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Eurecom, Sophia Antipolis
Complex Constrained CRB And Channel Estimation Applications to Semi-Blind MIMO/OFDM
Aditya K. Jagannatham,
Prof. Bhaskar D. Rao
University of California
SanDiego
Overview
Motivation and construction of complex constrained Cramer Rao Bound (CC-CRB).
Semi-Blind MIMO channel estimation.
Motivation, scheme, constrained estimators.
Semi-blind estimation for MRT.
Scheme and analysis.
Time Vs. Freq. domain OFDM channel estimation.
Parameter Estimation - Preliminaries
Observations
θparameter
nωωω ,,, 21 Kp (ω ; θ )
Estimator :
For instance - Estimation of the mean of a Gaussian
Estimator -
)1,(~);( θθω Np
),,,(ˆ21 nf ωωωθ K=
∑=
=n
iin
1
1ˆ ωθ
Cramer-Rao Bound (CRB)
Performance of an unbiased estimator is measured by its covariance as
CRB gives a lower bound on the achievable estimation error.
The CRB on the covariance of an un-biased estimator is given as
]) -ˆ )(ˆE[( HC θθθθθ −=
∂∂
∂∂
=θ
θωθ
θω ),(ln),(lnE ppJT
1- JC ≥θ
where,
Constrained Parameter Estimation
Most literature pertains to “unconstrained-real” parameter estimation.
Results for ‘complex’ parameter estimation ?
What are the corresponding results for “constrained” estimation ?
1||||)( 2 ==≡ θθθθ HhFor instance, estimation of a unit norm constrained singular vector i.e.
Estimator
CRB
Complex Cons. Par.
Complex Constrained Parameter CRB
Builds on work by Stoica’97 and VanDenBos’93
Define the extendedconstraint set f (θ)
≡
)()(
)( * θθ
θhh
f
∂∂
∂∂
=∂∂
≡ *
)()()()(γθ
γθ
θθθ fffF
With complex derivatives, define the matrix F (θ) as
= *γ
γθ
Define the extended parameter vector θ as
p(ω, θ ) be the likelihoodof the observation ωparameterized by θ
U span the NullSpace of F(θ).
0)( =UF θ
Let γ be an n - dim constrained complex parameter vector
The constraints on γare given by h(γ ) = 0
Constrained Parameter CRB (Contd …)
∂∂
∂∂
=*),(ln),(lnE
θθω
θθω ppJ
T
J is the complex un-constrained Fischer Information Matrix (FIM) defined as
CRB Result : The CRB for the estimation of the ‘complex-constrained’ parameter θis given as
HH UJUC 1)U(U −≥θ
MIMO System Model
t -
tran
s mi t
r-r e
c eiv
e
A MIMO system is characterized by multiple transmit (Tx) and receive (Rx) antennas
The channel between each Tx-Rx pair is characterized by a Complex fading Coefficient
hij denotes the channel between the ith receiver and jth transmitter.
This channel is represented by the Flat Fading Channel Matrix H
MIMO System Model
=
ry
yy
yM
2
1
MIMO
System
H
=
tx
xx
xM
2
1
)()()( kvkHxky +=
=
rtrr
t
t
hhh
hhhhhh
....
...
...
H
21
22221
11211
is the r x t complex channel matrixwhere
Estimating H is the problem of ‘Channel Estimation’
Issues in Channel Estimation
As the number of channels increases, employing entirely trainingdata to learn the channel would result in poorer spectral efficiency.
Calls for efficient use of blind and training information.
As the diversity of the MIMO system increases, the operating SNR decreases.
Diversity SNR
1 25
4 12
3102 −×=eP* Constellation Size = 4
Calls for more robust estimation strategies.
Estimation Strategies - Training Based
MIMO System
H
Training
Blind Outputs
pL Xxxx =},....,,{ 21
},....,,{ 21 NLL xxx ++
Data
pL Yyyy =},....,,{ 21Training Output
},....,,{ 21 NLL yyy ++
2|||| min Fpp H XY − Constraint
HXY pp =+Solution + denotes pseudo-inverse
Blind Estimation
MIMO System
HData
},...,,{ 21 Nxxx },...,,{ 21 Nyyy
Estimate channel from data.
No training necessary
Uses information in source statistics
Training Vs. Blind Estimation
Training
Blind Incre
asing
Effic
iency
Incre
asing
Sim
plicit
y
T e c h n iq u e A d v a n ta g e C o s tT ra in in g V e ry S im p le to
im p le m e n tIn e ffic ie n t B W
u sa g e .B lin d N o B W sa crifice d
fo r tra in in gC o m p u ta tio n a lly
C o m p le x
Semi-Blind MIMO Channel Estimation
MIMO System
H
Training
Blind Outputs
pL Xxxx =},....,,{ 21
},....,,{ 21 NLL xxx ++
Data
pL Yyyy =},....,,{ 21Training Output
},....,,{ 21 NLL yyy ++
Statistics:kls
Hlk xx δσ I][E 2=
klnHlk δσνν I][E 2=
Spatio-temporally un-correlated
(N-L), the number of blind “information” symbols can be large.
L, the pilot length is critical.
Semi-Blind Estimation
Goals :
Use as few training symbols as possible
Use total information
},...,,{},,....,,{ 2121 LL yyyxxx
},....,,{ 21 NLL yyy ++
Total Information
Training Data
Blind Data
Semi-Blind Estimation
Key Idea : H is decomposed as a matrix product, H= WQH
H= WQH
W is known as the “whitening” matrix
W can be estimated using only ‘Blind’ data.
QQH = I
Q is a ‘constrained’ matrix
Q is the non-minimum phase part and cannot be estimated employing Second Order Statistics.
Estimating W :
Output correlation :
Estimate output correlation
IHHR nH
sy22 σσ +=
∑+=
=N
Lk
Hy kyky
NR
1)()(1ˆ
H
nys
H
QWH
IRWW
ˆˆ
)ˆ(1ˆˆ 22
=
−= σσ
Estimate W by a matrix square root (Cholesky) factorization as
As # blind symbols grows ( i.e. N →∞), .
Assuming W is known, it remains to estimate Q
Q is the non-minimum phase part and cannot be estimated using Second Order Statistics
WW →ˆ
Q - Unconstrained Parameters
=
=
ntttt
t
t
qqq
qqqqqq
Q
θ
θθ
θ
θθθ
θθθθθθ
M
L
MOMM
K
L
2
1
21
22221
11211
where,
)()()(
)()()()()()(
has only ‘n’ un-constrained parameters, which can vary freely.
−θθθθ
cossinsincos
has only (n = ) 1 un-constrained parameter
t x t complex unitary matrix Q has only t2 un-constrained parameters.
Hence, if W is known, H = WQH has t2 un-constrained parameters.
Advantages
How to estimate Q
Solution : Estimate Q from the training sequence !
Q Matrix Estimation
Unitary matrix Q parameterized by a significantly lesser number of parameters than H.
r x r unitary - r2 parameters
r x r complex - 2r2 parameters
As the number of receive antennas increases, size of H increases while that of Q remains constant
- size of H is r x t
- size of Q is t x t
Estimation of constrained matrices
Let Nθ be the number of un-constrained parameters in H.
Xp be an orthogonal pilot. i.e. Xp XpH α I
θσσ N
LHH
s
nF 2
22
2]||ˆ[||E ≥−
Estimation is directly proportional to the number of un-constrained parameters.
E.g. For an 8 X 4 complex matrix H, Nθ = 64. The unitary matrix Q is 4 X 4 and has Nθ = 16 parameters. Hence, the ratio of semi-blind to training based MSE of estimation is given as
.dB) 6 (i.e. 41664
==sb
t
MSEMSE
Semi-Blind CRB
Let Q = [q1, q2,…., qt]. qi is thus a column of Q . The constraintson qi s are then given as:
Unit norm constraints : qiH qi = ||qi||2 = 1
Orthogonality Constraints : qiH qj = 0 for i ≠ j
−
=
M
M
13
12
32
21
11 1
)(
qqqq
qqqq
f
H
H
H
H
H
θConstraint Matrix :
Semi-Blind CRB (contd …)
F(θ ) and U for semi-blind estimation can be written down explicitly as
−−
−−=
OMMMMM
L
L
L
OMMMMM
L
L
L
*1
*2
*1
*2
*1
21
321
0000000000
00000000
00
qqq
qqqqq
U
U is 2t2 x t2
=
MMMMOMMM
LL
LL
LL
LL
LL
LL
00000000
0000000000000000
)(
31
13
22
12
21
11
TH
TH
TH
TH
TH
TH
qqqq
qqqq
qqqq
F θ
F(θ ) is t2 x 2t2
Variance of Estimation
Let SVD( H ) be given as PΣ QH.
Let the pilot Xp be orthogonal, i.e. Xp XpH α I,
The un-constrained FIM is given as
( )IILJn
s ⊗Σ⊗= ×2
222
2
σσ
CRB for the semi-blind estimation of vec(H) is given as
( ) ( ) ( )HttHH
tt IPUJUUUIP ×
−
× ⊗Σ⊗Σ1
∑∑= = +
=t
i
t
jijki
ji
i
s
nlk qp
LC
1 1
2222
2
2
2
, ||||σσ
σσσ
CRB on the variance of the (k,l)th element is
Constrained ML Estimation
2||||min pH
pQXWQY −Minimize the ‘True-Likelihood’
subject to : IQQH = Goal :
Orthogonal Pilot Maximum Likelihood - OPML
) SVD( where,ˆ HHHH YXWVUVUQ =Σ=Estimate:
Properties
Unbiased constrained estimator does NOT exist, hence does not achieve CRB.
Achieves CRB asymptotically in pilot length, L.
Also achieves CRB asymptotically in SNR.
Constrained ML Estimation
The unconstrained cost-function can be written using Lagrange multipliers λ, µ as
Define A = XpYpHΣ .
Let Tk-1 = A + (Lσs2 I - XpXp
H)Qk-1 Σ 2
The desired solution is then given as
{ }∑ ∑ ∑∑= = +==
+−+−=t
i
t
i
t
ij
Hjiij
Hiiip
Hii
t
ip qqqqXqiYQf
1 1 1
2
1 )( Re)1(||)(~||),,( µλσµλ
HTTk kk
VUQ11 −−
=
Iterative ML technique for a general pilot sequence - IGML
where is the SVD of Tk-1. H
TTT kkkVU
111 −−−Σ
Simulation Results
Semi-Blind is 6 dB lower in MSE.
Perfect W
MSE vs. L
r = 8, t = 4
Channel Estimation for MRT
)()()( kvkHxky +=
Transmit beamforming : We transmit a single data stream after passing through a beamforming vector w.
x(k) = w s(k) where w ε C tReceive beamforming :
s = zHy where z ε C rNormalization :
|| w || = || z || = 1
Consider the MIMO system
Maximum Ratio Transmission/Combining
Transmit Beamforming: Optimum transmit beamformer (MRT): w = v1, the dominant
eigenvector of HHH. (dominant right singular vector)v1 maximizes the received SNR and the mutual information
across the channel.
Receive Beamforming :Optimum receive beamformer (MRC): z = u1 the dominant
eigenvector of HHH.MRT requires feedback of v1 to the transmitter.For low dimensional (2 x 2, 3 x 3) systems, MRT/MRC achieves the
same capacity as tx-CSI systems, at least at low transmit powers.Goal: Estimate u1 and v1.
Key Results
Theoretical analysis of ``training only''-based conventional least squares estimation (CLSE)
Propose a Semi-Blind schemeClosed Form Semi-Blind (CFSB) solutionPropose a signal transmission scheme for CFSBTheoretically analyze performance of CFSB Demonstrate it asymptotically achieves the Cramer-Rao Bound
Study relative performance of CLSE and CFSB
Conventional Least Squares Est. (CLSE)
Pilot Beamformed Data
Conventional L
Employ L orthogonal training symbols Xp to find the ML estimate Hc of H, Estimate uc and vc via an SVD of Hc.
Input-output relationYp = H Xp + np
CLSE (contd …)
Orthogonal training:XpXp
H = It, gp = LPT/tEstimation problem:
Hc = arg min G ε Cr x t such that||Yp - G Xp||F2
Solution:Hc = Yp Xp
H /gp
Estimate uc and vc from singular value decomposition of Hc.By the invariance principle, uc and vc are the ML estimates
of u1 and v1
Channel Estimation for MRT
Semi-BlindPilotL N
Pilot Beamformed DataWhite Data
N unknown white data are information bearing symbols.
Step 1: Use N unknown data symbols to find estimate of U from:
rnyH IRUU 22 ˆˆˆ σ−=Σ
where ∑=
=N
i
Hy iyiy
NR
1)()(1ˆ
Closed-From Semi-Blind (CFSB) Est.
Step 2: Use L known orthogonal training symbols Xp and usto find vs:
sH
pp
sH
pps uYX
uYXv =
Result: If us =u1, then, vs is the constrained MMSE estimate ofv1 under ||vs|| = 1.
Why “White Data” ? If we beamforming using w at Tx, the output correlation will be
HwwHHH and not HHH
Performance of CFSB
Let # blind information bearing symbols N, be large.
Result: the Cramer-Rao Bound for the estimation of vswith perfect knowledge of u1
|| v1||2 = 1 ⇒ 1 constraint ⇒ (2t - 1) parameters.
This is also the asymptotic ML estimation error of v1under perfect knowledge of u1 ( N →∞).
{ } )12(2
1 E 21
21 −=− t
gvv
ps σ
CLSE vs. CFSB - Comparison
{ } ∑= −
+=−
t
i i
i
pc g
vv2
2221
2212
1 )(1 E
σσσσ
{ }
+
++
−+
−=− ∑
=2
21
221
2
2222
121
2
21
21 )(2
)12( Edd
iit
i i
i
ps g
NgNg
tvv σσσσσσσ
σσ
∑= −
−=t
i ipg 222
1
212
12
σσσσρ
( ) ( )
+++
−−
−−= ∑
=2
221
221
222
1
21
111
di
d
it
i ip gN
gNgt σσσσ
σσσρ
CLSE
CFSB
MSE in v1
Gain
MSE in v1
Gain
Simulation Results
SNR = 2dB and 10dB
Comparison of CLSE, CFSB and OPML schemes for estimation of v1
OPML performs best with perfect U followed by CFSB and CLSE.
Simulation Results
Bit error probability versus data SNR for the 2 x 2 system, withL = 2N = 16 pilot SNR = 2dB
OFDM Channel Estimation
# sub-carriers = K
# channel taps = L
TKaaap ],,,[ 110 −= K
TLhhhh ],,,[ 110 −= K
nahr +=
r is the received symbol vector.
n is AWGN.
System is cyclic prefix extended.
Time Vs. Freq. Domain channel estimation for OFDM systems.
Consider a multicarrier system with
Channel Model
Time Domain Est.
=
−−−−
−−−
+−−
+−−−
LKKKK
LLL
LKK
LKKK
aaaa
aaaa
aaaaaaaa
a
K
MOMMM
K
MOMMM
K
K
321
0321
2101
1210
Symbol matrix a is given as
ith column of p is 1st column cyclically shifted by i positions
Time domain Least-Squares estimate is given as
.)(ˆ 1 raaah HH −=
Freq. Domain Estimation
. , )( FnNFpdiagA ==
The frequency domain equivalent of the system is obtained as
,NAHR +=
where
,)1)(1(0)1(
)1(000
=−−−
−
KKK
K
WW
WWF
K
MOM
K
F is the K x K Fourier matrix.
Freq. Domain domain Least-Squares estimate is given as,
.)(ˆ 1 RAAAH HHf
−=
Constrained CRB
H = Fsh, where Fs is the left K x L submatrix of F.
When K > L, H is a constrained vector.
Infact, constraints on H are given as FrH H = 0, where Fr is the
right K x (K-L) submatrix of F.
.}ˆ {E
}ˆ {E2
2
LK
HH
HH
t
f=
−
−
Total # constrained parameters = K (i.e. dim. of H ).
# un-constrained parameters = L (i.e. dim. of h ).
Hence, from earlier result 1, if AHA = I (identical to orthogonal pilot)
Simulation Results
K = 40
L = 5
MSE vs SNR
Time Domain is 9dB better in terms of estimation.
Conclusions
Demonstrated the construction of Complex Constrained Cramer Rao Bound (CC-CRB)
Investigated applications of the CC-CRB to
Semi-Blind MIMO Channel Estimation
Semi-Blind estimation for MRT
Time Vs. Freq. domain OFDM channel estimation.
Conclusions