Application of Bayesian Hierarchical Prior Modeling to Sparse Channel Estimation

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1 Appication of Bayesian Hierarchica Prior Modeing to Sparse Channe Estimation Nies Lovmand Pedersen, Cares Navarro Manchón, Dmitriy Shutin and Bernard Henri Feury Department of Eectronic Systems, Aaborg University Nies Jernes Vej, DK-9 Aaborg, Denmark, Emai: Institute of Communications and Navigation, German Aerospace Center Oberpfaffenhofen, D-834 Wessing, Germany, Emai: arxiv:4.656v [stat.ml] 3 Apr Abstract Existing methods for sparse channe estimation typicay provide an estimate computed as the soution maximizing an objective function defined as the sum of the og-ikeihood function and a penaization term proportiona to the -norm of the parameter of interest. However, other penaization terms have proven to have strong sparsity-inducing properties. In this work, we design piot-assisted channe estimators for OFDM wireess receivers within the framework of sparse Bayesian earning by defining hierarchica Bayesian prior modes that ead to sparsity-inducing penaization terms. The estimators resut as an appication of the variationa message-passing agorithm on the factor graph representing the signa mode extended with the hierarchica prior modes. Numerica resuts demonstrate the superior performance of our channe estimators as compared to traditiona and state-of-the-art sparse methods. I. INTRODUCTION During the ast few years the research on compressive sensing techniques and sparse signa representations [], [] appied to channe estimation has received considerabe attention, see e.g., [3] [7]. The reason is that, typicay, the impuse response of the wireess channe has a few dominant mutipath components. A channe exhibiting this property is said to be sparse [3]. The genera goa of sparse signa representations from overcompete dictionaries is to estimate the sparse vector α in the foowing system mode: y = Φα+w. ( In this expression y C M is the vector of measurement sampes and w C M represents the sampes of the additive white Gaussian random noise with covariance matrix λ I and precision parameter λ >. The matrix Φ = [φ,...,φ L ] C M L is the overcompete dictionary with more coumns than rows (L > M and α = [α,...,α L ] T C L is an unknown sparse vector, i.e., α has few nonzero eements at unknown ocations. Often, a sparse channe estimator is constructed by soving the -norm constrained quadratic optimization probem, see c IEEE. Persona use of this materia is permitted. Permission from IEEE must be obtained for a other users, incuding reprinting/repubishing this materia for advertising or promotiona purposes, creating new coective works for resae or redistribution to servers or ists, or reuse of any copyrighted components of this work in other works. among others [4] [6]: α = argmin α { y Φα +κ α } with κ > and p, p, denoting the p vector norm. This method is aso known as Least Absoute Shrinkage and Seection Operator (LASSO regression [8] or Basis Pursuit Denoising [9]. The popuarity of the LASSO regression is mainy attributed to the convexity of the cost function, as we as to its provabe sparsity-inducing properties (see []. In [4] [6] the LASSO regression is appied to orthogona frequencydivision mutipexing (OFDM piot-assisted channe estimation. Various channe estimation agorithms that minimize the LASSO cost function using convex optimization are compared in [6]. Another approach to sparse channe estimation is sparse Bayesian earning (SBL [7], [] []. Specificay, SBL aims at finding a sparse maximum a posteriori (MAP estimate of α { α = argmin y Φα +λ Q(α } (3 α by specifying a priorp(α such that the penaty termq(α e ogp(α induces a sparse estimate α. Obviousy, by comparing ( and (3 the SBL framework reaizes the LASSO cost function by choosing the Lapace priorp(α exp( a α with κ = λ a. However, instead of working directy with the prior p(α, SBL modes this using a two-ayer (-L hierarchica structure. This invoves specifying a conditiona prior p(α γ and a hyperprior p(γ such that p(α = p(α γp(γdγ has a sparsity-inducing nature. The hierarchica approach to the representation of p(α has severa important advantages. First of a, one is free to choose simpe and anayticay tractabe probabiity density functions (pdfs. Second, when carefuy chosen, the resuting hierarchica structure aows for the construction of efficient yet computationay tractabe iterative inference agorithms with anaytica derivation of the inference expressions. In [3] we propose a -L and a three-ayer (3-L prior mode for α. These hierarchica prior modes ead to nove Here x e y denotes exp(x = exp(υexp(y, and thus x = υ+y, for some arbitrary constant υ. We wi aso make use of x y which denotes x = υy for some positive constant υ. (

2 sparsity-inducing priors that incude the Lapace prior for compex variabes as a specia case. This paper adapts the Bayesian probabiistic framework introduced in [3] to OFDM piot-assisted sparse channe estimation. We then propose a variationa message passing (VMP agorithm that effectivey expoits the hierarchica structure of the prior modes. This approach eads to nove channe estimators that make use of various priors with strong sparsity-inducing properties. The numerica resuts revea the promising potentia of our estimators with improved performance as compared to stateof-the-art methods. In particuar, the estimators outperform LASSO. Throughout the paper we sha make use of the foowing notation: ( T and ( H denote respectivey the transpose and the Hermitian transpose; the expression f(x q(x denotes the expectation of the function f(x with respect to the density q(x; CN(x a, B denotes a mutivariate compex Gaussian pdf with meanaand covariance matrixb; simiary, Ga(x a,b = ba Γ(a xa exp( bx denotes a Gamma pdf with shape parameter a and rate parameter b. II. SIGNAL MODEL We consider a singe-input singe-output OFDM system with N subcarriers. A cycic prefix (CP is added to preserve orthogonaity between subcarriers and to eiminate intersymbo interference between consecutive OFDM symbos. The channe is assumed static during the transmission of each OFDM symbo. The received (baseband OFDM signa r C N reads in matrix-vector notation r = Xh+n. (4 The diagona matrix X = diag(x,x,...,x N contains the transmitted symbos. The components of the vector h C N are the sampes of the channe frequency response at the N subcarriers. Finay, n C N is a zero-mean compex symmetric Gaussian random vector of independent components with variance λ. To estimate the vector h in (4, a tota of M piot symbos are transmitted at seected subcarriers. The piot pattern P {,...,N} denotes the set of indices of the piot subcarriers. The received signas observed at the piot positions r P are then divided each by the corresponding piot symbo X P = diag(x n : n P to produce the vector of observations: y (X P r P = h P +(X P n P. (5 We assume that a piot symbos hod unit power such that the statistics of the noise term (X P n P remain unchanged, i.e., y C M yieds the sampes of the true channe frequency response (at the piot subcarriers corrupted by additive compex white Gaussian noise with component variance λ. In this work, we consider a frequency-seective wireess channe that remains constant during the transmission of each OFDM symbo. The maximum reative deay τ max is assumed to be arge compared to the samping time T s, i.e., τ max /T s [3]. The impuse response of the wireess channe is modeed as a sum of mutipath components: g(τ = K β k δ(τ τ k. (6 k= In this expression, β k and τ k are respectivey the compex weight and the continuous deay of the kth mutipath component, and δ( is the Dirac deta function. The parameter K is the tota number of mutipath components. The channe parameters K, β k, and τ k, k =,...,K, are random variabes. Specificay, the weightsβ k, k =,...,K, are mutuay uncorreated zero-mean with the sum of their variances normaized to one. Additiona detais regarding the assumptions on the mode (6 are provided in Section VI. III. THE DICTIONARY MATRIX Our goa is to estimate h in (4 by appying the genera optimization probem (3 to the observation mode (5. For doing so, we must define a proper dictionary matrix Φ. In this section we give an exampe of such a matrix. As a starting point, we invoke the parametric mode (6 of the channe. Making use of this mode, (5 can be written as y = T(τβ +w (7 with h P = T(τβ, w = (X P n P, β = [β,...,β K ] T, τ = [τ,...,τ K ] T, andt(τ C M K depending on the piot pattern P as we as the unknown deays in τ. Specificay, the (m,kth entry of T(τ reads T(τ m,k exp( jπf m τ k, m =,,...,M k =,,...,K with f m denoting the frequency of the mth piot subcarrier. In the genera optimization probem (3 the coumns of Φ are known. However, the coumns of T(τ in (7 depend on the unknown deays in τ. To circumvent this discrepancy we foow the same approach as in [5] and consider a grid of uniformy-spaced deay sampes in the interva [,τ max ]: [ τ d =, T s ζ, T ] T s ζ,...,τ max (9 with ζ > such that ζτ max /T s is an integer. We now define the dictionary Φ C M L as Φ = T(τ d. Thus, the entries of Φ are of the form (8 with deay vector τ d. The number of coumns L = ζτ max /T s + in Φ is thereby inversey proportiona to the seected deay resoution T s /ζ. It is important to notice that the system mode ( with Φ defined using discretized deay components is an approximation of the true system mode (7. This approximation mode is introduced so that (3 can be appied to sove the channe estimation task. The estimate of the channe vector at the piot subcarriers is then ĥp = Φ α. In order to estimate the channe h in (4 the dictionary Φ is appropriatey expanded (row-wise to incude a N subcarrier frequencies. IV. BAYESIAN PRIOR MODELING In this section we specify the joint pdf of the system mode ( when it is augmented with the -L and the 3-L hierarchica (8

3 α.5.5 ǫ =.5 ǫ =. ǫ = α (a ˆα φ H y Fig.. -L hierarchica prior pdf for α C : (a Contour pot of the restriction to the Im{α } = Im{α } = pane of the penaty term Q(α,α ;ǫ,η e og(p(α ;ǫ,ηp(α ;ǫ,η. (b Restriction to Im{φ H y} = of the resuting MAP estimation rue (3 with ǫ as a parameter in the case when Φ is orthonorma. The back dashed ine indicates the hardthreshod rue and the back soid ine the soft-threshod rue (obtained with ǫ = 3/. The back dashed ine indicates the penaty term resuting when the prior pdf is a circuar symmetric Gaussian pdf. (b ˆα φ H y (a.5.5 ǫ =. ǫ =.5 ǫ = α Fig.. Three-ayer hierarchica prior pdf for α C with the setting a =, b =.: (a Restriction to Im{φ H y} = of the resuting MAP estimation rue (3 with ǫ as a parameter in the case when Φ is orthonorma. The back dashed ine indicates the hard-threshod rue and the back soid ine the soft-threshod rue. (b Contour pot of the restriction to the Im{α } = Im{α } = pane of the penaty term Q(α,α ;ǫ,a,b e og(p(α ;ǫ,a,bp(α ;ǫ,a,b. α (b prior mode. The joint pdf of ( augmented with the -L hierarchica prior mode reads p(y, α, γ, λ = p(y α, λp(λp(α γp(γ; η. ( The 3-L prior mode considers the parameter η specifying the prior of γ in ( as random. Thus, the joint pdf of ( augmented with this hierarchica prior mode is of the form p(y, α, γ, η, λ = p(y α, λp(λp(α γp(γ ηp(η. ( In ( and ( we have p(y α,λ = CN(y Φα,λ I due to (. Furthermore, we seect the conjugate prior p(λ = p(λ;c,d Ga(λ c,d. Finay, we et p(α γ = L = p(α γ with p(α γ CN(α,γ. In the foowing we show the main resuts and properties of these prior modes. We refer to [3] for a more detaied anaysis. A. Two-Layer Hierarchica Prior Mode The -L prior mode assumes that p(γ = L = p(γ with p(γ = p(γ ;ǫ,η Ga(γ ǫ,η. We compute the prior of α to be L p(α;ǫ,η = p(α γp(γ; ǫ, ηdγ = p(α ;ǫ,η ( with = p(α ;ǫ,η = πγ(ǫ η (ǫ+ α ǫ K ǫ ( η α. (3 In this expression, K ν ( is the modified Besse function of the second kind with order ν R. The prior (3 eads to the genera optimization probem (3 with penaty term Q(α;ǫ,η = L og ( α ǫ K ǫ ( η α. (4 = We now show that the -L prior mode induces the - norm penaty term and thereby the LASSO cost function as a specia case. Seecting ǫ = 3/ and using the identity K (z = π z exp( z [4], (3 yieds the Lapace prior p(α ;ǫ = 3/,η = η π exp( η α. (5 With the seection η = η, =,...,L, we obtain Q(α;η = η α. The prior pdf (3 is specified by ǫ and the reguarization parameter η. In order to get insight into the impact of ǫ on the properties of this prior pdf we consider the case α C. In Fig. (a the contour ines of the restriction to R of Q(α,α ;ǫ,η e og(p(α ;ǫ,ηp(α ;ǫ,η are visuaized; each contour ine is computed for a specific choice of ǫ. Notice that as ǫ decreases towards more probabiity mass accumuates aong the α-axes; as a consequence, the mode of the resuting posterior is more ikey to be ocated cose to the axes, thus promoting a sparse soution. The behavior of the cassica penaty term obtained for ǫ = 3/ can aso be ceary recognized. In Fig. (b we consider the case when Φ is orthonorma and compute the MAP estimator (3 with penaty term (4 for different vaues of ǫ. Note the typica soft-threshod-ike behavior of the estimators. As ǫ, more components of α are pued towards zero since the threshod vaue increases, thus encouraging a sparser soution. B. Three-Layer Hierarchica Prior Mode We now turn to the SBL probem with a 3-L prior mode for α eading to the joint pdf in (. Specificay, the goa is to incorporate the reguarization parameter η into the inference framework. To that end, we define p(η = L p(η with p(η = p(η ;a,b Ga(η a,b and compute the prior p(α. Defining a [a,...,a ] T and b [b,...,b L ] T we Let f denote a function defined on a set A. The restriction of f to a subset B A is the function defined on B that coincides with f on this subset.

4 λ α γ η f y f α f γ f η f λ Fig. 3. A factor graph that represents the joint pdf (. In this figure f y p(y α,λ, f α p(α γ, f γ p(γ η, f η p(η, and f λ p(λ. obtain p(α;ǫ,a,b = L p(α ;ǫ,a,b with p(α ;ǫ,a,b = = Γ(ǫ+a Γ(a + πb Γ(ǫΓ(a p(α γ p(γ dγ ( α ǫ U (ǫ+a ;ǫ; α b b. (6 In this expression, U( ; ; is the confuent hypergeometric function [4]. In Fig. (a we show the estimation rues produced by the MAP sover for different vaues of ǫ and fixed parameters a and b when Φ is orthonorma. It can be seen that the estimation rues obtained with the 3-L prior mode approximate the hard-threshoding rue. In Fig. (b, we depict the contour ines of the restriction to R of Q(α,α ;ǫ,a,b e og(p(α ;ǫ,a,bp(α ;ǫ,a,b. Observe that athough the contours behave quaitativey simiary to those shown in Fig. (a for the -L prior mode, the estimation rues in Fig. (a and Fig. (b are different. Naturay, the 3-L prior mode encompasses three free parameters, ǫ, a, and b. The choice ǫ = and b sma (practicay we et b = 6, =,...,L induces a weighted og-sum penaization term. This term is known to strongy promote a sparse estimate [], []. Later in the text we wi aso adopt this parameter setting. V. VARIATIONAL MESSAGE PASSING In this section we present a VMP agorithm for estimating h in (4 given the observation y in (5. Let Θ = {α,γ,η,λ} be the set of unknown parameters and p(y,θ be the joint pdf specified in (. The factor graph [5] that encodes the factorization of p(y,θ is shown in Fig. 3. Consider an auxiiary pdf q(θ for the unknown parameters that factorizes according to q(θ = q(αq(γq(ηq(λ. The VMP agorithm is an iterative scheme that attempts to compute the auxiiary pdf that minimizes the Kuback-Leiber (KL divergence KL(q(Θ p(θ y. In the foowing we summarize the key steps of the agorithm; the reader is referred to [6] for more information on VMP. From [6] the auxiiary function q(θ i, θ i Θ, is updated as the product of incoming messages from the neighboring factor nodes f n to the variabe node θ i : q(θ i f n N θi m fn θ i. (7 In (7 N θi is the set of factor nodes neighboring the variabe node θ i and m fn θ i denotes the message from factor node f n to variabe node θ i. This message is computed as m fn θ i = exp ( nf n j q(θj, θj N fn \{θi}, (8 where N fn is the set of variabe nodes neighboring the factor nodef n. After an initiaization procedure, the individua factors of q(θ are then updated iterativey in a round-robin fashion using (7 and (8. We provide two versions of the VMP agorithm: one appied to the -L prior mode (referred to as VMP-L and another one appied to the 3-L mode (VMP-3L. The messages corresponding to VMP-L are easiy obtained as a specia case of the messages computed for VMP-3L by assuming q(η = δ(η ˆη, where ˆη is some fixed rea number. Update of q(α: According to (7 and Fig. 3 the computation of the update of q(α requires evauating the product of messages m fy α and m fα α. Mutipying these two messages yieds the Gaussian auxiiary pdf q(α = CN (α ˆα, ˆΣ α with covariance matrix and mean given by ˆΣ α = ( λ q(λ Φ H Φ+V (γ, (9 ˆα = α q(α = λ q(λˆσα Φ H y. ( In the above expression we have defined V (γ = diag( γ q(γ,..., γ L q(γ. Update of q(γ: The update of q(γ is proportiona to the product of the messages m fα γ and m fγ γ: q(γ L = γ ǫ exp ( γ α q(α γ η q(η. ( The right-hand side expression in ( is recognized as the product of Generaized Inverse Gaussian (GIG pdfs [7] with order p = ǫ. Observe that the computation of V (γ in (9 requires evauating γ q(γ for a =,...,L. Luckiy, the moments of the GIG distribution are given in cosed form for any n R [7]: ( γ n α n ( q(α K p+n η q(η α q(α q(γ = (. η q(η K p η q(η α q(α ( 3 Update of q(η: The update of q(η is proportiona to the product of messages m fη η and m fγ η: q(η L = η ǫ+a exp ( ( γ q(γ +b η. (3 Ceary, q(η factorizes as a product of L gamma pdfs, one for each individua entry in η. The first moment of η used in ( is easiy computed as η q(η = ǫ+a γ q(γ +b. (4 Naturay, q(η is ony computed for VMP-3L. 4 Update of q(λ: It can be shown that q(λ = Ga(λ M + c, y Φα q(α+d. The first moment of λ used in (9

5 RWF SpaRSA RVM VMP-L VMP-3L RWF SpaRSA RVM VMP-L VMP-3L RWF SpaRSA RVM VMP-L VMP-3L BER MSE MSE Eb/N [db] (a 5 5 Eb/N [db] (b M (c Fig. 4. Comparison of the performance of the VMP-L, VMP-3L, RWF, RVM, and SparseRSA agorithms: (a BER versus E b /N, (b MSE versus E b /N, (c MSE versus number of avaiabe piots M with fixed L = and the ratio between received symbo power and noise variance set to 5 db. In (a,b we have M = and L =. In (a the dashed ine shows the BER performance when the true channe vector h in (4 is known. TABLE I PARAMETER SETTINGS FOR THE SIMULATIONS. THE CONVOLUTIONAL CODE AND DECODER HAS BEEN IMPLEMENTED USING [8]. Samping time, T s 3.55 ns CP ength 4.69 µs / 44 T s Subcarrier spacing 5 khz Piot pattern Equay spaced, QPSK Moduation QPSK Subcarriers, N Piots, M OFDM symbos Information bits 77 Channe intereaver Random Convoutiona code (33,7,65 8 Decoder BCJR agorithm [9] and ( is therefore λ q(λ = M +c y Φα q(α +d. (5 VI. NUMERICAL RESULTS We perform Monte Caro simuations to evauate the performance of the two versions of the derived VMP agorithm in Section V. We consider a scenario inspired by the 3GPP LTE standard [] with the settings specified in Tabe I. The mutipath channe (6 is based on the mode used in [] where, for each reaization of the channe, the tota number of mutipath components K is Poisson distributed with mean of K p(k = and the deays τ k, k =,...,K, are independent and uniformy distributed random variabes drawn from the continuous interva [, 44 T s ] (corresponding to the CP ength. Thekth nonzero componentβ k conditioned on the deay τ k has a zero-mean compex circuar symmetric Gaussian distribution with variance σ (τ k = β k p(βk τ k = uexp( τ k /v and parameters u,v >. 3 To initiaize the VMP agorithm we set λ q(λ and γ q(γ equa to the inverse of the sampe variance of y and 3 The parameter u is computed such that K k= β k(t p(β,τ,k =, where p(β,τ,k is the joint pdf of the parameters of the channe mode. In the considered simuation scenario, K p(k =, τ max = 44 T s, and v = T s (the decay rate. the inverse number of coumns L respectivey. Furthermore, we et c = d = in (5, which corresponds to the Jeffreys noninformative prior for λ. Once the initiaization is competed, the agorithm sequentiay updates the auxiiary pdfsq(α, q(γ, q(η, and q(λ unti convergence is achieved. Obviousy, q(η is ony updated for VMP-3L, whereas for VMP-L the entries in η are set to M. For both versions we seect ǫ = and for VMP-3L we set a = and b = 6, =,...,L. Finay, the dictionary Φ is specified by M piot subcarriers and a tota of L = coumns (corresponding to the choice τ max = 44 T s and ζ.4 in (9. The VMP is compared to a cassica OFDM channe estimator and two state-of-the-art sparse estimation schemes. Specificay, we use as benchmark the robusty-designed Wiener Fiter (RWF [], the reevance vector machine (RVM [], [], 4 and the sparse reconstruction by separabe approximation (SpaRSA agorithm [3]. 5 The RVM is an EM agorithm based on the -L prior mode of the student-t pdf over each α, whereas SpaRSA is a proxima gradient method for soving (. In case of the SpaRSA agorithm the reguarization parameter κ needs to be set. In a simuations, we et κ =, which eads to good performance in high signa-to-noise ratio (SNR regime. The performance is compared with respect to the resuting bit-error-rate (BER and mean-squared error (MSE of the estimate ĥ versus the SNR (E b/n. In addition, in order to quantify the necessary piot overhead, we evauate the MSE versus the number of avaiabe piots M. Hence, in this setup M is no onger fixed as in Tabe I. In Fig. 4(a we compare the BER performance of the different schemes. We see that VMP-3L outperforms the other schemes across a the SNR range considered. Specificay, at % BER the gain is approximatey db compared to VMP- L and RVM and 3 db compared to SpaRSA and RWF. Aso VMP-L achieves ower BER in the SNR range - db compared to RVM and across the whoe SNR range compared to SpaRSA and RWF. The superior BER performance of the VMP agorithm is we refected in the MSE performance shown in Fig. 4(b. 4 The software is avaiabe on-ine at 5 The software is avaiabe on-ine at

6 Again VMP-3L is a cear winner foowed by VMP-L. The bad MSE performance of the SpaRSA for ow SNR is due to the difficuty in specifying a suitabe reguarization parameter κ across a arge SNR range. We next fix the ratio between received symbo power and noise variance to 5 db 6 and evauate the MSE versus number of avaiabe piots M. The resuts are depicted in Fig. 4(c. Observe a noticeabe performance gain obtained with VMP-3L. In particuar, VMP-3L exhibits the same MSE performance as VMP-L and RVM using ony approximatey 85 piots, roughy haf as many as VMP-L and RVM. Furthermore, VMP-3L, using this number of piots, significanty outperforms SpaRSA and RWF using piots. VII. CONCLUSION In this paper, we proposed channe estimators based on sparse Bayesian earning. The estimators rey on Bayesian hierarchica prior modeing and variationa message passing (VMP. The VMP agorithm effectivey expoits the probabiistic structure of the hierarchica prior modes and the resuting sparsity-inducing priors. Our numerica resuts show that the proposed channe estimators yied superior performance in terms of bit-error-rate and mean-squared error as compared to other existing estimators, incuding the estimator based on the -norm constraint. They aso aow for a significant reduction of the amount of piot subcarriers needed for estimating a given channe. ACKNOWLEDGMENT This work was supported in part by the 4GMCT cooperative research project funded by Inte Mobie Communications, Agient Technoogies, Aaborg University and the Danish Nationa Advanced Technoogy Foundation. This research was aso supported in part by the project ICT Wireess Hybrid Enhanced Mobie Radio Estimators (WHERE. REFERENCES [] R. Baraniuk, Compressive sensing, IEEE Signa Processing Magazine, vo. 4, no. 4, pp. 8, Juy 7. [] E. J. Candes and M. B. Wakin, An introduction to compressive samping, IEEE Signa Process. Mag., vo. 5, no., pp. 3, Mar. 8. [3] W. Bajwa, J. Haupt, A. Sayeed, and R. 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