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Transcript of Channel Matrix
Supelec
Random Matrix Theoryfor
Wireless Communications
Merouane Debbahhttp://www.supelec.fr
February, 2008
Presentation
MIMO Channel Modelling and random matrices
1
Where do we stand on Channel Modelling
Google search: ”MIMO Wireless Channel Modelling”
• Over 15 000 publications on channel modelling• At a rate of 10 papers per day, 1 500 days (nearly 4 years)!• The models are different and many validated by measurements!
Three conflicting schools
• Geometry based channel models.• Stochastic channel models based on channel statistics• Do not model, use test measurements
Not even within each school, all experts agree on fundamental issues.
2
MIMO System Model
TxRx
The channel is linear, noise is additive
y(t) =
√ρ
nt
∫Hnr×nt(τ)x(t− τ)dτ + n(t)
Y(f, t) =
√ρ
nt
Hnr×nt(f, t)X(f) + N(f)
3
Why do we need a channel model?
Our Vision
Step 1: Collection of informationThe user (or base station) download information on his environment (dense, number ofbuildings,...) through a localization service process
Step 2: Model generationA statistical channel model is automatically created (at the base station or the mobile unit)integrating only that information and not more!
Step 3: High speed connectionThe coding scheme is adapted to the (statistical not realization) modelExample
• Additive Gaussian: Euclidean distance coding• Rayleigh i.i.d: rank and determinant criteria
This scenario could be called ”User customized channel model coding service” and is aviable scenario from a Soft Defined Radio perspective.
4
Why do we need a statistical channel model?
Ergodic Channel Capacity: (The receiver knows the channel and the transmitter knowsthe statistics)
C = maxQE(C(Q)) with C(Q) = log2det(Int + ρ
ntHHQH
)
Q = E(XXH) = I only with i.i.d zero mean Gaussian MIMO model!
The need to model: Statistical channel models stimulate creativity (patents!):
• to optimize the codes• to estimate the channel
in order to achieve the information theoretic limits.
This can not be performed with simulation or measurement based models!
5
Types of questions channel modelling will answer
Multiple versus Single Antenna
SISO AWGN Channel: C = log2(1 + ρ) ∼ log2(ρ) at high SNR.
MIMO Channel:
• Suppose that the channel matrix is deterministic with equal entries 1 (Pure Line of Sightcase):
C = log2 det(Inr +ρ
nt
HHH).
=
nr∑
i=1
log2(1 +ρ
nt
λi)
= log2(1 + ρnr)
→ log2(ρ)at high SNR
• i.i.d Rayleigh fading: C = min(nr, nt) log2(ρ) at high SNR.
Is there a Contradiction?
6
Let us start...
Model Construction
7
The i.i.d Gaussian model
The modeler would like to attribute a joint probability distribution to:
H(f) =
h11(f) . . . . . . h1nt(f)... . . . . . . ...... . . . . . . ...
hnr1(f) . . . . . . hnrnt(f)
(1)
Assumption 1: The modeler has no knowledge where the transmission took place (thefrequency, the bandwidth, the type of room, the nature of the antennas...)
Assumption 2: The only things the modeler knows:For all {i, j},
E(∑
i,j | hij |2) = nrntE
What distribution P (H) should the modeler assign to the channel based only on thatspecific knowledge?
8
The i.i.d Gaussian model
Principle of maximum entropy
Maximize the following expression:
−∫
dHP (H)logP (H) + γ[nrntE −∫
dHnr∑
i=1
nt∑
j=1
| hij |2 P (H)]
+β
[1−
∫dHP (H)
]
Solution:
P (H) =1
(πE)nrntexp{−
nr∑
i=1
nt∑
j=1
| hij |2E
}
Contrary to past belief, the i.i.d Gaussian model is not an assumption but the result offinite energy knowledge.
This method can be extensively used whenever additional information is provided in termsof expected values.
9
Knowledge of the covariance structure
In the general case, under the constraint that:∫
CNhih
∗jPH|Q(H)dH = qi,j
for (i, j) ∈ [1, . . . , N ]2 (N = nrnt). Then using Lagrangian multipliers,
L(PH|Q) =
∫
CN− log(PH|Q(H))PH|Q(H)dH
+ β
[1−
∫
CNPH|Q(H)dH
]
+∑
αi,j
[∫
CNhih
∗jPH|Q(H)dH− qi,j
].
we obtain:
PH|Q(H) =1
det(πQ)exp
(−(vec(H)
HQ−1vec(H)))
.
10
Existence of Correlation
Question
What to do if we know the existence of correlation but not its exact value?
Answer
P (H) =
∫P (H, Q)dQ =
∫P (H | Q)P (Q)dQ
1- Determine the a priori distribution of the covariance matrix based on limited informationat hand
2- Marginalize with respect to the a priori distribution
11
Construction of the a priori
Let us determine the a priori distribution of the covariance
Suppose that we only know that E(Trace(Q)) = nrntE (The covariance is not fixed butvaries due to mobility for example)
Result. The maximum entropy distribution for a covariance matrix Q under the constraintE(Trace(Q)) = nrntE is such as:
Q = UΛUH
where:
• U is haar unitary distributed matrix.• Λ = diag
(λ1, ..., λnrnt
)is diagonal matrix with independent Laplacian distributions.
P (Q)dQ =1
EnrntΠ
nrnr−1n=0 (n!(n + 1)!)e
−Trace(Q)E Πi>j(λi − λj)
2dUdΛ
Note that Q is nothing else than a Wishart matrix with nrnt degrees of freedom.
12
MIMO Channel distribution with correlation
What do we need to do?
P (H) =
∫P (H | Q)P (Q)dQ
=
∫ ∫1
π∏nrnt
i=1 λi
e−Trace(vec(H)vec(H)HUHΛ−1U)
1
EnrntΠ
nrnr−1n=0 (n!(n + 1)!)e
−Trace(Q)E Πi>j(λi − λj)
2dUdΛ
We need to integrate over U and Λ!
Difficult problem...but well known in statistical physics!
13
MIMO Channel distribution with correlation
Harish-Chandra, ”Differential Operator on a Semi-Simple Lie Algebra”, Amer. J. Math. 7987-120 (1957)
Harish-Chandra, 1923-1983
Harish-Chandra integral
∫
U∈U(m)
e−mTrace(Σ−1UΛU−1)
dU =det(e−σ−1
jλk)
∆(Σ−1)∆(Λ)
14
MIMO Channel distribution with correlation
”Maximum Entropy Analytical MIMO Channel Models”, M. Guillaud, M. Debbah and A.Moustakas, submitted to IEEE transactions on Information Theory, 2006.
Solution. P (H) is given by
P (H) =
nrnt∑n=1
2(Trace(HHH)
E)
n+nrnt−22 Kn+nrnt−2(2
√Trace(HHH)
E).
(−1)nnrnt
[(n− 1)!]2(nrnt − n)!
Kn(x) are bessel functions of order n.
We have therefore an explicit form that can be used for design when correlation exists in
the MIMO model but we are not aware of the explicit value of the correlation!
15
MIMO Channel distribution with correlation
What happens when we know of the existence of a channel covariance matrix with rank L?
Using the same methodology (and integration on a lower subspace), we obtain:
P(L)
(H) =2
Trace(HHH)
L∑
i=1
−L
√Trace(HHH)
E0
L+i
Ki+L−2
2L
√Trace(HHH)
E0
1
[(i− 1)!]2(L− i)!
.
Kn(x) are bessel functions of order n.
16
Some distributions
0 10 20 30 40 50 60−0.02
0
0.02
0.04
0.06
0.08
0.1
Energy x
PD
F o
f x=
||H|| F2
iid Gaussian (χ2)
MaxEnt P(16)x
(x)
MaxEnt P(12)x
(x)
MaxEnt P(8)x
(x)
MaxEnt P(4)x
(x)
MaxEnt P(2)x
(x)
Examples of the limited-rank covariance distribution for a 4× 4 MIMO matrix andL = 2, 4, 8, 12 and 16, and χ2 with 16 degrees of freedom, for E0 = 16.
17
Some distributions
0 5 10 15 20 250
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
mutual information (nats) at 15dB SNR, nr=n
t=4
CD
F
MaxEnt rank 1
MaxEnt rank 2
MaxEnt rank 3
MaxEnt rank 7
MaxEnt rank 16
iid Gaussian
CDF of the instantaneous mutual information of a 4× 4 flat-fading channel for the MaxEntmodel with various covariance ranks, at 15dB SNR.
18
Presentation
Asymptotic Analysis of MIMO systems
19
Representation of a multiple-antenna system: example
Rx Tx
Φnr×sr Ψst×ntΘsr×st
• Ψst×nt: matrix of direction of departure of size st × nt.• Φnr×sr: matrix of direction of arrival of size nr × sr.• Θsr×st i.i.d Gaussian matrix of size sr × st.
This model is known as the Maxent model (the Kronecker model, Sayeed’s virtual
representation and the key-hole model can be shown to be particular cases).
20
CLT for MIMO systems
M. Debbah and R. Muller, ”MIMO Channel Modelling and the Principle of MaximumEntropy,” IEEE Transactions on Information Theory, Vol. 51 , pp. 1667 - 1690, May, No.5-2005
y =
√ρ
nt
Hs + n
=
√ρ
nt
1√srst
Φnr×srΘsr×stΨst×nts + n
For many types of random matrices:
limnt→∞,nr
nt=α
log2det(
Inr +ρ
nt
HHH
)− ntµ → N (0, σ
2).
21
General result for H = 1√srst
ΦΘΨ
Result: Let η et ξ be the eigenvalues of matrices 1sr
ΦΦH and 1st
ΨHΨ respectively (Θ isan sr × st i.i.d zero mean Gaussian matrix):
µ =
nt∑
i=1
log2(1 + ρξir) +
nr∑
i=1
log2(1 + ρηiq)− nrqr
σ2= −2 log(1− g(r, q))
g(r, q) =
[1
nt
nt∑
i=1
(ρηi
1 + ηiρq)2
] [1
nt
nr∑
i=1
(ρξi
1 + ξiρr)2
]
r =1
nt
nt∑
i=1
ρηi
1 + ηiρq
q =1
nt
nr∑
i=1
ρξi
1 + ξiρr
22
Elements of Proof
µ =1
nt
log det(
Inr +ρ
nt
HHH
)
=1
nt
nr∑
i=1
log(1 + ρnr
nt
λi)
=nr
nt
1
nr
nr∑
i=1
log(1 + ρnr
nt
λi)
→ nr
nt
∫log(1 + ρ
nr
nt
λ)dF 1nrHHH(λ)
23
Elements of Proof
More specifically, let1
sr
ΦHΦ = VφΛφVHφ
1
st
ΨΨH= VψΛψVH
ψ
Vψ and Vφ are unitary matrices while Λφ and Λψ are diagonal matrices representing
respectively the eigenvalues of matrices 1sr
Pr12ΦHΦPr1
2 and 1st
Pt12ΨΨHPt
12 . The
non-zero eigenvalues of matrix 1nr
HHH = 1nrsrst
ΦPr12ΘPt
12ΨΨHPt
12ΘHPr1
2ΦH are the
same as Θ1ΘH1 = 1
nr[Λ
12Φ(VΦ
HΘVΨ)Λ12Ψ][Λ
12Ψ(V H
Ψ ΘHVΦ)Λ12Φ]. Without loss of generality,
we will suppose that sr ≤ nr. Therefore, the spectra of 1nr
HHH and Θ1ΘH1 are related
by:fHHH(x) = (1− sr
nr
)δ(x− 0) +sr
nr
fΘ1ΘH1(x)
and their Stieltjes transforms are related as:
mHHH(z) = (1− sr
nr
)1
z+
sr
nr
mΘ1ΘH1(z)
24
Elements of Proof
Matrix VφΘVΨ is an i.i.d zero mean Gaussian matrix with unit variance (only unitary
transforms are applied). Therefore, matrix Θ1 = 1√nr
[Λ12Φ(VΦΘVΨ)Λ
12Ψ] is a sr × st
random matrix composed of independent entries with zero mean and variances1
nrλi
φλjψ = 1
sr
λiΦλj
Ψ
γ . The weak convergence of the empirical eigenvalue distribution of
Θ1ΘH1 to a limiting distribution holds under certain assumptions and is an application of
the theorem of Girko.
25
Remarks on capacity of the double directional model
Proposition: In the high SNR regime, the mean mutual information of the doubledirectional model converges to:
min(
nt, nr, sr
∫
λ>0
dSdoa(λ), st
∫
λ>0
dSdod(λ)
)log2(ρ)
The factor min(nt, nr, sr
∫λ>0
dSdoa(λ), st
∫λ>0
dSdod(λ))
denotes the multiplexing gain.
∫λ>0
dSdoa(λ) and∫
λ>0dSdod(λ)) express the correlation factor of the sr and st
scatterers respectively.
The MIMO mutual information is limited by the scattering environment!
26
Mutual information compliance
For a given frequency f , a model will be called capacity complying if it minimizes:
∫ ∞
0
| F (C, f)− Fempirical(C, f) |2 dC (2)
Here Fempirical(C, f) is the empirical cdf given by measurements.
• How to derive the theoretical cdf F (C) of the capacity?
F (C) = 1−Q(C − ntµ
σ)
with
Q(x) =1√2π
∫ ∞
x
dte−t2
2
since C = log2det(Int + ρ
ntHHH
)is asymptotically Gaussian.
27
Measurements of channels
Measurements have been performed:
• at 2.1 GHz• at 5.2 GHz
In the following scenarios:
• Indoor• Atrium• Urban Open Place• Urban Regular High Antenna• Urban Regular Low Antenna
28
Sounder characteristics at 2.1 GHz
Measurement frequency 2.1 GHzMeasurement bandwidth 100 MHz
Delay resolution 10nsSounding signal linear frequency chirp
Transmitter antenna 8 element ULAElement spacing 71.4 mm (0.5 λ)Receiver antenna 32 (8× 4) elementElement spacing 73.0 mm (0.51 λ)
29
Sounder characteristics at 5.2 GHz
Measurement frequency 5.255 GHzMeasurement bandwidth 100 MHz
Sounding signal linear frequency chirpTransmitter antenna 8 element ULA
Element spacing 28.54 mm (0.5 λ)Receiver antenna 8 element ULA
30
Antennas
The receiver is an 8× 4 antenna whereas the transmitter is an 8 elements ULA.
31
Example: Urban Open Place at 2.1 GHz
32
Is mutual information Gaussian at 2.1 GHz
15 16 17 18 19 20 21 22 23 24 250
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
b/s/Hz
CD
F
Are the measured mutual information Gaussian?
MeasuredGaussian approximation
Urban RegularHIgh Antenna
Indoor
Urban Open Place
Atrium
Urban regular low Antenna
Mutual information has a Gaussian behavior for an 8× 8 system at 2.1Ghz!
33
Is mutual information Gaussian at 5.2 GHz
15 16 17 18 19 20 21 22 23 240
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
b/s/Hz
CD
F
Are the measured mutual information Gaussian?
MeasuredGaussian approximation
Urban Open Place
Indoor
Atrium
Urban Low Antenna
• Mutual information has a Gaussian behavior for 8× 8 system at 5.2Ghz!
34
Rice Model Case
Single Antenna/Multiple Antennas
SISO Channel: Capacity of the AWGN channel is higher than the ergodic capacity of aRayleigh fading channel (Jensen’s inequality).
Rice MIMO Channel: Suppose that the matrix representing the channel has deterministicentries equal to 1:
C = log2det(Inr +ρ
nt
HHH).
=
nr∑
i=1
log2(1 +ρ
nt
λi)
= log2(1 + ρnr)
→ log2(ρ)at high SNR
MIMO i.i.d zero mean Gaussian : C = min(nr, nt)log2(ρ) at high SNR.Is there a contradiction?
35
Rice Model Case
Diversity versus path loss trade-off
Free Space Propagation:ρ ∼ ρmax
r2
Propagation with reflections:ρ ∼ ρmax
r2n+2
(n is the number of reflections)
There is a trade-off between the gain in SNR due to the line of sight component and themultiplexing gain due to multiple reflections.
The analysis of MIMO Rice channels is important to determine the interval of interest.
36
Rice Model Case
Rice versus Rayleigh
Rayleigh fading:
• Rayleigh channels have been studied extensively in the case of non-line of sight.• In the case of i.i.d entries, ergodic capacity increase is min(nr, nt) bits per second per
hertz for every 3dB increase at high SNR.• Highly scattered environnements increase ergodic capacity.
Rice fading:
• Many propagation scenarios have line of sight components (Inter-base stationtransmissions,..).
• Capacity studies limited to rank 1 Rice fading or based on very loose bounds.• The impact of line sight with scattering is still an issue!
37
Rice Model Case
General Model
H =
√K
K + 1A +
√1
K + 1B
A is the deterministic line of sight component part of the matrix such as ‖A‖2F = ntnr
B is a gaussian unit variance zero mean i.i.d matrix.
The average energy of the channel is normalized according to: E(tr(HHH))nrnt
= 1.
Assumption: The matrix size 1nt
AAH grows large with β = nrnt
remaining fixed such asthe empirical eigenvalue distribution of 1
ntAAH converges in distribution to a deterministic
limit function F A√nt
.
Remark: The rank of A increases at the same rate as the number of antennas.
38
Rice Model Case
Remember the useful result
Let mH(z) be the Stieljes transform of the limiting eigenvalues FH(λ) of HHHnt
:
mH(z) =
∫dFH(λ)
λ− z
As nr = βnt →∞,
mH(z) =
∫d FA(λ)
KλβmH(z)+K+1 − z
(βmH(z)
K+1 + 1)
+ 1−βK+1
The asymptotic eigenvalue distribution of HHHnt
is completely determined knowing onlyFA(λ), β and K and not the particular fluctuations of the fading!
39
Rice Model Case
Perfect Channel Knowledge at the transmitter
The channel capacity per receiving antenna converges almost surely, as nr = βnt →∞,to:
C1 =1
ln 2
∫ +∞
1µ∗
ln λµ∗d FH(λ)
β
∫ +∞
1µ∗
(µ∗ − 1
λ
)d FH(λ) = ρ
Remark:
• The capacity is achieved through waterfilling power allocation.• The channel should change slowly enough in order to have complete channel
knowledge at the transmitter.
40
Rice Model Case
Channel knowledge of the limiting singular value distribution of A
L. Cottatellucci, M. Debbah, ”The Effect of Line of Sight Components on the AsymptoticCapacity of MIMO Systems,” 2004 IEEE International Symposium on InformationTheory,Chicago, June 27 - July 2, 2004, USA
Since the transmitter has no knowledge of the eigenvector structure, the transmittedpower is equally distributed among the antennas.
The channel mutual information per receiving antenna converges almost surely, asnr = βnt →∞, to:
C =1
ln 2
∫ ρ
0
1
x
(1− 1
xmH
(−1
x
))d x
Remark:
• If K → 0, mH(z) = 1−z(βmH(z)+1)+(1−β) (Marchenko Pastur Distribution)
• If K →∞, mH(z) =∫ d FA(λ)
λ−z (distribution of the mean)
41
Rice Model Case
Perfect Knowledge of line of sight matrix A
D. Hosli and A. Lapidoth, The Capacity of a MIMO Ricean Channel is Monotonic in theSingular Values of the Mean, 5th International ITG Conference on Source and ChannelCoding (SCC), Erlangen, Nuremberg, jan, 2004
The optimal covariance matrix Q is shown to have the same eigenvectors as AHA.
42
Rice Model Case
Perfect Knowledge of line of sight matrix A
The capacity is then given by:
J(H, UQUH) =
1
nr
log2 det(
Inr +ρ
nt
HUQUHHH)
=1
nr
log2 det(
Inr +ρ
nt
VHQHHVH)
=1
nr
log2 det(
Inr +ρ
nt
HQHH
)
H = VHHU =√
KK+1A +
√1
K+1VHBU
VHBU is a random matrix with i.i.d zero mean Gaussian entries
The line of sight matrix√
KK+1A is diagonal.
43
Rice Model Case
Perfect Knowledge of line of sight matrix A
The asymptotic channel capacity per receive antenna converges almost surely to:
C2(H) = maxQ(λ,FA(λ),K,β)
J(H, UQUH)
J(H, UQUH) =
1
ln 2
∫ ρ
0
1
x
(1− 1
xm2
(−1
x
))d x
m1(z) =∫ β(m1(z)− z(K + 1))q(K + 1) d FA(λ)
(K + 1 + βm2(z)q)(m1(z)− z(K + 1)) + (K + 1)Kλq+
(1− β)q0(K + 1)
(K + 1 + βm2(z)q0)
m2(z) =∫ (K + 1 + qβm2(z))(K + 1) d FA(λ)
(K + 1 + βm2(z)q)(m1(z)− z(K + 1)) + λq(K + 1)K
q = q(λ, F (λ), K, β) denotes the diagonal entries of Q and q0 = q(0, F (λ), K, β).
Remark: The solution is not waterfilling on the mean matrix!
44
Rice Model Case
Simulations: general resultsFA = 1
4δ(λ− 3) + 34δ(λ− 1
3), β = 1, ρ = 5dB.
10−2
10−1
100
101
102
103
104
1.6
1.65
1.7
1.75
1.8
1.85
Rice channel capacity for FA
(λ)=δ(λ−3)/4+3 δ(λ−1/3)/4, β=1, ρ=5 dB
Rice factor K
Cap
acity
(bi
t/s/H
z)
C1: Asymptotic evaluationC1: Simulation with 8 × 8 MIMO systemC2: Asymptotic evaluationC2: Simulation with 8 × 8 MIMO systemC3: Asymptotic evaluationC3: Simulation with 8 × 8 MIMO system
Rayleigh channel
• Close match with ergodic capacities for a system with 8× 8 antennas.• Mean feedback is sufficient for values of K > 10.
45
Rice Model Case
Simulations: Low and high SNR regimeFA = 1
4δ(λ− 3) + 34δ(λ− 1
3), β = 1.
10−2
10−1
100
101
102
103
104
0
0.05
0.1
0.15
0.2
0.25
Rice channel with FA(λ)=1/4 δ(λ−3)+3/4 δ(λ−1/3), β=1.
Rice factor K
(C1−
C3)/
C3
ρ=0 dB
1 dB
2 dB
4 dB
6 dB
8 dB
10 dB
10−2
10−1
100
101
102
103
104
−0.02
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
Rice channel with FA(λ)=1/4 δ(λ−3)+3/4 δ(λ−1/3), β=1.
Rice factor K
(C2−
C3)/
C3
ρ=0 dB
1 dB
2 dB
4 dB
6 dB 8 dB 10 dB
• At high SNR, the effect of feedback on the capacity diminishes.• For low SNR, the effect of feedback is quite important and depends mainly on the Ricean
factor K.
46
Rice Model Case
Simulations: reduced rank, mean matrix known at the transmitter.FA =
nt8 δ(λ) +
8−ntnt
δ(λ− 88−nt
), β = 1, ρ = 10.
10−2
10−1
100
101
102
103
104
1
1.5
2
2.5
3
3.5
Rice factor K
Cap
acity
C2, (
bits
/s/H
z)
Rice channel capacity C2 for F
A(λ)=p
0 δ(λ)+p
1δ(λ−1/p
1), β=1, ρ=10.
p0= 0
1/8
2/8
3/8
4/8
5/8
6/8
7/8
• Rank reduction yields a dramatric effect on the capacity.• Transition phase with respect to K around [1,10].
47
Rice Model Case
Simulations: rank 1, perfect channel knowledge at the transmitter.FA =
nt−1nt
δ(λ) + 1nt
δ(λ− nt), β = 1, ρ = 10.
10−2
10−1
100
101
102
103
104
0.5
1
1.5
2
2.5
3
Rice factor K
Cap
acity
C1, (
bits
/s/H
z)
Rice channel capacity C1 for F
A(λ)=(n
t−1)/n
t δ(λ)+δ(λ−n
t)/n
t, β=1, ρ=10.
4
5
678
10
16
23
nt= 2
3
• Asymptotic analysis able to capture the rank 1 case.• Close match with a small number of antenna elements.
48
Rice Model Case
Rice or Rayleigh?
Asymptotic analysis of Ricean MIMO channels show that Rice fading is not always betterthan Rayleigh fading.
The result depends in fact on:
• the limiting behavior of the mean matrix.• the ratio β = nr
nt.
• the SNR ρ.• the Ricean factor K.• through various fixed point equations depending on the channel state of knowledge
available at the transmitter.
49
Rice Model Case
More Reading
J. Dumont, P. Loubaton, S. Lasaulce and M. Debbah, ”On the Asymptotic Performance ofMIMO Correlated Ricean Channels,” in Proc. IEEE Int. Conf. on Acoustics, Speech andSig. Proc. (ICASSP), Montreal, Canada, Mar. 2005, pp. 813-816.
J. Dumont, W. Hachem, P. Loubaton and J. Najim, ”On the asymptotic Analysis of mutualinformation of MIMO Rician correlated channels”, IEEE-EURASIP Int. Sym. on Control,Commun. and Sig. Proc. (ISCCSP 2006), Marrakech, Morocco, Mar. 2006
A. L. Moustakas and S.H. Simon, ”Random matrix theory of multi-antennacommunications: the ricean case”, J. Phys. A: Math. Gen. 38 (2005), 10859-10872.
J. Dumont, P. Loubaton and S. Lasaulce, ”On the capacity Achieving Transmit CovarianceMatrices of MIMO Rician Channels: A Large System Approach”, GlobeCom 2006,San-Francisco, USA, FIRST PATENT DUE TO FRANCE TELECOM based on Randommatrices.
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Presentation
Asymptotic design of receivers
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Model
y = W P12 s + n
Received signal code matrix Amplitude Gains emitted signal AWGN
N × 1 N ×K K ×K K × 1 ∼ N (0, σ2IN)
The goal is to detect s.
Linear MMSE Filter:LMMSE = P
12
H
WH(R + σ
2I)−1
R = WPWH
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Matrix Inversion
Let R be non-singular.Let λi be the eigenvalues of R.
Then, ∏Kk=1 (R− λkI) = 0 ⇒ −I +
∑Kk=1 αkRk−1 = 0
Cayley − Hamilton Theoreom with appropriate αk′s.
Solution to matrix inversion problem given the eigenvalues
R−1=
K∑
k=1
αkRk
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Linear Multi-Stage Detection
M. Honig and W. Xiao, ”Performance of Reduced Rank Linear Interference Suppression”,IEEE Transactions on Information Theory, vol. 47, No.5, July 2001.
Linear MMSE Filter: LMMSE =(R + σ2I
)−1
Approximation by Power Series:
Cayley-Hamilton Theorem yields:
(R + σ
2I)−1
=
K−1∑
i=0
θiRi
'D−1∑
i=0
θiRi for D < K
For a N ×K matrix W with i.i.d random spreading and R = WWH, the o¯ptimum
weights converge almost surely as K, N →∞ with α = KN , and can be given in c
¯losed
form.
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Semi-Universal Weights
R. Muller and S. Verdu, ”Design and Analysis of Low-Complexity Interference Mitigationon Vector Channels”, IEEE Journal on Selected Areas in Communications, vol. 19, no. 8,pp. 1429-1441, August 2001.
Filter shall be independent from the realization of the random matrix W, but may use itsstatistics.
For most large random matrices, as K = αN →∞, many finite dimensional functions ofthe eigenvalues, e.g, the filter coefficients free.
The asymptotic limits depend only on parts of the statistics of the random matrix.
The weights can be calculated off-line with the help of random matrix theory and freeprobability theory.
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Weight Design
is given by the Yule-Walker equations:
m1
m2...
mD+1
=
m2 + σ2m1 m3 + σ2m2 . . . mD+2 + σ2mD+1
m3 + σ2m2 m4 + σ2m3 . . . mD+3 + σ2mD+2... ... . . . ...
mD+2 + σ2mD+1 mD+3 + σ2mD+2 . . . m2D+2 + σ2m2D+1
w0
w1...
wD+1
with the total moments:mn = E(λn
) = Trace(WWH)n
.
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Example for Weight Design
Random with i.i.d entries
D=2 w0 = −σ2m1 + 2 + 2β
w1 = −1
D=3 w0 = −σ2m1 + 3 + 4β + 3β2
w1 = −σ2m2 − 3− 3β
w2 = 1
D=4 w0 = −σ2m1 + 4 + 6β + 6β2 + 4β3
w1 = −σ2m2 − 6− 9β − 6β2
w2 = −σ2m3 + 4 + 4β
w3 = −1
mn =1
n
n∑
i=1
Cni C
ni−1β
i
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Rate of Convergence
Ph. Loubaton and W. Hachem, ”Asymptotic Analysis of Reduced Rank Wiener Filters”,Information Theory Workshop 2003, Paris, France
Theorem: Suppose that the elements of the matrix are i.i.d zero-mean random variablewith finite fourth moment. Then the multi-user efficiencies η of all users c
¯onverge almost
surely as N, K →∞ but KN fixed to:
ηD+1 =1
1 + β
σ2+ηD
with η0 = 0 for optimally chosen weights.
The approximation converges to the exact MMSE performance as a continued fraction.For optimally weights wi, the approximation error ε decreases exponentially with thenumber of stages D.
ε < Const(1 + SNR)−D
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Individual Weight Design
Allow for different weight for different users:
(R + σ
2I)−1
=
K−1∑
i=1
wiRi
≈D−1∑
i=0
WiRi for D < K and all Wi diagonal
Weight design by the same Yule-Walker equations, but with the k-partial moments:
mkn =
[(SSH
)n]
kk
For users with different powers, individual weight design is better.
Do the k-partial moments converge asymptotically?
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Convergence of partial moments
Let the random matrix S fulfill the same conditions as before. Let A be an K ×K
diagonal matrix such as its singalur value distribution converges almost surely asK →∞ to a non-random limit distribution. Let R = ASSHAH
Then Rlkk, the k-th diagonal element of Rl converges, conditioned on akk, the k-th
diagonal element of A, almost surely, as K = βN →∞ to
Rlkk =| akk |2 β
∑lq=1 Rq−1
kk mRl−q with mR
q = Trace(Rq)
The total moments of R are conveniently given by the recursion:
mRl = β
l∑q=1
mRl−q lim
K→∞1
K
K∑
k=1
| akk |2 Rq−1kk
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Model
The system is described by a virtual NR×K spreading matrix
S =
h11s1 h12s2 . . . h1KsK
h21s1 h22s2 . . . h2KsK... ... . . . ...
hR1s1 hR2s2 . . . hRKsK
Note that with the Kronecker product ⊗:
sk = hk ⊗ sk
Note also that the entries of S are not jointly independent even if those ones of S and H
are.
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Ressource Pooling Result
S. Hanly and D. Tse, ”Resource Pooling and Effective Bandwidths in CDMA Systems withMultiuser Receivers and Spatial Diversity”, IEEE Transactions on Information Theory, vol.47(4), May 2001, pp. 1328-1351
Theorem Let the chips of any user be i.i.d zero-mean random variables with finite 6thmoment and the antenna array channel hrk follow the i.i.d Complex Gaussian Model.Then the multiuser efficiency ηMMSE of the linear MMSE detector c
¯onverges for all users
almost surely as N, K →∞ but β = KN and R fixed to the deterministic unique positive
solution of the fixed point equation:
1
ηMMSE= 1 +
β
R
∫x
σ2 + ηMMSExdPA2(x)
if the power of the users converge weakly to the limit distribution PA2 with:
| Ak |2=| Ak |2R∑
r=1
| hrk |2
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Ressource Pooling result for correlated MIMO Channels
L. Cottatellucci, R. Muller, ”A Generalized Resource Pooling Result for CorrelatedAntennas with Applications to Asynchronous CDMA” International Symposium onInformation Theory and its Applications, Parma, Italy, October 2004
Theorem Let the chips of any user be i.i.d zero-mean random variables with finite 6thmoment and the empirical distribution of the channel gains hrk across the usersconverge, jointly for all receive antennas r to an R-dimensional joint limit distribution PH.Then, with linear MMSE detection, the SINR converges as N, K →∞ but β = K
N and Rfixed, conditioned on the channel gains of user k to:
hHk Ahk
σ2
where A is the deterministic unique positive definite solution of the matrix valued fixedpoint equation
A−1
= I + β
∫xxH
σ2 + xHAxdPH(x)
Asymptotic Performance is characterized by an R× R matrix.
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Multi-stage Detection for Correlated Ressource Pooling
As the dimension of S grows, the following result hold:
the k partial moments conditioned on hk converge
Explicit Recursive expressions for them are now available
The proof follow along the same lines as before.
The system is sensitive to the correlation at the receiving side.
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Last Slide
THANK YOU!
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