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Wireless Communication Elec 534 Set IV October 23, 2007. Behnaam Aazhang. Reading for Set 4. Tse and Viswanath Chapters 7,8 Appendices B.6,B.7 Goldsmith Chapters 10. Outline. Channel model Basics of multiuser systems Basics of information theory - PowerPoint PPT Presentation
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Wireless CommunicationElec 534
Set IVOctober 23, 2007
Behnaam Aazhang
Reading for Set 4
• Tse and Viswanath– Chapters 7,8– Appendices B.6,B.7
• Goldsmith– Chapters 10
Outline• Channel model• Basics of multiuser systems• Basics of information theory• Information capacity of single antenna single
user channels– AWGN channels– Ergodic fast fading channels– Slow fading channels
• Outage probability• Outage capacity
Outline
• Communication with additional dimensions– Multiple input multiple output (MIMO)
• Achievable rates• Diversity multiplexing tradeoff• Transmission techniques
– User cooperation• Achievable rates• Transmission techniques
Dimension
• Signals for communication– Time period T– Bandwidth W– 2WT natural real dimensions
• Achievable rate per real dimension
)1log(21
2navP
Communication with Additional Dimensions: An Example
• Adding the Q channel– BPSK to QPSK
• Modulated both real and imaginary signal dimensions• Double the data rate• Same bit error probability
)2(0N
EQP be
Communication with Additional Dimensions
• Larger signal dimension--larger capacity– Linear relation
• Other degrees of freedom (beyond signaling)– Spatial– Cooperation
• Metric to measure impact on– Rate (multiplexing)– Reliability (diversity)
• Same metric for– Feedback– Opportunistic access
Multiplexing Gain
• Additional dimension used to gain in rate• Unit benchmark: capacity of single link AWGN
• Definition of multiplexing gain
Hertzper secondper bit )1log()( SNRSNRC
)log()(lim
SNRSNRCr
SNR
Diversity Gain
• Dimension used to improve reliability• Unit benchmark: single link Rayleigh fading
channel
• Definition of diversity gain
SNRout1
)log())(log(lim
SNRSNRd out
SNR
Multiple Antennas
• Improve fading and increase data rate• Additional degrees of freedom
– virtual/physical channels– tradeoff between diversity and multiplexing
Transmitter Receiver
Multiple Antennas
• The model
where Tc is the coherence time
cRcTTRcR TMTMMMTM nbHr
Transmitter Receiver
Basic Assumption
• The additive noise is Gaussian
• The average power constraint
)2
,0(Gaussian~ 0RRR MMM INn
avHMM Pbb
TT]}Trace{E[
Matrices
• A channel matrix
• Trace of a square matrix
**
*1
*11
1
111
1
,
RMTMRM
T
RT
RTT
R
TR
hh
hhH
hh
hhH
MH
MM
MMM
M
MM
M
iiiMM hH
1
]Trace[
Matrices
• The Frobenius norm
• Rank of a matrix = number of linearly independent rows or column
• Full rank if
][Trace][Trace HHHHH HHF
},min{][Rank TR MMH
},min{][Rank TR MMH
Matrices
• A square matrix is invertible if there is a matrix
• The determinant—a measure of how noninvertible a matrix is!
• A square invertible matrix U is unitary if
IAA 1
IUU H
Matrices• Vector X is rotated and scaled by a matrix A
• A vector X is called the eigenvector of the matrix and lambda is the eigenvalue if
• Then
with unitary and diagonal matrices
Axy
xAx
HUUA
Matrices• The columns of unitary matrix U are
eigenvectors of A• Determinant is the product of all eigenvalues • The diagonal matrix
N
0
01
Matrices
• If H is a non square matrix then
• Unitary U with columns as the left singular vectors and unitary V matrix with columns as the right singular vectors
• The diagonal matrix
HMMMMMM TTTMRMRRTR
VUH
00
00or
000
0
1
1
R
TTR
M
MMM
Matrices
• The singular values of H are square root of eigenvalues of square H
)(eigenvalue
)singular(
iH
MMMM
MMi
RTTR
TR
HH
H
MIMO Channels• There are channels
– Independent if• Sufficient separation compared to carrier wavelength• Rich scattering
– At transmitter– At receiver
• The number of singular vectors of the channel
• The singular vectors are the additional (spatial) degrees of freedom
RT MM
},min{ RT MM
Channel State Information
• More critical than SISO– CSI at transmitter and received– CSI at receiver– No CSI
• Forward training• Feedback or reverse training
Fixed MIMO Channel
• A vector/matrix extension of SISO results• Very large coherence time
)log()]det()log[(
)log()|(
)()|(
),|()|(
)|;(
0*
0
0
eNMHQHINe
eNMHrh
nhHrh
HbrhHrh
HbrI
RMMM
RMMM
MMMM
MMMMMMM
MMMM
RR
R
TRR
RTRR
TRTRTRR
TRTR
Exercise
• Show that if X is a complex random vector with covariance matrix Q its differential entropy is largest if it was Gaussian
Solution
• Consider a vector Y with the covariance as X
0log
loglog
loglog)()(
dYf
ff
dYffdYff
dXffdYffXhYh
Y
GaussianY
GaussianYYY
GaussianGaussianYY
Solution
• Since X and Y have the same covariance Q then
dYff
dYQYYf
dXQXXfdXff
GaussianY
Y
GaussianGaussianGaussian
log
][
][log*
*
Fixed Channel
• The achievable rate
with • Differential entropy maximizer is a complex
Gaussian random vector with some covariance matrix Q
)det(log
)log()]det()log[();(max
0
*
0*
0
NHQHI
eNMHQHINerbI
RR
RR
R
b
MM
RMMM
p
avMM PbbEbbEQTT
][ and ][ *
Fixed Channel
• Finding optimum input covariance• Singular value decomposition of H
• The equivalent channel
},min{
1
**TR MM
mmmm vuVUH
bVbrUrnbrRTTRR MMMMM
** ~ and ~ with ~~~
Parallel Channels
• At most parallel channels
• Power distribution across parallel channels
},M{M,,,mnbr TRmmmm min21; ~~~
},min{ RT MM
avMM PVQVQbbEbbEQTT
)(tr)(tr][ and ][ **
Parallel Channels
• A few useful notes
)~1(log
)~
det(log
)det(log
)det(log)det(log
2
0
*
0
**
0
*
0
*
mmmm
MM
MMMMMMMM
MMMMMMMMMM
Q
NQI
NQVV
I
NHHQ
IN
HQHI
RR
RTTTTR
RR
TRRTTT
TTRR
Parallel Channels
• A note
diagonal is ~hen equality wwith
)~1()~
det( 2
0
*
Q
QNQI mmmmMM RR
Fixed Channel
• Diagonal entries found via water filling• Achievable rate
with power
)1log();(},min{
1 0
2*
TR MM
m
mm
NPbrI
mavm
mm PPNP *
20* with )(
Example
• Consider a 2x3 channel
• The mutual information is maximized at
2/12/163/13/13/1
111111
23
H
2/][ with )61log();( *
0
PbbENPbrI ji
Example
• Consider a 3x3 channel
• Mutual information is maximized by
100010001
H
330 3
with )3
1log(3);( IPQNPbrI
Ergodic MIMO Channels
• A new realization on each channel use• No CSI• CSIR • CSITR?
Fast Fading MIMO with CSIR
• Entries of H are independent and each complex Gaussian with zero mean
• If V and U are unitary then distribution of H is the same as UHV*
• The rate
]|;([)|;(
)|;();();,(
hHbrIEHbrI
HbrIbHIbrHITRTR MMMM
MIMO with CSIR• The achievable rate
since the differential entropy maximizer is a complex Gaussian random vector with some covariance matrix Q
)log()]det()log[();(max 0*
0 eNMHQHINebrI RMMM
p RR
R
b
Fast Fading and CSIR• Finally,
with
• The scalar power constraint• The capacity achieving signal is circularly
symmetric complex Gaussian (0,Q)
)]det([log);(0N
HQHIErbIRR MM
][ bbEQ
TT MM
avPbbE ][
MIMO CSIR
• Since Q is non-negative definite Q=UDU*
• Focus on non-negative definite diagonal Q• Further, optimum
)])()(det([log)]det([log);(0
*
0 NHUDHUIE
NHQHIErbI
TT MMIQ
)]det([log);(0NM
HHPIErbIT
av
Rayleigh Fading MIMO
• CSIR achievable rate
• Complex Gaussian distribution on H• The square matrix W=HH*
– Wishart distribution– Non negative definite– Distribution of eigenvalues
)]det([log);(0NM
HHPIErbIT
av
Ergodic / Fast Fading
• The channel coherence time is• The channel known at the receiver
• The capacity achieving signal b must be
circularly symmetric complex Gaussian
1cT
)}det({log0
HH
NMPIECT
avMM RR
))/(,0(TT MMTav IMP
Slow Fading MIMO
• A channel realization is valid for the duration of the code (or transmission)
• There is a non zero probability that the channel can not sustain any rate
• Shannon capacity is zero
Slow Fading Channel
• If the coherence time Tc is the block length
• The outage probability with CSIR only
with and
)det(log);(0
*
NQHH
IbrI RTTR
RR
MMMMMM
])det(Pr[loginf),(0
RN
HQHIPRRR MMQavout
avPbbE ][ ][ bbEQ
Slow Fading
• Since
• Diagonal Q is optimum• Conjecture: optimum Q is
])det(Pr[log])det(Pr[log0
*
0
RN
HHUQUIRN
HQHIRRRR MMMM
00
11
1
mPQ av
opt
Example
• Slow fading SIMO, • Then and
• Scalar
avopt PQ
])1Pr[log(])det(Pr[log0
*
0
RN
HHPRN
HQHI avMM RR
1TM
ddistribute is 2* HH
)(),(
)1(
0
1
out
0
R
PeN
uM
av M
dueuPR
av
R
R
Example
• Slow fading MISO,• The optimum
• The outage
1RM
Tmmav
opt MmImPQ somefor
)(])1Pr[log(
)1(
0
1
0
*
0
m
dueuR
mNHHP
av
R
PemN
um
av
Diversity and Multiplexing for MIMO
• The capacity increase with SNR
• The multiplexing gain
)1log(k
SNRkC
)log()(lim
SNRSNRCr
SNR
Diversity versus Multiplexing
• The error measure decreases with SNR increase
• The diversity gain
• Tradeoff between diversity and multiplexing– Simple in single link/antenna fading channels
dSNR
)log())(log(lim
SNRSNRd out
SNR
Coding for Fading Channels
• Coding provides temporal diversity
or
• Degrees of freedom – Redundancy– No increase in data rate
dcSNRgFER
dcSNRgECP )(
M versus D
(0,MRMT)
(min(MR,MT),0)
Multiplexing Gain
Div
ersi
ty G
ain