Semi-Blind (SB) Multiple-Input Multiple-Output (MIMO) Channel Estimation

1
Semi-Blind (SB) Multiple-Input Multiple-Output
(MIMO) Channel Estimation
ArrayComm Presentation

• Aditya K. Jagannatham
• DSP MIMO Group, UCSD

2
Overview of Talk

• 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.

3
MIMO System Model

• 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

4
MIMO System Model

• where,
• is the r x t complex channel matrix
• Estimating H is the problem of Channel
  Estimation
• Parameters 2.r.t (real parameters)

5
MIMO Channel Estimation

• 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.

6
Training Based Estimation

• One can formulate the Least-Squares cost
  function,
• The estimate of H is given as
• Training symbols convey no information.

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Blind Estimation

• Uses information in source statistics.
• Statistics
• - Source covariance is known, E(x(k)x(k)H)
  ss2It
• - Noise covariance is known, E(v(k)v(k)H) sn2Ir
• Estimate channel entirely from blind information
  symbols.
• No training necessary.

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Channel Estimation Schemes
Increasing Complexity
Decreasing BW Efficiency

• 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 ?

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Semi-Blind Estimation
N symbols
H(z)
Training inputs
Blind data inputs
Training outputs
Blind data outputs

• Training information
• - Xp x(1), x(2),, x(L) , Yp y(1),
  y(2),, y(L)
• Blind information
• - E (x(k)x(k)H) ss2It, E (v(k)v(k)H) sn2Ir
• (N-L), the number of blind information symbols
  can be large.
• L, the pilot length is critical.

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Whitening-Rotation

• H is decomposed as a matrix product, H WQH.
• For instance, if SVD(H) P? QH, W P?.

W is known as the whitening matrix
W can be estimated using only Blind data.
H WQH
QQH I Q is a constrained matrix
Q , the unitary matrix, cannot be estimated from
Second Order Statistics.
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Estimating Q

• How to estimate Q ?
• Solution Estimate Q from the training sequence
  !

Advantages
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

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Estimating W

• 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.

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Constrained Estimation

• 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.

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Parameter Estimation

• Estimator
• For instance - Estimation of the mean of a
  Gaussian
• Estimator

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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
• where

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Constrained Estimation

• 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.

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Complex-Constrained Estimation

• Builds on work by Stoica97 and VanDenBos93

Let ? be an n - dim constrained complex
parameter vector
The constraints on ? are given by h(? ) 0
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Constrained Estimation(Contd.)
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Semi-Blind CC-CRB

• 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
  qi2 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

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Unconstrained Parameters

• 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.

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Semi-Blind CRB

• 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

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Simulation Results

• Perfect W, MSE vs. L.
• r 8, t 4.

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OFDM Channel Estimation

• 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
  ).

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FIR-MIMO System

• H(0),H(1),,H(L-1) to be estimated.
• r receive antennas, t transmit antennas (r
  gt t).
• Parameters 2.r.t.L (L complex r X t matrices)

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Fisher Information Matrix (FIM)

• Let p(??) be the p.d.f. of the observation
  vector ?.
• The FIM (Fisher Information Matrix) of the
  parameter ? is given as
• Result Rank of the matrix J? equal to the number
  of identifiable parameters.
• In other words, the dimension of its null space
  is precisely the number of un-identifiable
  parameters.

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SB Estimation for MIMO-FIR

• FIM based analysis yields insights in to SB
  estimation.
• Let the channel be parameterized as ?2rtL.
• Application to MIMO Estimation
• J? 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 gtt), rank(JB) is given
  as

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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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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(J?).
• J? is full rank for identifiability.
• If Lt is the number of pilot symbols,
• Lt t for full rank, i.e. rank(J?) 2rtL.

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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
  Tugnait00
• 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.

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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 a t2 (indeterminate parameters)
• MSEt a 2.r.t.L (total parameters).
• Hence the ratio of the limiting MSEs is given as

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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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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 a 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.

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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.