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ArrayComm Presentation. Semi-Blind (SB) Multiple-Input Multiple-Output (MIMO) Channel Estimation. Aditya K. Jagannatham DSP MIMO Group, UCSD. Overview of Talk. Semi-Blind MIMO flat-fading Channel estimation. Motivation Scheme: Constrained Estimators.

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semi blind sb multiple input multiple output mimo channel estimation

ArrayComm Presentation

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

Aditya K. Jagannatham

DSP MIMO Group, UCSD

overview of talk
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.
mimo system model

Rx

r- receive

TX

t - transmit

Receiver

Transmitter

=Antenna

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
  • hijdenotes the channel between theithreceiver and jth transmitter.
  • This channel is represented by the Flat-Fading Channel MatrixH
mimo system model1

MIMO System

H

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)
mimo channel estimation
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.

training based estimation

H(z)

Training inputs

Training outputs

Outputs

Inputs

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

H(z)

‘Blind’ data inputs

‘Blind’ data outputs

Blind Estimation
  • 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.
channel estimation schemes

Training

Blind

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 ?

Increasing Complexity

Decreasing BW Efficiency

semi blind estimation
Semi-Blind Estimation

N symbols

  • 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) = σn2Ir

  • (N-L),the number of blind “information” symbols can be large.
  • L, the pilot length is critical.

H(z)

Training inputs

‘Blind’ data inputs

Training outputs

‘Blind’ data outputs

whitening rotation
Whitening-Rotation
  • His decomposed as a matrix product,H= WQH.
  • For instance, if SVD(H) = P QH, W = P.

Wis known as the “whitening” matrix

Wcan be estimated using only ‘Blind’ data.

H= WQH

QQH = I

Qis a ‘constrained’ matrix

Q , the unitary matrix, cannot be estimated from Second Order Statistics.

estimating q
Estimating Q
  • How to estimateQ ?
  • Solution : EstimateQfrom the training sequence !

Advantages

Unitary matrixQparameterized by a significantly lesser number of parameters thanH.

r x r unitary - r2 parameters

r x r complex - 2r2 parameters

  • As the number of receive antennas increases, sizeofHincreases while that ofQremains constant
  • size of H is r x t
  • size of Q is t x t
estimating w
Estimating W
  • Output correlation :
  • Estimate output correlation
  • EstimateW by a matrix square root (Cholesky) factorization as,
  • As # blind symbols grows ( i.e. N ), .
  • AssumingWis known, it remains to estimateQ.
constrained estimation
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.

parameter estimation

p(;)

parameter

Observations

Parameter Estimation
  • Estimator :
  • For instance - Estimation of the mean of a Gaussian
  • Estimator
cramer rao bound crb
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

constrained estimation1

CRB

Complex Cons. Par.

Estimator

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.
complex constrained estimation

p(,  )be thelikelihoodof the observationparameterized by

Define the extended parameter vector as

With complexderivatives, define the matrixF ()as

Define the extended constraint setf ()

Uspan the NullSpace ofF().

Complex-Constrained Estimation
  • Builds on work by Stoica’97 and VanDenBos’93

Letbe ann- dim constrained complex parameter vector

The constraints onare given byh( ) = 0

constrained estimation contd

Jis 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

Constrained Estimation(Contd.)
semi blind cc crb
Semi-Blind CC-CRB
  • LetQ = [q1, q2,…., qt].qiis thus a column ofQ . The constraints onqisare given as:
  • Unit norm constraints:qiHqi = ||qi||2 = 1
  • Orthogonality Constraints :qiHqj = 0 for i  j
  • Constraint Matrix :
  • Let SVD( H )be given asP QH.
  • CRB on the variance of the(k,l)thelement is
unconstrained parameters
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 t2un-constrained parameters.
  • Hence, ifWis known,H = WQH hast2un-constrained parameters.
semi blind crb
Semi-Blind CRB
  • LetNbe the number of un-constrained parameters inH.
  • Also, Xpbe an orthogonal pilot.i.e.Xp XpHI
  • Estimation is directly proportional to the number of un-constrained parameters.
  • E.g. For an8 X 4complex matrixH, N= 64. The unitary matrixQis 4 X 4and hasN= 16parameters. Hence, the ratio of semi-blind to training based MSE of estimation is given as
simulation results
Simulation Results
  • Perfect W, MSE vs. L.
  • r = 8, t = 4.
ofdm channel estimation
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,whereFs is the leftK x Lsubmatrix ofF (Fourier Matrix).
  • Total # constrained parameters =K(i.e. dim. of H ).
  • # un-constrained parameters =L(i.e. dim. of h ).
fir mimo system

x(k)

D

D

D

+

+

+

H(0)

H(1)

H(2)

H(L-1)

y(k)

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)
fisher information matrix fim
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 Jθequal to the number of identifiable parameters.
    • In other words, the dimension of its null space is precisely the number of un-identifiable parameters.
sb estimation for mimo fir
SB Estimation for MIMO-FIR
  • FIM based analysis yields insights in to SB estimation.
  • Letthe 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
implications for estimation
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
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.
sb estimation scheme
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 Lttransmitted pilot symbols.
semi blind crb1
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 α2.r.t.L (total parameters).
  • Hence the ratio of the limiting MSEs is given as
simulation
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
talk summary
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.
references
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.