Wireless Communications: An Information Theoretic Perspective

40 Pages·2012·2.05 MB·English

1/31/2012
Wireless Communications: An
Information Theoretic Perspective
Ajit Kumar Chaturvedi
Department of EE
IIT Kanpur
[email protected]
Main References
• David Tse and Pramod Viswanath, Fundamentals of
WirelessCommunication, Cambridge, 2005
• Andrea Goldsmith et al, Breaking Spectrum Gridlock
With Cognitive Radios: An Information Theoretic
Perspective, ProceedingsIEEE, May 2009, 894-914.
2
1
1/31/2012
Outline
• Information Theory
– Introduction and some key results
• Fading Channel
– Slow Fading
– Receive and Transmit Diversity
– Time and Frequency Diversity
– Fast Fading
• Multiuser Channel
– Uplink
– Capacity Region
• Cognitive Radio
3
Information Theory
• Whatis information theory ?
– A way of measuring/weighing/quantifying information in terms of
thequantityentropy
– Provides a way of characterizing information flow from source to
destinationintermsofthequantitymutualinformation
• Whydo we need to study information theory ?
– We need to „weigh‟ or „measure‟ or „quantify‟ the amount of
information or rather the information rate produced by a source,
sayavideocameraoraspeaker orafile……….
– We need to find out that given a fidelity criterion how much a bit
sequencecanbecompressed
– Weneed tofindoutthemaximum rateatwhich information can be
transmittedinapracticalchannel whichcanbemodeledasaBSC,
erasure,AWGN,fading,multiuser,multi-cell,……
– Weneedtofindways (i.e.codes) toachievethemaximum
4
2
1/31/2012
Key Qualitative Results of Information Theory
• There is a limit beyond which a bit sequence cannot be
compressed. Given the distribution of the sequence, this
limit can be computed. Compression methods for
achievingthat limit are known.
• A communication channel can be characterized in terms
of a number called capacity which is expressed in
bits/s/Hz.
• Even in the presence of noise and fading, coding
techniques are known for achieving reliable
communication over the channel as long as the rate of
informationflow does not exceed its capacity.
5
Information Theory
• What are the most significant contributions of information
theory?
– Lempel-Zivfamilyofalgorithms
– TurboandLDPCcodes
– MIMOormultipletransmitandmultiplereceiveantennas
– CapacityoftheAWGNchannel
• Whichmajor goals elude(failures)information theory ?
– Usefulinsightsintonetworks
– IncontrasttotraditionalShannontheory,sourceisbursty
– In contrast to traditional Shannon theory, end-to-end delay is
finite
6
3
1/31/2012
Some Queries
• We know of Nyquist‟s result regarding an upper limit on the
data rate for zero inter symbol interference in a given finite
bandwidth.
• How does Shannon‟s capacityresult reconcilewith that ?
• What is the fundamental question that one should ask
regardingdata rate through a communication channel ?
7
Limit on Spectral Efficiency
• Let R be the information rate, E be the energy per
b
informationbit and P be the transmitted power
• ThenP=E R
b
• Definer=R/W as the spectral efficiency
• Usingthe fact C > R and the relation
• We obtain the following condition for reliable
communication
4
1/31/2012
Performance of QAM and QPSK
• As we let spectral efficiency r → 0, we enter a power-
limited regime
Reliable rate of communication and capacity
• Reliable communication at rate R bits/symbol means that
one can design codes at that rate with arbitrarily small error
probability.
• To get reliable communication, one must code over a long
block; this is to exploit the law of large numbers to average
outthe randomness of the noise.
• Repetition coding over a long block can achieve reliable
communication, but the corresponding data rate goes to
zerowith increasingblock length.
10
5
1/31/2012
Capacity of Fading Channels
• Consider the complex baseband representation of a flat
fadingchannel
• Assume that the receiver can perfectly track the fading
processi.e. coherent reception
• This can be done by a pilot signal if the channel varies
slowlyrelativeto the symbol rate
• Let us first look at the situation when the channel gain is
random but remains constant i.e. h[m]=m for all m. This is
calledthe slow fadingor the quasi staticscenario.
11
Capacity of Fading Channels Contd..
• Conditional on a realization of the channel h, this is an
AWGNchannel with received SNR as |h|2 SNR
• What is the ultimate performance limit when information can
be coded over a sequence of symbols?
• The maximum rate of reliable communication supported by
thischannel is log(1+|h|2SNR) bits/s/Hz.
• If the channel realization h is such that log(1+|h|2SNR) < R,
then whatever the code used by the transmitter, the
decodingerror probabilitycannot be made arbitrarilysmall.
• The system is said to be in outage, and the outage
probabilityis
12
6
1/31/2012
Slow Fading Channel Contd..
• Density of log(1+|h|2SNR) , for Rayleigh fading and SNR = 0 dB.
Foranytarget rate R,there isa non-zerooutageprobability.
• the best the transmitter can do is to encode the data assuming
that the channel gain is strong enough to support the desired
rateR.
• Reliable communication can be achieved whenever that
happens,andoutageoccursotherwise. 13
Slow Fading Channel Contd..
• For Rayleigh fading (i.e., h is CN(0 1)), the outage probability
is
• At high SNR,
• Outageprobabilitydecays as 1/SNR.
• Coding cannot significantly improve the error probability in
a slow fading scenario.
• The reason is that while coding can average out the
Gaussian white noise, it cannot average out the channel
fade,which affects all the coded symbols.
14
7
1/31/2012
Slow Fading Channel Contd..
• There is a conceptual difference between the AWGN channel and
theslowfadingchannel.
• In the AWGN channel, one can send data at a positive rate (any
rate less than C) while making the error probability as small as
desired.
• This cannot be done for the slow fading channel as long as the
probabilitythatthe channelis indeepfadeis non-zero.
• Thus, the capacityof the slow fading channel in the strict sense is
zero.
• Analternativeperformancemeasureis the ϵ-outagecapacityC .
ϵ
• This is the largest rate of transmission R such that the outage
probabilityp (R) is less thanϵ.
out
where, F is the complementary cumulative distribution function of
|h|2 i.e. F(x)=P[|h|2 > x] 15
Slow Fading Channel Contd..
• ϵ-outage capacity as a fraction of AWGN capacity under
Rayleigh fading, for ϵ= 0.1 and ϵ= 0.01.
16
8
1/31/2012
Slow Fading Channel Contd..
• It is clear that the impact is much more significant in the low
SNR regime.
• At high SNR,
• At low SNR,
• Intuitively, the impact of the randomness of the channel is in
the received SNR, and the reliable rate supported by the
AWGN channel is much more sensitive to the received
SNR at low SNR than at high SNR.
17
Receive Diversity
• Let us increase the diversity of the channel by having L
receiveantennas insteadof one.
• Outage occurs whenever:
or
• Under independent Rayleigh fading, ||h||2 is a sum of the
squares of 2L independent Gaussian random variables and
isdistributedas Chi-squarewith 2Ldegrees of freedom.
18
9
1/31/2012
Receive Diversity Contd..
• Hence at high SNR, the outage probabilityis given by
• We see a diversity gain of L since the outage probability
now decays like1/(SNR)L
• This parallels the performance of uncoded transmission,
thuscoding cannot increase the diversitygain.
• At low SNR and smallϵ
• The loss with respect to the AWGN capacity is by a factor of
(ϵ)1/Lrather than by ϵ when there is no diversity.
19
Receive Diversity Contd..
• ϵ-outage capacity with L-fold receive diversity, as a fraction
of the AWGN capacity log(1+LSNR) for ϵ = 0.01 and
differentL.
• At ϵ=0.01 and L=2, the outage capacity is increased to 14%
of the AWGN capacity.
20
10