Eurecom, Sophia Antipolis

Complex Constrained CRB And Channel Estimation Applications to Semi-Blind MIMO/OFDM

Aditya K. Jagannatham,

Prof. Bhaskar D. Rao

University of California

SanDiego

Overview

Motivation and construction of complex constrained Cramer Rao Bound (CC-CRB).

Semi-Blind MIMO channel estimation.

Motivation, scheme, constrained estimators.

Semi-blind estimation for MRT.

Scheme and analysis.

Time Vs. Freq. domain OFDM channel estimation.

Parameter Estimation - Preliminaries

Observations

θparameter

nωωω ,,, 21 Kp (ω ; θ )

Estimator :

For instance - Estimation of the mean of a Gaussian

Estimator -

)1,(~);( θθω Np

),,,(ˆ21 nf ωωωθ K=

∑=

=n

iin

1

1ˆ ωθ

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

]) -ˆ )(ˆE[( HC θθθθθ −=

∂∂

∂∂

=θ

θωθ

θω ),(ln),(lnE ppJT

1- JC ≥θ

where,

Constrained Parameter Estimation

Most literature pertains to “unconstrained-real” parameter estimation.

Results for ‘complex’ parameter estimation ?

What are the corresponding results for “constrained” estimation ?

1||||)( 2 ==≡ θθθθ HhFor instance, estimation of a unit norm constrained singular vector i.e.

Estimator

CRB

Complex Cons. Par.

Complex Constrained Parameter CRB

Builds on work by Stoica’97 and VanDenBos’93

Define the extendedconstraint set f (θ)

≡

)()(

)( * θθ

θhh

f

∂∂

∂∂

=∂∂

≡ *

)()()()(γθ

γθ

θθθ fffF

With complex derivatives, define the matrix F (θ) as

= *γ

γθ

Define the extended parameter vector θ as

p(ω, θ ) be the likelihoodof the observation ωparameterized by θ

U span the NullSpace of F(θ).

0)( =UF θ

Let γ be an n - dim constrained complex parameter vector

The constraints on γare given by h(γ ) = 0

Constrained Parameter CRB (Contd …)

∂∂

∂∂

=*),(ln),(lnE

θθω

θθω ppJ

T

J is the complex un-constrained Fischer Information Matrix (FIM) defined as

CRB Result : The CRB for the estimation of the ‘complex-constrained’ parameter θis given as

HH UJUC 1)U(U −≥θ

MIMO System Model

t -

tran

s mi t

r-r e

c eiv

e

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

MIMO System Model

=

ry

yy

yM

2

1

MIMO

System

H

=

tx

xx

xM

2

1

)()()( kvkHxky +=

=

rtrr

t

t

hhh

hhhhhh

....

...

...

H

21

22221

11211

is the r x t complex channel matrixwhere

Estimating H is the problem of ‘Channel Estimation’

Issues in Channel Estimation

As the number of channels increases, employing entirely trainingdata 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.

Diversity SNR

1 25

4 12

3102 −×=eP* Constellation Size = 4

Calls for more robust estimation strategies.

Estimation Strategies - Training Based

MIMO System

H

Training

Blind Outputs

pL Xxxx =},....,,{ 21

},....,,{ 21 NLL xxx ++

Data

pL Yyyy =},....,,{ 21Training Output

},....,,{ 21 NLL yyy ++

2|||| min Fpp H XY − Constraint

HXY pp =+Solution + denotes pseudo-inverse

Blind Estimation

MIMO System

HData

},...,,{ 21 Nxxx },...,,{ 21 Nyyy

Estimate channel from data.

No training necessary

Uses information in source statistics

Training Vs. Blind Estimation

Training

Blind Incre

asing

Effic

iency

Incre

asing

Sim

plicit

y

T e c h n iq u e A d v a n ta g e C o s tT ra in in g V e ry S im p le to

im p le m e n tIn e ffic ie n t B W

u sa g e .B lin d N o B W sa crifice d

fo r tra in in gC o m p u ta tio n a lly

C o m p le x

Semi-Blind MIMO Channel Estimation

MIMO System

H

Training

Blind Outputs

pL Xxxx =},....,,{ 21

},....,,{ 21 NLL xxx ++

Data

pL Yyyy =},....,,{ 21Training Output

},....,,{ 21 NLL yyy ++

Statistics:kls

Hlk xx δσ I][E 2=

klnHlk δσνν I][E 2=

Spatio-temporally un-correlated

(N-L), the number of blind “information” symbols can be large.

L, the pilot length is critical.

Semi-Blind Estimation

Goals :

Use as few training symbols as possible

Use total information

},...,,{},,....,,{ 2121 LL yyyxxx

},....,,{ 21 NLL yyy ++

Total Information

Training Data

Blind Data

Semi-Blind Estimation

Key Idea : H is decomposed as a matrix product, H= WQH

H= WQH

W is known as the “whitening” matrix

W can be estimated using only ‘Blind’ data.

QQH = I

Q is a ‘constrained’ matrix

Q is the non-minimum phase part and cannot be estimated employing Second Order Statistics.

Estimating W :

Output correlation :

Estimate output correlation

IHHR nH

sy22 σσ +=

∑+=

=N

Lk

Hy kyky

NR

1)()(1ˆ

H

nys

H

QWH

IRWW

ˆˆ

)ˆ(1ˆˆ 22

=

−= σσ

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

Q is the non-minimum phase part and cannot be estimated using Second Order Statistics

WW →ˆ

Q - Unconstrained Parameters

=

=

ntttt

t

t

qqq

qqqqqq

Q

θ

θθ

θ

θθθ

θθθθθθ

M

L

MOMM

K

L

2

1

21

22221

11211

where,

)()()(

)()()()()()(

has only ‘n’ un-constrained parameters, which can vary freely.

−θθθθ

cossinsincos

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.

Advantages

How to estimate Q

Solution : Estimate Q from the training sequence !

Q Matrix Estimation

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

Estimation of constrained matrices

Let Nθ be the number of un-constrained parameters in H.

Xp be an orthogonal pilot. i.e. Xp XpH α I

θσσ N

LHH

s

nF 2

22

2]||ˆ[||E ≥−

Estimation is directly proportional to the number of un-constrained parameters.

E.g. For an 8 X 4 complex matrix H, Nθ = 64. The unitary matrix Q is 4 X 4 and has Nθ = 16 parameters. Hence, the ratio of semi-blind to training based MSE of estimation is given as

.dB) 6 (i.e. 41664

==sb

t

MSEMSE

Semi-Blind CRB

Let Q = [q1, q2,…., qt]. qi is thus a column of Q . The constraintson qi s are then given as:

Unit norm constraints : qiH qi = ||qi||2 = 1

Orthogonality Constraints : qiH qj = 0 for i ≠ j

−

=

M

M

13

12

32

21

11 1

)(

qqqq

qqqq

qq

f

H

H

H

H

H

θConstraint Matrix :

Semi-Blind CRB (contd …)

F(θ ) and U for semi-blind estimation can be written down explicitly as

−−

−−=

OMMMMM

L

L

L

OMMMMM

L

L

L

*1

*2

*1

*2

*1

21

321

0000000000

00000000

00

qqq

qq

qqqqq

U

U is 2t2 x t2

=

MMMMOMMM

LL

LL

LL

LL

LL

LL

00000000

0000000000000000

)(

31

13

22

12

21

11

TH

TH

TH

TH

TH

TH

qqqq

qqqq

qqqq

F θ

F(θ ) is t2 x 2t2

Variance of Estimation

Let SVD( H ) be given as PΣ QH.

Let the pilot Xp be orthogonal, i.e. Xp XpH α I,

The un-constrained FIM is given as

( )IILJn

s ⊗Σ⊗= ×2

222

2

σσ

CRB for the semi-blind estimation of vec(H) is given as

( ) ( ) ( )HttHH

tt IPUJUUUIP ×

−

× ⊗Σ⊗Σ1

∑∑= = +

=t

i

t

jijki

ji

i

s

nlk qp

LC

1 1

2222

2

2

2

, ||||σσ

σσσ

CRB on the variance of the (k,l)th element is

Constrained ML Estimation

2||||min pH

pQXWQY −Minimize the ‘True-Likelihood’

subject to : IQQH = Goal :

Orthogonal Pilot Maximum Likelihood - OPML

) SVD( where,ˆ HHHH YXWVUVUQ =Σ=Estimate:

Properties

Unbiased constrained estimator does NOT exist, hence does not achieve CRB.

Achieves CRB asymptotically in pilot length, L.

Also achieves CRB asymptotically in SNR.

Constrained ML Estimation

The unconstrained cost-function can be written using Lagrange multipliers λ, µ as

Define A = XpYpHΣ .

Let Tk-1 = A + (Lσs2 I - XpXp

H)Qk-1 Σ 2

The desired solution is then given as

{ }∑ ∑ ∑∑= = +==

+−+−=t

i

t

i

t

ij

Hjiij

Hiiip

Hii

t

ip qqqqXqiYQf

1 1 1

2

1 )( Re)1(||)(~||),,( µλσµλ

HTTk kk

VUQ11 −−

=

Iterative ML technique for a general pilot sequence - IGML

where is the SVD of Tk-1. H

TTT kkkVU

111 −−−Σ

Simulation Results

Semi-Blind is 6 dB lower in MSE.

Perfect W

MSE vs. L

r = 8, t = 4

Channel Estimation for MRT

)()()( kvkHxky +=

Transmit beamforming : We transmit a single data stream after passing through a beamforming vector w.

x(k) = w s(k) where w ε C tReceive beamforming :

s = zHy where z ε C rNormalization :

|| w || = || z || = 1

Consider the MIMO system

Maximum Ratio Transmission/Combining

Transmit Beamforming: Optimum transmit beamformer (MRT): w = v1, the dominant

eigenvector of HHH. (dominant right singular vector)v1 maximizes the received SNR and the mutual information

across the channel.

Receive Beamforming :Optimum receive beamformer (MRC): z = u1 the dominant

eigenvector of HHH.MRT requires feedback of v1 to the transmitter.For low dimensional (2 x 2, 3 x 3) systems, MRT/MRC achieves the

same capacity as tx-CSI systems, at least at low transmit powers.Goal: Estimate u1 and v1.

Key Results

Theoretical analysis of ``training only''-based conventional least squares estimation (CLSE)

Propose a Semi-Blind schemeClosed Form Semi-Blind (CFSB) solutionPropose a signal transmission scheme for CFSBTheoretically analyze performance of CFSB Demonstrate it asymptotically achieves the Cramer-Rao Bound

Study relative performance of CLSE and CFSB

Conventional Least Squares Est. (CLSE)

Pilot Beamformed Data

Conventional L

Employ L orthogonal training symbols Xp to find the ML estimate Hc of H, Estimate uc and vc via an SVD of Hc.

Input-output relationYp = H Xp + np

CLSE (contd …)

Orthogonal training:XpXp

H = It, gp = LPT/tEstimation problem:

Hc = arg min G ε Cr x t such that||Yp - G Xp||F2

Solution:Hc = Yp Xp

H /gp

Estimate uc and vc from singular value decomposition of Hc.By the invariance principle, uc and vc are the ML estimates

of u1 and v1

Channel Estimation for MRT

Semi-BlindPilotL N

Pilot Beamformed DataWhite Data

N unknown white data are information bearing symbols.

Step 1: Use N unknown data symbols to find estimate of U from:

rnyH IRUU 22 ˆˆˆ σ−=Σ

where ∑=

=N

i

Hy iyiy

NR

1)()(1ˆ

Closed-From Semi-Blind (CFSB) Est.

Step 2: Use L known orthogonal training symbols Xp and usto find vs:

sH

pp

sH

pps uYX

uYXv =

Result: If us =u1, then, vs is the constrained MMSE estimate ofv1 under ||vs|| = 1.

Why “White Data” ? If we beamforming using w at Tx, the output correlation will be

HwwHHH and not HHH

Performance of CFSB

Let # blind information bearing symbols N, be large.

Result: the Cramer-Rao Bound for the estimation of vswith perfect knowledge of u1

|| v1||2 = 1 ⇒ 1 constraint ⇒ (2t - 1) parameters.

This is also the asymptotic ML estimation error of v1under perfect knowledge of u1 ( N →∞).

{ } )12(2

1 E 21

21 −=− t

gvv

ps σ

CLSE vs. CFSB - Comparison

{ } ∑= −

+=−

t

i i

i

pc g

vv2

2221

2212

1 )(1 E

σσσσ

{ }

+

++

−+

−=− ∑

=2

21

221

2

2222

121

2

21

21 )(2

)12( Edd

iit

i i

i

ps g

NgNg

tvv σσσσσσσ

σσ

∑= −

−=t

i ipg 222

1

212

12

σσσσρ

( ) ( )

+++

−−

−−= ∑

=2

221

221

222

1

21

111

di

d

it

i ip gN

gNgt σσσσ

σσσρ

CLSE

CFSB

MSE in v1

Gain

MSE in v1

Gain

Simulation Results

SNR = 2dB and 10dB

Comparison of CLSE, CFSB and OPML schemes for estimation of v1

OPML performs best with perfect U followed by CFSB and CLSE.

Simulation Results

Bit error probability versus data SNR for the 2 x 2 system, withL = 2N = 16 pilot SNR = 2dB

OFDM Channel Estimation

# sub-carriers = K

# channel taps = L

TKaaap ],,,[ 110 −= K

TLhhhh ],,,[ 110 −= K

nahr +=

r is the received symbol vector.

n is AWGN.

System is cyclic prefix extended.

Time Vs. Freq. Domain channel estimation for OFDM systems.

Consider a multicarrier system with

Channel Model

Time Domain Est.

=

−−−−

−−−

+−−

+−−−

LKKKK

LLL

LKK

LKKK

aaaa

aaaa

aaaaaaaa

a

K

MOMMM

K

MOMMM

K

K

321

0321

2101

1210

Symbol matrix a is given as

ith column of p is 1st column cyclically shifted by i positions

Time domain Least-Squares estimate is given as

.)(ˆ 1 raaah HH −=

Freq. Domain Estimation

. , )( FnNFpdiagA ==

The frequency domain equivalent of the system is obtained as

,NAHR +=

where

,)1)(1(0)1(

)1(000

=−−−

−

KKK

K

WW

WWF

K

MOM

K

F is the K x K Fourier matrix.

Freq. Domain domain Least-Squares estimate is given as,

.)(ˆ 1 RAAAH HHf

−=

Constrained CRB

H = Fsh, where Fs is the left K x L submatrix of F.

When K > L, H is a constrained vector.

Infact, constraints on H are given as FrH H = 0, where Fr is the

right K x (K-L) submatrix of F.

.}ˆ {E

}ˆ {E2

2

LK

HH

HH

t

f=

−

−

Total # constrained parameters = K (i.e. dim. of H ).

# un-constrained parameters = L (i.e. dim. of h ).

Hence, from earlier result 1, if AHA = I (identical to orthogonal pilot)

Simulation Results

K = 40

L = 5

MSE vs SNR

Time Domain is 9dB better in terms of estimation.

Conclusions

Demonstrated the construction of Complex Constrained Cramer Rao Bound (CC-CRB)

Investigated applications of the CC-CRB to

Semi-Blind MIMO Channel Estimation

Semi-Blind estimation for MRT

Time Vs. Freq. domain OFDM channel estimation.

Conclusions