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FIM Regularity for Gaussian Semi-Blind(SB) MIMO FIR

Channel Estimation

Aditya K. JagannathamDSP MIMO Group, UCSD

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MIMO Channel Estimation

• CSI (Channel State Information) is critical in MIMO Systems.

• Number of training symbols increases with transmit antennas resulting in Bandwidth inefficiency.

• Low SNR in MIMO systems aggravates the problem.

• Channel estimation holds key to MIMO gains.

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FIR-MIMO System• r = #receive antennas, t = #transmit antennas (r > t)

• H(0),H(1),…,H(L-1) to be estimated.• #Parameters = 2.r.t.L (L complex r X t matrices)

)1()1(...)1()1()()0()( LnxLHnxHnxHny

D

+

D Dx(n)

H(1) H(2) H(L-1)

+ y(n)+

H(0)

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Channel Estimation Schemes

Blind Estimation:

Training based Estimation:

H(z)Training inputs

Training outputs

‘Blind’ data inputs

‘Blind’ data outputs

H(z)

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Channel Estimation Schemes

• Semi-blind schemes trade off BW efficiency for algorithmic simplicity and complete estimation.

• How much information can be obtained from blind symbols ?– In other words, how many of the 2rtL parameters can be

estimated blind ?• How does one quantify the performance of an SB

Scheme ?

Training

Blind

Increasing Complexity

Decreasing BW Efficiency

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Fisher Information Matrix (FIM)

• Let p(ω,θ) be the p.d.f. of the observation vector ω.• The FIM of the parameter θ is given as

• Result: Rank of the matrix Jθ equal to the number of identifiable parameters, or in other words, the dimension of its null space is precisely the number of un-identifiable parameters.

H

pEJ

);(ln2

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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.

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Blind FIM, JB

• Let the input-output blind symbols be stacked as

• Gaussianity on input symbols, X(k) ~ N(0,IPt)

• The blind likelihood Lb is given as,

where• The blind FIM JB is given as,

)()(

)1(

)2(

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)1()0(0

)2()1()0(

)2(

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kNy

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Nky

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kY

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.)( 2IHHHR nH

Y

HbB HYL

EJ

))(;(2

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Rank Properties

• Null space of JB can be shown to be given as

• It can then be demonstrated that for MIMO-FIR channels, rank(JB) is given as

where r= #receive antennas, t = #transmit antennas.

,2)( 2trtLJrank B

21

*1

*2

*2

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*1

*3

*2

000

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00

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))1((

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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 ?

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Training FIM, Jt

• The t2 indeterminate parameters have to be estimated from pilot symbols.

• How many pilot symbols are needed for identifiability?

• Again, answer is found from rank(Jθ), total SB FIM.

• Jθ = JB + Jt, is full rank for identifiability.

• Let xp(1), xp(2),…, xp(Lt) be Lt transmitted pilots.

• Jt, the training FIM is given as

tL

i

tt iJJ1

)(

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Training FIM, Jt (Contd.)

• Jt(i) can be shown to be given as

• If Lt is the number of pilot symbols,

• Lt = t for full rank, i.e. rank(Jθ) = 2rtL

1 ),2(2)()( 22 tLLtLtrtLJJrankJrank ttttB

Ijxix

IjxixV

VVV

VVV

VVV

iJ Tpp

Hpp

ji

LiLi

iLi

iLi

Lii

ii

ii

Lii

ii

ii

n

t

)()(0

0)()( ,

1)( *

11

111

11

111

11

2

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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.

• This requires a ‘Constrained’ Maximum-Likelihood (ML) estimation procedure.

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Constrained Estimation

• Let xp(1), xp(2),…, xp(Lt) be Lt transmitted pilots.

• In addition, let Xp be orthogonal, i.e. XpXpH =k I.

• The ‘Constrained’ ML cost function is given as

• The ML estimate of Q is given as

))()(( where,,ˆ1

0

L

i

HHH iWYiXSVDVUUVQ

)()1()()( ,

)1(

)1(

)0(

iLxiLxiLxiX

LX

X

X

X tpppp

p

p

p

p

IQQtosubjectiXQiWY HL

ip

Hp

, )()(21

0

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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 is prop. to t2 (indeterminate parameters)

– MSEt is prop. to 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

dBLMSE

MSEtr

SB

t 32

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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.

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End

• FIM based analysis provides framework tool to study SB estimation of FIR-MIMO channels.

• The blind indeterminacy in a FIR-MIMO system corresponds to a unitary matrix of t2 parameters.

• The unitary matrix has much fewer parameters and is estimated through a constrained ML procedure.

References• A. K. Jagannatham and B. D. Rao, “Semi-Blind MIMO FIR Channel Estimation:

Regularity and Algorithms”, Submitted to IEEE Transactions on Signal Processing.

• A. K. Jagannatham and B. D. Rao, “FIM Regularity for Gaussian Semi-Blind MIMO FIR Channel Estimation”, Asilomar Conference on Signals, Systems, and Computers, 2005.