Capacity of correlated MIMO channels.ppt
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Transcript of Capacity of correlated MIMO channels.ppt
1
Capacity of Correlated MIMO Channels: Channel Power and
Multipath Sparsity
Akbar Sayeed(joint work with Vasanthan Raghavan,
UIUC)
Random Matrix Theory and Wireless Communications WorkshopBoulder, CO, July 14 2008
Wireless Communications Research LaboratoryDepartment of Electrical and Computer Engineering
University of Wisconsin-Madison
[email protected] , [email protected]
http://dune.ece.wisc.edu
2
Sergio Verdu – Hard Act to Follow!
Shannon (belly) Dance! (ISIT 2006, Seattle)
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Sergio Verdu – Model Incognito?
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Multipath Wireless Channels
• Multipath signal propagation over spatially distributed paths due to signal scattering from multiple objects – Necessitates statistical channel modeling– Accurate and analytically tractable Understanding the physics!
• Fading – fluctuations in received signal strength• Diversity – statistically independent modes of communication
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Antenna Arrays: Multiplexing and Energy Capture
Multiplexing – Parallel spatial channels
sec/bitsW
1logW),W(C 2
sec/bits1logNN
N1logN~),N(C 22
Array aperture: Energy capture
Wideband (W):
Multi-antenna (N):
Dramatic linear increase in capacity with number of antennas
6
Key Elements of this Work• Sparse multipath
– i.i.d. model – rich multipath– Seldom true in practice– Physical channels exhibit sparse multipath
• Modeling of sparse MIMO channels – Virtual channel representation (beamspace)– Physically meaningful channel power
normalization– Sparse degrees of freedom– Spatial correlation/coherence
• The Ideal MIMO Channel– Fastest (sub-linear) capacity scaling with N– Capacity-maximization with SNR for fixed N– Multiplexing gain versus received SNR tradeoff– Simple capacity formula for all SNRs (RMT)
• Creating the Ideal MIMO Channel in Practice– Reconfigurable antenna arrays– Three canonical configurations: near-optimum
performance over entire SNR range – Source-channel matching– New capacity formulation
Capa
city
MUX IDEAL BF
qC p, p log 1p
C(N)
sparse C N O Di.i.d.
C N O N
Correlated C N O N
N
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Virtual Channel Modeling
Spatial sampling commensurate with signal space resolution
Channel statistics induced by the physical scattering environment
Abstract statistical models
Physical models
Virtual Model
Tractable Accurate
Accurate & tractable
Interaction between the signal space and the physical channel
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Narrowband MIMO ChannelHsx
TR N
2
1
TRRR
T
T
N
2
1
s
ss
)N,N(H)2,N(H,)1,N(H
)N,2(H)2,2(H,)1,2(H)N,1(H)2,1(H,)1,1(H
x
xx
Received signal Transmitted signalTN
RNTransmit antennas
Receive antennas
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Uniform Linear Arrays
RR
R
)1N(2j
2j
RR
e
e
1
)(
aReceive response veector
/)sin(d RRR
= Rx antenna spacingRd
TT
T
)1N(2j
2j
TT
e
e
1
)(
aTransmit steeringvector
/)sin(d TTT
= Tx antenna spacingTd
2/2/
Spatial sinusoids: angles frequencies
)sin(d
d
10
Physical Model
)()( n,THTn,RR
N
1nn
path
aaH
pathN
}{ n,R }{ n,T
}{ n: number of paths : complex path gains
: Angles of Arrival (AoA’s) : Angles of Departure (AoD’s)
Non-linear dependence of H on AoA’s and AoD’s
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Virtual Modeling
T
N
1i
N
1k
HT
RRVn,T
N
1n
HTn,RRn N
kNik,iH
R Tpath
aaaaH
RR
TT N
1,N1
Spatial array resolutions:
Physical Model Virtual Model
(AS ’02)Virtual model is linear -- virtual beam angles are fixed
12
Antenna Domain and Beamspace
HTVR AHAH T
HRV HAAH
RRR NN: A
TTT NN: A
Two-dimensional unitary (Fourier) transform
Generalization to non-ULA’s (Kotecha & AS ’04 ; Weichselberger et. al. ’04; Tulino, Lozano, Verdu ’05;)
HRRR
HR
RR
E UΛUHHΣ
UA
HTTT
HT
TT
E UΛUHHΣ
UA
Unitary (DFT) matrices
13
Virtual Imaging of Scattering Geometry
2 point scatterers 2 scattering clusters
Diagonal scattering Rich scattering
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Virtual Path Partitioning
pathk,Ti,Rk,i
k,Tk
i,Ri
N,,2,1]SS[SS
T
R
RRn,Ri,R N
)2/1i(,N
)2/1i(:nS
RN1
TN1
TTn,Tk,T N
)2/1k(,N
)2/1k(:nS
k,Ti,R SSnnV )k,i(H
Distinct virtual coefficients disjoint sets of paths
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Virtual Coefficients are Approximately Independent
'kk'ii*VV k,i'k,'iHk,iHE
k,Ti,R SSnnV )k,i(H
k,Ti,R SSn
2n
2V Ek,iHEk,i
Channel power matrix: joint angular power profile
T
R
TN1
RN1
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Joint and Marginal Statistics
k,Ti,R SSn
2n
2V Ek,iHEk,i
Joint distribution of channel power as a function of transmit and receive virtual angles
Joint statistics:
Marginal statistics:
HvvR E HHΛ vHvT E HHΛ
RN
T Ti 1
k k,k i, k
Λ TN
R Rk 1
i i, i i, k
Λ
Transmit Receive
(diagonal)
V Vvec( )h H
H
V V VE[ ]
diag i, k
R h h
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Kronecker Product Model
RTVRT ΛΛRΣΣR
Independent transmit and receive statistics
Separable angular scattering function (angular power profile)
2V R T| H |i, k E i, k (i) (k) Separable
arbitrary kronecker
i, k R T{ (i) (k)}
2/1Tiid
2/1RV
2/1Tiid
2/1R ΛHΛHΣHΣH
parameters1NN RT
parametersNN RT
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Communication in Eigen (Beam) Space
nHsx
vvvv nsHx xAx
sAsHRv
HTv
Multipath PropagationEnvironmen
tTA
VsHRA
x Vx
is an image of the far-field of the RX
VxImage of is created in the far-field of TX
Vs
s
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Capacity Maximizing Input
)det(logEmax)det(logEmaxC HVVV)(tr
H)(tr
VHQHIHQHI QQ
nHsx
HToptTopt
opt,V
UΛUQQ
Optimal input covariance matrix is diagonal in the virtual domain:
- Beamforming optimal at low SNR (rank-1 input)- Uniform power input optimal at high SNR (full-rank input)- Uniform power input optimal for regular channels (all SNRs)
2H
E
][E
s
Inn
Veeravalli, Liang, Sayeed (2003); Tulino, Lozano, Verdu (2003); Kotecha and Sayeed (2003)
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Degrees of Freedom
R ,i T ,k
2 2V n
n S S
H i,ki, k E E
Dominant (large power) virtual coefficients
Statistically independent Degrees of Freedom (DoF)
DoF’s are ultimately limited by the number of resolvable paths
D DoF (i, k) : (i, k) 0
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Channel Power and Degrees of Freedom
D(N) = number of dominant non-vanishing virtual coefficients = Degrees of Freedom (DoF) in the channel
L(N)H H 2
c V V ni,k n 1
(N) (i, k) trace(E[ ]) trace(E[ ]) E[| | ]
H H H H
c (N) ~ O(D(N))
The D non-vanishing virtual coefficients are O(1)
Simplifying assumption:
Assume equal number of transmit and receive antennas – RT NNN
Channel power:
22
Prevalent Channel Power Normalization
2c N D(N) ~ O N
The channel power/DoF grow quadratically with N
2c T RN N N
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Quadratic Channel Power Scaling? 2
c N D(N) ~ O(N ) is physically impossible indefinitely(received power < transmit power)
N
Total TX power
Total RX powerQuadratic growth in channel power
Linear growth in total received power
Linear capacity scaling
Increasing power coupling between the TX and RX due to increasing array apertures
2TX EP s2
RX TX TXD(N)P P P NEN
Hs
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Sparse (Resolvable) Multipath
2rich max R TD D O N N O N
2R TN N N
2sparse R TD o N N o N N , 0,2
Degr
ees o
f Fre
edom
(D)
Rich (linear)
Sparse (sub-linear)
Channel Dimension
c (N) ~ O(D(N))
Sub-quadratic power scaling dictates sparsity of DoF
25
Capacity Scaling: Sparse MIMO Channels
c ~ D ~ O(N ) , 0 2
For a given channel power/DoF scaling law
what is the fastest achievable capacity scaling?
New scaling result: coherent capacity cannot scale faster than
2/c NO)N(O)N(DO~)N(C
and this scaling rate is achievable (Ideal channel)
(AS, Raghavan, Kotecha ITW 2004)
26
MIMO Capacity Scaling
N
C(N)
Correlated channels (kronecker model)Chua et. al. ’02
i.i.d. modelTelatar ’95Foschini ’96 physical channels
(virtual representation)Liu et. al. ’03
RT NNN 2,NOD
2/NODONC AS et. al. ’04
27
Sparse Virtual Channels
• Sub-quadratic power scaling dictates sparse virtual channels:
• Capacity scaling depends on the spatial distribution of the D(N) channel DoF in the possible channel dimensions
2D(N) N
2N
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Simple Model for Sparse MIMO Channels
D N
NNN RT iidv )D( HMH
0/1 mask matrix with D non-zero
entries)D(MΨ
Sparsity in virtual (beam) domain correlation/coherence in the antenna (spatial) domain
29
Three Canonical (Regular) Configurations
)1(ND
Beamforming
p = number of parallel channels (multiplexing gain)q = D/p = DoF’s per parallel channel
Ideal NDqp
Nqq1pp
max
min
Multiplexing
1qqNpp
min
max
qpD Consider
p transmit dimensions; r = max(q , p) receive dimensions
Multiplexing gain = p increasesReceived SNR = q/p increases
Received SNR = q/p
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Simple Capacity Formula:Multiplexing Gain vs Received-SNR
rx 2
q DC ~ p log 1 p log 1 p log 1p p
2
crx
(N) q(N)E(N)p(N) p(N)p(N)
Hs Received SNR per parallel channel
bf
rx
C (N) ~ log(1 N)(N) N
id
rx
C (N) ~ N log(1 )(N)
mux
rx
C (N) ~ N log(1 / N)(N) / N 0
Beamforming (BF) Ideal Multiplexing (MUX)
31
Morphing Between the Configurations
qpND:]2,0(
:)1,0(
:)2,1[
0min
1min
beamforming)2/,[ min
ideal2/
multiplexing],2/( max)0,1max(min )1,min(max
]1,0[,Nq,Np
max
1max
:2 1min 1max
32
Fastest Capacity Scaling: The Ideal Configuration
2rx p 2
p
C(N) p log 1 N log 1 N
pp
D 1D N , p N , q Np
rxqp
)N(;NlogNO~)N(C)2/,[ rx2
bfmin
2prx
2/idid /)N(;NO~)N(C2/
0)N(;NO~)N(C],2/( rxmuxmax
BF regime:
Ideal regime:
MUX regime:
bf mux
id id
C (N) C (N)0 , 0C (N) C (N)
/ 2id cC (N) ~ O N ~ O (N)
33
Impact of Transmit SNR on Capacity Scaling
BF:
MUX:
Ideal:
N)N(q)N(p
1)N(q,N)N(p
N)N(q,1)N(p
0min
2/1id
1;N)N(D
1max C(
N)
N
34
Accuracy of Asymptotic Expressions
C(N)
N
BF and MUX tight atall SNRs
Ideal tight in the low-
or high-SNR regimes
35
Capacity Formula Proofs: RMT
• If H is r x p, coherent ergodic is given by
• If, in addition, H is regular
36
Capacity Formula Proofs: RMT If under broad
assumptions on entries of H, the empirical spectral distribution function (Fp) of (normalized by p) converges to a deterministic limit (F)
Limit capacity computation Approach 1: Sometimes this limit can be characterized explicitly
Approach 2: Often, the limit can only be characterized implicitly via the Stieltjes transform. The limit capacity formula is the solution to a set of recursive equations
37
Beamforming Configuration
• Two cases:
• In either case,
• Thus, with
38
Ideal Configuration
• Two cases:
Case i) reduces to a q x p i.i.d. channel Case ii) reduces to a q-connected p-dimensional channel [LRS
2003]
Case i) Case ii)
39
Ideal Configuration
• In either case, empirical density of (normalized by q) converges to Case i) Case ii)
• Thus,
Case i) Case ii)
40
Multiplexing Configuration
• Previous result due to [Grenander & Silverstein 1977] not applicable • Two cases: • In either case, empirical density is unknown • The implicit characterization of [Tulino, Lozano & Verdu 2005] based
on results due to [Girko] can be easily extended here • Exploiting the regular nature of H
41
The Ideal MIMO Channel: Fixed N
iidv )D( HMH
2ND
NNN RT
0/1 mask matrix with D non-zero
entries)D(MΨ
2NSpatial distribution of the D channel DoF in the possible dimensions (“resolution bins”) that yields the highest capacity
),D(Cmaxarg),D()D(ideal MM
M
42
Optimum Input Rank versus SNR
5.235.34diag5.05.05.015.05.0115.01111111
T
Correlated channels:- beamforming (rank-1 input) optimal at low SNR- uniform power (full-rank, i.i.d.) input optimal at high SNR
1234
SNR
rank
Vs
s
TAi.i.d. channels: equal power (i.i.d.) input optimal at all SNRs
Loss of precious channel power!
43
Ideal Channel: Optimum MG vs SNR tradeoff
Np
)p(1logppD1logp
pq1logp,pC rx2
Capa
city
Beamforming:
Ideal:
Multiplexing:
Nq,1p
Nqp
1q,Np
NpqD
high,max
highlow
lowmin
ideal
p
,,2D
,p
)(p
low
BF
Ideal
high
MUX
Np
2/1
44
Impact of Antenna Spacing on Beamstructure
)sin(d
maxmux dd
Nbeams#
N1beamwidth
Ndd mux
ideal
Nbeams#
N1beamwidth
1beams#
Ndd mux
bf
1beamwidth
45
Adaptive-resolution Spatial SignalingIdeal
Medium resolution TX and RX
Multiplexing gain and spatial coherence: Fewer independent streams with wider beamwidths at lower SNRs.
High resolution TX and RX
Multiplexing Beamforming
Low resolutionTX and High-Res. RX
46
Wideband/Low-SNR Capacity Gain
MUXmin,o
b
IDEALmin,o
b
BFmin,o
b
NE
N1
NE
N1
NE
N-fold increase in capacity (or reduction in ) via BF configuration at low SNR
minob )N/E(
47
Source-Channel MatchingAdapting the multiplexing gain p via array configuration:
matching the rank of the inputrank of the input to the rank of the effective rank of the effective channel channel
Multiplexing
Full-rank channel
Full-rank input
RX
TX
48
Source-Channel Matching
Ideal
“Square root” rank channel
“Square root” rank input
RX
TX
Adapting the multiplexing gain p via array configuration: matching the rank of the inputrank of the input to the rank of the effective rank of the effective
channel channel
49
Source-Channel Matching
Beamforming
Rank-1 channel
Rank-1 input
TX
RX
Adapting the multiplexing gain p via array configuration: matching the rank of the inputrank of the input to the rank of the effective rank of the effective
channel channel
50
New Capacity Formulation for Reconfigurable MIMO Channels )det(logEmaxmaxC H
VVVpqD:)(tr VV
HQHIHQ
To achieve O(N) MIMO capacity gain at all SNRs
Optimal channel configuration realizable with reconfigurable antenna arrays
Optimum number of antennas: N ~ DCa
pacit
y
pq1logp,pC
MUX IDEAL BF
51
Summary• Sparse multipath
– i.i.d. model – rich multipath– Seldom true in practice– Physical channels exhibit sparse multipath
• Modeling of sparse MIMO channels – Virtual channel representation (beamspace)– Physically meaningful channel power
normalization– Sparse degrees of freedom– Spatial correlation/coherence
• The Ideal MIMO Channel– Fastest (sub-linear) capacity scaling with N– Capacity-maximization with SNR for fixed N– Multiplexing gain versus received SNR tradeoff– Simple capacity formula for all SNRs (RMT)
• Creating the Ideal MIMO Channel in Practice– Reconfigurable antenna arrays– Three canonical configurations: near-optimum
performance over entire SNR range – Source-Channel Matching– New capacity formulation
Capa
city
MUX IDEAL BF
qC p, p log 1p
C(N)
sparse C N O Di.i.d.
C N O N
Correlated C N O N
N
52
Extensions: Implications of Sparsity
• Relaxing the 0-1 sparsity model
• Non-uniform sparsity
• Wideband MIMO channels/doubly-selective MIMO channels
• Space-time coding
• Reliability (error exponents)
• Impact of TX CSI (full or partial)
• Channel estimation (compressed sensing) and feedback
• Network implications (learning the network CSI)
53
REFERENCES Beamforming Channel:
Bai & Yin: “Convergence to the semicircle law,” Annals Prob., vol. 16, pp. 863-875, 1988 Ideal Channel:
Marcenko & Pastur: “Distribution of eigenvalues for some sets of random matrices,” Math-USSR-Sb., vol. 1, pp. 457-483, 1967
Bai: “Methodologies in spectral analysis of large dimensional random matrices: A review,” Statistica Sinica, vol. 9, pp. 611-677, 1999
Silverstein & Bai: “On the empirical distribution of eigenvalues of a class of large dimensional random matrices,” Journal of Multivariate Analysis, vol. 54, no. 2, pp. 175-192, 1995
Grenander & Silverstein: “Spectral analysis of networks with random topologies,” SIAM Journal on Appl. Math., vol. 32, pp. 499-519, 1977
Liu, Raghavan & Sayeed, “Capacity and spectral efficiency of wideband correlated MIMO channels,” IEEE Trans. Inform. Theory, vol. 49, no. 10, pp. 2504-2526, Oct. 2003
Multiplexing Channel: Girko: Theory of random determinants, Springer Publishers, 1st edn, 1990 Tulino, Lozano & Verdu: “Impact of antenna correlation on the capacity of multiantenna
channels,” IEEE Trans. Inform. Theory, vol. 51, no. 7, pp. 2491-2509, July 2005
• Multi-antenna Capacity of Sparse Multipath Channels, V. Raghavan and A. Sayeed. http://dune.ece.wisc.edu