Channel shortening has been studied in the context of ISI and MIMO channels as a means to compute a posteriori probabilities with a BCJR algorithm at a reduced computational complexity. This is done by considering an approximate channel response of reduced length. In a turbo receiver, soft a priori information can be linearly combined with the received sequence to form a new input to the BCJR trellis-based processing. In this letter, we provide closed-form expressions for the channel shortening filters using a generalized mutual information objective function. The proposed receiver allows a complexity reduction with respect to numerical optimization approaches which may also present stability, precision and convergence issues

Abelló Barberán, Albert

Freixe, Jean-Marie

Roque, Damien

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											 	   		 	  	     	 			 									  an author's https://oatao.univ-toulouse.fr/19908http://dx.doi.org/10.1109/LCOMM.2018.2830329Abello Barberan, Albert and Freixe, Jean-Marie and Roque, Damien Closed-Form Expressions for ChannelShortening Receivers Using A Priori Information. (2018) IEEE Communications Letters. ISSN 1089-7798Closed-Form Expressions for Channel ShorteningReceivers Using A Priori InformationAlbert Abello´, Jean-Marie Freixe and Damien Roque, Member, IEEEAbstract—Channel shortening has been studied in the contextof ISI and MIMO channels as a means to compute a posterioriprobabilities with a BCJR algorithm at a reduced computationalcomplexity. This is done by considering an approximate channelresponse of reduced length. In a turbo receiver, soft a prioriinformation can be linearly combined with the received sequenceto form a new input to the BCJR trellis-based processing. In thisletter, we provide closed-form expressions for the channel short-ening filters using a generalized mutual information objectivefunction. The proposed receiver allows a complexity reductionwith respect to numerical optimization approaches which mayalso present stability, precision and convergence issues.Index Terms—inter-symbol interference, turbo equalization,channel shortening, iterative receivers.I. INTRODUCTIONChannel shortening (CS) was first explored in [1] as a meansto reduce the complexity of trellis-based detection in the pres-ence of inter-symbol interference (ISI). This approach consid-ers a linear preprocessing followed by a trellis-based nonlinearprocessing. A complexity reduction with respect to the optimalapproach is obtained by considering an approximate channelresponse (the target response) of reduced length. The receivingfilter and the target response are properly optimized. Recentwork on channel shortening considers the mutual informationas the optimization criterion [2] and closed-form expressionsfor the receiving and target response filters have been obtainedby assuming that input symbols follow a Gaussian distribution.Channel shortening has been extended to the optimization ofthe transmit filters [3] and a channel shortening receiver withan additional return filter using a priori information has beenintroduced in [4], [5] although no closed-form expressionshave been provided for the return filter and the target responseso far. In [6], the authors observe that further constraints mustbe put on the return channel shortening filter in order forthe receiver to perform interference cancellation in a turboequalization scheme. These additional constraints do not allowa closed-form derivation of the channel shortening solutionwith a priori information. In this letter, we consider a moregeneral case in which we do not restrict the choice of thesymbol estimator and we consider a Gaussian model for aThis work was supported by the French armament procure-ment agency under Grant 10/2015/DGA. The associate editor coordinatingthe review of this paper and approving it for publication was C. Zhong.A. Abello´ and J-M. Freixe are with Eutelsat S.A., 75015 Paris, FRANCE.email: aabellob@eutelsat.com.D. Roque is with the Institut Supe´rieur de l’Ae´ronautique et de l’Espace(ISAE-SUPAERO), Universite´ de Toulouse, 31055 Toulouse, FRANCE.Digital Object Identifier 10.1109/LCOMM.2018.2830329priori information. We first provide both the preprocessing andreturn filters in closed-form. This result allows us to providethe target response in closed-form by applying the method in[2] to our resulting generalized mutual information. In a turboreceiver, this implies that all filters are obtained in closed-form at each iteration step thus avoiding the use of morecomplex numerical optimization methods. We validate theproposed closed-form solution by comparing it to a standardnumerical optimization approach and we demonstrate practicalperformance results by considering a typical Gaussian modelfor a priori information.II. CAPACITY APPROACHING CHANNEL SHORTENINGThe received signal at the output of the additive, white,Gaussian noise (AWGN) channel can be described by thefollowing discrete-time modelx =Hs+ n (1)where s = [s0, . . . , sN−1]T is the sequence of trans-mitted symbols with (·)T the transpose operator, x =[x0, . . . , xN−1]T is the sequence of received symbols and n =[n0, . . . , nN−1]T is the sequence of circularly-symmetric in-dependent complex Gaussian noise samples with variance σ2n.The ISI channel is described by a Toeplitz matrix H whosefirst column has L+1 nonzero elements h = [h0, h1, . . . , hL]Tforming the channel impulse response. The conditional prob-ability density function of x is given by [7]p(x|s) ∝ exp(1σ2n(2<{sHHHx}− sHGs))(2)where G =HHH is the channel response after the matchedfilter with (·)H the conjugate transpose operator.On the receiver side, optimal turbo equalization algorithmsneed for the computation of symbol a posteriori probabilitiesp(sk|x), k = 0, . . . , N − 1 [8] . A suitable factorization of(2) can be used by the BCJR algorithm to compute exacta posteriori symbol probabilities [9] although this approachyields a complexity that is exponential in L. To overcome thiscomplexity limitation, channel shortening [2], [4] considers amodified channel lawp˜(x|s) ∝ exp(1σ2n(2<{sH (Fx−Bsˆ)}− sHGrs)) (3)where F acts as a linear preprocessing of the observation xand the band matrix G is replaced by a target response Gr ofreduced length ν ≤ L(Gr)i,j = 0 | i− j |> ν (4)with (Gr)i,j the (i, j)th term of the target response. If softa priori information sˆ is available to the channel shorteningreceiver [4], it can be further preprocessed by B forming theinput sequencey = Fx−Bsˆ. (5)F−BCJRBxsˆp(sk|x)yFig. 1. Channel shortening receiver with soft a priori information for thecomputation of symbol a posteriori probabilities (APP).The modified channel law (3) along with (5) are used by theBCJR algorithm to compute approximate a posteriori symbolprobabilities. The channel shortening receiver is depicted inFigure 1. In [4], the channel shortening filters are obtainedwith standard numerical optimization methods by consideringa generalized mutual information objective functionIG = E {log (p˜ (x|s))} − E {log (p˜ (x))} (6)withp˜ (x) =∫ ∞−∞p˜ (x|s) p (s) dsand E {·} the expectation operator. F and B are subject to anoptimization on (6) of the form(F cs,Bcs) = argmaxF ,BIG. (7)To obtain a closed-form solution, we assume that inputsymbols follow a circularly-symmetric Gaussian distribution[2] with variance σ2s . We assume that input symbols and apriori symbols have a correlation matrix E{ssˆH}= C1and that a priori symbols have a correlation matrixE{sˆsˆH}= C0. We further assume that a priori symbolsare uncorrelated to the channel noise samples E{nsˆH}= 0.The optimization process is as follows. First, F and B areobtained by maximizing (6) for a given target response.Proposition 1: the optimization (7) yields optimal filtersF cs = (Gr + I)(HHH + Γ)−1HH , (8)Bcs = (Gr + I)((HHH + Γ)−1HHH − I)C1C−10(9)withΓ = σ2n(I −C1C−10 CH1)−1.Proof is given in Appendix A. Due to the Hermitian propertyof Gr and the optimization (7), F cs and Bcs are band Toeplitzmatrices. A filter processing can thus be implemented on thereceiver side with a complexity quasi linear in the length ofthe receiving filters. With optimum F cs and Bcs, the objectivefunction becomesIG = log (det {Gr + I}) +N +Tr {(Gr + I)A} , (10)A = σ2nHHΓ−1H(HHH + Γ)−1. (11)The optimal Gr can be provided in closed-form by applyingthe method proposed in [2] to the resulting generalized mutualinformation (10) which takes into account a priori information.This is done by taking the derivative of (10) and by consideringa Cholesky factorization of the term Gr+I . The optimizationprocess is summarized in Appendix B.It can be shown that the linear minimum mean square error(MMSE) approach using a priori information is a particularcase of rate maximizing channel shortening reception. TheMMSE filters Fmse and Bmse are well known [10] and canbe found by setting(Fmse,Bmse) = argminF ,BE{‖y − s‖2}.The optimum filters can be put on the formFmse =(HHH + Γ)−1HH ,Bmse =((HHH + Γ)−1HHH − I)C1C−10 .Therefore, the channel shortening filters equal the MMSEfilters weighted by Gr+I . The two approaches are equivalentwhen ν = 0 (i.e., Gr = I , no trellis-based processing isperformed). The full complexity receiver is found by settingν = L (i.e., Gr = HHH) which yields the matched filterF cs =HH and Bcs = −ΓC1C−10 .If no a priori information is available to the receiver, thenC1 = C0 = 0, Γ = σ2nI . In this case, F cs equals theclassical MMSE receiver. On the other hand, with perfect apriori information sˆ = s and if we consider σ2s = 1 thenC1 = C0 = I and Γ = ∞I . In this situation, F cs = 0and Bcs = − (Gr + I). The noiseless case σ2n = 0 yieldsthe zero forcing solution weighted by the target responseF cs = GrH−1 and Bcs = 0.III. NUMERICAL RESULTSThe transmitted uncoded symbols are obtained using abinary phase-shift keying (BPSK) mapping. For numericalsimulation, we assume C1 = c1I and C0 = c0I so thatΓ = γI with γ = σ2n(1− |c1|2/c0)−1. Performance of theproposed channel shortening receiver has been evaluated byassuming an additive white Gaussian noise model for soft apriori information sˆ = s+ e with e a sequence of circularly-symmetric independent complex Gaussian noise samples withTABLE ISE BETWEEN CLOSED-FORM AND NUMERICAL FILTER OPTIMIZATIONν = 2γ gr f cs bcs0.2 1.5832e-06 1.3836e-06 1.8215e-060.4 3.0729e-06 2.3371e-06 3.3412e-060.6 7.1155e-06 2.3204e-06 2.7197e-060.8 2.2225e-05 3.6139e-06 2.6077e-061.0 7.1042e-05 8.3007e-06 4.7152e-06ν = 30.2 2.7235e-06 7.5011e-06 8.7205e-060.4 4.5422e-06 1.7081e-05 2.0377e-050.6 4.2465e-06 1.5438e-05 1.8783e-050.8 7.5227e-06 1.3169e-05 1.5589e-051.0 2.4116e-05 1.3224e-05 1.3622e-05variance σ2e . This yields correlation coefficients c1 = 1,c0 = 1+σ2e . In a turbo receiver [8], a priori information variesat each iteration step and the receiving filters F cs, Bcs and Grcan be recalculated accordingly in closed-form as described inSection II.Table I shows the squared error between the receiving filtersobtained in closed-form and obtained iteratively using thegradient projection method [11] on the generalized mutualinformation derived in (14) as a function of γ and ν forσ2n = 2. The receiving filters f cs, bcs and gr correspond toa row of F cs, Bcs and Gr respectively. We define the error asSE = k−1‖f cs − f ics‖2 where the superscript (·)i means thatthe iterative method was considered and k is the filter length.Results confirm that a numerical optimization of F , B andGr converges toward the optimal closed-form solution.To better illustrate the complexity reduction obtained withthe closed-form solution, please note that we consider ingeneral N×N band Toeplitz matrices and we focus on the caseC1 = c1I and C0 = c0I . The direct method for computingthe optimal shortening filters yields a complexity in O(N2).On the other hand, the gradient projection method, that we2 4 6 8 1010−410−310−210−1‖h‖2/σ2n (dB)BERσ2e=1σ2e=0.5σ2e=0.3σ2e=0.2σ2e=0.15Fig. 2. Performance results of the channel shorteningreceiver for ν = 2 over the partial response channelh = [0.033, -0.131, -0.037, 0.446, 0.750, 0.446, -0.037, -0.131, 0.033]T .2 4 6 8 10 1210−510−410−310−210−1‖h‖2/σ2n (dB)BERσ2e=1, ν=1σ2e=1, ν=2σ2e=1, ν=3σ2e=0.3, ν=1σ2e=0.3, ν=2σ2e=0.3, ν=3Fig. 3. Performance results of the channel shortening receiver over the EPR4channel h = [0.5, 0.5,−0.5,−0.5]T .used to validate the closed-form solution, has a complexity inO( · N2) with  the number of iterations performed by thealgorithm. The number of iterations needed depends on γ, σ2nand ν. In the worst case simulation scenario, we needed 2000iterations to achieve convergence. For a variety of numericaloptimization methods the complexity remains O(·N2). Otherthan the complexity advantage, the proposed closed-formsolution avoids stability, convergence and precision issues thatmust be dealed with when numerical optimization approachesare used.Receiver performance is depicted in Figures 2 and 3. Thelines correspond to the optimal closed-form solution. Themarkers correspond to the iterative solution after convergence.The squared errors obtained between the closed-form and theiterative approaches prove to be sufficiently low in terms ofbit error rate degradation.IV. CONCLUSIONIn this letter, we have provided a generalization of the rate-maximizing channel shortening receiver using a priori infor-mation. The receiving filters have been provided in closed-form at each turbo iteration step thus avoiding the use ofmore complex numerical optimization methods. Additionally,this result enables the derivation of the linear MMSE turboapproach as a particular case of channel shortening when a pri-ori information is available to the receiver. Numerical resultsshow that standard iterative optimization methods convergetoward the optimal closed-form solution. Future work maystudy a suitable soft-symbol estimator for the optimal closed-form channel shortening solution.APPENDIXA. Proof of proposition 1We assume s ∼ CN (0, σ2sI). For ease of reading andwithout loss of generality, we take σ2s = 1 and we introduceσ2n in F ,B and Gr respectively, yieldingp˜(x) =K(x)piN∫sexp(2<{sHy}− sH (Gr + I) s)ds,K(x) =1(piσ2n)Nexp(−xHxσ2n).We restrict Gr to being Hermitian so that rearranging intosquared forms yieldsp˜(x) = K(x) det{(Gr + I)−1}exp(yH (Gr + I)−1y).The mutual information isIG =E {log (p˜ (x|s))} − E {log (p˜ (x))}=2<{E{sHy}}− E{sHGrs}+ log (det {Gr + I})− E{yH (Gr + I)−1y}, (12)E{sHy}= Tr{FH −BCH1},E{sHGrs}= Tr {Gr} ,E{yH (Gr + I)−1y}= Tr{(F(HHH + σ2nI)FH+BC0BH − 2<{FHC1BH})(Gr + I)−1}. (13)Introducing (13) in (12) yieldsIG = log (det {Gr + I})− Tr {Gr}+ 2<{Tr{FH −BCH1}}− Tr{(F(HHH + σ2nI)FH+BC0BH − 2<{FHC1BH})(Gr + I)−1}. (14)The optimal filter F cs is obtained from (14) by setting∇F cs IG = 0 which yields after some calculationsF cs =(BCH1 + (Gr + I))HH(HHH + σ2nI)−1. (15)We now set ∇Bcs IG = 0 which yields after some calculationsBcs = (FH − (Gr + I))C1C−10 . (16)Introducing (16) in (15) yields optimum filters (8) and (9).B. Optimization of GrGr+I is assumed to be a Hermitian positive-definite matrixallowing the Cholesky factorization Gr + I = UUH with Uan upper triangular matrix(U)m,n ={um,n if n ≥ m0 otherwise.Then, (10) can be rewritten asIG =(2N∑n=1log(un,n)− Tr{UAUH}+N). (17)Since the first term in (17) does not contain off-diagonalvalues, IG can be optimized over the diagonal and off-diagonalelements separately [2]U = argmaxun,n(2N∑n=1log(un,n) +N− argmin{um,n}m+1≤n≤min(m+ν,N)Tr{UAUH}). (18)By defining the submatrixAνn = an+1,n+1 · · · an+1,min(N,n+ν)... . . . ...amin(N,n+ν),n+1 · · · amin(N,n+ν),min(N,n+ν)and the row vectors aνn =[an,n+1, · · · , an,min(N,n+ν)]anduνn =[un,n+1, · · · , un,min(N,n+ν)]thenTr{UAUH}=N∑n=1(un,n uνn)(an,n aνnaνnH Aνn)(un,nuνnH).Setting ∇uνn Tr{UAUH}= 0 yields uνn =−un,naνn(Aνn)−1. Introducing uνn in (18) yieldsU = argmaxun,n(2N∑n=1log(un,n) +N −N∑n=1u2n,ncn)(19)where cn = an,n − aνn(Aνn)−1aνnH . Setting the derivative of(19) to zero yields un,n =√1/cn.REFERENCES[1] D. D. Falconer and F. R. Magee, “Adaptive channel memory truncationfor maximum likelihood sequence estimation,” Bell Syst. Techn. J.,vol. 52, no. 9, pp. 1541–1562, Nov. 1973.[2] F. Rusek and A. Prlja, “Optimal channel shortening for MIMO and ISIchannels,” IEEE Trans. Wireless Commun., vol. 11, pp. 810–818, Feb.2012.[3] A. Modenini, F. Rusek, and G. Colavolpe, “Optimal transmit filters forconstrained complexity channel shortening detectors,” in 2013 IEEE Int.Conf. Commun. (ICC), Jun. 2013, pp. 3095–3100.[4] F. Rusek, N. Al-Dhahir, and A. Gomaa, “A rate-maximizing channel-shortening detector with soft feedback side information,” in 2012 IEEEGlobal Commun. Conf. (GLOBECOM), Dec. 2012, pp. 2256–2261.[5] S. Hu, H. Kro¨ll, Q. Huang, and F. Rusek, “Optimal channel shortenerdesign for reduced- state soft-output viterbi equalizer in single-carriersystems,” IEEE Trans. Commun., vol. 65, no. 6, pp. 2568–2582, Jun.2017.[6] S. Hu and F. Rusek, “On the design of channel shortening demodu-lators for iterative receivers in MIMO and ISI channels,” CoRR, vol.abs/1506.07331, 2015.[7] G. Ungerboeck, “Adaptive maximum-likelihood receiver for carrier-modulated data-transmission systems,” IEEE Trans. Commun., vol. 22,no. 5, pp. 624–636, May 1974.[8] M. Tu¨chler and A. Singer, “Turbo equalization: An overview,” IEEETrans. Inf. Theory, vol. 57, no. 2, pp. 920–952, Feb. 2011.[9] G. Colavolpe and A. Barbieri, “On MAP symbol detection for ISIchannels using the Ungerboeck observation model,” IEEE Commun.Lett., vol. 9, no. 8, pp. 720–722, Aug. 2005.[10] M. Tu¨chler, A. C. Singer, and R. Koetter, “Minimum mean squared errorequalization using a priori information,” IEEE Trans. Signal Process.,vol. 50, no. 3, pp. 673–683, Mar. 2002.[11] J. Rosen, “The gradient projection method for nonlinear programming.part I. linear constraints,” J. Soc. Ind. Appl. Mathematics, vol. 8, no. 1,pp. 181–217, Mar. 1960.

Closed-Form Expressions for Channel Shortening Receivers Using A Priori Information

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Closed-Form Expressions for Channel Shortening Receivers Using A Priori Information

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