ARTICLE IN PRESS Signal Processing 89 (2009) 111–115

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Bayesian Cramer–Rao bounds for complex gain parameters estimation of slowly varying Rayleigh channel in OFDM systems Hussein Hijazi , Laurent Ros GIPSA-lab, Image and Signal Department, BP 46, 38402 Saint Martin d’He`res, France

a r t i c l e in fo

abstract

Article history: Received 28 March 2008 Received in revised form 23 June 2008 Accepted 15 July 2008 Available online 3 August 2008

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.

Keywords: Bayesian Cramer–Rao bound OFDM Rayleigh complex gains

1. Introduction Dynamic estimation of frequency selective and timevarying channel is a fundamental function [1] for orthogonal frequency division multiplexing (OFDM) mobile communication systems. In radio-frequency transmissions, channel estimation can be generally obtained by estimating only some physical propagation parameters, such as multi-path delays and multi-path complex gains [2–4]. Moreover, in slowly varying channels, the number of paths and time delays can be easily obtained [2], since delays 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 with channel estimation methods? Tools to face this problem are available from parameters estimation theory [5] in form of the Cramer–Rao Bounds (CRBs), which give fundamental lower limits of the mean square error (MSE) achievable by any unbiased estimator. A modified CRB (MCRB), easier to evaluate than the Standard CRB (SCRB), has been introduced in [6,7]. The MCRB is effective

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E-mail addresses: [email protected] (H. Hijazi), [email protected] (L. Ros). 0165-1684/$ - see front matter & 2008 Published by Elsevier B.V. doi:10.1016/j.sigpro.2008.07.017

when, in addition to the parameter to be estimated, the observed data also depend on other unwanted parameters. More recently, the problem of deriving CRBs, suited to time-varying parameters, has been addressed throughout the Bayesian context. In [8], the authors propose a general framework for deriving analytical expression of on-line CRBs. In [9], the authors introduce a new asymptotic bound, namely the asymptotic Bayesian CRB (ABCRB), for non-data-aided (NDA) scenario. This bound is closer to the classical BCRB than the Modified BCRB (MBCRB) and it is easier to be evaluated than BCRB. In this paper, we investigate the BCRB related to the estimation of the complex gains of a Rayleigh channel, assuming negligible time variation within one OFDM symbol and, both channel delay and Doppler frequency related information. Explicit expressions of the BCRB and its variants, MBCRB and ABCRB, are provided for NDA and DA 4-QAM on-line scenarios. Notations: ½xk denotes the kth entry of the vector x, and ½Xk;m the ½k; mth entry of the matrix X. As in Matlab, X½k1 : k2 ; m1 : m2  is a submatrix extracted from rows k1 to k2 and from columns m1 to m2 of X. diagfxg is a diagonal matrix with x on its diagonal, diagfXg is a vector whose elements are the elements of the diagonal of X and blkdiagfX; Yg is a block diagonal matrix with the matrices X and Y on its diagonal. Ex;y ½ denotes the expectation over

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

x and y. J 0 ðÞ is the zeroth-order Bessel function of the first kind. rx and Dxy represent the first- and the second-order partial derivatives operator, i.e., rx ¼ ½q=qx1 ; . . . ; q=qxN T  T and Dxy ¼ ry rx .

unbiased estimator of a ¼ ½aTð1Þ ; . . . ; aTðKÞ T using the set of measurements y ¼ ½yTð1Þ ; . . . ; yTðKÞ T . In the on-line scenario, the receiver estimates aðnÞ based on the current and previous observations only, i.e., y ¼ ½yTð1Þ ; . . . ; yTðnÞ T .

2. System model 3.1. Bayesian Cramer–Rao bound Consider an OFDM system with N sub-carriers, and a cyclic prefix of length Ng . The duration of an OFDM symbol is T ¼ vT s , where T s is the sampling time and v ¼ N þ Ng . Let xðnÞ ¼ ½xðnÞ ½N=2; xðnÞ ½N=2 þ 1; . . . ; xðnÞ ½N=2  1T be the nth transmitted OFDM symbol, where fxðnÞ ½bg are normalized 4-QAM symbols. It is assumed that the transmission is over a multi-path Rayleigh channel, with negligible variation within one OFDM symbol, characterized by the impulse response hðnT; tÞ ¼

L X

alðnÞ dðt  tl T s Þ

(1)

l¼1

where L is the total number of propagation paths, al is the P lth complex gain of variance s2al (with Ll¼1 s2al ¼ 1), and tl  T s is the lth delay (tl is not necessarily an integer, but tL oN g ). The L individual elements of faðnÞ g are uncorrel lated with respect to each other. They are wide-sense stationary narrow-band complex Gaussian processes, with the so-called Jakes’ power spectrum [10] with Doppler frequency f d. It means that aðnÞ are correlated complex l Gaussian variables with zero-means and correlation coefficients given by 

ðnÞ ðnpÞ RðpÞ  ¼ s2al J 0 ð2pf d TpÞ al ¼ E½al al

(2)

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

(3)

where wðnÞ is a N  1 zero-mean complex Gaussian noise vector with covariance matrix s2 IN , and HðnÞ is a N  N diagonal matrix with diagonal elements given by [2,3]: ½HðnÞ k;k ¼

L h X

1

aðnÞ  ej2pððk1Þ=N2Þtl l

i

This coefficients are the Fourier Transform of (1) evaluated at the discrete frequency f k ¼ ðk  1  N=2Þ1=NT s with k 2 ½1; N. Using (4), the observation model in (3) for the nth OFDM symbol can be re-written as

ðnÞ 1 ;...;

(5)

ðnÞ T L 

where aðnÞ ¼ ½a a is a L  1 vector and F is the N  L Fourier matrix defined by ½Fk;l ¼ ej2pððk1Þ=N1=2Þtl

Ey;a ½ða^ ðyÞ  aÞða^ ðyÞ  aÞH XBCRBðaÞ

(7)

where XXY is interpreted as meaning that the matrix X  Y is positive semidefinite. The BCRB is the inverse of the Bayesian information matrix (BIM), which can be written as B ¼ Ea ½FðaÞ þ Ea ½Daa lnðpðaÞÞ

(8)

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

FðaÞ ¼ Eyja ½Daa lnðpðyjaÞÞ

(9)

where pðyjaÞ is the conditional pdf of y given a. The online BCRB associated to observation vector y ¼ ½yTð1Þ ; . . . ; yTðKÞ T will be obtained [9] by BCRBðaðKÞ Þon-line ¼ TrðBCRBðaÞ½iðKÞ;iðKÞ Þ

(10)

where iðnÞ is a sequence of indices defined by iðnÞ ¼ 1 þ ðn  1ÞL : nL with n 2 ½1; K. Definition (10) will stand for the closed-form BCRBs. (1) Computation of Ea ½Daa lnðpðaÞÞ: a is a complex Gaussian vector with zero mean and covariance matrix Ra ¼ EfaaH g of size KL  KL defined as ( ðpp0 Þ 0 Ral for l ¼ l 2 ½1; L; p; p0 2 ½0; K  1 ½Ra iðl;pÞ;iðl0 ;p0 Þ ¼ 0 0 for l al; p; p0 2 ½0; K  1 (11)

(4)

l¼1

yðnÞ ¼ diagfxðnÞ gF aðnÞ þ wðnÞ

The BCRB is particularly suited when a priori information is available. The BCRB has been proposed in [5] such that

(6)

3. Bayesian Cramer–Rao bounds (BCRBs) In this section, we present a general formulation for BCRB which is related to the estimation of the multi-path complex gains. In NDA context, we derive a closed-form expression of a BCRB, i.e., the asymptotic BCRB or the modified BCRB. In DA context, we will find that the true BCRB is equal to the MBCRB in NDA. a^ ðyÞ denotes an

ðpÞ

where iðl; pÞ ¼ 1 þ ðl  1Þ þ pL and Ral is defined in (2). For example, if K ¼ L ¼ 2 then, Ra is given by 2 3 0 Rð1Þ 0 Rð0Þ a1 6 a1 7 6 0 7 0 Rð1Þ Rð0Þ 6 a2 a2 7 7 Ra ¼ 6 (12) ð0Þ 6 Rð1Þ 0 R a1 0 7 6 a1 7 4 5 0 Rð0Þ 0 Rð1Þ a2 a2 Thus, the pdf pðaÞ is defined as pðaÞ ¼

1 H 1 ea Ra a jpRa j

(13)

Taking the second derivative of the natural logarithm of (13) with respect to a and making the expectation over a, hence Ea ½Daa lnðpðaÞÞ ¼ R1 a

(14)

(2) Computation of Ea ½FðaÞ: Using the whiteness of the noise and the independence of the transmitted OFDM symbols, one obtains from the observation model

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

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

in (5) that

Daa lnðpðyjaÞÞ ¼

K X Daa lnðpðyðnÞ jaðnÞ ÞÞ

(15)

n¼1

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

Daa lnðpðyðnÞ jaðnÞ ÞÞ½iðnÞ;iðnÞ ¼ DaaðnÞ lnðpðyðnÞ jaðnÞ ÞÞ ðnÞ

(16)

As a direct consequence, Daa lnðpðyjaÞÞ is a block diagonal matrix with the nth diagonal block given by (16). Moreover, because of the circularity of the observation noise, the expectation of (16) with respect to yðnÞ and aðnÞ does not depend on aðnÞ . One then obtains Ea ½FðaÞ ¼ blkdiagfJ; J; . . . ; Jg

(17)

where J is a L  L matrix defined as J¼

Ey;a ½DaaðnÞ ðnÞ

lnðpðyðnÞ jaðnÞ ÞÞ

(18)

3.2. Asymptotic BCRB (1) High-SNR BCRB asymptote: From the definition of BIM (8), only the first term (i.e., Ea ½FðaÞ) depends on the SNR, which is fully characterized by J. Hence, we focus on the behavior of J. At high SNR (i.e., s2 ! 0), the pffiffiffitanhfunction in (22) can be approximated as tanhð 2=s2 xÞ  sgnðxÞ. Hence, we obtain the high-SNR asymptote of J, which is Jh ¼

1

s2

FH F

(23)

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

DaaðnÞ lnðpðyðnÞ jaðnÞ ÞÞ   ðnÞ

lnðpðyðnÞ jaðnÞ ÞÞ ¼ ln

X

! pðyðnÞ jxðnÞ ; aðnÞ ÞpðxðnÞ Þ

(19)

xðnÞ

The vector yðnÞ for given xðnÞ and aðnÞ is complex Gaussian with mean vector mðnÞ ¼ diagfxðnÞ gFaðnÞ and covariance matrix s2 IN . Thus, the conditional pdf is pðyðnÞ jxðnÞ ; aðnÞ Þ ¼

H 1 2 e1=s ðyðnÞ mðnÞ Þ ðyðnÞ mðnÞ Þ jps2 IN j

(20)

 2 H H H 1 e1=s ðyðnÞ yðnÞ þaðnÞ F FaðnÞ Þ lnðpðyðnÞ jaðnÞ ÞÞ ¼ ln jps2 IN j ! !# pffiffiffi pffiffiffi N Y 2 2 cosh Reða ðkÞÞ cosh Imða ðkÞÞ  n n 2 2

s

s

(21) 

where an ðkÞ ¼ ½yðnÞ k gTk aðnÞ and gTk is the kth row of the matrix F. The result of the second derivative of (21) with respect to aðnÞ is given by N  X 1 1  lnðpðyðnÞ jaðnÞ ÞÞ ¼  2 FH F þ ½y  ½y  g gT 2s4 ðnÞ k ðnÞ k k k s k¼1 ! pffiffiffi 2 2 Reða ðkÞÞ  2  tanh n 2

s

tanh

2

pffiffiffi 2

s2

Imðan ðkÞÞ

2

s

FH F þ

N  X 1 k¼1

s8

ðs  an ðkÞan ðkÞÞ



½yðnÞ k ½yðnÞ k gk gTk



(24)

Plugging (24) into (18), we obtain the low-SNR asymptote of J, which is (see Appendix B): ! b 8b2 6b3 H Jl ¼ þ þ (25) F F 4 6 8

s

s

s

PL

where b ¼ 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 1 ABCRBðaÞ ¼ ðblkdiagfJmin ; . . . ; Jmin g þ R1 a Þ

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

k¼1

1

4

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

DaaðnÞ ðnÞ

113

H

(26) 4

2

where Jmin ¼ minðvl ; vh ÞF F, with vl ¼ b=s þ 8b =s6 þ 3 6b =s8 and vh ¼ 1=s2 . 3.3. Modified BCRB The analytical computation of FðaÞ is quite tedious in case of NDA context because of the OFDM symbols x ¼ ½xTð1Þ ; . . . ; xTðKÞ T , which are ‘‘nuisance parameters’’. In order to circumvent this kind of problem, a Modified BCRB (MBCRB) has been proposed in [6]. This MBCRB is the inverse of the following information matrix: C ¼ Ea ½GðaÞ þ Ea ½Daa lnðpðaÞÞ

(27)

where GðaÞ is the modified FIM defined as GðaÞ ¼ Ex Eyjx;a ½Daa lnðpðyjx; aÞÞ

(28)

Hence, following the same reasoning as before, we have Ea ½GðaÞ ¼ blkdiagfJm ; Jm ; . . . ; Jm g

(29)

where Jm is a L  L matrix defined as

!!# (22)

The expectation of (22) with respect to yðnÞ jaðnÞ does not have any simple analytical solution. Hence, we have to resort to either numerical integration methods or some approximations. In the following, we present both the

Jm ¼ Ey;x;a ½DaaðnÞ lnðpðyðnÞ jxðnÞ ; aðnÞ ÞÞ ðnÞ

(30)

By taking the second derivative of the natural logarithm (ln) of (20) with respect to aðnÞ , one easily obtains that   1 1 H Jm ¼ Ex 2 FH diagfxH (31) ðnÞ gdiagfxðnÞ gF ¼ 2 F F

s

s

ARTICLE IN PRESS 114

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

since the QAM-symbols are normalized and uncorrelated with respect to each other. The MBCRB for the estimation of a is given by: 1 MBCRBðaÞ ¼ ðblkdiagfJm ; Jm ; . . . ; Jm g þ R1 a Þ

10−2

We see that Jh ¼ Jm hence, the high-SNR asymptote of the BCRB is equal to the MBCRB. This corroborates the result of [11] for a scalar parameter in non-Bayesian case. Notice that the term DaaðnÞ lnðpðyðnÞ jxðnÞ ; aðnÞ ÞÞ ¼ s12 FH F ðnÞ does not depend on the transmitted data sequence x. Hence, in the case of ‘‘time-invariant’’, the true BCRB in data-aided (DA) context is equal to the MBCRB in nondata-aided (NDA) context.

Performance Bounds

(32)

SCRB −3 On−line ABCRB fdT = 5*10 −3 On−line ABCRB fdT = 10 −4 On−line ABCRB fdT = 10 −5 On−line ABCRB fdT = 10

10−3

10−4

10−5 4. Discussion and conclusion

10

In this section, we illustrate the behavior of the previous 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 normalized Rayleigh channel contains L ¼ 6 paths and others parameters given in [3]. Fig. 1 presents the on-line BCRB (evaluated by MonteCarlo trials), ABCRB and MBCRB versus SNR ¼ 1=s2 , for a block-observation length K ¼ 20 and a normalized Doppler frequency f d T ¼ 103. We also plot as reference the SCRB (i.e., the prior information is not used). We observe that both ABCRB and the MBCRB are lower than SCRB since the prior information of the complex gains is considered. We also verify that MBCRBpABCRBpBCRB, as in [9]. At high SNR, the MBCRB and the ABCRB are very close, as predicted by our theoretical analysis. Fig. 2 presents the on-line ABCRB versus time index K for different normalized Doppler frequencies 105 pf d Tp 5  103 and SNR ¼ 10 dB. When the number of observations increases, the estimation can be significantly improved when the estimator takes also into account the previous information; the bound thus decreases and converges to an asymptote. The estimation gain is larger using previous symbols with slow channel variations (low f d T). In brief, our contribution permits to measure the benefit of using additional previous OFDM symbols for channel estimation process of the current symbol, whereas most methods use only the current symbol [1].

Performance Bounds

102

80

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

Appendix A. Derivation of expression (21) and (22) Plugging (20) into (19), we obtain lnðpðyðnÞ jaðnÞ ÞÞ ¼ 

1

s2

H ðyH ðnÞ y ðnÞ þ mðnÞ mðnÞ Þ

pðxðnÞ Þ X 22 ReðyHðnÞ mðnÞ Þ es jps2 IN j xðnÞ

þ ln

! (33)

since the 4QAM-symbols are equiprobable (i.e., pðxðnÞ Þ ¼ 1=4N ). However, mðnÞ ¼ diagfxðnÞ gFaðnÞ then, yH ðnÞ mðnÞ ¼ PN k¼1 an ðkÞ½xðnÞ k , where an ðkÞ is defined in Section 3.1. Hence, one obtains 0 1 N X 2=s2 ReðyH m Þ Y X 2 2= s Reða ðkÞ½x  Þ ðnÞ n ðnÞ k A @ ðnÞ (34) e ¼ e xðnÞ

k¼1

½xðnÞ k

pffiffiffi Since ½xðnÞ k ¼ ð1= 2Þð1  jÞ (4QAM-symbol), we obtain ! pffiffiffi X 2 2 e2=s Reðan ðkÞ½xðnÞ k Þ ¼ 4 cosh Reða ðkÞÞ cosh n 2

s

½xðnÞ k



pffiffiffi 2

s2

! Imðan ðkÞÞ

(35)

Appendix B. Evaluation of Jl in (25)

10−2 10−4

Inserting the definition of an ðkÞ into (24) and plugging the result into (18), one obtains

10−6 10−8 10

70

Inserting this result into (33), we obtain the expression in (21). Taking the second derivative of (21) with respect to  aðnÞ and using raðnÞ Reðan ðkÞÞ ¼ 12½yðnÞ k gk and raðnÞ Imðan ðkÞÞ  1 ¼ 2j ½yðnÞ k gk , we obtain finally the expression in (22).

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

100

20 30 40 50 60 Observation Block Length K

Jl ¼

−10

−30 −20 −10

0

10

20 30 SNR

40

50

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

60

70

1

s

2

FH F 

N 1 X

s

4



gk Ea Eyja ½½yðnÞ k ½yðnÞ k gTk þ

k¼1 

2  T ½aðnÞ aH ðnÞ gk Eyja ½ð½yðnÞ k ½yðnÞ k Þ gk

N 1 X

s8 k¼1

gk gTk Ea (36)

Using that ½yðnÞ k ¼ ½xðnÞ k gTk aðnÞ 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

results below

Using that gTk Dgk ¼ TrðVk DÞ ¼

E½xðnÞ k ½½xðnÞ 2k  ¼ E½wðnÞ k ½½wðnÞ 2k  ¼ 0 E½wðnÞ k ½½wðnÞ 2k ½wðnÞ k 2 

and

4

¼ 2s

(37)

Eyja ½½yðnÞ k ½yðnÞ k  ¼

 gTk ðnÞ H ðnÞ gk

a a

 T H  þ gTk aðnÞ aH ðnÞ gk gk aðnÞ aðnÞ gk

(38)

Hence, Jl becomes



Vk DVk þ

k¼1 N X

N 4 X

s6 k¼1

Vk Ea ½T1 Vk þ

1

s8

Vk Ea ½T2 Vk

(39)

k¼1 H H where Vk ¼ gk gTk , T1 ¼ aðnÞ aH ðnÞ Vk aðnÞ aðnÞ , T2 ¼ aðnÞ aðnÞ V k aðnÞ

aHðnÞ Vk aðnÞ aHðnÞ and D ¼ Ea ½aðnÞ aHðnÞ  ¼ diagfs2a1 ; . . . ; s2aL g. The elements of T1 and T2 are given by ½T1 l;l0 ¼

L X L X

½Vk l1;l2 ½aðnÞ l ½aðnÞ l2 ½aðnÞ l0 ½aðnÞ l1

l1 ¼1 l2 ¼1

½T2 l;l0 ¼

¼ b , DVk DVk D ¼ bDVk D, and inserting these results into (39), we obtain the expression of Jl in (25).

þs



s

s2al ¼ b, TrðVk DVk DÞ

2

 Eyja ½ð½yðnÞ k ½yðnÞ k Þ2  ¼ 2s4 þ 4s2 gTk aðnÞ aH ðnÞ gk

4

l¼1

References 

N 1 X

PL

2

we obtain

Jl ¼

115

L X L X L X L X

½Vk l1;l2 ½Vk l3;l4 ½aðnÞ l ½aðnÞ l2 ½aðnÞ l4 ½aðnÞ l0

l1 ¼1 l2 ¼1 l3 ¼1 l4 ¼1

½aðnÞ l1 ½aðnÞ 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 Ea ½T1  ¼ DVk D þ TrðVk DÞD Ea ½T2  ¼ 2DVk DVk D þ 2TrðVk DÞDVk D þ TrðVk DVk DÞD þ ðTrðVk DÞÞ2 D

(41)

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