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Semi-Blind (SB) Multiple-Input Multiple-Output (MIMO) Channel
Estimation
Aditya K. JagannathamDSP MIMO Group, UCSD
ArrayComm Presentation
• Semi-Blind MIMO flat-fading Channel estimation.
- Motivation
- Scheme: Constrained Estimators.
- Construction of Complex Constrained Cramer Rao Bound (CC-CRB).
- Additional Applications: Time Vs. Freq. domain OFDM channel estimation.
• Frequency selective MIMO channel estimation.
- Fisher information matrix (FIM) based analysis
- Semi-blind estimation.
Overview of Talk
• 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
Rx
Transmitter
Receiver
= Antenna
TX
t -
t ran
smit
r - r
ece
ive
where,
is the r x t complex channel matrix
• Estimating H is the problem of ‘Channel Estimation’
• #Parameters = 2.r.t (real parameters)
MIMO System Model
rtrr
t
t
hhh
hhh
hhh
...
.
...
...
H
21
22221
11211
ry
y
y
y2
1
MIMO
System
H
tx
x
x
x2
1
)()()( kvkHxky
• CSI (Channel State Information) is critical in MIMO Systems.- Detection, Precoding, Beamforming, etc.
• Channel estimation holds key to MIMO gains.
• As the number of channels increases, employing
entirely training data 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.- Calls for more robust estimation strategies.
MIMO Channel Estimation
• One can formulate the Least-Squares cost function,
• The estimate of H is given as
• Training symbols convey no information.
Training Based Estimation
.|||| min 2Fpp H XY
.ˆ ppLS XYH
H(z)Training inputs
Training outputs
Inputs
)](),....,2(),1([ LxxxX p pYLyyy )](),....,2(),1([
Outputs
• Uses information in source statistics.
• Statistics:
- Source covariance is known, E(x(k)x(k)H) = σs2It
- Noise covariance is known, E(v(k)v(k)H) = σn2Ir
• Estimate channel entirely from blind information symbols.
• No training necessary.
Blind Estimation
‘Blind’ data inputs
‘Blind’ data outputs
H(z)
Channel Estimation Schemes
• Is there a way to trade-off BW efficiency for algorithmic simplicity and complete estimation.
• How much information can be obtained from blind data?– In other words, how many of the 2rt parameters can be
estimated blind ?
• How does one quantify the performance of an SB Scheme ?
Training
Blind
Increasing Complexity
Decreasing BW Efficiency
• Training information
- Xp = [x(1), x(2),…, x(L)] , Yp = [y(1), y(2),…, y(L)]
• Blind information
- E (x(k)x(k)H) = σs2It, E (v(k)v(k)H) = σn
2Ir
• (N-L), the number of blind “information” symbols can be large.
• L, the pilot length is critical.
Semi-Blind Estimation
H(z)Training inputs
‘Blind’ data inputs
Training outputs
‘Blind’ data outputs
N symbols
• H is decomposed as a matrix product, H= WQH.
• For instance, if SVD(H) = P QH, W = P.
Whitening-Rotation
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 , the unitary matrix, cannot be estimated from Second Order Statistics.
• How to estimate Q ?
• Solution : Estimate Q from the training sequence !
Estimating Q
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
Advantages
• Output correlation :
• Estimate output correlation
• 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.
Estimating W
IHHR nH
sy22
N
k
Hy kyky
NR
1
)()(1ˆ
IRWW nys
H 22
ˆ1ˆˆ
WW ˆ
• Orthogonal Pilot Maximum Likelihood – OPML
• Goal - Minimize the ‘True-Likelihood’
subject to :
• Estimate:
• Properties
1. Achieves CRB asymptotically in pilot length, L.
2. Also achieves CRB asymptotically in SNR.
Constrained Estimation
) SVD( where,ˆ Hpp
HHH XYWVUVUQ
2||||min pH
pQ
XWQY
IQQH
• Estimator :
• For instance - Estimation of the mean of a Gaussian
• Estimator
Parameter Estimation
Observations
parameter
n ,,, 21 p( ; )
),,,(ˆ21 nf
)1,(~);( Np
n
iin
1
1ˆ
• 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
where
Cramer-Rao Bound (CRB)
]) -ˆ )(ˆE[( HC
),(ln),(lnE
ppJ
T
1- JC
• Most literature pertains to “unconstrained-real” parameter estimation.
• Results for ‘complex’ parameter estimation ?
• What are the corresponding results for “constrained” estimation?
• For instance, estimation of a unit norm constrained singular vector i.e.
Constrained Estimation
1||||)( 2 Hh
Complex Cons. Par.
Estimator
CRB
Builds on work by Stoica’97 and VanDenBos’93
Let be an n - dim constrained complex parameter vector
The constraints on are given by h( ) = 0
Define the extended constraint set f ()
)(
)()( *
h
hf
*
)()()()(
fff
F
With complex derivatives, define the matrix F () as
*
Define the extended parameter vector as
p(, ) be the likelihood of the observation parameterized by
U span the Null Space of F().
0)( UF
Complex-Constrained Estimation
*
),(ln),(lnE
pp
JT
J is the complex un-constrained Fischer Information Matrix (FIM) defined as
CRB Result : The CRB for the estimation of the ‘complex-constrained’ parameter is given as HH UJUC 1)U(U
Constrained Estimation(Contd.)
• Let Q = [q1, q2,…., qt]. qi is thus a column of Q . The constraints on qi s are given as:
• Unit norm constraints : qiH
qi = ||qi||2 = 1
• Orthogonality Constraints : qiH
qj = 0 for i j
• Constraint Matrix :
• Let SVD( H ) be given as P QH.
• CRB on the variance of the (k,l)th element is
13
12
32
21
11 1
)(
f
H
H
H
H
H
t
i
t
jijki
ji
i
s
nlk qp
LC
1 1
2222
2
2
2
, ||||
Semi-Blind CC-CRB
• has only ‘n’ un-constrained parameters, which can vary freely.
• 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.
Unconstrained Parameters
ntttt
t
t
qqq
qqq
qqq
Q
2
1
21
22221
11211
where,
)()()(
)()()(
)()()(
cossin
sincos
• Let N be the number of un-constrained parameters
in H.
• Also, Xp be an orthogonal pilot. i.e. Xp XpH I
• Estimation is directly proportional to the number of un-constrained parameters.
• E.g. For an 8 X 4 complex matrix H, N = 64. The
unitary matrix Q is 4 X 4 and has N = 16 parameters. Hence, the ratio of semi-blind to training based MSE of estimation is given as
Semi-Blind CRB
NL
HHs
nF 2
22
2]||ˆ[||E
.dB) 6 (i.e. 416
64
sb
t
MSE
MSE
Simulation Results
• Perfect W, MSE vs. L.
• r = 8, t = 4.
• Time Vs. Freq. Domain channel estimation for OFDM systems.
• Consider a multicarrier system with
# channel taps = L (10), # sub-carriers = K (32,64)
• h is the channel vector.
• g = Fsh, where Fs is the left K x L submatrix of F (Fourier Matrix).
• Total # constrained parameters = K (i.e. dim. of H ).
• # un-constrained parameters = L (i.e. dim. of h ).
OFDM Channel Estimation
.}ˆ {E
}ˆ {E2
2
L
K
Hg
gg
t
f
FIR-MIMO System
• H(0),H(1),…,H(L-1) to be estimated.• r = #receive antennas, t = #transmit antennas (r > t).• #Parameters = 2.r.t.L (L complex r X t matrices)
)()1()1(...)1()1()()0()( kLkxLHkxHkxHky
D
+
D Dx(k)
H(1) H(2) H(L-1)
+ y(k)+
H(0)
Fisher Information Matrix (FIM)
• Let p(ω;θ) be the p.d.f. of the observation vector ω.• The FIM (Fisher Information Matrix) of the parameter θ
is given as
• Result: Rank of the matrix Jθ equal to the number of identifiable parameters.– In other words, the dimension of its null space is precisely
the number of un-identifiable parameters.
H
pEJ
);(ln2
SB Estimation for MIMO-FIR
• FIM based analysis yields insights in to SB estimation.
• Let the channel be parameterized as θ2rtL.
Application to MIMO Estimation:• Jθ = JB + Jt, where JB, Jt are the blind and training CRBs respectively.• It can then be demonstrated that for irreducible MIMO-FIR channels with (r >t), rank(JB) is given as
))0((
))0(( ,
)
*)(
)1(
)1(
)0(
Hvec
HveciH
LH
H
H
22)( trtLJrank B
Implications for Estimation
• Total number of parameters in a MIMO-FIR system is 2.r.t.L . However, the number of un-identifiable parameters is t2.
• For instance, r = 8, t = 2, L = 4. – Total #parameters = 128. – # blindly unidentifiable parameters = 4.
• This implies that a large part of the channel, can be identified blind, without any training.
• How does one estimate the t2 parameters ?
Semi-Blind (SB) FIM
• The t2 indeterminate parameters are estimated from pilot symbols.
• How many pilot symbols are needed for identifiability?
• Again, answer is found from rank(Jθ).
• Jθ is full rank for identifiability.
• If Lt is the number of pilot symbols,
• Lt = t for full rank, i.e. rank(Jθ) = 2rtL.
1 ),2(2)()( 22 tLLtLtrtLJJrankJrank ttttB
SB Estimation Scheme
• The t2 parameters correspond to a unitary matrix Q.
• H(z) can be decomposed as H(z) = W(z) QH.
• W(z) can be estimated from blind data [Tugnait’00]
• The unitary matrix Q can be estimated from the pilot symbols through a ‘Constrained’ Maximum-Likelihood (ML) estimate.
• Let x(1), x(2),…,x(Lt) be the Lt transmitted pilot symbols.
))()(( where,,ˆ1
0
L
i
HHH iWYiXSVDVUUVQ
Semi-Blind CRB
• Asymptotically, as the number of data symbols increases, semi-blind MSE is given as
• Denote MSEt = Training MSE, MSESB = SB MSE.
– MSESB α t2 (indeterminate parameters)
– MSEt α 2.r.t.L (total parameters).
• Hence the ratio of the limiting MSEs is given as
SBt
nF
L
CRBLimitingtL
HHELimb
2
}||ˆ{|| 22
2
)(log10 32 10 LdBLMSE
MSEtr
SB
t
Simulation
• SB estimation is 32/4 i.e. 9dB lower in MSE
• r = 4, t = 2 (i.e. 4 X 2 MIMO system). L = 2 Taps.
• Fig. is a plot of MSE Vs. SNR.
Talk Summary
• Complex channel matrix H has 2rt parameters.– Training based scheme estimates 2rt parameters.– SB scheme estimates t2 parameters.– From CC-CRB theory, MSE α #Parameters.– Hence,
• FIR channel matrix H(z) has 2rtL parameters.– Training scheme estimates 2rtL parameters.– From FIM analysis, only t2 parameters are unknown.– Hence, SB scheme can potentially be very efficient.
dBMSE
MSEtr
SB
t 32
References
Journal • Aditya K. Jagannatham and Bhaskar D. Rao, "Cramer-Rao Lower
Bound for Constrained Complex Parameters", IEEE Signal Processing Letters, Vol. 11, no. 11, Nov. 2004.
• Aditya K. Jagannatham and Bhaskar D. Rao, "Whitening-Rotation Based Semi-Blind MIMO Channel Estimation" - IEEE Transactions on Signal Processing, Accepted for publication.
• Chandra R. Murthy, Aditya K. Jagannatham and Bhaskar D. Rao, "Semi-Blind MIMO Channel Estimation for Maximum Ratio Transmission" - IEEE Transactions on Signal Processing, Accepted for publication.
• Aditya K. Jagannatham and Bhaskar D. Rao, “Semi-Blind MIMO FIR Channel Estimation: Regularity and Algorithms”, Submitted to IEEE Transactions on Signal Processing.
