ARTICLE IN PRESS

Contents lists available at ScienceDirect

Signal Processing

Signal Processing 89 (2009) 111–115

0165-16

doi:10.1

� Cor

E-m

laurent.

journal homepage: www.elsevier.com/locate/sigpro

Fast communication

Bayesian Cramer–Rao bounds for complex gain parametersestimation of slowly varying Rayleigh channel in OFDM systems

Hussein Hijazi �, Laurent Ros

GIPSA-lab, Image and Signal Department, BP 46, 38402 Saint Martin d’Heres, France

a r t i c l e i n f o

Article history:

Received 28 March 2008

Received in revised form

23 June 2008

Accepted 15 July 2008Available online 3 August 2008

Keywords:

Bayesian Cramer–Rao bound

OFDM

Rayleigh complex gains

84/$ - see front matter & 2008 Published by

016/j.sigpro.2008.07.017

responding author. Tel.: +33 4 76 82 7178; fax

ail addresses: hussein.hijazi@gipsa-lab.inpg.fr

ros@gipsa-lab.inpg.fr (L. Ros).

a b s t r a c t

This paper deals with on-line Bayesian Cramer–Rao (BCRB) lower bound for complex

gains dynamic estimation of time-varying multi-path Rayleigh channels. We propose

three novel lower bounds for 4-QAM OFDM systems in case of negligible channel

variation within one symbol, and assuming both channel delay and Doppler frequency

related information. We derive the true BCRB for data-aided (DA) context and, two

closed-form expressions for non-data-aided (NDA) context.

& 2008 Published by Elsevier B.V.

1. Introduction

Dynamic estimation of frequency selective and time-varying channel is a fundamental function [1] fororthogonal frequency division multiplexing (OFDM) mo-bile communication systems. In radio-frequency trans-missions, channel estimation can be generally obtained byestimating only some physical propagation parameters,such as multi-path delays and multi-path complex gains[2–4]. Moreover, in slowly varying channels, the numberof paths and time delays can be easily obtained [2], sincedelays are quasi-invariant over a large number of symbols.Assuming full availability of delay related information,which is the ultimate accuracy that can be achieved withchannel estimation methods? Tools to face this problemare available from parameters estimation theory [5] inform of the Cramer–Rao Bounds (CRBs), which givefundamental lower limits of the mean square error(MSE) achievable by any unbiased estimator. A modifiedCRB (MCRB), easier to evaluate than the Standard CRB(SCRB), has been introduced in [6,7]. The MCRB is effective

Elsevier B.V.

: +33 4 76 82 63 84.

(H. Hijazi),

when, in addition to the parameter to be estimated, theobserved data also depend on other unwanted para-meters. More recently, the problem of deriving CRBs,suited to time-varying parameters, has been addressedthroughout the Bayesian context. In [8], the authorspropose a general framework for deriving analyticalexpression of on-line CRBs. In [9], the authors introducea new asymptotic bound, namely the asymptotic BayesianCRB (ABCRB), for non-data-aided (NDA) scenario. Thisbound is closer to the classical BCRB than the ModifiedBCRB (MBCRB) and it is easier to be evaluated than BCRB.In this paper, we investigate the BCRB related to theestimation of the complex gains of a Rayleigh channel,assuming negligible time variation within one OFDMsymbol and, both channel delay and Doppler frequencyrelated information. Explicit expressions of the BCRB andits variants, MBCRB and ABCRB, are provided for NDA andDA 4-QAM on-line scenarios.

Notations: ½x�k denotes the kth entry of the vector x,and ½X�k;m the ½k;m�th entry of the matrix X. As in Matlab,X½k1: k2;m1:m2� is a submatrix extracted from rows k1 tok2 and from columns m1 to m2 of X. diagfxg is a diagonalmatrix with x on its diagonal, diagfXg is a vector whoseelements are the elements of the diagonal of X andblkdiagfX;Yg is a block diagonal matrix with the matricesX and Y on its diagonal. Ex;y½�� denotes the expectation over

ARTICLE IN PRESS

H. Hijazi, L. Ros / Signal Processing 89 (2009) 111–115112

x and y. J0ð�Þ is the zeroth-order Bessel function of the firstkind. rx and Dx

y represent the first- and the second-orderpartial derivatives operator, i.e., rx ¼ ½q=qx1; . . . ;q=qxN �

T

and Dxy ¼ r

�

yrTx.

2. System model

Consider an OFDM system with N sub-carriers, and acyclic prefix of length Ng . The duration of an OFDMsymbol is T ¼ vTs, where Ts is the sampling time andv ¼ N þ Ng . Let xðnÞ ¼ ½xðnÞ½�N=2�; xðnÞ½�N=2þ 1�; . . . ; xðnÞ½N=2� 1��T be the nth transmitted OFDM symbol, wherefxðnÞ½b�g are normalized 4-QAM symbols. It is assumed thatthe transmission is over a multi-path Rayleigh channel,with negligible variation within one OFDM symbol,characterized by the impulse response

hðnT; tÞ ¼XL

l¼1

aðnÞl dðt� tlTsÞ (1)

where L is the total number of propagation paths, al is thelth complex gain of variance s2

al(with

PLl¼1 s2

al¼ 1), and

tl � Ts is the lth delay (tl is not necessarily an integer, buttLoNg). The L individual elements of faðnÞl g are uncorre-lated with respect to each other. They are wide-sensestationary narrow-band complex Gaussian processes, withthe so-called Jakes’ power spectrum [10] with Dopplerfrequency f d. It means that aðnÞl are correlated complexGaussian variables with zero-means and correlationcoefficients given by

RðpÞal¼ E½aðnÞl aðn�pÞ

l

�� ¼ s2

alJ0ð2pf dTpÞ (2)

Hence, the nth received OFDM symbol yðnÞ ¼ ½yðnÞ½�N=2�;yðnÞ½�N=2þ 1�; . . . ; yðnÞ½N=2� 1��T is given by [2,3]:

yðnÞ ¼ HðnÞxðnÞ þwðnÞ (3)

where wðnÞ is a N � 1 zero-mean complex Gaussian noisevector with covariance matrix s2IN , and HðnÞ is a N � N

diagonal matrix with diagonal elements given by [2,3]:

½HðnÞ�k;k ¼XL

l¼1

aðnÞl � e�j2pððk�1Þ=N�12Þtl

h i(4)

This coefficients are the Fourier Transform of (1) evaluatedat the discrete frequency f k ¼ ðk� 1� N=2Þ1=NTs withk 2 ½1;N�. Using (4), the observation model in (3) for thenth OFDM symbol can be re-written as

yðnÞ ¼ diagfxðnÞgF aðnÞ þwðnÞ (5)

where aðnÞ ¼ ½aðnÞ1 ; . . . ;aðnÞL �T is a L� 1 vector and F is the

N � L Fourier matrix defined by

½F�k;l ¼ e�j2pððk�1Þ=N�1=2Þtl (6)

3. Bayesian Cramer–Rao bounds (BCRBs)

In this section, we present a general formulation forBCRB which is related to the estimation of the multi-pathcomplex gains. In NDA context, we derive a closed-formexpression of a BCRB, i.e., the asymptotic BCRB or themodified BCRB. In DA context, we will find that the trueBCRB is equal to the MBCRB in NDA. aðyÞ denotes an

unbiased estimator of a ¼ ½aTð1Þ; . . . ;a

TðKÞ�

T using the set of

measurements y ¼ ½yTð1Þ; . . . ;y

TðKÞ�

T. In the on-line scenario,

the receiver estimates aðnÞ based on the current and

previous observations only, i.e., y ¼ ½yTð1Þ; . . . ; y

TðnÞ�

T.

3.1. Bayesian Cramer–Rao bound

The BCRB is particularly suited when a priori informa-tion is available. The BCRB has been proposed in [5] suchthat

Ey;a½ðaðyÞ � aÞðaðyÞ � aÞH�XBCRBðaÞ (7)

where XXY is interpreted as meaning that the matrix X�Y is positive semidefinite. The BCRB is the inverse of theBayesian information matrix (BIM), which can be writtenas

B ¼ Ea½FðaÞ� þ Ea½�Daa lnðpðaÞÞ� (8)

where pðaÞ is the prior probability density function (pdf)and FðaÞ is the Fisher information matrix (FIM) definedas

FðaÞ ¼ Eyja½�Daa lnðpðyjaÞÞ� (9)

where pðyjaÞ is the conditional pdf of y given a. The on-line BCRB associated to observation vector y ¼ ½yT

ð1Þ; . . . ;

yTðKÞ�

T will be obtained [9] by

BCRBðaðKÞÞon-line ¼ TrðBCRBðaÞ½iðKÞ;iðKÞ�Þ (10)

where iðnÞ is a sequence of indices defined by iðnÞ ¼

1þ ðn� 1ÞL : nL with n 2 ½1;K�. Definition (10) will standfor the closed-form BCRBs.

(1) Computation of Ea½�Daa lnðpðaÞÞ�: a is a complex

Gaussian vector with zero mean and covariance matrixRa ¼ EfaaHg of size KL� KL defined as

½Ra�iðl;pÞ;iðl0 ;p0 Þ ¼Rðp�p0 Þal

for l0 ¼ l 2 ½1; L�; p; p0 2 ½0;K � 1�

0 for l0al; p; p0 2 ½0;K � 1�

(

(11)

where iðl; pÞ ¼ 1þ ðl� 1Þ þ pL and RðpÞalis defined in (2). For

example, if K ¼ L ¼ 2 then, Ra is given by

Ra ¼

Rð0Þa10 Rð�1Þ

a10

0 Rð0Þa20 Rð�1Þ

a2

Rð1Þa10 Rð0Þa1

0

0 Rð1Þa20 Rð0Þa2

26666664

37777775

(12)

Thus, the pdf pðaÞ is defined as

pðaÞ ¼1

jpRaje�aHR�1

a a (13)

Taking the second derivative of the natural logarithm of(13) with respect to a and making the expectation over a,hence

Ea½�Daa lnðpðaÞÞ� ¼ R�1

a (14)

(2) Computation of Ea½FðaÞ�: Using the whitenessof the noise and the independence of the transmittedOFDM symbols, one obtains from the observation model

ARTICLE IN PRESS

H. Hijazi, L. Ros / Signal Processing 89 (2009) 111–115 113

in (5) that

Daa lnðpðyjaÞÞ ¼

XK

n¼1

Daa lnðpðyðnÞjaðnÞÞÞ (15)

Each term of the sum (15) is a KL� KL block diagonalmatrix with only one nonzero L� L block matrix, namely

Daa lnðpðyðnÞjaðnÞÞÞ½iðnÞ;iðnÞ� ¼ DaðnÞ

aðnÞlnðpðyðnÞjaðnÞÞÞ (16)

As a direct consequence, Daa lnðpðyjaÞÞ is a block diagonal

matrix with the nth diagonal block given by (16). More-over, because of the circularity of the observation noise,the expectation of (16) with respect to yðnÞ and aðnÞ doesnot depend on aðnÞ. One then obtains

Ea½FðaÞ� ¼ blkdiagfJ; J; . . . ; Jg (17)

where J is a L� L matrix defined as

J ¼ Ey;a½�DaðnÞaðnÞ

lnðpðyðnÞjaðnÞÞÞ� (18)

The log-likelihood function in (18) can be expanded as

lnðpðyðnÞjaðnÞÞÞ ¼ lnXxðnÞ

pðyðnÞjxðnÞ;aðnÞÞpðxðnÞÞ

!(19)

The vector yðnÞ for given xðnÞ and aðnÞ is complex Gaussianwith mean vector mðnÞ ¼ diagfxðnÞgFaðnÞ and covariancematrix s2IN . Thus, the conditional pdf is

pðyðnÞjxðnÞ;aðnÞÞ ¼1

jps2INje�1=s2ðyðnÞ�mðnÞÞ

HðyðnÞ�mðnÞÞ (20)

Since each element of the vector mðnÞ depends on only oneelement of xðnÞ then, using the Gaussian nature of thenoise and the equiprobability of the normalized QAMsymbols, one finds (see Appendix A) that

lnðpðyðnÞjaðnÞÞÞ ¼ ln1

jps2INje�1=s2ðyH

ðnÞyðnÞþaH

ðnÞFHFaðnÞ Þ

�

�YNk¼1

cosh

ffiffiffi2p

s2ReðanðkÞÞ

!cosh

ffiffiffi2p

s2ImðanðkÞÞ

!#

(21)

where anðkÞ ¼ ½yðnÞ��

kgTkaðnÞ and gT

k is the kth row of thematrix F. The result of the second derivative of (21) withrespect to aðnÞ is given by

DaðnÞaðnÞ

lnðpðyðnÞjaðnÞÞÞ ¼ �1

s2FHFþ

XN

k¼1

1

2s4½yðnÞ�k½yðnÞ�

�

kg�kgTk

�

� 2� tanh2

ffiffiffi2p

s2ReðanðkÞÞ

!

�tanh2

ffiffiffi2p

s2ImðanðkÞÞ

!!#(22)

The expectation of (22) with respect to yðnÞjaðnÞ does nothave any simple analytical solution. Hence, we have toresort to either numerical integration methods or someapproximations. In the following, we present both the

high-SNR and the low-SNR approximations of the BCRB, asdefined in [9].

3.2. Asymptotic BCRB

(1) High-SNR BCRB asymptote: From the definition ofBIM (8), only the first term (i.e., Ea½FðaÞ�) depends onthe SNR, which is fully characterized by J. Hence, we focuson the behavior of J. At high SNR (i.e., s2 ! 0), the tanh-function in (22) can be approximated as tanhð

ffiffiffi2p

=s2xÞ

� sgnðxÞ. Hence, we obtain the high-SNR asymptote of J,which is

Jh ¼1

s2FHF (23)

(2) Low-SNR BCRB asymptote: Following the samereasoning as before, at low SNR (i.e., s2 !þ1), we havetanhðxÞ � x around x ¼ 0. Hence, we obtain

DaðnÞaðnÞ

lnðpðyðnÞjaðnÞÞÞ � �1

s2FHFþ

XN

k¼1

1

s8½yðnÞ�k½yðnÞ�

�

kg�kgTk

�

�ðs4 � anðkÞa�nðkÞÞ

�(24)

Plugging (24) into (18), we obtain the low-SNR asymptoteof J, which is (see Appendix B):

Jl ¼bs4þ

8b2

s6þ

6b3

s8

!FHF (25)

where b ¼PL

l¼1 s2al

is the total channel energy.The asymptotic BCRB (ABCRB) defined in [9] leads to a

lower bound on the MSE. This ABCRB is given by

ABCRBðaÞ ¼ ðblkdiagfJmin; . . . ; Jming þ R�1a Þ�1 (26)

where Jmin ¼ minðvl; vhÞFHF, with vl ¼ b=s4 þ 8b2=s6 þ

6b3=s8 and vh ¼ 1=s2.

3.3. Modified BCRB

The analytical computation of FðaÞ is quite tedious incase of NDA context because of the OFDM symbolsx ¼ ½xT

ð1Þ; . . . ;xTðKÞ�

T, which are ‘‘nuisance parameters’’. Inorder to circumvent this kind of problem, a Modified BCRB(MBCRB) has been proposed in [6]. This MBCRB is theinverse of the following information matrix:

C ¼ Ea½GðaÞ� þ Ea½�Daa lnðpðaÞÞ� (27)

where GðaÞ is the modified FIM defined as

GðaÞ ¼ ExEyjx;a½�Daa lnðpðyjx;aÞÞ� (28)

Hence, following the same reasoning as before, we have

Ea½GðaÞ� ¼ blkdiagfJm; Jm; . . . ; Jmg (29)

where Jm is a L� L matrix defined as

Jm ¼ Ey;x;a½�DaðnÞaðnÞ

lnðpðyðnÞjxðnÞ;aðnÞÞÞ� (30)

By taking the second derivative of the natural logarithm(ln) of (20) with respect to aðnÞ, one easily obtains that

Jm ¼ Ex1

s2FHdiagfxH

ðnÞgdiagfxðnÞgF

� �¼

1

s2FHF (31)

ARTICLE IN PRESS

10 20 30 40 50 60 70 8010−5

10−4

10−3

10−2

Observation Block Length K

Per

form

ance

Bou

nds

SCRBOn−line ABCRB fdT = 5*10−3

On−line ABCRB fdT = 10−3

On−line ABCRB fdT = 10−4

On−line ABCRB fdT = 10−5

Fig. 2. BCRBs vs. number of observations, for SNR ¼ 10 dB.

H. Hijazi, L. Ros / Signal Processing 89 (2009) 111–115114

since the QAM-symbols are normalized and uncorrelatedwith respect to each other. The MBCRB for the estimationof a is given by:

MBCRBðaÞ ¼ ðblkdiagfJm; Jm; . . . ; Jmg þ R�1a Þ�1 (32)

We see that Jh ¼ Jm hence, the high-SNR asymptote of theBCRB is equal to the MBCRB. This corroborates the resultof [11] for a scalar parameter in non-Bayesian case.

Notice that the term DaðnÞaðnÞ

lnðpðyðnÞjxðnÞ;aðnÞÞÞ ¼ �1s2FHF

does not depend on the transmitted data sequence x.Hence, in the case of ‘‘time-invariant’’, the true BCRB indata-aided (DA) context is equal to the MBCRB in non-data-aided (NDA) context.

4. Discussion and conclusion

In this section, we illustrate the behavior of theprevious bounds for the complex gains estimation.A 4-QAM OFDM system with normalized symbols, N ¼

128 subcarriers and Ng ¼ N=8 is used. The normalizedRayleigh channel contains L ¼ 6 paths and others para-meters given in [3].

Fig. 1 presents the on-line BCRB (evaluated by Monte-Carlo trials), ABCRB and MBCRB versus SNR ¼ 1=s2, for ablock-observation length K ¼ 20 and a normalized Dop-pler frequency f dT ¼ 10�3. We also plot as reference theSCRB (i.e., the prior information is not used). We observethat both ABCRB and the MBCRB are lower than SCRBsince the prior information of the complex gains isconsidered. We also verify that MBCRBpABCRBpBCRB,as in [9]. At high SNR, the MBCRB and the ABCRB are veryclose, as predicted by our theoretical analysis.

Fig. 2 presents the on-line ABCRB versus time index K

for different normalized Doppler frequencies 10�5pf dTp5� 10�3 and SNR ¼ 10 dB. When the number of observa-tions increases, the estimation can be significantlyimproved when the estimator takes also into accountthe previous information; the bound thus decreases andconverges to an asymptote. The estimation gain is largerusing previous symbols with slow channel variations(low f dT). In brief, our contribution permits to measurethe benefit of using additional previous OFDM symbols forchannel estimation process of the current symbol,whereas most methods use only the current symbol [1].

−30 −20 −10 0 10 20 30 40 50 60 7010−10

10−8

10−6

10−4

10−2

100

102

SNR

Per

form

ance

Bou

nds

SCRBOn−line MBCRB K = 20On−line ABCRB K = 20On−line BCRB K = 20 (Monte Carlo)

Fig. 1. SCRB and BCRBs vs. SNR for f dT ¼ 0:001.

Appendix A. Derivation of expression (21) and (22)

Plugging (20) into (19), we obtain

lnðpðyðnÞjaðnÞÞÞ ¼ �1

s2ðyHðnÞyðnÞ þmH

ðnÞmðnÞÞ

þ lnpðxðnÞÞ

jps2INj

XxðnÞ

e2s2ReðyH

ðnÞmðnÞÞ

!(33)

since the 4QAM-symbols are equiprobable (i.e., pðxðnÞÞ ¼1=4N). However, mðnÞ ¼ diagfxðnÞgFaðnÞ then, yH

ðnÞmðnÞ ¼PNk¼1 anðkÞ½xðnÞ�k, where anðkÞ is defined in Section 3.1.

Hence, one obtains

XxðnÞ

e2=s2ReðyHðnÞmðnÞÞ ¼

YNk¼1

X½xðnÞ �k

e2=s2ReðanðkÞ½xðnÞ �kÞ

0@

1A (34)

Since ½xðnÞ�k ¼ ð1=ffiffiffi2pÞð�1� jÞ (4QAM-symbol), we obtain

X½xðnÞ �k

e2=s2ReðanðkÞ½xðnÞ �kÞ ¼ 4 cosh

ffiffiffi2p

s2ReðanðkÞÞ

!cosh

�

ffiffiffi2p

s2ImðanðkÞÞ

!(35)

Inserting this result into (33), we obtain the expression in(21). Taking the second derivative of (21) with respect toaðnÞ and using raðnÞReðanðkÞÞ ¼ 1

2½yðnÞ��

kgk and raðnÞ ImðanðkÞÞ

¼ 12j ½yðnÞ�

�

kgk, we obtain finally the expression in (22).

Appendix B. Evaluation of Jl in (25)

Inserting the definition of anðkÞ into (24) and pluggingthe result into (18), one obtains

Jl ¼1

s2FHF�

1

s4

XN

k¼1

g�kEaEyja½½yðnÞ�k½yðnÞ��

k�gTk þ

1

s8

XN

k¼1

g�kgTkEa

�½aðnÞaHðnÞg�kEyja½ð½yðnÞ�k½yðnÞ�

�

kÞ2��gT

k (36)

Using that ½yðnÞ�k ¼ ½xðnÞ�kgTkaðnÞ þ ½wðnÞ�k, the independence

between the QAM-symbols and the noise, and these

ARTICLE IN PRESS

H. Hijazi, L. Ros / Signal Processing 89 (2009) 111–115 115

results below

E½xðnÞ �k ½½xðnÞ�2k � ¼ E½wðnÞ �k ½½wðnÞ�

2k � ¼ 0 and

E½wðnÞ �k ½½wðnÞ�2k ½wðnÞ�

�k

2� ¼ 2s4 (37)

we obtain

Eyja½½yðnÞ�k½yðnÞ��

k� ¼ gTkaðnÞa

HðnÞg�k þ s

2

Eyja½ð½yðnÞ�k½yðnÞ��

kÞ2� ¼ 2s4 þ 4s2gT

kaðnÞaHðnÞg�k

þ gTkaðnÞa

HðnÞg�kgT

kaðnÞaHðnÞg�k (38)

Hence, Jl becomes

Jl ¼1

s4

XN

k¼1

VkDVk þ4

s6

XN

k¼1

VkEa½T1�Vk þ1

s8

�XN

k¼1

VkEa½T2�Vk (39)

where Vk ¼ g�kgTk , T1 ¼ aðnÞaH

ðnÞVkaðnÞaHðnÞ, T2 ¼ aðnÞaH

ðnÞVkaðnÞ

aHðnÞVkaðnÞa

HðnÞ and D ¼ Ea½aðnÞaH

ðnÞ� ¼ diagfs2a1; . . . ;s2

aLg. The

elements of T1 and T2 are given by

½T1�l;l0 ¼XL

l1¼1

XL

l2¼1

½Vk�l1;l2½aðnÞ�l½aðnÞ�l2 ½aðnÞ��

l0 ½aðnÞ��l1

½T2�l;l0 ¼XL

l1¼1

XL

l2¼1

XL

l3¼1

XL

l4¼1

½Vk�l1;l2½Vk�l3;l4½aðnÞ�l½aðnÞ�l2 ½aðnÞ�l4 ½aðnÞ��

l0

�½aðnÞ��l1½aðnÞ�

�l3

(40)

Using that E½cðnÞ �l ½½cðnÞ�2l � ¼ 0 and the definitions of fourth

and sixth order moments for complex Gaussian variables,we obtain

Ea½T1� ¼ DVkDþ TrðVkDÞD

Ea½T2� ¼ 2DVkDVkDþ 2TrðVkDÞDVkDþ TrðVkDVkDÞD

þ ðTrðVkDÞÞ2D (41)

Using that gTkDg�k ¼ TrðVkDÞ ¼

PLl¼1 s2

al¼ b, TrðVkDVkDÞ

¼ b2, DVkDVkD ¼ bDVkD, and inserting these results into(39), we obtain the expression of Jl in (25).

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