1. FIM Regularity for Gaussian Semi-Blind(SB) MIMO FIR Channel Estimation
Aditya K. Jagannatham
DSP MIMO Group, UCSD
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2. 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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3. 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) D +
x(n)
H(1)
H(2)
H(L-1)
y(n)
H(0)
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4. Channel Estimation Schemes
Training based Estimation:
H(z)
Training inputs
Training outputs
Blind Estimation:
‘Blind’ data inputs
‘Blind’ data outputs
H(z)
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5. Channel Estimation Schemes
Training
Increasing Complexity
Decreasing BW Efficiency
Blind
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 ?
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6. 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.
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7. 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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8. 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,
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9. 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.
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10. 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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11. 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
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12. 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
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13. 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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14. 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
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15. 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
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16. Simulation r = 4, t = 2 (i.e. 4 X 2 MIMO system). L = 2 Taps.
Fig. is a plot of MSE Vs. SNR.
SB estimation is 32/4 i.e. 9dB lower in MSE
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17. 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.
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