SITUACIÓN ACTUAL

A.6

Conclusions

In this paper we proposed a turbo OFDM receiver which employs sparsity- based parametric channel estimation. The iterative receiver is derived using the BP-MF framework for approximate Bayesian inference. Unlike state-of- the-art sparse channel estimators, our scheme does not restrict multipath de- lays of the estimated channel impulse response to a grid. As a result it can truly exploit parsimony of the channel impulse response, without resorting to approximate sparsity (as in [11, 23, 24]).

We have presented a numerical evaluation that compares our algorithm with state-of-the-art methods, i.e., Turbo-GAMP [11] and LMMSE BP-MF [9]. This study demonstrated that restricting the multipath delays to a baud- spaced grid (e.g., as in Turbo-GAMP) is not a viable approach because the resulting equivalent vector of channel taps is only approximately sparse.

The numerical evaluation also shows that our proposed scheme can effec- tively exploit the structure of wireless channels. Even though our receiver is derived assuming the channel to be specular (as in Scenario A), it also per- forms very well in diffuse channels (Scenario B). Thus, we expect that it will also perform well in any channel which shows a mixture of both characteris- tics.

A

The Region-Based Free Energy Approximation

At the heart of the derivation of our algorithm lies the RBFE as defined by [8, Eq. (17)], [45]. In this paper we use the RBFE of the probability distribution corresponding to the factor graph depicted in Fig. A.2. For convenience, we give here the complete expression of the RBFE:

FBP-MF=FBP+FMF (A.35) with FBP=

∑

k∈Ku_k∈{0,1}

∑

buk(uk)ln buk(uk) p(uk) +

∑

i∈D x_i

∑

∈AD c(i)_∈{0,1}Q bMi(xi, c (i)_)lnbMi(xi, c (i)₎ p(xi|c(i)) +

∑

c_∈{0,1}K/R u∈{0,1}K bC(c, u)ln bC(c, u) p(c|u) −_k

∑

_∈Ku

∑

k∈{0,1} q(uk)ln q(uk) −

∑

i∈Dm∈{1,...,Q}

∑

[c(i)_]

∑

m∈{0,1} q([c(i)]m)ln q([c(i)]m),

FMF=

∑

l∈L Z q(αl)ln q(αl)dαl−ln p(y|xD, α, ˆτ; ˆβ)_x D,α −

∑

l∈L hln p(αl|ˆzl; ˆη)p(ˆzl; ˆρ)p(ˆτl)i_αl, where bC(c, u), bMi(xi, c

(i)_{)for i∈ Dand b}

uk(uk)for k∈ Kare factor beliefs.

With abuse of notation we let q(·) denote variable beliefs and h·ia denote expectation with respect to the belief density q(a).

B

Efficient Calculation of

µˆ

_A

When ˆL is Large

In this appendix we present a computationally efficient method for evaluating ˆ

µ_Aˆ as defined by (A.31). We first note that direct evaluation and inversion of Qhas time complexityO(ˆL2_{N), where ˆL,|A|. The method is iterative andˆ} has complexityO(ˆLN√N)provided Conjecture A.1 (below) holds. It is thus beneficial to use it when ˆL grows faster than√N.

We first use the Woodbury matrix identity to write ˆµas

ˆ µ= ˆβ−1ˆη  I−ˆβ−1_ˆηΨH_(ˆ τ_Aˆ)C−1Ψ(τˆ_Aˆ) ΨH(τˆ_Aˆ)hXiHxDy, where C=DXHXE−1 x_D + ˆβ −1_ˆηΨ(τˆ ˆ A)ΨH(τˆ_Aˆ).

We immediately recognize that the computationally dominating part is to solve a system of N linear equations of the form Cz=a. Since C is Hermitian and positive-definite, we can solve this system via the conjugate-gradient (CG) method (Alg. 2.1 in [57]), which is an iterative method for solving systems of linear equations. In the following we show that the number of iterations of the CG method isO(√N).

We first need a conjecture on the eigenvalues of the (Hermitian-Toeplitz) matrix T= ˆβ−1ˆηΨ(τˆ_Aˆ)ΨH(τˆ_Aˆ).

Conjecture A.1. There exists an upper bound on the largest eigenvalue of T which grows linearly with N, i.e.,

λmax(T) =O(N).

To justify this conjecture we refer to Fig. A.8, where the largest eigenvalue is shown for varying N.

We also need a number of lemmas.

Lemma A.2. There exists constants c1>0 and c2<∞, such that c1≤

|xi|2



x_D ≤

References 256 512 _{1,024 1,536 2,048} 0.5 1 1.5 2 ·104 Number of subcarriers (N) 256 512 _{1,024 1,536 2,048} 4,000 5,000 6,000 7,000 Number of subcarriers (N)

Fig. A.8:Average of the largest eigenvalue of the matrix T encountered during one execution of Parametric BP-MF for Scenario A (left) and B (right). Average obtained from 1000 Monte-Carlo trials. Both plots were generated at 20 dB SNR. A dashed line depicts the least-squares linear fit.

Proof: Observe that the data and pilot modulation symbol alphabets AD and AP only contain finite, non-zero values. We can thus take c1 = minx∈AP∪AD|x|

2_{and c}

2=maxx∈AP∪AD|x|

2_{to complete the proof.}

Lemma A.3. Assume that Conjecture A.1 holds. The largest and smallest eigenval- ues of C obey

λmax(C) =O(N), λmin(C)≥c−11 .

Proof: By the Weyl inequality for Hermitian matrices C, T andXH_X−1

x_D we have λmax(C)≤λmax _D XHXE−1 x_D  +λmax(T).

The first inequality follows directly from Conjecture A.1 and Lemma A.2. Similarly by the dual Weyl inequality

λmin(C)≥λmin _D XHXE−1 x_D  +λmin(T).

Since ˆL<N, the matrix T is singular and λmin(T) =0. The second inequality now follows from Lemma A.2.

By Theorem 2.2 in [57] the number of iterations required by the CG method to achieve a desired accuracy in the solution ofa=CzisO

r

λmax(C) λmin(C)

 .

By Lemma A.3 the number of iterations is thus O(√N). Each iteration has time complexityO(ˆLN)and the overall complexity of solving (A.31) via this method is thereforeO(ˆLN√N).

References

[1] Evolved Universal Terrestrial Radio Access (E-UTRA), Physical channels and modulation, 3GPP Std. 36.211, Rev. 12.8.0, Dec. 2015.

[2] C. Douillard, M. Jézéquel, C. Berrou, A. Picart, P. Didier, and A. Glavieux, “Iterative correction of intersymbol interference: Turbo- equalization,” European Trans. Telecommun., vol. 6, no. 5, pp. 507–511, 1995.

[3] M. Tüchler and A. C. Singer, “Turbo equalization: An overview,” IEEE Trans. Inf. Theory, vol. 57, no. 2, pp. 920–952, Feb. 2011.

[4] S. Park, Y. G. Kim, and C. G. Kang, “Iterative receiver for joint detection and channel estimation in OFDM systems under mobile radio channels,” IEEE Trans. Veh. Technol., vol. 53, no. 2, pp. 450–460, Mar. 2004.

[5] A. P. Worthen and W. E. Stark, “Unified design of iterative receivers using factor graphs,” IEEE Trans. Inf. Theory, vol. 47, no. 2, pp. 843–849, Feb. 2001.

[6] S. Wu, L. Kuang, Z. Ni, J. Lu, D. David Huang, and Q. Guo, “Expec- tation propagation approach to joint channel estimation and decoding for OFDM systems,” in Proc. IEEE Int. Conf. Acoust., Speech and Signal Process., May 2014, pp. 1941–1945.

[7] Y. Zhu, D. Guo, and M. L. Honig, “A message-passing approach for joint channel estimation, interference mitigation, and decoding,” IEEE Trans. Wireless Commun., vol. 8, no. 12, pp. 6008–6018, Dec. 2009.

[8] E. Riegler, G. E. Kirkelund, C. N. Manchón, M. Badiu, and B. H. Fleury, “Merging belief propagation and the mean field approximation: A free energy approach,” IEEE Trans. Inf. Theory, vol. 59, no. 1, pp. 588–602, Jan. 2013.

[9] M.-A. Badiu, G. E. Kirkelund, C. N. Manchón, E. Riegler, and B. H. Fleury, “Message-passing algorithms for channel estimation and decod- ing using approximate inference,” in Proc. IEEE Int. Symp. Inform. Theory, Jul. 2012, pp. 2376–2380.

[10] P. Schniter, “Belief-propagation-based joint channel estimation and de- coding for spectrally efficient communication over unknown sparse channels,” Physical Communication, vol. 5, no. 2, pp. 91–101, Jun. 2012. [11] ——, “A message-passing receiver for BICM-OFDM over unknown

clustered-sparse channels,” IEEE J. Select. Topics in Signal Process., vol. 5, no. 8, pp. 1462–1474, Dec. 2011.

[12] Guidelines for evaluation of radio interface technologies for IMT-Advanced, ITU-R Std. M.2135-1, Dec. 2008.

References

[13] A. F. Molisch, D. Cassioli, C.-C. Chong, S. Emami, A. Fort, B. Kannan, J. Karedal, J. Kunisch, H. G. Schantz, K. Siwiak et al., “A comprehen- sive standardized model for ultrawideband propagation channels,” IEEE Trans. Antennas Propag., vol. 54, no. 11, pp. 3151–3166, Nov. 2006. [14] M. Feder and J. A. Catipovic, “Algorithms for joint channel estimation

and data recovery-application to equalization in underwater communi- cations,” IEEE J. Ocean. Eng., vol. 16, pp. 42–55, Jan. 1991.

[15] V. Lottici, A. D’Andrea, and U. Mengali, “Channel estimation for ultra- wideband communications,” IEEE J. Sel. Areas Commun., vol. 20, pp. 1638–1645, Dec. 2002.

[16] I. Maravic, J. Kusuma, and M. Vetterli, “Low-sampling rate UWB chan- nel characterization and synchronization,” J. of Commun. and Networks, vol. 5, pp. 319–327, Dec. 2003.

[17] A. F. Molisch, A. Mammela, and D. P. Taylor, Eds., Wideband Wireless Digital Communication. Prentice Hall, 2000.

[18] W. Bajwa, A. Sayeed, and R. Nowak, “Compressed channel sensing: A new approach to estimating sparse multipath channels,” Proc. IEEE, vol. 98, pp. 1058–1076, Jun. 2010.

[19] J. Paredes, G. Arce, and Z. Wang, “Ultra-wideband compressed sensing: Channel estimation,” IEEE J. Sel. Topics Signal Process., vol. 1, no. 3, pp. 383–395, Oct. 2007.

[20] C. Berger, S. Zhou, J. Preisig, and P. Willett, “Sparse channel estimation for multicarrier underwater acoustic communication: From subspace methods to compressed sensing,” IEEE Trans. Signal Process., vol. 58, no. 3, pp. 1708 – 1721, Mar. 2010.

[21] G. Taubock, F. Hlawatsch, D. Eiwen, and H. Rauhut, “Compressive esti- mation of doubly selective channels in multicarrier syst.: Leakage effects and sparsity-enhancing process.” IEEE J. Sel. Topics Signal Process., vol. 4, no. 2, pp. 255–271, Apr. 2010.

[22] N. L. Pedersen, C. N. Manchón, D. Shutin, and B. H. Fleury, “Applica- tion of Bayesian hierarchical prior modeling to sparse channel estima- tion,” in Proc. IEEE Int. Conf. Commun., Jun. 2012, pp. 3487–3492. [23] R. Prasad, C. R. Murthy, and B. D. Rao, “Joint approximately sparse

channel estimation and data detection in OFDM systems using sparse Bayesian learning,” IEEE Trans. Signal Process., vol. 62, no. 14, pp. 3591– 3603, Jun. 2014.

[24] ——, “Joint channel estimation and data detection in MIMO-OFDM sys- tems: A sparse Bayesian learning approach,” IEEE Trans. Signal Process., vol. 63, no. 20, pp. 5369–5382, Jun. 2015.

[25] J.-J. Van de Beek, O. Edfors, M. Sandell, S. K. Wilson, and P. O. Börjes- son, “On channel estimation in OFDM systems,” in Proc. IEEE 45th Veh. Technology Conf., vol. 2, Jul. 1995, pp. 815–819.

[26] Y. Chi, L. Scharf, A. Pezeshki, and A. Calderbank, “Sensitivity to basis mismatch in compressed sensing,” IEEE Trans. Signal Process., vol. 59, pp. 2182–2195, May 2011.

[27] E. J. Candès and C. Fernandez-Granda, “Towards a mathematical theory of super-resolution,” Commun. Pure and Appl. Math., vol. 67, no. 6, pp. 906–956, 2014.

[28] B. N. Bhaskar, G. Tang, and B. Recht, “Atomic norm denoising with ap- plications to line spectral estimation,” IEEE Trans. Signal Process., vol. 61, no. 23, pp. 5987–5999, Dec. 2013.

[29] S. Pejoski and V. Kafedziski, “Estimation of sparse time dispersive chan- nels in pilot aided OFDM using atomic norm,” IEEE Wireless Commun. Lett., vol. 4, pp. 397–400, Apr. 2015.

[30] Y. Barbotin and M. Vetterli, “Fast and robust parametric estimation of jointly sparse channels,” IEEE J. on Emerging and Select. Topics in Circuits and Syst., vol. 2, pp. 402–412, Sep. 2012.

[31] T. L. Hansen, M. A. Badiu, B. H. Fleury, and B. D. Rao, “A sparse Bayesian learning algorithm with dictionary parameter estimation,” in Proc. IEEE 8th Sensor Array and Multichannel Signal Process. Workshop, Jun. 2014, pp. 385–388.

[32] M.-A. Badiu, T. L. Hansen, and B. H. Fleury, “Variational Bayesian in- ference of line spectra,” IEEE Trans. Signal Process., vol. 65, no. 9, pp. 2247–2261, May 2017.

[33] D. Shutin, W. Wang, and T. Jost, “Incremental sparse Bayesian learn- ing for parameter estimation of superimposed signals,” in Proc. 10th Int. Conf. Sampling Theory and Applicat., Jul. 2013.

[34] L. Hu, Z. Shi, J. Zhou, and Q. Fu, “Compressed sensing of complex sinusoids: An approach based on dictionary refinement,” IEEE Trans. Signal Process., vol. 60, no. 7, pp. 3809–3822, Jul. 2012.

[35] J. J. Kormylo and J. Mendel, “Maximum likelihood detection and esti- mation of Bernoulli-Gaussian processes,” IEEE Trans. Inf. Theory, vol. 28, no. 3, pp. 482–488, May 1982.

References

[36] O. Edfors, M. Sandell, J.-J. Van de Beek, S. K. Wilson, and P. O. Bör- jesson, “OFDM channel estimation by singular value decomposition,” IEEE Trans. Commun., vol. 46, no. 7, pp. 931–939, Jul. 1998.

[37] C. Berrou, A. Glavieux, and P. Thitimajshima, “Near Shannon limit error-correcting coding and decoding: Turbo-codes,” in Proc. IEEE Int. Conf. Commun., vol. 2, May 1993, pp. 1064–1070.

[38] R. Gallager, “Low-density parity-check codes,” Inform. Theory, IRE Trans., vol. 8, no. 1, pp. 21–28, January 1962.

[39] M. Luise, R. Reggiannini, and G. M. Vitetta, “Blind equaliza- tion/detection for OFDM signals over frequency-selective channels,” IEEE J. Sel. Areas Commun., vol. 16, no. 8, pp. 1568–1578, Oct. 1998. [40] O.-E. Barbu, N. L. Pedersen, C. N. Manchón, G. Monghal, C. Rom, and

B. H. Fleury, “Sparse channel estimation including the impact of the transceiver filters with application to OFDM,” in Proc. 15th Int. Workshop on Signal Process. Advances in Wireless Commun., Jun. 2014, pp. 424–428. [41] R. J.-M. Cramer, R. A. Scholtz, and M. Z. Win, “Evaluation of an ultra-

wide-band propagation channel,” IEEE Trans. Antennas Propag., vol. 50, no. 5, pp. 561–570, May 2002.

[42] A. F. Molisch, “Ultrawideband propagation channels – theory, measure- ment, and modelling,” IEEE Trans. Veh. Technol., vol. 54, no. 5, pp. 1528– 1545, Oct. 2004.

[43] M. Stojanovic and J. Preisig, “Underwater acoustic communication chan- nels: Propagation models and statistical characterization,” IEEE Com- mun. Mag., vol. 47, no. 1, pp. 84–89, Jan. 2009.

[44] C. R. Berger, Z. Wang, J. Huang, and S. Zhou, “Application of compres- sive sensing to sparse channel estimation,” IEEE Commun. Mag., vol. 48, no. 11, pp. 164–174, Nov. 2010.

[45] J. S. Yedidia, W. T. Freeman, and Y. Weiss, “Constructing free-energy approximations and generalized belief propagation algorithms,” IEEE Trans. Inf. Theory, vol. 51, no. 7, pp. 2282–2312, 2005.

[46] J. M. Winn and C. M. Bishop, “Variational message passing,” J. Mach. Learning Research, vol. 6, pp. 661–694, Apr. 2005.

[47] M. J. Beal, “Variational algorithms for approximate bayesian inference,” Ph.D. dissertation, University College London, 2003.

[48] J. Pearl, Probabilistic reasoning in intelligent systems: networks of plausible inference. Morgan Kaufmann, 1988.

[49] F. R. Kschischang, B. J. Frey, and H.-A. Loeliger, “Factor graphs and the sum-product algorithm,” IEEE Trans. Inf. Theory, vol. 47, no. 2, pp. 498–519, Feb. 2001.

[50] R. McEliece, D. MacKay, and J.-F. Cheng, “Turbo decoding as an instance of Pearl’s "belief propagation" algorithm,” IEEE J. Sel. Areas Commun., vol. 16, no. 2, pp. 140–152, Feb 1998.

[51] F. R. Kschischang and B. J. Frey, “Iterative decoding of compound codes by probability propagation in graphical models,” IEEE J. Sel. Areas Com- mun., vol. 16, no. 2, pp. 219–230, Feb. 1998.

[52] L. Bahl, J. Cocke, F. Jelinek, and J. Raviv, “Optimal decoding of linear codes for minimizing symbol error rate,” IEEE Trans. Inf. Theory, vol. 20, no. 2, pp. 284–287, Mar 1974.

[53] G. H. Golub and C. F. V. Loan, Matrix Computations, 3rd ed. The Johns Hopkins Univ. Press, 1996.

[54] T. Heskes, “Stable fixed points of loopy belief propagation are local min- ima of the Bethe free energy,” in Advances in neural information processing systems, 2002, pp. 343–350.

[55] J. M. Mooij and H. J. Kappen, “Sufficient conditions for convergence of the sum–product algorithm,” IEEE Trans. Inf. Theory, vol. 53, no. 12, pp. 4422–4437, Dec. 2007.

[56] Q. Su and Y.-C. Wu, “On convergence conditions of Gaussian belief propagation,” IEEE Trans. Signal Process., vol. 63, no. 5, pp. 1144–1155, May 2015.

[57] M. K. Ng, Iterative Methods for Toeplitz Systems. Oxford University Press, 2004.

Paper B

Superfast Line Spectral Estimation

Thomas L. Hansen, Bernard H. Fleury and Bhaskar D. Rao

The paper has been accepted for publication in the IEEE Transactions on Signal Processing

B.1. Introduction

Abstract

A number of recent works have proposed to solve the line spectral estimation problem by applying off-the-grid extensions of sparse estimation techniques. These methods are preferable over classical line spectral estimation algorithms because they inher- ently estimate the model order. However, they all have computation times which grow at least cubically in the problem size, thus limiting their practical applicability in cases with large dimensions. To alleviate this issue, we propose a low-complexity method for line spectral estimation, which also draws on ideas from sparse estima- tion. Our method is based on a Bayesian view of the problem. The signal covariance matrix is shown to have Toeplitz structure, allowing superfast Toeplitz inversion to be used. We demonstrate that our method achieves estimation accuracy at least as good as current methods and that it does so while being orders of magnitudes faster.

B.1

Introduction

The problem of line spectral estimation (LSE) has received significant atten- tion in the research community for at least 40 years. The reason is that many fundamental problems in signal processing can be recast as LSE; examples include direction of arrival estimation using sensor arrays [1, 2], bearing and range estimation in synthetic aperture radar [3], channel estimation in wire- less communications [4] and simulation of atomic systems in molecular dy- namics [5].

In trying to solve the LSE problem, classical approaches include subspace methods [6] such as MUSIC [7] or ESPRIT [8] which estimate the frequencies based on an estimate of the signal covariance matrix. These approaches must be augmented with a method for estimation of the model order. Popular choices include generic information theoretic criteria (e.g. AIC, BIC) or more specialized methods, such as SORTE [9] which is based on the eigenvalues of the estimated signal covariance matrix. Subspace methods typically perform extremely well if the model order is known, but their estimation accuracy can degrade significantly if the model order is unknown.

The stochastic maximum likelihood (ML) method is known to be asymp- totically efficient (it attains the Cramér-Rao bound as the problem size tends to infinity) [2]. Unfortunately it also requires knowledge of the model order. Inspired by the ideas of sparse estimation and compressed sensing, many papers on sparsity-based LSE algorithms have appeared in recent years, e.g. [1, 10]. In particular, the LSE problem is simplified to a finite sparse re- construction problem by restricting the frequencies to a grid. Such meth- ods inherently estimate the model order, alleviating the issues arising from separate model order and frequency estimation in classical methods. The granularity of the grid leads to a non-trivial trade-off between accuracy and

computational requirements. To forego the use of a grid, so-called off-the- grid compressed sensing methods have been proposed [11–13]. These meth- ods provably recover the frequencies in the noise-free case under a minimum separation condition. They suffer from prohibitively high computational re- quirements even for moderate problem sizes, see Sec B.6.

In [14–16] a Bayesian view is taken on the LSE problem. The model used in stochastic ML is extended with a sparsity-promoting prior on the coeffi- cients of the sinusoid components. Thereby inherent estimation of the model order is achieved. These algorithms generally have high estimation accuracy. Their per-iteration computational complexity is cubic in the number of sinu- soidal components, meaning that their runtime grows rapidly as the number of components increases.

In this work we introduce the Superfast LSE algorithm for solving the LSE problem in scenarios where the full measurement vector is available (com- plete data case). The modelling and design of the basic algorithm which we present in Sec. B.2 is based upon the ideas in [14–16]. The main novelty resides in the computational aspects of Superfast LSE. The derived method is based upon several techniques: a so-called superfast Toeplitz inversion al- gorithm [17, 18] (thereof the name of our algorithm), low-complexity Capon beamforming [19], the Gohberg-Semencul formula [20] and non-uniform fast Fourier transforms [21, 22]. The Superfast LSE algorithm has the following virtues: It inherently estimates all model parameters such as the noise vari- ance and model order and it has low per-iteration computational complexity. Specifically it scales as O(N log2_{N) where N is the length of the observed} vector. We show empirically that it converges after a few iterations (typi- cally less than 20). This means that for large problem sizes our algorithm can have computation time orders of magnitude lower than that of current methods. It does so without any penalty in estimation accuracy. Our numeri- cal experiments show that Superfast LSE has high estimation accuracy across a wide range of scenarios, being on par with or better than state-of-the-art algorithms.

Synergistically and computationally efficiently combining the steps in the algorithm might appear easy after the fact. This is however not the case. Some other LSE algorithms can benefit in terms of computational effort from our approach, yet not to the extent achieved with the proposed algorithm. For instance, the computational methods in Sec. B.3 can be embedded in VALSE [16]. The resulting scheme will have high computational complexity due to the variational estimation of the posterior on the frequencies. Note that our algorithm performs on par with VALSE, but at a significantly reduced computational effort.

For completeness we also present a semifast version of the algorithm which works when only a subset of entries in the measurement vector are available. The Semifast LSE algorithm has per-iteration complexityO(N ˆK2₊

In document MEMORIA LOTE 2 LA PALOMA PLANTA DE BIOMETANIZACIÓN (página 5-11)