12/2019 Notes 2013a
10/142
Basic concepts Model of a digital communication system
8/12/2019 Notes 2013a
11/142
Basic concepts Model of a digital communication system
Model of a digital communication system
TX ENCODER MODULATOR
CHA
NNEL
DEMODULATORDECODERRX
user section interface section channel section
7 c Prof. Giorgio Taricco c WIRELESS COMMUNICATIONS
Basic concepts Model of a digital communication system
8/12/2019 Notes 2013a
12/142
p g y
Model of a digital communication system
TX ENCODER MODULATOR
CHA
NNEL
DEMODULATORDECODERRX
user section interface section channel section
D D W
WDD
D=digital
W=waveform
7 c Prof. Giorgio Taricco c WIRELESS COMMUNICATIONS
Basic concepts Model of a digital communication system
8/12/2019 Notes 2013a
13/142
p g y
Model of a digital communication system
ENCODER
Implementssource encodingto limit the amount oftransmitted data (for example, voice can be encoded at 4kbit/s or sent at 64 kbit/s with conventional telephony).
Implementschannel encodingto limit the effect of channeldisturbances
MODULATOR
Converts the digital signal into a waveform to be transmittedover the channel
CHANNEL
Reproduces the transmitted waveform at the receiver
Its operation is affected by frequency distortion, fading,additive noise, and other disturbances
8 c Prof. Giorgio Taricco c WIRELESS COMMUNICATIONS
8/12/2019 Notes 2013a
14/142
Basic concepts Model of a digital communication system
8/12/2019 Notes 2013a
15/142
Model of a digital communication system
ENCODER
Implementssource encodingto limit the amount oftransmitted data (for example, voice can be encoded at 4kbit/s or sent at 64 kbit/s with conventional telephony).
Implementschannel encodingto limit the effect of channeldisturbances
MODULATOR
Converts the digital signal into a waveform to be transmittedover the channel
CHANNEL
Reproduces the transmitted waveform at the receiver
Its operation is affected by frequency distortion, fading,additive noise, and other disturbances
8 c Prof. Giorgio Taricco c WIRELESS COMMUNICATIONS
Basic concepts Model of a digital communication system
8/12/2019 Notes 2013a
16/142
Model of a digital communication system
DEMODULATOR
Converts the received waveform into a sequence of samples tobe processed by the decoder
DECODER
Implements channel decoding to limit the effect of the errorsintroduced by the channel
Implements source decoding
9 c Prof. Giorgio Taricco c WIRELESS COMMUNICATIONS
Basic concepts Model of a digital communication system
8/12/2019 Notes 2013a
17/142
Model of a digital communication system
DEMODULATOR
Converts the received waveform into a sequence of samples tobe processed by the decoder
DECODER
Implements channel decoding to limit the effect of the errorsintroduced by the channel
Implements source decoding
9 c Prof. Giorgio Taricco c WIRELESS COMMUNICATIONS
Basic concepts Band-pass signalling
8/12/2019 Notes 2013a
18/142
Band-pass signals
A band-pass signal has spectral components in a limited range
of frequencies f (f2, f1) (f1, f2) provided that0< f1< f2.
A certain frequency in the range (f1, f2) (usually the middlefrequency) is called carrier frequency and denoted by fc.
The signal bandwidth is Bx =f2 f1.
f
Bx
f1 f1
f2 f2
fc fc
It is often convenient to representband-pass signalsasequivalent complex signals with low-pass frequency spectrum(i.e., including the zero frequency).
10 c Prof. Giorgio Taricco c WIRELESS COMMUNICATIONS
Basic concepts Band-pass signalling
8/12/2019 Notes 2013a
19/142
The analytic signal
A real band-pass signal x(t) can be mapped to a complexanalytic signal x(t) by passing through a linear filter withtransfer function 2u(f) = 2 1f>0:
x(t) 2u(f) x(t)
The indicator function 1A= 1 ifA is true, 0 otherwise.
Summarizing:
The analytic signal is a complex representation of a real signal.
It is used to simplify the analysis of modulated signals.It generalizes the concept of phasor used in electronics.
The basic properties of the analytic signal derive from theFourier transform.
11 c Prof. Giorgio Taricco c WIRELESS COMMUNICATIONS
Basic concepts Band-pass signalling
8/12/2019 Notes 2013a
20/142
The analytic signal (cont.)
Ifx(t) is a real signal, then its Fourier transform is aHermitian functionsince:
X(f) =
x(t)e +j 2ftdt=
x(t)ej 2(f)tdt
= X(f)
Therefore, the spectrum is completely determined by its
positive frequency (or negative frequency) part.
12 c Prof. Giorgio Taricco c WIRELESS COMMUNICATIONS
Basic concepts Band-pass signalling
8/12/2019 Notes 2013a
21/142
Hilbert transform
Since:
X(f) = 2u(f)X(f) =X(f) +sgn(f)X(f) X(f) + jX(f),applyingF1 yields:
x(t) =x(t) +jx(t).The signalx(t) is called theHilbert transformofx(t):x(t) x(t) 1
t=
1
x()
t d
13 c Prof. Giorgio Taricco c WIRELESS COMMUNICATIONS
Basic concepts Band-pass signalling
( )
8/12/2019 Notes 2013a
22/142
Hilbert transform (cont.)
Here, the Cauchy principal part of the integral has been taken,namely,
lim0,T
tT
+ Tt+
x()t d
14 c Prof. Giorgio Taricco c WIRELESS COMMUNICATIONS
Basic concepts Band-pass signalling
S
8/12/2019 Notes 2013a
23/142
Spectral properties
Assuming x(t) a zero-mean stationary real random process,
we have
Gx(f) = |2u(f)|2Gx(f) = 4u(f)Gx(f)
Therefore,
Gx(f) =1
4[Gx(f) + Gx(f)]
Moreover,
E[|x(t)|2] =
04Gx(f)df= 2
Gx(f)df= 2E[x(t)2]
15 c Prof. Giorgio Taricco c WIRELESS COMMUNICATIONS
Basic concepts Band-pass signalling
B d i lli
8/12/2019 Notes 2013a
24/142
Band-pass signalling
Assume that x(t) is a zero-mean stationary band-pass randomprocess with bandwidth Bx and carrier frequency fc so thatits power density spectrum is nonzero over the frequencies
f
(
fc
Bx/2,
fc+ Bx/2)
(fc
Bx/2, fc+ Bx/2)
where fc> Bx/2> 0.
We define the basebandcomplex envelopeofx(t) as
x(t) = x(t) ej 2fct
The complex envelope is sometimes calledbasebandequivalent signal.
16 c Prof. Giorgio Taricco c WIRELESS COMMUNICATIONS
Basic concepts Band-pass signalling
Th l l (t)
8/12/2019 Notes 2013a
25/142
The complex envelope x(t)
Then, we derive the autocorrelation function and the powerdensity spectrum ofx(t):
Rx() = E
x(t + ) ej 2fc(t+)x(t) ej 2fct
= Rx() ej 2fc= Gx(f) =Gx(f+ fc).
Then, the power density spectrum ofx(t) is nonzero over thefrequencies f (Bx/2, Bx/2), i.e., it is a baseband signalwith bandwidth Bx/2.
17 c Prof. Giorgio Taricco c WIRELESS COMMUNICATIONS
Basic concepts Band-pass signalling
I h d d t t
8/12/2019 Notes 2013a
26/142
In-phase and quadrature components
The real and imaginary parts ofx(t) =xc(t) + j xs(t) arecalledin-phaseandquadraturecomponents of the signal.
They can be expressed in terms of the signal itself and of itsHilbert transform:
xc(t) = Re[x(t)ej 2fct] =x(t) cos(2fct) +x(t) sin(2fct)xs(t) =Im[x(t)ej 2fct] =x(t) cos(2fct) x(t) sin(2fct)
The previous relationships can be inverted and yield:
x(t) = Re[x(t)e j 2fct] =xc(t) cos(2fct) xs(t) sin(2fct)x(t) =Im[x(t)e j 2fct] =xs(t) cos(2fct) + xc(t) sin(2fct)
18 c Prof. Giorgio Taricco c WIRELESS COMMUNICATIONS
Basic concepts Band-pass signalling
F i d l ti
8/12/2019 Notes 2013a
27/142
Frequency up-conversion = modulation
Themodulation(or frequency up-conversion) of a real signal
x(t) consists in the following operation:
x(t) x(t) cos(2fct).
The modulation of a couple of real signals xc(t) and xs(t)consists in the following operation:
[xc(t), xs(t)] xc(t) cos(2fct) xs(t) sin(2fct).
In the analytic signal domain, modulation can be representedas follows:
x(t) x(t) = x(t)e +j 2fct.
19 c Prof. Giorgio Taricco c WIRELESS COMMUNICATIONS
Basic concepts Band-pass signalling
F d si d d l ti
8/12/2019 Notes 2013a
28/142
Frequency down-conversion = demodulation
Therefore, demodulation(or frequency down-conversion) inthe analytic signal domain is represented as follows:
x(t) x(t) = x(t)ej 2fct.
Correspondingly, in the real signal domain, demodulation canbe represented by:
xc(t) =x(t) cos(2fct) +x(t) sin(2fct)xs(t) =x(t) cos(2fct) x(t) sin(2fct)
20 c Prof. Giorgio Taricco c WIRELESS COMMUNICATIONS
Basic concepts Band-pass signalling
Frequency down conversion demodulation (cont )
8/12/2019 Notes 2013a
29/142
Frequency down-conversion = demodulation (cont.)
In a real system, demodulation can be implemented byobserving that
MULTIPLICATION BY IN-PHASE CARRIER
x(t) cos(2fct + ) 2 cos(2fct + )= x(t) + x(t) cos(4fct + 2)
In other words, multiplication of the signal x(t) cos(2fct + )by the phase-coherent sinusoid 2 cos(2fct + ) returns thesuperposition of
the modulating signal x(t);another modulated signal with carrier frequency 2fc.
Low-pass filtering with bandwidth Bx eliminates themodulated signal with carrier frequency 2fc.
21 c Prof. Giorgio Taricco c WIRELESS COMMUNICATIONS
Basic concepts Band-pass signalling
Demodulator
8/12/2019 Notes 2013a
30/142
Demodulator
The following picture illustrates the block diagram of ademodulator with input:
x(t) =xc(t) cos(2fct) xs(t) sin(2fct).
x(t)
2 sin(2fct) = 2 cos(2fct + 2 )
2 cos(2fct)
LOW-PASSFILTER
LOW-PASSFILTER
xs(t)
xc(t)
22 c Prof. Giorgio Taricco c WIRELESS COMMUNICATIONS
Basic concepts Problem set 1
Problem set 1
8/12/2019 Notes 2013a
31/142
Problem set 1
1 Calculate the analytic signal corresponding to
x(t) = cos(2fct + ).
x(t) = sinc(t) cos(20t).
x(t) = sinc(t)2
cos(20t).2 Calculate the baseband equivalent signal corresponding to
x(t) = cos(41t) + 2 sin(39t), fc= 20.
x(t) = sinc(t) cos(20t), fc= 10.
x(t) = sinc(t) cos(20t), fc= 9.
23 c Prof. Giorgio Taricco c WIRELESS COMMUNICATIONS
Basic concepts Probability
Topics on Probability
8/12/2019 Notes 2013a
32/142
Topics on Probability
Probability is based on the concept ofprobability space.
A probability space consists of three parts:
1 A set of all possible outcomes.2 A set ofevents,F, which aresets of outcomes.3 A probability function P : F [0, 1], assigning a probability to
every event.
The probability function is a normalized measure:
24 c Prof. Giorgio Taricco c WIRELESS COMMUNICATIONS
8/12/2019 Notes 2013a
33/142
Basic concepts Probability
Topics on Probability (cont )
8/12/2019 Notes 2013a
34/142
Topics on Probability (cont.)
3 In all cases, the followingnormalizationholds:
P() = 1,
that is, the probability of the set of all possible outcomes is 1.Given two events A, B, we can build the union A B and theintersection A B:
A B = set of outcomes in A orin B.A
B = set of outcomes in A andin B.
26 c Prof. Giorgio Taricco c WIRELESS COMMUNICATIONS
Basic concepts Probability
Topics on Probability (cont )
8/12/2019 Notes 2013a
35/142
Topics on Probability (cont.)
The probability of the intersection is called joint probability
and allows to define the conditional probability as
P(A | B) = P(A B)P(B)
.
Commonly, we write P(A, B) P(A B).Conditional probabilities satisfy the Bayes rule:
P(A | B) = P(A, B)P(B)
=P(B, A)
P(A)
P(A)
P(B)
= P(B| A) P(A)
P(B) .
27 c Prof. Giorgio Taricco c WIRELESS COMMUNICATIONS
Basic concepts Probability
Topics on Probability (cont.)
8/12/2019 Notes 2013a
36/142
Topics on Probability (cont.)
A useful result is thetotal probability law:
If the events Bi, i= 1, 2, . . . form apartitionof (i.e.,iBi= and Bi Bj = fori =j), then:
P(A) =
iP(A, Bi) =
iP(A | Bi)P(Bi).
By the total probability law, one can obtain the conditionalprobabilitiesP(Bi| A) from the P(A | Bi):
P(Bi| A) = P(A | Bi)P(Bi)jP(A | Bj)P(Bj)
.
28 c Prof. Giorgio Taricco c WIRELESS COMMUNICATIONS
Basic concepts Probability
Topics on Probability (cont.)
8/12/2019 Notes 2013a
37/142
Topics on Probability (cont.)
The above result finds application in the design of digitalcommunication receivers where the event A represents the
received signal and the events Bi represent all the possibletransmitted data in a given framework.
29 c Prof. Giorgio Taricco c WIRELESS COMMUNICATIONS
Basic concepts Probability
Topics on Probability (cont.)
8/12/2019 Notes 2013a
38/142
p b b y ( )
The probability of the union has also some special properties.
For illustration, we interpret events as two-dimensional regionsand their probabilities as the areas of the regions.
A
B
P(A) P(A B)P(B) P(A B)P(A B) P(A) + P(B)
=P(A) + P(B) P(A B)
The inequalities derive from the fact that the area of the unionis always greater than or equal to the areas of each event.
30 c Prof. Giorgio Taricco c WIRELESS COMMUNICATIONS
Basic concepts Probability
Topics on Probability (cont.)
8/12/2019 Notes 2013a
39/142
p y ( )
Moreover, the sum of the areas is equal to the area of theunion plus that of the intersection, which is counted twice.This yields the last inequality.
The previous results can be generalized to the case ofmevents:
Lower and upper union bounds
Given a set of events{A1, . . . , Am}, the following inequalities hold:
max1im
P(Ai)
Pi Ai
m
i=1 P(Ai). (1)
31 c Prof. Giorgio Taricco c WIRELESS COMMUNICATIONS
Basic concepts Probability
Random variables
8/12/2019 Notes 2013a
40/142
A discrete real random variable X is characterized by itsprobability distribution
pX(xn) =P(X=xn)
forn= 1, 2, . . . , N (where Nmay become infinity).
Theexpectation operator E[] is defined by
E[(X)] = n (xn)pX(xn)for an arbitrary function ().
32 c Prof. Giorgio Taricco c WIRELESS COMMUNICATIONS
Basic concepts Probability
Random variables (cont.)
8/12/2019 Notes 2013a
41/142
( )
Since the expected value of every constant is the constantitself, we obtain by definition:
E[1] =n
pX(xn) = 1.
This property holds for all probability distributions.
ThemeanofX is X= E[X] =
n xnpX(xn).
Thesecond momentofX is (2)X = E[X
2] =
n x
2npX(xn).
ThevarianceofX is 2X= E[(X X)2] =(2)X 2X.
The square root of the variance is calledstandard deviation.
33 c Prof. Giorgio Taricco c WIRELESS COMMUNICATIONS
Basic concepts Probability
Random variables (cont.)
8/12/2019 Notes 2013a
42/142
( )
Continuous random variables are characterized by a pdffX(x)defining the expectation operator as
E[(X)] =
I
(x)fX(x)dx,
whereI is the support of the random variable, i.e., the set ofvalues where fX(x)>0.
Again, the expected value of every constant is the constantitself, so that:
E[1] = IfX(x)dx= 1.This property holds for all pdfs.
34 c Prof. Giorgio Taricco c WIRELESS COMMUNICATIONS
Basic concepts Probability
Random variables (cont.)
8/12/2019 Notes 2013a
43/142
The mean, second moment, and variance are given by
X =
I
xfX(x)dx
(2)
X = Ix2fX(x)dx2X =
(2)X 2X,
respectively.
Random variables can also be complex. Their propertiesderive from the properties of real random variables.
35 c Prof. Giorgio Taricco c WIRELESS COMMUNICATIONS
Basic concepts Probability
Random variables (cont.)
8/12/2019 Notes 2013a
44/142
A complex random variable Z=X+ j Y has mean
Z= E[X] + j E[Y]
and variance
2Z E[|Z Z|
2
]= E[(X X)2 + (Y Y)2]= 2X+
2Y
= E[X2 + Y2] (2X+ 2Y)= E[|Z|2] |Z|2.
36 c Prof. Giorgio Taricco c WIRELESS COMMUNICATIONS
Basic concepts Gaussian random variables
Gaussian random variables
8/12/2019 Notes 2013a
45/142
We will be particularly concerned with Gaussian randomvariables whose distribution is given by
fX(x) = 1
22e(x)
2/(22)
and denoted byN(, 2).The parameters and are the mean and the standarddeviation of a Gaussian random variable with distributionN(, 2).
37 c Prof. Giorgio Taricco c WIRELESS COMMUNICATIONS
Basic concepts Gaussian random variables
Gaussian random variables (cont.)
8/12/2019 Notes 2013a
46/142
We will often be interested in calculating the probability
P(N(, 2)> x), i.e., the probability that a Gaussian randomvariable with mean and standard deviation exceeds thereal value x.
These probability can be calculated by using the function
Q(x) (referred to as Q-function), which is the countercumulative probability distribution function of the normalizedGaussian random variableN(0, 1).The Q-function is defined as:
Q(x) P(N(0, 1)> x) = x
12
eu2/2du. (2)
38 c Prof. Giorgio Taricco c WIRELESS COMMUNICATIONS
Basic concepts Gaussian random variables
Gaussian random variables (cont.)
8/12/2019 Notes 2013a
47/142
By using the Q-function we can see that
P(N(, 2)> x) = P(N(0, 2)> x )= PN(0, 1)>
x
= Qx
.
The Q-function cannot be calculated in terms of elementaryfunctions (such as exp, ln, and trigonometric functions).
39 c Prof. Giorgio Taricco c WIRELESS COMMUNICATIONS
Basic concepts Gaussian random variables
Gaussian random variables (cont.)
8/12/2019 Notes 2013a
48/142
However, by applying integration by parts
(udv=uv vdu), we obtain the following simpleinequalities:
ex2/22x
1 1x2
Q(x) ex2/2
2x
.
The upper bound is a good approximation for x 3 andyields the asymptotic behavior:
Q(x) ex2
/22x
.
40 c Prof. Giorgio Taricco c WIRELESS COMMUNICATIONS
Basic concepts Gaussian random variables
Gaussian random variables (cont.)
8/12/2019 Notes 2013a
49/142
In some cases, the following crude approximation is used:
Q(x) ex2/2.
The following diagram compares the Q-function and the two
approximations, which are plotted as thered( ex2/2
2x) and
green(ex2/2) dashed curves, respectively.
We can see that the approximation of the red curve is betterthan 10% forx 3.
41 c Prof. Giorgio Taricco c WIRELESS COMMUNICATIONS
Basic concepts Gaussian random variables
Gaussian random variables (cont.)
8/12/2019 Notes 2013a
50/142
4 2 0 2 4
0.2
0.4
0.6
0.8
1.0
42 c Prof. Giorgio Taricco c WIRELESS COMMUNICATIONS
Basic concepts Complex Gaussian random variables
Complex Gaussian random variables
8/12/2019 Notes 2013a
51/142
We will also considercomplexGaussian random variables withaspecial property, namely, that of having zero mean andindependent and identically distributed (iid) real and
imaginary parts.
In other words, ifZ=X+ j Y, we assume thatX N(0, 2/2) and Y N(0, 2/2), where X and Y arestatistically independent.
43 c Prof. Giorgio Taricco c WIRELESS COMMUNICATIONS
Basic concepts Complex Gaussian random variables
Complex Gaussian random variables (cont.)
8/12/2019 Notes 2013a
52/142
These assumptions lead to the following equivalent pdf
expressions:
fZ(z) = fXY(x, y)
= 1
2ex
2/2 12
ey2/2
= 1
2e(x
2+y2)/2
= 1
2e|z|
2/2 .
The concept can be extended to vectors of complex Gaussianrandom variables.
44 c Prof. Giorgio Taricco c WIRELESS COMMUNICATIONS
Basic concepts Complex Gaussian random variables
Complex Gaussian random variables (cont.)
8/12/2019 Notes 2013a
53/142
Ifz = (Z1, . . . , Z n)T is a vector of complex Gaussian random
variables with zero mean and covariance matrixz = E[zz
H], then the pdf ofz can be expressed as follows:
fz(z) = det(z)1ez
H1z z.
In the special case of iid components ofz, corresponding toz =
2In, the pdf simplifies to
fz(z) = (2)nez
2/2 ,
which is the product of the individual (marginal) pdfs of theZis: (
2)1e|zi|2/2 .
45 c Prof. Giorgio Taricco c WIRELESS COMMUNICATIONS
Basic concepts Signal spaces
Signal spaces
8/12/2019 Notes 2013a
54/142
Signal spacesare linear (or vector) spaces built upon theconcept ofHilbert space, i.e., finite or infinite-dimensionalcomplete inner product spaces.
The elements of a signal space are real or complex signals x(t)
defined over a support intervalI, for exampleI= (0, T).The inner product of two elements (signals) xandy is definedas
(x, y) = Ix(t)y(t)dt. (3)
46 c Prof. Giorgio Taricco c WIRELESS COMMUNICATIONS
Basic concepts Signal spaces
Signal spaces (cont.)
8/12/2019 Notes 2013a
55/142
Correspondingly, theinduced normofx is given by
x (x, x)1/2. (4)
Accordingly, the set
L2(I) = {x: x < }
is defined as asignal space.
47 c Prof. Giorgio Taricco c WIRELESS COMMUNICATIONS
Basic concepts Signal spaces
Signal spaces (cont.)
I d i f h C h S h i li
8/12/2019 Notes 2013a
56/142
Inner products satisfy theCauchy-Schwarzinequality
|(x, y)
| x
y
.
If|(x, y)| = x y, then the two signals are proportional,i.e., y(t) =x(t) for some C.A signal x(t) H L2(I) ifx is finite.The squared norm of a signal x(t) is thesignal energy:
E(x) (x, x) =
I
|x(t)|2dt= x2.
48 c Prof. Giorgio Taricco c WIRELESS COMMUNICATIONS
8/12/2019 Notes 2013a
57/142
Basic concepts Signal spaces
Signal spaces (cont.)
Th ffi i t i thi i b l l t d b
8/12/2019 Notes 2013a
58/142
The coefficients xn in this expansion can be calculated by
xn = (x, n) = Ix(t)n(t)dt.In many cases, a signal spaceH is defined as the set of allpossible linear combinations of a set of signals:
H = L(s1, . . . , sM)= {x(t) =1s1(t) + + MsM(t),
(1, . . . , M) CM}.
The setL
(s1, . . . , sM)is calledlinear spanofs1(t), . . . , sM(t).
In general, the signal set{s1, . . . , sM} is not a base, but abase can be found by using the Gram-Schmidt algorithm.
50 c Prof. Giorgio Taricco c WIRELESS COMMUNICATIONS
Basic concepts Signal spaces
Signal spaces (cont.)
Th G S h idt l ith fi d b ( )N f
8/12/2019 Notes 2013a
59/142
The Gram-Schmidt algorithm finds a base (n)Nn=1 of
H= L(s1, . . . , sM) by the following set of iterative equations:For k= 1, . . . , n:
dk = sk k1i=1
(sk, i)i (projection step)
k =
dk
dk (normalization step)
51 c Prof. Giorgio Taricco c WIRELESS COMMUNICATIONS
Basic concepts Signal spaces
Signal spaces (cont.)
8/12/2019 Notes 2013a
60/142
At every projection step such that dk = 0 the correspondingk is not assigned and not accounted for in the remaining
steps.
The number of signals in the base is the number ofdimensions ofH.
52 c Prof. Giorgio Taricco c WIRELESS COMMUNICATIONS
Basic concepts Signal spaces
Signal spaces (cont.)
The Gram Schmidt algorithm works since at every step the
8/12/2019 Notes 2013a
61/142
The Gram-Schmidt algorithm works since, at every step, thesignal dk(t) is orthogonal to all previously generated signals
i(t), i= 1, . . . , k 1. In fact,
(dk, i) =
sk
k1=1
(sk, ), i
= (sk, i) k
1
=1
(sk, )(, i)
= (sk, i) (sk, i) = 0.
53 c Prof. Giorgio Taricco c WIRELESS COMMUNICATIONS
Basic concepts Signal spaces
Signal spaces (cont.)
8/12/2019 Notes 2013a
62/142
The projection of a signal x(t) H over the subspaceY= L(1, . . . , N) H is a signal xY(t) with the followingproperties:
It can be expressed through the base ofYas follows:
xY(t) =N
n=1(x, n)n(t).It is theclosestsignal inY to x(t):
xY= arg minyY
x y.The minimum distance is:
minyY
x y2 = x2 xY2.
54 c Prof. Giorgio Taricco c WIRELESS COMMUNICATIONS
8/12/2019 Notes 2013a
63/142
8/12/2019 Notes 2013a
64/142
Basic concepts Signal spaces
Signal spaces (cont.)
8/12/2019 Notes 2013a
65/142
The minimization over|
yn|
is straightforward and gives|yn| = |(x, n)|. In this case, n= |(x, n)|2.Summarizing, the yn minimizingx y2 overy Y is:
yn= |yn|ejyn = |(x, n)|ej(x,n) = (x, n).
The minimum is:
minyY
x y2 = x2 N
n=1|(x, n)|2.
57 c Prof. Giorgio Taricco c WIRELESS COMMUNICATIONS
Basic concepts Signal spaces
A matrix version of the GS algorithm
8/12/2019 Notes 2013a
66/142
An alternative approach to the Gram-Schmidt algorithm is
based on standard linear algebra methods.It can be noticed that the GS algorithm leads to expressionsof the orthogonal base signals of the following type:
i(t) = ji Cijsj(t).These equations can be written in matrix form as follows:
=Cs,
where = (1(t), . . . , N(t))T and s= (s1(t), . . . , sM(t))
T.
C is a lower triangular matrix.
58 c Prof. Giorgio Taricco c WIRELESS COMMUNICATIONS
Basic concepts Signal spaces
A matrix version of the GS algorithm (cont.)
8/12/2019 Notes 2013a
67/142
From the previous equation we get
(,T) = (Cs, sTCT) =C(s, sT)CT.
The lhs is the identity matrix IN since (i, j) =ij.
The rhs can be written as CsCT where the matrix s isthe Gram matrixof the signals in s(t).
The elements ofs are(s)ij = (si, sj).
If a signal si(t) is linearly dependent from the signals
s1(t), . . . , si1(t), the corresponding row and column ofsmust be eliminated because linear combinations of theprevious row and columns.
59 c Prof. Giorgio Taricco c WIRELESS COMMUNICATIONS
Basic concepts Signal spaces
A matrix version of the GS algorithm (cont.)
A reduced Cholesky factorization can be applied to s leading
8/12/2019 Notes 2013a
68/142
y pp s gto the following matrix expression:
EsET =LLT
where the N M matrix Eremoves the redundant rows andcolumns and L is a square nonsingular lower triangularmatrix.
The previous result leads to C=L1E.As an example, if we have four signals but only s3 is a linearcombination ofs1, s2,
E= 1 0 0 00 1 0 00 0 0 1
The matrix product E(s1, s2, s3, s4)
T = (s1, s2, s4)T
eliminates s3.
60 c Prof. Giorgio Taricco c WIRELESS COMMUNICATIONS
Basic concepts Problem set 2
Problem set 2
8/12/2019 Notes 2013a
69/142
1 Calculate the mean and variance of the discrete random
variableXwith probability distributionpX(1) = 0.5, pX(2) = 0.25, pX(4) = 0.25.
2 Calculate the mean and variance of the continuous randomvariableXwith probability distribution fX(x) = e
x1x>0.
3 Calculate the mean and variance of the continuous randomvariableXwith probability distribution fX(x) = 0.5 1|x| a) for a Gaussian random
variableX N(, 2).5 Show that the variance identity holds: 2
X= E[X2]
E[X]2.
6 Assume support intervalI= (0, 1) from now on.LetHbe the linear span ofcos(2t) and sin(2t). Determineif the signal cos(2t + /4) belongs toH.
61 c Prof. Giorgio Taricco c WIRELESS COMMUNICATIONS
Basic concepts Problem set 2
Problem set 2 (cont.)
8/12/2019 Notes 2013a
70/142
7 Given the signals s1(t) =u(t) u(t 0.6),s2(t) =u(t 0.4) u(t 1), and s3(t) =u(t) u(t 1),apply the Gram-Schmidt algorithm to find a base ofL(s1, s2, s3) [u(t) = 0 fort 0 is the unitstep function].
8 Given the signalss1(t) = cos(2t)ands2(t) = sin(3t), apply
the Gram-Schmidt algorithm to find a base ofL(s1, s2).9 Check Schwarzs inequality for the signals
s1(t) =u(t) u(t 0.6), s2(t) =u(t 0.4) u(t 1), ands3(t) =u(t) u(t 1).
10 LetY= L(s1= sin(t), s2= cos(3t)). Find the projectionofx(t) =u(t) u(t 1) overYand calculatex xY2(notice that s1 and s2 are orthogonal).
62 c Prof. Giorgio Taricco c WIRELESS COMMUNICATIONS
Basic concepts Problem set 2
Problem set 2 (cont.)
8/12/2019 Notes 2013a
71/142
11 Apply the matrix GS algorithm to find the matrix Cdetermining the orthogonal base to the set of signals
s1= 10
8/12/2019 Notes 2013a
72/142
8/12/2019 Notes 2013a
73/142
Digital modulations over the AWGN channel Additive White Gaussian Noise (AWGN) Channel
AWGN channel
Thi h l d l i ifi d b th ti
8/12/2019 Notes 2013a
74/142
This channel model is specified by the equation
y(t) =Ax(t) + z(t) (6)
where:
Channel parameters
A is the real channel gain.
x(t) and y(t) are the channel input and output signals.
z(t) is the zero-mean additive white Gaussian noise process. It hasautocorrelation function and power density spectrum:
Rz() = E[z(t + )z(t)] = N02 (),
Gz(f) = N0
2 .
66 c Prof. Giorgio Taricco c WIRELESS COMMUNICATIONS
Digital modulations over the AWGN channel Linear digital modulation
Linear modulations
8/12/2019 Notes 2013a
75/142
We consider the following modulated signal:
x(t;a) =N
n=1
ann(t) (7)
where
N is the number ofdimensionsof the modulation scheme.The vector a= (an)
Nn=1 represents amodulation symbol
vectorand is taken from a finite setA = {1, . . . ,M}.A is calledmodulation alphabetorsignal constellation.n(t) is the nthshaping pulseof the modulated signal.We assume that each n(t) = 0 only for t (0, T).We also assume that (m, n) =mn= 1 ifm= n and 0otherwise (Kroneckers delta).
67 c Prof. Giorgio Taricco c WIRELESS COMMUNICATIONS
Digital modulations over the AWGN channel Linear digital modulation
Linear modulations (cont.)
8/12/2019 Notes 2013a
76/142
The signal x(t;a) allows us to send one symbol vector everyTtime units so that T is calledsymbol time,symbol interval,orsignalling interval.
The signalx(t;a)belongs to the Hilbert spaceHgenerated byall possible linear combinations of the base signals (n)
Nn=1.
The corresponding received signal
y(t) =AN
n=1
ann(t) + z(t) (8)
does not necessarily belong toH= L(1, . . . , N).
68 c Prof. Giorgio Taricco c WIRELESS COMMUNICATIONS
Digital modulations over the AWGN channel Linear digital modulation
Linear modulations (cont.)
8/12/2019 Notes 2013a
77/142
The projection ofy(t) onHis given by:
yH(t) =N
n=1
ynn(t) =N
n=1
(Aan+ zn)n(t). (9)
In this expression we find the nth received signal and noise
components:yn= (y, n) =
T0
y(t)n(t)dt and
zn= (z, n) =T0
z(t)n(t)dt
Here, zn, is a Gaussian random variable.
A receiver calculating the vector y= (y1, . . . , yN) fromy(t) iscalledcorrelation receiver.
69 c Prof. Giorgio Taricco c WIRELESS COMMUNICATIONS
Digital modulations over the AWGN channel Digital receiver design
Receiver design
8/12/2019 Notes 2013a
78/142
The goal of a digital receiver is to recover the transmittedsymbol vector a from the received signal y(t).
The correlation receiver projects y(t) onHto obtain(9)andoutputs the coefficients yn=Aan+ zn forn= 1, . . . , N .
In the absence of noise, the correlation receiver outputs ascaled version (by the channel gain A) of the transmittedsymbol vector.
When noise is present, the receiverguesseswhich symbolvector from
Awas transmitted with the goal of minimizing
theerror probability.This process is calleddetectionordecision.
70 c Prof. Giorgio Taricco c WIRELESS COMMUNICATIONS
Digital modulations over the AWGN channel Digital receiver design
Receiver design (cont.)
8/12/2019 Notes 2013a
79/142
The correlation receiver can be interpreted as amatched filterby observing that:
yn =
T0
y(t)n(t)dt
= T0
y(t)hn(T t)dt= [y(t) hn(t)]t=T,
where we defined the impulse response of the matched filter
as hn(t) =n(T t).
71 c Prof. Giorgio Taricco c WIRELESS COMMUNICATIONS
Digital modulations over the AWGN channel Digital receiver design
Sequential receiver design
8/12/2019 Notes 2013a
80/142
The signalling model described can be repeated over many
symbol times.
We can write the sequential modulated signal as:
x(t;a0, . . . ,aL) =L
i=0N
n=1 ai,nn(t iT).The corresponding received signal over the AWGN channel is:
y(t) =A
Li=0
Nn=1
ai,nn(t iT) + z(t).
72 c Prof. Giorgio Taricco c WIRELESS COMMUNICATIONS
Digital modulations over the AWGN channel Digital receiver design
Sequential receiver design (cont.)
The matched filter structure lends itself to a sequential
8/12/2019 Notes 2013a
81/142
The matched filter structure lends itself to a sequentialimplementation accounting for the transmission of successive
modulated symbols in time.
The bank of matched filters receiver is illustrated as follows:
y(t) 1(T
t)
2(T t)
...
N(T t)
yi,1
yi,2
yi,N
t= (i + 1)T
73 c Prof. Giorgio Taricco c WIRELESS COMMUNICATIONS Digital modulations over the AWGN channel Digital receiver design
Sequential receiver design (cont.)
The output of the nth matched filter at time t= (i + 1)T isgiven by:
8/12/2019 Notes 2013a
82/142
given by:
yi,n = [y(t) n(T t)]t=(i+1)T=
y(t1)n(T (i + 1)T+ t1)dt1
= T
0 y(t2+ iT)n(t2)dt2
= AL
j=0
Nm=1
T0
aj,mm(t2+ iTjT)n(t2)dt2
+ T0
z(t2)n(t2)dt2
= A ai,n+ zi,n
74 c Prof. Giorgio Taricco c WIRELESS COMMUNICATIONS Digital modulations over the AWGN channel Digital receiver design
Sequential receiver design (cont.)
8/12/2019 Notes 2013a
83/142
The previous expression holds since the shaping pulses areorthogonal and time-limited to the signalling interval (0, T).Therefore,
T
0 m(t2+ iTjT)n(t2)dt2=m,ni,j.
This is just an extension of the digital receiver operationdescribed over the first signalling interval (0, T).
75 c Prof Giorgio Taricco c WIRELESS COMMUNICATIONS
8/12/2019 Notes 2013a
84/142
Digital modulations over the AWGN channel Baseband digital modulation
Baseband digital modulation (cont.)
8/12/2019 Notes 2013a
85/142
Another example (same data symbols).
x(t)
t
+1 +1 1 1 1 +1 1 +1 +1 1
Baseband digital modulations are represented in aone-dimensional space.
77 c Prof Giorgio Taricco c WIRELESS COMMUNICATIONS Digital modulations over the AWGN channel Band-pass digital modulation
Band-pass digital modulation
Next, we deal with band-pass digital modulations.
8/12/2019 Notes 2013a
86/142
We start from the baseband equivalent signal:
x(t; a) =a g(t),
where g(t) is a real baseband signal of bandwidth lower thanfc and we assume that a
A=
{a1, . . . , aM
} is acomplex
modulation symbol.
Then, we obtain the corresponding band-pass signal as:
x(t; a) = Re
x(t; a)ej 2fct
= Re(a) [g(t) cos(2fct)]+ Im(a) [g(t) sin(2fct)],
78 c Prof Giorgio Taricco c WIRELESS COMMUNICATIONS Digital modulations over the AWGN channel Band-pass digital modulation
Band-pass digital modulation (cont.)
8/12/2019 Notes 2013a
87/142
The signal x(t; a) can be interpreted as a two-dimensionallinear modulation.
In fact, it can be represented as a linear combination of:
1(t) =g(t) cos(2fct)
2(t) = g(t) sin(2fct) (10)
with coefficientsRe(a) andIm(a).Now, assume that the bandwidth ofg(t) is Bg < fc, i.e., itsFourier transform G(f) =
F[g(t)] is equal to 0 for every
f fc.Then, G2(f) F[g(t)2] =G(f) G(f) has bandwidth2Bg
8/12/2019 Notes 2013a
88/142
G2(2fc) = g(t)2ej 4fctdt= 0.Using the above result,
1,2
2 =
g(t)2
1 cos(4fct)2
dt=1
2g
2
and
(1, 2) = 12
g(t)2 sin(4fct)dt= 0.
Thus, ifg2 = 2, the signals (1, 2) are orthogonal andform the base of a two-dimensional signal spaceHprovidedthat the bandwidth ofg(t) is smaller than the carrierfrequency fc.
80 c Prof Giorgio Taricco c WIRELESS COMMUNICATIONS Digital modulations over the AWGN channel Signal detection
Detection of transmitted symbols
The receiver outputs an estimate of the transmitted symbol a
8/12/2019 Notes 2013a
89/142
based on the received signal y(t) over t (0, T).The first stage of the receiver converts y(t) into the vectory=Aa + z
where a= (a1, . . . , aN) and z = (z1, . . . , zN).
We define a generic decision rule (or detection rule):
a(y) = (a1(y), . . . , aN(y)). (11)
a(y) mapsH into the modulation alphabetA.The decision rule can be optimized according to somegoodness criterion.
Typically, the goal isminimizing the error probability.
81 c Prof Giorgio Taricco c WIRELESS COMMUNICATIONS Digital modulations over the AWGN channel Signal detection
Optimum digital receiver
We can write the (average) error probability as follows:
8/12/2019 Notes 2013a
90/142
( g ) p y
P(e) =M
m=1
P(m)P(e | m) (12)
where
P(m) is thea priori probability of transmitting m.P(e | m) is the probability of error conditioned on thetransmission ofm.
We notice that
P(e | m) =Pa(y=Am+ z) =m,i.e., the probability that the decision rule returns a symboldifferent from the transmitted one.
82 c Prof Giorgio Taricco c WIRELESS COMMUNICATIONS Digital modulations over the AWGN channel Signal detection
Optimum digital receiver (cont.)
8/12/2019 Notes 2013a
91/142
It is plain to see that minimizing the (average) errorprobability is equivalent to maximizing the (average)probability of correct decision P(c) since P(c) = 1 P(e).Let us define
The pdf ofy given the transmitted symbol : f(y
|).
The decision regions
Rm {y:a(y) =m), m= 1, . . . , M .Notice that there is a one-to-one correspondence between the
set of decision regions and the decision rule.
83 c Prof Giorgio Taricco c WIRELESS COMMUNICATIONS Digital modulations over the AWGN channel Signal detection
Optimum digital receiver (cont.)
8/12/2019 Notes 2013a
92/142
Since the decision rulea(y) is a well defined (i.e.,single-valued) function for all y H= RN (the signal space),the decision regions do not intersect and their union fillsHitself:
Mm=1
Rm= H
(
denotes the union of disjoint sets).
84 c Prof Giorgio Taricco c WIRELESS COMMUNICATIONS Digital modulations over the AWGN channel Signal detection
Optimum digital receiver (cont.)
Now, we can write the probability of correct decision as af f h b b l ( ) f ( ) d h
8/12/2019 Notes 2013a
93/142
function of the a prioriprobabilities P(m), f(y
|), and the
decision rule. Sincea(y) =m fory Rm, we get:P(c) =
Mm=1
P(m)P(
a(y) =m| m)
=
Mm=1
yRm
P(m)f(y|m)dy
=Mm=1
yRm
P(
a(y))f(y|
a(y))dy
= H=RN
P(a(y))f(y|a(y))dy (13)
85 c Prof Giorgio Taricco c WIRELESS COMMUNICATIONS Digital modulations over the AWGN channel Signal detection
Optimum digital receiver (cont.)
Maximizing P(c) requires to maximize the integrand in(13),which can be accomplished by selecting the symbol A
8/12/2019 Notes 2013a
94/142
that maximizes
P()f(y| )for all possible received vectors y.
The resulting optimum decision rule is:
aopt(y) = arg maxA P()f(y| ). (14)
86 c Prof Giorgio Taricco c WIRELESS COMMUNICATIONS Digital modulations over the AWGN channel Signal detection
Optimum digital receiver (cont.)
Since applying the Bayes rule, we have
8/12/2019 Notes 2013a
95/142
P(m| y) = P(m)f(y| m)f(y)
,
the optimum decision rule is equivalent to maximizing the aposteriori probability P(m
|y).
Thus, the optimum decision rule is calledmaximuma-posteriori (MAP)decision.
87 c Prof Giorgio Taricco c WIRELESS COMMUNICATIONS Digital modulations over the AWGN channel Signal detection
Optimum digital receiver (cont.)
When transmitted symbols are equiprobable, i.e.,P ( ) 1/M h MAP l d i
8/12/2019 Notes 2013a
96/142
P(m) = 1/M, the MAP rule reduces to amaximum
likelihood (ML)rule:
a(y) = arg maxm
f(y| m)
This name comes from the name of the functions f(y| m)(likelihood functionsin radar theory).
88 c Prof Giorgio Taricco c WIRELESS COMMUNICATIONS Digital modulations over the AWGN channel Signal detection
Optimum digital receiver (cont.)
8/12/2019 Notes 2013a
97/142
The decision regions can be represented as follows:
Rm= {y: P(m)f(y| m)> P(n)f(y| n)n =m} MAP{y: f(y| m)> f(y| n)n =m} ML
89 P f Gi i T i WIRELESS COMMUNICATIONS Digital modulations over the AWGN channel Signal detection
Special case: The AWGN channel
Proposition. The additive noise components of an AWGN
8/12/2019 Notes 2013a
98/142
channel areiid Gaussian random variables with zero mean andvarianceN0/2.
Proof.We have, by definition,
zn= T0
z(t)n(t)dt
forn= 1, . . . , N .
Then,
E[zn] = T0
E[z(t)]n(t)dt= 0
since the additive noise random process has zero mean.
90 P f Gi i T i WIRELESS COMMUNICATIONS Digital modulations over the AWGN channel Signal detection
Special case: The AWGN channel (cont.)
Moreover,
8/12/2019 Notes 2013a
99/142
E[znzn ] = E T0
z(t)n(t)dt T0
z(t)n(t)dt=
T0
T0
E[z(t)z(t)]n(t)n(t)dtdt
= T0
T0
N02
(t t)n(t)n(t)dtdt
=
T0
N02
n(t)n(t)dt
= N02 n,n .
91 c
P f Gi i T i c
WIRELESS COMMUNICATIONS Digital modulations over the AWGN channel Signal detection
Special case: The AWGN channel (cont.)
In other words, different components of the noise vector z areuncorrelated (and hence independent since Gaussian), and
h h i N /2
8/12/2019 Notes 2013a
100/142
each one has variance N0/2.
As a result, the conditional pdf of the received vector y is
f(y| ) = fz(y A)= (N0)
N/2eyA2/N0 . (15)
It is worth noting that the joint pdf(15)depends only on thedistance of the received signal from the transmitted onescaled by the channel gain A.
Using(15), the logarithms of the likelihood functions arereadily obtained as follows:
ln f(y| m) = N2
ln(N0) 1N0
y Am2.
92 c
P f Gi i T i c
WIRELESS COMMUNICATIONS Digital modulations over the AWGN channel Signal detection
Special case: The AWGN channel (cont.)Since these functions depend on a distance, they are calleddecision metrics.
Th MAP d ML d i i l b d i f
8/12/2019 Notes 2013a
101/142
The MAP and ML decision rules can be expressed in terms ofdecision metrics for the AWGN channel as follows:
a(y) =
arg minm
y Am2 N0ln P(m)
MAP
arg minm y Am2 ML
As a result, the ML decision rule for the AWGN channel isoften referred to asminimum distance decision.
93 c Prof. Giorgio Taricco c WIRELESS COMMUNICATIONS
Digital modulations over the AWGN channel Signal detection
Special case: The AWGN channel (cont.)
The decision regions on the AWGN channel can bet d f ll
8/12/2019 Notes 2013a
102/142
represented as follows:
Rm =
{y: y Am2 N0ln P(m) P(m)f(y| m) | m
.
forn= 1, . . . , M and n
=m.
Notice that all the pairwise error events contain theconditioning clause | m. This clause is equivalent to theassumption that m was transmitted.
97 c Prof. Giorgio Taricco c WIRELESS COMMUNICATIONS
Digital modulations over the AWGN channel Error probability
Error probability (cont.)
Thus, the error probability, conditioned on the transmission ofm, is given by
8/12/2019 Notes 2013a
106/142
m, is given by
P(e | m)= P
n=m
P(n)f(y| n)> P(m)f(y| m)
m
whererepresents the union of events.
98 c Prof. Giorgio Taricco c WIRELESS COMMUNICATIONS
Digital modulations over the AWGN channel Error probability
Error probability (cont.)
The above expression of the error probability is too complex
8/12/2019 Notes 2013a
107/142
to calculate analytically, whereas thepairwise errorprobabilities (PEPs)
Pairwise Error Probability
P(m n) P
P(n)f(y| n)> P(m)f(y| m)
m
can be calculated very simply!
Thus, lower and upper bounds are used to approximateP(e|m).
99 c Prof. Giorgio Taricco c WIRELESS COMMUNICATIONS
Digital modulations over the AWGN channel Error probability
Error probability (cont.)Applying the bounds(1)to the conditional probabilitiesP(e|m), we obtain
8/12/2019 Notes 2013a
108/142
Error probability lower and upper bounds
Mm=1
P(m)maxn=m
P(m n) P(e) Mm=1
P(m)n=m
P(m n)
(16)
100 c Prof. Giorgio Taricco c WIRELESS COMMUNICATIONS
Digital modulations over the AWGN channel Error probability
Error probability (cont.)
8/12/2019 Notes 2013a
109/142
Assuming theMAP decision ruleover theAWGN channel, andletting A= 1, the PEPs are given by
Pairwise error probability
P(m n) =Qm n2 + N0ln[P(m)/P(n)]2N0 m n .
(17)
Equation(17)is based on the Q-function(2)and will bederived in detail in a problem.
101 c Prof. Giorgio Taricco c WIRELESS COMMUNICATIONS
Digital modulations over the AWGN channel Error probability
Error probability of binary modulations
8/12/2019 Notes 2013a
110/142
The inequalities(16)yield the exact error probability in thecase of for binary modulations (M= 2):
P(e) = P(1)P(1 2) + P(2)P(2 1)
= P(1)Q1 22 + N0ln[P(1)/P(2)]2N0 1 2
+ P(2)Q
1 22 N0ln[P(1)/P(2)]2N0 1 2
.
102 c Prof. Giorgio Taricco c WIRELESS COMMUNICATIONS
Digital modulations over the AWGN channel Error probability
Error probability of binary modulations (cont.)
With equiprobable signals i e P (m) = 1/M inequalities
8/12/2019 Notes 2013a
111/142
With equiprobable signals, i.e., P(m) 1/M, inequalities(16)yield:
1
M
M
m=1maxn=m
P(m n) P(e)
1M
Mm=1
n=m
P(m n)
Here, P(m
n) =Q(
m
n
/
2N0).
103 c Prof. Giorgio Taricco c WIRELESS COMMUNICATIONS
Digital modulations over the AWGN channel Error probability
High SNR approximation
In most situations, one is mostly interested to the high SNR(and then low N0) case.
8/12/2019 Notes 2013a
112/142
Since the Q function decreases very quickly, we can keep inthe bounds only the terms with minimum distance:
dmin minm=n
m n (18)
and disregard the others which are very small.
To be conservative, we use the upper bound to P(e) andobtain this approximation:
P(e) NminQ dmin2N0 (19)where Nmin=
1M
m
n 1mn=dmin.
104 c Prof. Giorgio Taricco c WIRELESS COMMUNICATIONS
8/12/2019 Notes 2013a
113/142
Digital modulations over the AWGN channel Error probability
Bit error probability
In some applications, it is better considering the bit errorprobability Pb(e) than the symbol error probability P(e).
8/12/2019 Notes 2013a
114/142
Typically, modulation symbols are assigned to bit vectors (bitmapping), so that a symbol error corresponds to having areceived bit vector different from the transmitted one.
The bit error probability is the average number of errors in the
received bit vector divided by the vector size:
Pb(e) = E[Nb]
log2 M,
where Nb
denotes the number of bit errors.Of course, Pb(e) depends on the bit mapping.
Assuming high SNR, most errors occur between minimumdistance symbols.
106 c Prof. Giorgio Taricco c WIRELESS COMMUNICATIONS
8/12/2019 Notes 2013a
115/142
Digital modulations over the AWGN channel Standard digital modulations
PAM = Pulse Amplitude Modulation
The alphabet ofM-PAM is = (2m M 1) Mm=1.
8/12/2019 Notes 2013a
116/142
A { }For example, the constellation of8-PAM is as follows:
7
5
3
+ +3 +5 +7
The error probability ofM-PAM is:
P(e) = 2M 1
M
Q6log2 M
M2
1Eb
N0. (20)
108 c Prof. Giorgio Taricco c WIRELESS COMMUNICATIONS
8/12/2019 Notes 2013a
117/142
Digital modulations over the AWGN channel Standard digital modulations
QAM = Quadrature Amplitude Modulation (cont.)
8/12/2019 Notes 2013a
118/142
The error probability ofM-QAM is:
P(e)
4
M 1
MQ
3log2 M
M 1
Eb
N0. (21)
110 c Prof. Giorgio Taricco c WIRELESS COMMUNICATIONS
Digital modulations over the AWGN channel Standard digital modulations
PSK = Phase Shift Keying
The alphabet ofM-PSK isA = {Esej (2m1)/M}Mm=1.
8/12/2019 Notes 2013a
119/142
For example, the constellation of8-PSK is as follows:
111 c Prof. Giorgio Taricco c WIRELESS COMMUNICATIONS
Digital modulations over the AWGN channel Standard digital modulations
PSK = Phase Shift Keying (cont.)
The error probability ofM-PSK is:
8/12/2019 Notes 2013a
120/142
P(e) 2 Q
2sin2
M log2 M
EbN0
. (22)
In the special case ofM= 4 we have:
P(e) 2 Q
2EbN0
.
112 c Prof. Giorgio Taricco c WIRELESS COMMUNICATIONS
Digital modulations over the AWGN channel Standard digital modulations
Orthogonal modulations
The alphabet of an orthogonal modulation consists ofMvectors in RM with a single nonzero coordinate equal to
Es.
8/12/2019 Notes 2013a
121/142
For example, a quaternary orthogonal modulation isrepresented by the following four signals:
1= (
Es, 0, 0, 0), 2= (0,
Es, 0, 0),
3= (0, 0,
Es, 0), 4= (0, 0, 0,
Es).
The error probability of an M-ary orthogonal modulation is:
P(e) (M 1)Q
log2 MEbN0
. (23)
113 c Prof. Giorgio Taricco c WIRELESS COMMUNICATIONS
Digital modulations over the AWGN channel Standard digital modulations
Orthogonal modulations (cont.)
Two examples of orthogonal modulations are given as follows.
8/12/2019 Notes 2013a
122/142
1 Pulse position modulation (M-PPM): Given the signal pulse(t), the modulated signals are:
xm(t) =
M (M t (m 1)T), (24)
i.e.,(t) is contracted in time to (0,T/M) and shifted by(m 1)T /M.
2 Frequency shift keying (M-FSK):
xm(t) =
2 cos[2(fc+ mf)t] (25)
where fc T and f Tare integer numbers.
114 c Prof. Giorgio Taricco c WIRELESS COMMUNICATIONS
Digital modulations over the AWGN channel Standard digital modulations
Asymptotic comparison of digital modulations
Consider two digital modulation schemes with approximateunion bounds to the error probability
8/12/2019 Notes 2013a
123/142
P(e) iQ
iEbN0
fori= 1, 2.
The asymptotic behavior of the error probability (when Eb/N0is very large) is dominated by the Q-function term and can beapproximated by
P(e) e(i/2)Eb/N0
fori= 1, 2 (we used Q(x) exp(x2/2)).If1> 2, the first modulation is better than the secondsince its error probability is smaller at the same Eb/N0.
115 c Prof. Giorgio Taricco c WIRELESS COMMUNICATIONS
Digital modulations over the AWGN channel Standard digital modulations
Asymptotic comparison of digital modulations (cont.)
The same asymptotic error probability is obtained when
Eb Eb
8/12/2019 Notes 2013a
124/142
1EbN01 =2EbN02,disregarding thei.
Hence, we define theasymptotic gainof the first modulation
with respect to the second one as the dB-difference between(Eb/N0)2 and (Eb/N0)1, which are the Eb/N0 ratios requiredto have the same asymptotic error probability:
G= 10 log10EbN02 10log10EbN01 = 10log10 12
116 c Prof. Giorgio Taricco c WIRELESS COMMUNICATIONS
Digital modulations over the AWGN channel Problem set 3
Problem set 3
1 Derive the PEP(17).
2 Derive the union bound approximation (19)along with the
8/12/2019 Notes 2013a
125/142
expression ofNmin by keeping only those terms from theupper bound
1
M
M
m=1 n=mP(m n)
corresponding to minimum distance errors, i.e., such thatm n =dmin.
3 Derive the error probability in(20).
4
Derive the error probability in(21).5 Derive the error probability in(22).
6 Check the orthogonality of the signals in(24)and(25).
117 c Prof. Giorgio Taricco c WIRELESS COMMUNICATIONS
Digital modulations over the AWGN channel Problem set 3
Problem set 3 (cont.)7 Derive the error probability in(23).
8 Calculate the error probability of the 32-QAM modulationcharacterized by the following signal set:
8/12/2019 Notes 2013a
126/142
5 3 1 +1 +3 +55
3
1
+1
+3
+5
118 c Prof. Giorgio Taricco c WIRELESS COMMUNICATIONS
Digital modulations over the AWGN channel Problem set 3
Problem set 3 (cont.)
9 Find the error probability of the binary modulation whosesignals are
8/12/2019 Notes 2013a
127/142
s1(t) = 10
8/12/2019 Notes 2013a
128/142
s1(t) = A[u(t) u(t T)]s2(t) = A[u(t) u(t T /4) + u(t T /2) u(t 3T /4)]s3(t) = A[u(t) u(t T /4) u(t T /2) + u(t 3T /4)]
s4(t) = A[u(t) 2u(t T /2) + u(t T)]Calculate i) the average energy per bit Eb, ii) the minimumdistance d2min, and iii) the average symbol error probability(high-SNR approximation) in the form Q(
Eb/N0).
120 c Prof. Giorgio Taricco c WIRELESS COMMUNICATIONS
Digital modulations over the AWGN channel Problem set 3
Problem set 3 (cont.)
11 Calculate the error probability of a 4-PSK signal set assumingthat the receiver has a constant phase offset that rotates the
decision regions b an angle
8/12/2019 Notes 2013a
129/142
decision regions by an angle .12 Calculate the error probability of an octonary signal set whose
signals are located over two concentric circles with rays 1 and0.5 +
1.5. The signals are equally spaced over each circle
and have a phase offset of/4 radians between thecorresponding signals over different circles.
13 Calculate the error probability of the digital modulation basedon the following four signals:
sm(t) = sin 52Tt (m 1) T5 1|tmT/5|T/5form= 1, 2, 3, 4.
121 c Prof. Giorgio Taricco c WIRELESS COMMUNICATIONS
Digital modulations over the AWGN channel Power density spectrum of digital modulated signals
Power density spectrum of digital modulations
The power density spectrum ofx(t) = n an(t nT),where an is a wide-sense stationary sequence with
autocorrelation function Ra(p) E[an+pan] can be expressed
8/12/2019 Notes 2013a
130/142
autocorrelation function Ra(p) = E[an+pan], can be expressedas the product of two terms:
Power density spectrum: Gx(f) =Sa(f) G(f)
S
a(f) p Ra(p)ej 2pfT (data spectrum)
G(f) 1
T|(f)|2 (pulse spectrum)
In many circumstances, the bandwidth of a digital signal isapproximated by an expression depending only on thesignalling interval T.
122 c Prof. Giorgio Taricco c WIRELESS COMMUNICATIONS
Digital modulations over the AWGN channel Power density spectrum of digital modulated signals
Shannon bandwidth
A common approximation to the bandwidth of a digital signal
is the Shannon bandwidth:
8/12/2019 Notes 2013a
131/142
is theShannon bandwidth:
Wsh Nd 12T
where Nd is the signal space dimension and T is the symbol
interval.It can be shown that this approximation is very good whenthe number of dimensions is large.
However, even with Nd= 1, the bandwidth overhead is
limited for suitably chosen pulses, as illustrated in thefollowing example.
123 c Prof. Giorgio Taricco c WIRELESS COMMUNICATIONS
8/12/2019 Notes 2013a
132/142
8/12/2019 Notes 2013a
133/142
Digital modulations over the AWGN channel Power density spectrum of digital modulated signals
Bandwidth of antipodal signals (cont.)
0.8
1
8/12/2019 Notes 2013a
134/142
0 0.5 1 1.5 20
0.2
0.4
0.6
0.8
WT
(W)
square
sine
126 c Prof. Giorgio Taricco c WIRELESS COMMUNICATIONS
Digital modulations over the AWGN channel Power density spectrum of digital modulated signals
Bandwidth of antipodal signals (cont.)One way to limit the bandwidth occupation of a digitalmodulation signal consists of extending the duration of themodulation pulse beyond the signalling interval (0, T).
When the signalling pulse is limited to the signalling interval
8/12/2019 Notes 2013a
135/142
When the signalling pulse is limited to the signalling interval,the modulation signal is calledfull response. When itsduration exceeds T, it is calledpartial response.
Stretching in time the signalling pulse by a factor
corresponds to an equivalent stretching in the frequencydomain by the inverse of:
(t) (t/) (f) (f) (W) (W).
The price to be payed is related to the generation ofintersymbol interference.
127 c Prof. Giorgio Taricco c WIRELESS COMMUNICATIONS
Digital modulations over the AWGN channel Comparison of digital modulations
Key parameters
The performance of different modulation schemes is describedby three system parameters:
1 Error probability (symbol or bit).
2 Spectral efficiency, i.e., the ratio between the bit rate Rb and
8/12/2019 Notes 2013a
136/142
Spectral efficiency, i.e., the ratio between the bit rate R andthe occupied bandwidth W.
3 The signal-to-noise ratio Eb/N0.
For Nd-dimensional signal sets, the occupied bandwidth isapproximately equal to the Shannon bandwidth
Wsh=Nd 12T
= Nd Rb2log2 M
Hence, the spectral efficiency is given by
b RbWsh
= 2log2 M
Nd.
128 c Prof. Giorgio Taricco c WIRELESS COMMUNICATIONS
Digital modulations over the AWGN channel Comparison of digital modulations
Spectral efficiency
The spectral efficiency b grows slowly (logarithmically) with
8/12/2019 Notes 2013a
137/142
The spectral efficiency grows slowly (logarithmically) withthe constellation size Mand decreases rapidly (linearly) withthe number of dimensions Nd.
For a fixed M, PAM modulations have higher spectralefficiency than orthogonal modulations. Therefore,
PAM modulations are used in channels with limited bandwidth(bandwidth limited channels) and high power.
Orthogonal modulations are used in channels with limitedpower (power limited channels) and large bandwidth.
129 c Prof. Giorgio Taricco c WIRELESS COMMUNICATIONS
Digital modulations over the AWGN channel Comparison of digital modulations
Shannons bound
Shannons theorem yields the maximum bit rate that can be
sustained with arbitrarily low error probability by an
8/12/2019 Notes 2013a
138/142
sustained with arbitrarily low error probability by anNd-dimensional digital modulation with symbol interval Tover an AWGN channel:
Rb =
Nd
2T log21 + SN. (26)Here, S is the received power, N is the noise power, and S/Nis calledsignal-to-noise ratio.
We assume that the signal bandwidth is Wsh=Nd/(2T).
130 c Prof. Giorgio Taricco c WIRELESS COMMUNICATIONS
Digital modulations over the AWGN channel Comparison of digital modulations
Shannons bound (cont.)Since the noise power is N=N0Wsh and the signal power isS=Eb/Tb =RbEb,(26)can be written as:
Rb Wshlog21 + RbEbWshN0.
8/12/2019 Notes 2013a
139/142
g + WshN0Since the spectral efficiency is b=Rb/Wsh, we obtain:
b log21 + b EbN0 EbN0 2b 1b .Finally, for b
0, we have
EbN0
ln 2
EbN0
dB
10log10(ln2) 1.6 dB.
131 c Prof. Giorgio Taricco c WIRELESS COMMUNICATIONS
Digital modulations over the AWGN channel Comparison of digital modulations
Shannons bound (cont.)
50
60
1024-PAMShannon's bound
Pb(e)=1e-4
Pb(e)=1e-6
8/12/2019 Notes 2013a
140/142
10-2
10-1
100
101
0
10
20
30
40
Rb/W [bit/s/Hz]
Eb
/N0
[dB]
4-PPM
1024-PPM
2-PAM
-1.6 dB
b( )
Pb(e)=1e-8
132 c Prof. Giorgio Taricco c WIRELESS COMMUNICATIONS
Digital modulations over the AWGN channel Problem set 4
Problem set 4
1 Derive the power density spectrum formulaGx(f) =Sa(f) G(f) for the signal
(t)
(t T ) (27)
8/12/2019 Notes 2013a
141/142
x(t) =n
an(t nT), (27)
where
an is a wide-sense stationary sequence with autocorrelationfunction Ra(p) = E[an+pan];
Sa(f)
p Ra(p)ej 2pfT is the data spectrum;
G(f) 1T|(f)|2 is the pulse spectrum.
Hint: Consider the (randomly delayed and stationary) signal
x(t ), with uniformly distributed in (0, T), and calculatethe Fourier transform of its autocorrelation function to obtainthe power density spectrum.
133 c Prof. Giorgio Taricco c WIRELESS COMMUNICATIONS
Digital modulations over the AWGN channel Problem set 4
Problem set 4 (cont.)
2 Calculate the power density spectrum of the signal (27)
assuming that the symbols an are uncorrelated with mean ad i 2
8/12/2019 Notes 2013a
142/142
g y and variance 2a.
3 Calculate the power density spectrum of the signal (27)assuming that the transmitted symbols an have zero meanand correlation Ra(m) =
|m| (where
(0, 1)), (t) hasunit energy, and the average signal power is P.
4 Calculate the power density spectrum of(27)assuming thatthe transmitted symbols are iid and taken from a 4-PSKsignal set with probabilities (0.7, 0.1, 0.1, 0.1).
134 c Prof. Giorgio Taricco c WIRELESS COMMUNICATIONS