A Semi-Blind Technique for MIMO Channel Matrix EstimationAditya Jagannatham and Bhaskar D. Rao

•The proposed algorithm performs well compared to its training based parallel for low pilot lengths.•Its performance improves as the accuracy of computation of the whitening matrix increases. •Typically the algorithm performs well in • 1. Low SNR environments• 2. When there are more receive than transmit antennas

Summary:Summary:

Semi-BlindSemi-Blind

•What ?•Semi-Blind refers to using a “pilot” signal of known input symbols along with “blind” outputs from unknown data symbols to estimate the channel matrix.

•Why ?•In most practical scenarios training symbols are available•Avoids “Convergence problems”•Makes complete use of the training and blind information available.•Robust algorithms for low SNR scenarios.

•Then, HTS = Y X † is the training based least-squares estimate of the channel matrix.•It is the MLE and hence is optimal.•However, it does NOT use blind information.

Training only based estimationTraining only based estimation

)](),....,2(),1([

)](),....,2(),1([

LyyyY

LxxxX

• Let N channel outputs

be observed.

• Of the inputs, are known “training” symbols.

• are “blind” outputs to unknown data symbols

• It is desired to make best use of complete available data to estimate H.

)}(),....,1(),(),...,2(),1({ NyLyLyyy

)}(),....,2(),1({ Lxxx

)}(),.....,2(),1({ NyLyLy )}(),.....,2(),1({ NxLxLx

• If is the number of parameters required to parameterize a matrix G then,

Since is defined by a much lesser number of parameters than , it can be determined with greater accuracy from the limited pilot data.

• As the number of receive antennas r increases, the size of increases while the size of (and hence, its complexity) remains constant.

G#

2# tQ rtH 2#

QH

QH

Advantages:Advantages:

When the whitening matrix is perfectly known, the error bound for estimating by the SB algorithm is:

The error bound for the computation of employing the pilot signal based procedure is :

2

1,

TS

SB

e

etrWhen

L

te

s

nSB 2

22 22

F 2]||H-HE[||

L

rtes

nTS 2

22F ]||H-HE[||

W

H

H

Simulation Results:Simulation Results:

Figure shows the error after the optimization of the ‘True-Likelihood’ cost function. H is 4X4, Data is 16-QAM, SNR =13dB

The proposed procedure always performs better than the Training based one.

• Problem motivation• Training only based estimation• Semi-blind estimation • Algorithm• Performance bounds• Simulations and observations• Conclusions and future work

OutlineOutline

Tx R

# transmit antennas = t # receive antennas = r

System Model: )()()( knkxHky

)(

)(

)(

)(

)(

)(

)(

)( 2

1

2

1

kx

kx

kx

kx

ky

ky

ky

ky

tr

Multi-Input Multi-Output SystemMulti-Input Multi-Output System

University Of California, San Diego

- Flat Fading Channel Matrix - Input Symbol Vector - Channel Output Vector - Additive White Gaussian Noise

- Input is “white” =

- Noise Covariance =

M is unknown. “Channel Estimation” refers to determining this channel matrix.

trCH tCx rCy rCn

lknlnkn ,rr2I)]()([E

lkslxkx ,tt2I)]()([E

Plot shows the estimation error Vs pilot for an 8X4 matrix H, SNR = 13 dB and W is perfectly known. The proposed algorithm performs 4 dB better. This is expected because the parameter ratio is 64:16.

Our work focuses on developing optimization procedures for steps 3,4 and 5. The Optimal matrix is first estimated as a solution to a simple modified likelihood.

Using this as an initial estimate, the optimal for the actual cost function is determined, followed by a procedure for computing the jointly optimal estimates of

Compute “blind”from output data.

yR

Factorize Find

“Whiten” OutputCompute initial by SVD Q

“Refine” Estimate of Q

Find jointly optimalW and Q

2[ - ]y nR I

Semi-Blind AlgorithmSemi-Blind Algorithm

2||ˆ||min ppQ

QXWY Minimize the ‘True-Likelihood’ :subject to IQQH

Goal :

2||ˆ||min ppQ

QXYW ‘Modified-Likelihood’ :

L

jjj

Hjj

n

N

Liin

HHin

H WQXYWQXYYIWWYIWWLN1

21

122 )()(1

)()]ln[det(2

1)(

Step 5: Minimize True Likelihood

W

QW and

This has a ‘Closed-Form’ Solution

The above likelihood has to be optimized for ‘JOINTLY’ optimal solution of W and Q

1.

2.

3.

4.

5.

Step 3.

Step 4.

• can be determined employing only output• Assuming that noise power is known, can be computed “blind” from the output, without any knowledge of input.• can then be computed using training data.

yRW2

n

Q

However, since is a unitary matrix, the optimal solution has to be determined through a “constrained” optimization procedure.

Q

• is a “whitening” matrix and • is a unitary “rotation” matrix.

WQ

ttH IQQ

Then IWWyyER nHH

y2)(

Every matrix can be decomposed as:)( trtr MWQH

The semi-blind techniqueThe semi-blind technique

tr tt