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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
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
Pav
1
log( 1  2 )
2
n
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 Eb
Pe  Q(
)
N0
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
C ( SNR)  log( 1  SNR) bit per second per Hertz
• Definition of multiplexing gain
C ( SNR)
r  lim
SNR  log( SNR)
Diversity Gain
• Dimension used to improve reliability
• Unit benchmark: single link Rayleigh fading
channel
 out
1

SNR
• Definition of diversity gain
log(  out ( SNR))
d   lim
SNR 
log( 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
rM R Tc  H M R M T bM T Tc  nM R Tc
where Tc is the coherence time
Transmitter
Receiver
Basic Assumption
• The additive noise is Gaussian
nM R
N0
~ Gaussian (0,
I M R M R )
2
• The average power constraint
Trace{E[ bMT b ]}  Pav
H
MT
Matrices
• A channel matrix
H M R M T
 h11*
 h11  h1M R 


 H
 

 , H M T M R   
 h*
hM 1  hM M 
T
R 
 T
 1M R
• Trace of a square matrix
M
Trace[ H M M ]   hii
i 1
 hM* T 1 


 
*
 hM M 
T R 
Matrices
• The Frobenius norm
H
 Trace[ HH ]  Trace[ H H ]
H
F
H
• Rank of a matrix = number of linearly
independent rows or column
Rank [ H ]  min{ M R , MT }
• Full rank if
Rank [ H ]  min{ M R , MT }
Matrices
• A square matrix is invertible if there is a matrix
1
AA  I
• The determinant—a measure of how
noninvertible a matrix is!
• A square invertible matrix U is unitary if
UU H  I
Matrices
• Vector X is rotated and scaled by a matrix A
y  Ax
• A vector X is called the eigenvector of the matrix
and lambda is the eigenvalue if
Ax  x
• Then
A  U U
H
with unitary and diagonal matrices
Matrices
• The columns of unitary matrix U are
eigenvectors of A
• Determinant is the product of all eigenvalues
• The diagonal matrix
 1  0 


    
0   
N 

Matrices
• If H is a non square matrix then
H M R M T  U M R M R  M RMT V
H
M T M T
• Unitary U with columns as the left singular vectors
and unitary V matrix with columns as the right
singular vectors
• The diagonal matrix 1  0 
 M R M T

 0

0


  1  0


  M T or    
 
 0   0  M R
 

0 




0 
Matrices
• The singular values of H are square root of
eigenvalues of square H
i  singular( H M
R M T
)
 i  eigenvalue ( H M R M T H MHT M R )
MIMO Channels
• There are M T  M R channels
– Independent if
• Sufficient separation compared to carrier wavelength
• Rich scattering
– At transmitter
– At receiver
• The number of singular vectors of the channel
 min{ MT , M R }
• The singular vectors are the additional (spatial)
degrees of freedom
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
I (rM R ; bM T | H M R M T )
 h(rM R | H M R M T )  h(rM R | bM T , H M R M T )
 h(rM R | H M R M T )  h(nM R )
 h(rM R | H M R M T )  M R log( eN 0 )
 log[( e) M R det( N 0 I M R M R  HQH * )]  M R log( eN 0 )
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
h(Y )  h( X )    fY log fY dY   f Gaussian log f GaussiandX
   fY log fY dY   fY log f GaussiandY

f Gaussian
fY log
dY  0
fY
Solution
• Since X and Y have the same covariance Q
then
f
log f GaussiandX   f Gaussian[ X QX ]dX 
*
Gaussian
  fY [Y QY ]dY
*
  fY log f GaussiandY
Fixed Channel
• The achievable rate
max I (b; r )  log[( e) M R det( N 0 I M R M R  HQH * )]  M R log( eN 0 )
pb
 log det( I M R M R
HQH *

)
N0

*
Q

E
[
bb
]
and
E
[
b
b]  Pav
with MT MT
• Differential entropy maximizer is a complex
Gaussian random vector with some covariance
matrix Q
Fixed Channel
• Finding optimum input covariance
• Singular value decomposition of H
H  UV * 
min{ M R , M T }
*

u
v
 mmm
m 1
• The equivalent channel
~
~
*
~
~
~
rM R   M R M T bM T  nM R with r  U r and b  V *b
Parallel Channels
• At most min{ MT , M R } parallel channels
~ ~
~
rm  mbm  nm ; m  1,2,, min {M R ,M T }
• Power distribution across parallel channels
QMT MT  E[bb ] and E[b*b]  tr (Q)  tr (VQV * )  Pav
Parallel Channels
• A few useful notes
*
log det( I M R M R
HQH

)  log det( I M T M T 
N0
 log det( I M R M R 
 log det( I M R M R
QM T M T H M* T M R H M R M T
N0
 M R M T VM T M T QV **M T M R
~ *
Q

)
N0
~ 2
 log  (1  Qmmm )
m
)
N0
)
Parallel Channels
• A note
~ *
Q
~ 2
det( I M R M R 
)  (1  Qmmm )
m
N0
~
with equality w hen Q is diagonal
Fixed Channel
• Diagonal entries found via water filling
• Achievable rate
I ( r ; b) 
min{ M R , M T }

m 1
P
log( 1 
)
N0
* 2
m m
with power
P  ( 
*
m
N0

2
m

) with
P
*
m
m
 Pav
Example
• Consider a 2x3 channel
H 32
1 1  1 / 3 


 1 1   1 / 3  6
1 1  1 / 3 

 

  1 / 2
1/ 2
• The mutual information is maximized at
6P
*
I (r; b)  log( 1 
) with E[bi b j ]  P / 2
N0

Example
• Consider a 3x3 channel
1 0 0


H   0 1 0
0 0 1


• Mutual information is maximized by
P
P
I (r ; b)  3 log( 1 
) with Q  I 33
3N 0
3
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
I ( H M R M T , rM R ; bM T )  I ( H ; b)  I (r; b | H )
 I (r; b | H )  E[ I (r; b | H  h]
MIMO with CSIR
• The achievable rate
max I (r; b)  log[( e) M R det( N 0 I M R M R  HQH * )]  M R log( eN 0 )
pb
since the differential entropy maximizer is a
complex Gaussian random vector with some
covariance matrix Q
Fast Fading and CSIR
• Finally,
I (b; r )  E[log det( I M R M R
HQH 

)]
N0

with Q

E
[
bb
]
M T M T

E
[
b
b]  Pav
• The scalar power constraint
• The capacity achieving signal is circularly
symmetric complex Gaussian (0,Q)
MIMO CSIR
• Since Q is non-negative definite Q=UDU*
HQH 
( HU ) D( HU )*
I (b; r )  E[log det( I 
)]  E[log det( I 
)]
N0
N0
• Focus on non-negative definite diagonal Q
• Further, optimum Q  I M T  M T

Pav HH
I (b; r )  E[log det( I 
)]
M T N0
Rayleigh Fading MIMO
• CSIR achievable rate

Pav HH
I (b; r )  E[log det( I 
)]
M T N0
• Complex Gaussian distribution on H
• The square matrix W=HH*
– Wishart distribution
– Non negative definite
– Distribution of eigenvalues
Ergodic / Fast Fading
• The channel coherence time is Tc  1
• The channel known at the receiver
C  E{log det( I M R M R
Pav

HH  )}
M T N0
• The capacity achieving signal b must be
circularly symmetric complex Gaussian
(0, ( Pav / M T ) I M T M T )
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
I (r ; b)  log det( I M R M R 
H M R M T QH M* T M R
N0
)
• The outage probability with CSIR only
 out ( R, Pav )  inf Pr[log det( I M R M R
Q
with E[bb]  Pav and Q  E[bb ]
HQH 

)  R]
N0
Slow Fading
• Since
Pr[log det( I M R M R
HQH 
HUQU * H 

)  R]  Pr[log det( I M R M R 
)  R]
N0
N0
• Diagonal Q is optimum
• Conjecture: optimum Q is
Qopt
1



 1


1
Pav 



m 



0




0


Example
• Slow fading SIMO, M T  1
• Then Qopt  Pav and
Pr[log det( I M R M R
Pav H * H
HQH 

)  R]  Pr[log( 1 
)  R]
N0
N0
• Scalar H * H is  2 distribute d
N 0 ( e R 1)
Pav
M R 1 u
 out ( R, Pav ) 
u
e du
0
( M R )
Example
• Slow fading MISO, M R  1
• The optimum
Qopt
Pav

I mm for some m  M T
m
• The outage
N 0 m ( e R 1)
Pa v
m 1 u
*
Pav HH
Pr[log( 1 
)  R] 
mN0
u
e du
0
 ( m)
Diversity and Multiplexing for MIMO
• The capacity increase with SNR
SNR
C  k log( 1 
)
k
• The multiplexing gain
C ( SNR)
r  lim
SNR  log( SNR)
Diversity versus Multiplexing
• The error measure decreases with SNR
increase  SNR  d
• The diversity gain
log(  out ( SNR))
d   lim
SNR 
log( SNR)
• Tradeoff between diversity and multiplexing
– Simple in single link/antenna fading channels
Coding for Fading Channels
• Coding provides temporal diversity
FER  g c SNR
d
or
P(C  E )  g c SNR  d
• Degrees of freedom
– Redundancy
– No increase in data rate
M versus D
Diversity Gain
(0,MRMT)
(min(MR,MT),0)
Multiplexing Gain
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