OFDM for Broadband Communication Course Reader (EIT 140)

167 Pages·2013·3.2 MB·English

OFDM for Broadband Communication
Course Reader (EIT 140)
(cid:13)
2013
Edited by Thomas Magesacher
Department of Electrical and Information Technology
Lund University
Box 118, S-221 00 LUND
SWEDEN
ISBN 91-7167-036-X
Second Edition (V2.0.6)
c authors of individual chapters
(cid:13)
Contents
Contents iii
Preface v
Chapter I — Multicarrier Modulation 1
Chapter II — Channels and Channel Models 25
Chapter III — Data Detection, Channel Equalisation, Channel Estimation 45
Chapter IV — Bit- and Power-Loading 55
Chapter V — Coding for Multicarrier Modulation 65
Chapter VI — Synchronisation for OFDM(A) 95
Chapter VII — Reduction of PAR 113
Chapter VIII — Spectrum efficiency in OFDM(A) 119
Chapter IX — MIMO OFDM and Multiuser OFDM 141
Problems 147
iv
Preface
This course reader is a compilation of contributions from several authors. Additionally, the following
literature is recommended for broadening and deepening the reader’s view on the subject:
J.B. Anderson, Digital Transmission Engineering, IEEE Press, ISBN 0-7803-3457-4, 1999.
•
E. Biglieri and G. Taricco, Transmission and Reception with Multiple Antennas: Theoretical Foun-
•
dations, now Publishers Inc., ISBN 1933019018, 2004.
J. M. Cioffi, “Advanced Digital Communication,” class reader EE379C, Stanford University,
•
Available from http://www.stanford.edu/class/ee379c/
R. Johannesson and K. Zigangirov, Fundamentals of Convolutional Coding, IEEE Press, ISBN
•
0-7803-3483-3, 1999.
S.LinandD.J.Costello, ErrorControlCoding, PrenticeHall,ISBN0-13-042672-5,secondedition,
•
2004.
J. L. Massey, Applied Digital Information Theory—Lecture Notes, ETH Zurich, Available from
•
www.isi.ee.ethz.ch/education/public/pdfs/aditI.pdf.
A. Paulraj, R. Nabar, and D. Gore, Introduction to Space-Time Wireless Communications, Cam-
•
bridge, ISBN 0521826152, 2003.
J. G. Proakis, Digital Communications, Mc Graw Hill, ISBN 0-07-232111-3, fourth edition, 2001.
•
T.Starr,J.M.Cioffi,andP.Silverman, UnderstandingDigitalSubscriberLineTechnology, Prentice
•
Hall, Englewood Cliffs, 1998.
D.TseandP.Viswanath,FundamentalsofWirelessCommunications,Cambridge,ISBN0521845270,
•
2005.
R.VanNeeandR.Prasad, OFDM for Wireless Multimedia Communications, ArtechHouse,ISBN
•
0890065306, 2003.

Part I
The Elegance of OFDM and DMT
1 Sinusoids, multiplexing and... a miracle?
Eversincetheearlydaysofradiocommunicationsoneoftheguidingnotionshasbeenthattheamplitude
and the phase of a sinusoidal radio wave can carry information from a transmitter to a receiver. Why
sinusoids? Sinusoids were easy to generate with existing hardware components, their phase and ampli-
tude could be simply modulated and demodulated using available components, and, moreover, collision
of signals from different radio transmitters could be simply avoided by a proper choice of the carrier
frequencies – by proper frequency planning. Sinusoids have been the key signal components in radio
communications ever since.
The availability of hardware components that could generate sinusoids has played an important role
in the popularity of sinusoids. Another reason for their popularity builds on a key appearance of the
radio environment: its linearity. This linearity is quite remarkable; many phenomena in nature appear
to be non-linear. Fluid mechanics, for instance, is governed by non-linear differential equations, the
weather forecast cannot be made reliably for more than three days ahead because of the chaotic nature
of atmospheric mechanics, and mechanics of materials does not follow linear rules. Non-linear systems
havebeendifficulttoaddressmathematicallyandconsequentlyhavebeenhardertoexploit. Radiowaves,
however,closelyfollowMaxwell’slinearrelationsandcanbeaddressedwithwell-developedmathematical
tools from linear algebra.
Mathematically, infinitely long sinusoids have a typical relation to linear environments: only the
amplitudeandthephaseofasinusoidalradiowaveformareaffectedbytheenvironment,notitsfrequency.
In other words, an infinitely long sinusoidal radio wave of a certain frequency will, after propagation
through the linear radio channel, be observed at the receiver as an infinitely long sinusoidal radio wave
with the same frequency – only its amplitude and its phase are changed. Because of this property,
sinusoids are generally being referred to as eigenfunctions of linear systems. A combination of the
availabilityofsuitablehardwarecomponents, thewell-developedmathematicalfieldoflinearalgebraand
theabove-discussedeigen-propertymakessinusoidsparticularlysuitableforuseinradiocommunications.
In today’s digital communication systems, sinusoids play an important role, too. Based on a stream
of information bits, Quadrature Amplitude Modulation (QAM), which is one of the most fundamental
modulation schemes, changes the amplitude and/or the phase of a sinusoid every T seconds, which
sym
we call the symbol period. The information bits are mapped consecutively onto pairs of real numbers
x(i),x(q) , referred to as transmit symbols. This mapping, implemented for example by means of a
{ k k }
look-up table, has a strong impact on the performance of the scheme. A QAM modulator generates a
continuous-time signal
s(t)=p(t kT )(x(i)cos(2πF t) x(q)sin(2πF t)), kT t<(k+1)T ,
− sym k c − k c sym ≤ sym
where F is the so called carrier frequency in Hz—a sinusoidal signal with frequency F carries the
c c
information we want to transmit. The waveform p(t) shapes each transmit symbol in time domain and
ChapterwrittenbyP.O¨dling,J.J.vandeBeek,D.Landstro¨m,P.O.Bo¨rjesson,andT.Magesacher.
2 Part I The Elegance of OFDM and DMT
thus determines the frequency characteristic of the s(t). Using Euler’s formula, the modulator output
s(t) can also be written as
s(t)=Re p(t kT ) x(i)+jx(q) ej2πFct , kT t<(k+1)T , (1)
− sym k k sym ≤ sym
n (cid:16) (cid:17) o
which is a frequently used description. The complex notation is a mathematical convenience and serves
as a means to describe two orthogonal real-valued dimensions. Amplitude and phase of s(t) are two
independent parameters, which are implicitly controlled by the two independent symbols x(i),x(q) .
{ k k }
The left plot of Figure 1 shows an exemplary waveform s(t) of three transmitted symbols. For the
sakeofsimplicity,wechoosearectangularpulsep(t)=1,0 t T . Introducingthecomplexnotation
sym
≤ ≤
x(i)+jx(q), each symbol can be interpreted as a point in the complex plane. The right plot illustrates
k k
the set of valid transmission symbols x(i),x(q) as points in the complex plane, which is commonly
{ k k }
referred to as constellation plot. The three symbols modulated onto the waveform are marked by plus
signs (+). The term ej2πFct represents a pointer rotating counter-clockwise in the complex plane and
changing its length and its phase (angle between the pointer and the real axis) every T seconds to
sym
(x(i))2+(x(q))2 andarctan(x(q)/x(i)),respectively. TheoperationRe in(1),i.e.,extractingthereal
k k k k {·}
pqartofthisrotatingpointer,correspondstotheprojectionofthepointerontotherealaxisofthecomplex
plane. A receiver tunes its local oscillator to the same radio frequency and tracks the phase/amplitude
of the received sinusoid, an operation commonly referred to as demodulation.
Since amplitude and phase of the modulated sine wave now change with the symbol period, the
resulting signal is not mathematically an eigenfunction of the linear channel. After all, only infinitely
long sinusoids with constant amplitude and phase have the above described invariance property. The
fact that the practically used signals are not eigenfunctions of the linear channel has some important
practicalimplications. Toillustratethis,wefocusnowononeoftheimmediatelymeasurableconsequences
appearing when we transmit two modulated sinusoids in parallel. This concept, known as frequency-
divisionmultiplexing inthecontextofaccommodatingseveraluserscommunicatingoverthesamechannel,
isameanstoincreasethedataratewithouthavingtoshortenthesymbolperiod—atthecostofincreasing
bandwidth.
Opposite to how true infinitely long sinusoids would behave, any two such modulated sinusoids now
generally interfere with each other. In other words, both before and after passing through the linear
channel, one sinusoid can actually be measured at other frequencies. This undesirable interference limits
the reliable detection of the amplitudes/phases and needs to be avoided. One effective way to avoid this
interferenceistochoosethefrequenciesofthesinusoidssufficientlyfarapart. Theinterferenceontoother
frequencies is generally largest on frequencies close to that of the transmitted carrier and smaller when
the two frequencies are chosen far from each other.
We now describe an experiment in order to investigate how much interference occurs and answer the
question how far apart the frequencies need to be chosen. We modulate five sinusoids of different (but
equally spaced) frequencies and transmit them in parallel over a linear radio channel. The amplitude in
eachsymbolintervalcantaketwovalues,thefirstamplitudelevelrepresentinga‘one’andtheotherlevel
a ‘zero’. Thus we transmit 5 bits per symbol interval. At the receiver, we demodulate the sinusoids (we
5
x(i),x(q)
3 { 1 1 }
x(i),x(q)
1 { 0 0 }
0
1
− x(i),x(q)
3 { 2 2 }
−
5
−
0 T 2T 3T 3 1 1 3
sym sym sym
− −
Figure 1: Amplitude/phase modulation of a sinusoid.
Part I The Elegance of OFDM and DMT 3
0
2 0 2 4 6 8
−
0
2 0 2 4 6 8
−
0
2 0 2 4 6 8
−
0
2 0 2 4 6 8
−
0
2 0 2 4 6 8
−
0
2 0 2 4 6 8
−
Figure 2: Multicarrier transmission.
extracttheamplitudeineachofthesymbolintervals)andwemakedecisionsforeachsymbolintervaland
foreachofthemodulatedsinusoidsastowhethertheamplituderepresenteda‘zero’ora‘one’. Theheart
of the experiment is that we vary the frequency-spacing of the sinusoids. We expect the bit-decisions at
the receiver to become more and more erroneous as the frequencies are chosen closer together, because
we expect the interference to increase.
Figure 2illustrateswhathappenstothereceptionofthebits(hereassociatedwithablackorawhite
pixel) for various choices of the frequency spacing. The plots on the left show the Fourier transforms,
which reveal the frequency characteristics, of the transmitted signals; in the plots on the right, black
pixels illustrate erroneous detections at the receiver. As we anticipated, when we choose the frequencies
of the two modulated sinusoids far apart, very little interference occurs, and, consequently, the receiver’s
detector hardly makes any errors (topmost figures). When we choose other frequencies, closer together,
the receiver decisions become less reliable. The number of erroneous decisions gradually increases, again
in line with our expectations. We again decrease the frequency spacing of the carriers and suddenly... a
miracle occurs! Just when the level of interference seems to prevent any reliable decisions at the receiver
we obtain error-free transmission! Is this possible? Is something wrong with experimental set-up? With
4 Part I The Elegance of OFDM and DMT
the equipment? We immediately choose an even smaller frequency difference and see: the interference
and the associated bit-errors are back again (lowermost figures). Changing back to the previous setting
confirms what we just observed: no bit-errors. Something very special is happening for this particular
choice of carrier frequencies. The reason for this seemingly miraculous behaviour will be revealed at the
end of this chapter.
OuraboveexperimentillustratestheeleganceofOrthogonalFrequencyDivisionMultiplex(OFDM)and
Discrete Multi-Tone (DMT) as it was first experienced and recognised in the sixties. In OFDM and
DMT the transmitted sinusoids are packed together to such a degree that our intuition does not allow
the thought of interference-free transmission. Yet this is exactly what is happening as the result of a
subtle choice of the sinusoid frequencies. The way in which OFDM and DMT avoid the inter-sinusoidal
interference is both ingenious and extremely efficient.
2 A desirable channel partitioning
One of the key aspects of digital communications is that there is no relation between the transmitted
waveforms and the signal representing the message to be transmitted. The message is assumed to be a
sequence of zeros and ones. How these bits represent the message is not relevant in the context of this
book. If we are not limited to any of the characteristics of the message signal, how should we choose the
transmitted waveforms? We could, for instance, choose waveforms that are simple to transmit, simple
to receive, or waveforms that have neat properties in the sense that they do not disturb other users.
The choice is enormous. By carefully adapting the waveforms to characteristics of the communications
channel, we can accomplish features that are desirable in a certain situation. In this book, we choose a
set of waveforms that, on one hand, is particularly simple to receive in linear channels, and, on the other
hand, uses the frequency spectrum in an efficient way.
Wewanttotransmitinformation-carryingreal-valuednumbersx ,representingthemessage. Forthis
k
purpose, we let each number x modulate one finite-length real-valued transmit signal component s (n).
k k
The word ‘modulate’ means that x scales the amplitude of s (n). We then transmit the sum of these
k k
modulated signal components, the multiplex
s(n)= x s (n). (2)
k k
k
X
Now we pass the signal multiplex s(n) through a linear time-invariant dispersive channel, characterised
by its impulse response h(n) = 0,n = 0,...,M, as schematically depicted in Figure 3. The length of
6
the channel impulse response is M +1. An input sample s(n ) fed into the channel at an arbitrary
0
time instant n does not just affect the output sample r˜(n ) but also influences M subsequent samples
0 0
r˜(n),n=n +1...n +M. Hence, we refer to M as the dispersion of the channel. Each transmit signal
0 0
components (n)producesbyitself, afterpassingthroughthechannel, areceivesignalcomponentr˜ (n).
k k
Due to the linearity property of the channel, we observe that the received signal r˜(n) is a multiplex of
the receive signal components r˜ (n),
k
r˜(n)=h(n) s(n)= x (h(n) s (n))= x r˜ (n), (3)
k k k k
∗ ∗
k k
X X
where denotes linear convolution.
∗
Thisequationsuggeststhatthetransmitandreceivesignalcomponentsappearinfixedpairs s (n),r˜ (n) .
k k
{ }
However, there are two problems that equation (3) does not immediately reveal. The first is that we
somehowneedtoseparatethesumofthereceivedwaveformsatthereceiverinordertoretrievethedata.
We will return to this problem in a moment. The second problem appears if we transmit subsequent
blocks of length N, one after another, which is necessary in order to convey information. Due to the
dispersive channel, a fixed relation between s (n) and r˜ (n) is not quite obvious. After passing through
k k
s(n) r˜(n)
h(n)
Figure 3: The discrete-time linear channel.