A maximum entropy approach to OFDM channel estimation

A maximum entrop y approach to OFDM channel estimation Romain Couillet NXP Semicondu ctors, Supelec 505 Rou te d es Lucio les 06560 Sophia Antipo lis, France Email: r omain.cou illet@nxp.com M ´ erouane Debb ah Alcatel-Lucen t Chair , Supele c Plateau d e Moulon, 3 r ue Jolio t-Curie 91192 Gif sur Yvette, Fran ce Email: m erouane. debbah@sup elec.fr Abstract In this work, a ne w B ayesian framew ork f or OF DM channel estimati on is proposed. Using Jaynes’ maximum entropy principle to deriv e prior information, we successi v ely tackle the situations when only t he channel delay spread is a priori known , then when it is not kno wn. Exploitation of the time-frequency dimensions are also considered in this framew ork, to derive the optimal channel estimation associated to some performance measure under any state of kno wledge. Simulations corroborate the optimality claim and always prov e as good or better in performance than classical esti mators. I . I N T RO D U C T I O N Modern high rate wireless comm unication systems, such as IEEE-Wimax [1] or 3GPP-Lon g T erm Evolution ( L TE) [2], usu- ally come along with large band widths. I n multipath fadin g ch annels, th is entails high f requency selectivity , which theoretically is beneficial for it pr ovides system diversity . But in practice, this constitutes a strong challen ge f or e qualization. T he orthogo nal frequen cy division mu ltiplexing (OFDM ) mod ulation [3], considered as the schem e for most future w ireless systems, allows for simplified equalization through pilot sequences scatter ed in th e time-freq uency g rid and p ossibly over the space dimension when multiple antenn as are used. The ch allenge in channel estima tion with a limited nu mber of pilots lies in the optim al way to exploit all th e info rmation the r eceiv er is pr ovided with . Classical methods co nsider that only d ata received from pilot positions are informative. As a consequen ce, b etween p ilot po sitions, the estimated ch annel mu st be reconstru cted u sing inter polation tech niques that would prove r obust ( whatever one means b y robustness) in simulation s [4]. Th en it ap peared that a Bayesian min imum mean squ are error (MMSE) [5] can be deri ved wh en not only the p ilot sequen ces but also the channe l c ov ariance matrix are k nown. This solution c oincides with the linear MMSE (LMMSE) estimator and thu s provides o ptimal p erform ance wh en the state of knowledge on th e system is limited to those pilo t d ata and to th e c hannel covariance matrix [6]. However , wh en the chan nel covariance matrix is unkn own, then again, only ad-h oc techniques were derived to cope with the lack of knowledge. The defin ite choice of a prior co rrelation matrix for the chan nel is one of the classical ap proach (identity matrix or exponen tially decaying matrix for instance) [7]. But all those ap proach es are only justified b y good perform ance ar ising in selected simu lations an d do not p rovide any pr oof as fo r their overall p erforman ce. In the fo llowing work, we tackle chan nel estimation for OFDM as a prob lem of inductive reasoning b ased on th e av ailable prior informatio n and the received p ilots. Especially , to r ecover missing inform ation, we extensiv ely use the maximum entr opy principle , introd uced by Shann on [8], extended by Jaynes [ 9] and accu rately proven by Shore and Joh nson [10] to be th e desirable ma thematical tool to cope with lack of in formation . Som e of the aforemen tioned classical results shall be fou nd anew and proven optimal in o ur in formation theo retic fr amew ork, while new results will be provid ed which sh ow to p erform better than classical a pproach es. The remain der of this paper is structured as follows: In Section II, we in troduce th e chan nel pilot-aided OFDM system, then in Sectio n III, we ca rry o ut the Bayesian channel estimation study based o n different lev els o f knowledge. Simu lations are then pro posed in Section IV an d a tho rough discussion on the re sults, limitations and extendability of o ur sch eme is ha ndled in Sectio n V. T hen w e draw our conclu sions in Section VI. Notations : In the following, b oldface lower case sy mbols represent vectors, capital bo ldface c haracters denote matrice s ( I N is th e N × N identity matrix ). The transposition o peration is d enoted ( · ) T . Th e Herm itian transpo se is deno ted ( · ) H . Th e operator diag ( x ) turns the vector x into a diag onal matrix . Th e sy mbol det( X ) is the d eterminan t of m atrix X . The sy mbol E[ · ] deno tes expectation. Th e Kron ecker delta function is denoted δ x that eq uals 1 if x = 0 and equa ls 0 otherwise. I I . S Y S T E M M O D E L W e consider here a single cell OFDM system with N subcarriers. The cyclic prefix (CP) length is N C P samples. In th e time-frequ ency OFDM sy mbol grid, pilots are fo und in the symbol position s ind exed by the fu nction φ t ( n ) ∈ { 0 , 1 } wh ich equals 1 if a pilot symb ol is present at sub carrier n and 0 otherwise; the subscript index t denotes the OFDM symbol time. Th is is depicted in Figure 1. Both d ata an d pilo ts are gath ered at time t in a frequ ency-domain vector s t ∈ C N with pilot entr ies of One resso urce blo c k, 180 kHz One subframe (2 slots), 1 ms s 1 , 12 s 1 , 6 s 5 , 3 s 5 , 9 s 8 , 12 s 8 , 6 s 12 , 3 s 12 , 9 Fig. 1. Time-fre quenc y OFDM grid with pilot positions enhanced amplitude | s t,k | 2 = 1 , for all t, k such th at φ t ( k ) = 1 . They are then sent throug h a ch annel of frequency resp onse h t ∈ C N with entries of m ean power E[ | h t,k | 2 ] = 1 , and backgrou nd n oise n t ∈ C N with entries of m ean p ower E[ | n t,k | 2 ] = σ 2 . The time-dom ain represen tation o f h t is denoted ν t ∈ C L with L the chann el leng th, i.e. the number of time-dom ain samples which suffer inter symbo l interferen ce. The OFDM frequ ency signal y t ∈ C N received at time t at the r eceiv er is y t = diag( h t ) s t + n t (1) This work aims at op timally estimating the vector h t for some per formanc e measure defined her eafter . T he estimate shall be d enoted ˆ h t ∈ C N . Different states of knowledge at the receiver are co nsidered in th e subseq uent work, • the ch annel length L is either kn own o r un known, • at discrete time t , th e pilots at time ( t − k ) fo r k ∈ { 1 , . . . , K } as well as the cha nnel time-corre lation, ar e either kn own or u nknown. On top of those parameter s, classical system parameter s are supposed to be known (some were explicitly a lready used ), • signal m ean p ower • noise m ean p ower • channel m ean p ower . In each of the subsequent deriv ations, the exact quan tity of knowledge will b e clearly stated, since it is essential to the inductive in ference we will p erform on the ch annel h . I I I . C H A N N E L E S T I M AT I O N A. Th e cha nnel length is known W e first con sider th e simp le scenario in wh ich the ch annel power delay profile, i.e. the diagonal elements of th e time-dom ain channel covariance matrix, is unk nown but the channel length L is k nown. Only pilot sequen ces r eceiv ed at discrete time t are k nown to the receiver . This is, we assume that the receiver is n ot able to register eithe r past received signals, n or p ast estimates of th e channe l. This hypo thesis will be re lie ved in subsequ ent consider ations. The amou nt of prior info rmation, i.e. noise power , signal p ower and L , is deno ted I . The amou nt of information that can be inferred abo ut some entity E fr om the prior inf ormation I will be deno ted ( E | I ) . For ease of reading, we will remove the ind ex t in the notations whe n unn ecessary . W e will also, ∀ k ∈ { 1 , . . . , N } , deno te h ′ k = y k /s k = h k + n k /s k and h ′ = ( h ′ 1 , . . . , h ′ N ) T . The only knowledge o n the additio nal noise vector in th e system is the mea n power σ 2 of its entries. The max imum entropy principle [ 13] req uires then that the noise p rocess b e assigned a Gau ssian indep endent and identically d istributed (i.i.d.) de nsity: n ∼ C N (0 , σ 2 I N ) . From the chan nel mo del (1 ), equivalently , the multip ath ch annel of len gth L is only known to be of unit mean p ower . Again, th e max imum entr opy principle de mands ν ∼ C N (0 , 1 L I L ) . Sin ce ν is a Gaussian vector with i.i.d. e ntries, h , its d iscrete Fourier tran sform, is a cor related Gaussian vector, h ∼ C N (0 , Q ) (2) with, f or any coup le ( n, m ) ∈ { 1 , . . . , N } 2 , Q nm = E " L − 1 X k =0 L − 1 X l =0 ν k ν ∗ l e − 2 π i kn − lm N # (3) = 1 L L − 1 X k =0 e − 2 π ik n − m N (4) T o derive the optimal ch annel estimator, a decision regarding the targeted erro r function to be minimized h as to be mad e. Follo wing mo st o f previous contributions in this respect, we propo se to take ˆ h as th e estimator th at minimizes th e mean quadra tic estimation e rror (MM SE e stimator), given the received signal y . This is [5] ˆ h = E [ h | y ] (5) = Z C N h P ( h ) P ( y | h ) P ( y ) d h (6) = Z C N h P ( h ) P ( y | h ) R C N P ( h ) P ( y | h ) d h  d h (7) = lim ˜ Q → Q 1 π N + M σ 2 M det ˜ Q Z C N h · e − h H ˜ Q − 1 h · e − 1 σ 2 ( h − h ′ ) H P H P ( h − h ′ ) d h  1 π N + M σ 2 M det ˜ Q R C N e − h H ˜ Q − 1 h · e − 1 σ 2 ( h − h ′ ) H P H P ( h − h ′ ) d h  (8) = lim ˜ Q → Q Z C N h · e − h H ˜ Q − 1 h · e − 1 σ 2 ( h − h ′ ) H P H P ( h − h ′ ) d h  R C N e − h H ˜ Q − 1 h · e − 1 σ 2 ( h − h ′ ) H P H P ( h − h ′ ) d h  (9) in which the limiting proce ss is taken over a set of invertible matrices ˜ Q (w hich tends to Q that is by definition of ran k L < N ), and P is a pr ojection matr ix over the set of pilo t freq uency carriers ( P ij = δ i − j δ φ ( i ) ). Let us note at th is poin t that, fr om the r eceiv ed data y , we on ly use the sym bols indexed at pilot position s in equ ation (8), hence the introdu ction of th e pro jectors P . This seems to go again st our o ptimality claim. Ind eed, on e migh t object that data outside the pilot positions somehow carry informatio n abou t the cha nnel a nd should be taken into accou nt. I n th e same manner, we could also claim that po tential interfer ers ar e not correc tly dealt with. Still, o ur problem is correctly form ulated sin ce I does not car ry any info rmation ab out in formative data apar t f rom pilots, nor does it even sug gest the presen ce of interf erers. This epistemolog ical discussion is further d ebated in Sectio n V. The p roduc t of th e exp onential terms in (9) can be written (by expan sion and id entification) − h H ˜ Q − 1 h − 1 σ 2 ( h − h ′ ) H P H P ( h − h ′ ) = − ( h − ˜ k ) H ˜ M ( h − ˜ k ) − ˜ C (10) with    ˜ M = ˜ Q − 1 + 1 σ 2 P H P ˜ k = 1 σ 2 ˜ M − 1 P H Ph ′ ˜ C = 1 σ 2 h ′ H P H Ph ′ − ˜ k H ˜ M ˜ k (11) This allows to isolate the d umb v ariable h in the integrals and leads then to comp ute the first o rder moment o f a multiv ariate Gaussian d istribution, ˆ h = lim ˜ Q → Q Z C N h · e − ( h − ˜ k ) H ˜ M ( h − ˜ k ) d h R C N e − ( h − ˜ k ) H ˜ M ( h − ˜ k ) d h (12) = lim ˜ Q → Q ˜ k (13) in which ˜ k d epends on ˜ Q throu gh ˜ M . Noting that ˜ M − 1 = ( ˜ Q − 1 + 1 σ 2 P H P ) − 1 = ( I N + 1 σ 2 ˜ QP H P ) − 1 ˜ Q , ˆ h is then ˆ h = lim ˜ Q → Q ( σ 2 I N + ˜ QP H P ) − 1 ˜ QP H Ph ′ (14) = ( σ 2 I N + QP H P ) − 1 QP H Ph ′ (15) This is exactly the well-kn own L MMSE solution [5]. Howe ver , this result is n ot yet anoth er demon stration of LM MSE as it is classically d erived. W e used her e the maxim um entropy pr inciple to observe that, at the e nd, the limited kn owledge o n the channel length L mathema tically gives the same estimation as when on e “ assumes” an a priori covariance matrix Q . Theref ore the intuitive classical solu tion is the co rrect estimate in th e sense of max imum entr opy [13]. B. Un known chann el length If L is only known to b e in an interval { L min , . . . , L max } , th e m aximum entro py princip le assigns a uniform prio r distribution for L ; otherwise one would add non desirable implicit informatio n to the cur rent state of knowledge. The channel MMSE estimator is th en given b y ˆ h = E [ h | y ] (16) = Z C N h ( P L P ( h | L ) P ( L )) P ( y | h )  R C N ( P L P ( h | L ) P ( L )) P ( y | h ) d h  d h (17) = lim ˜ Q L min → Q L min ··· ˜ Q L max → Q L max L max X L = L min 1 det ˜ Q L Z C N h · e − h H ˜ Q − 1 L h · e − 1 σ 2 ( h − h ′ ) H P H P ( h − h ′ ) d h  P L max L = L min 1 det ˜ Q L R C N · e − h H ˜ Q − 1 L h · e − 1 σ 2 ( h − h ′ ) H P H P ( h − h ′ ) d h  (18) where Q k is the c hannel covariance m atrix for a c hannel length k ∈ { L min , . . . , L max } and ˜ Q k are taken in a set of invertible matrices in the neig hborh ood of Q k . Using the same tran sformation s as in (1 0) and the fact that the numerato r a nd deno minator constants in (18) do not simplify any lon ger , we en d u p with ˆ h = lim ˜ Q L min → Q L min ··· ˜ Q L max → Q L max P L max L = L min det( ˜ M ( L ) ˜ Q L ) − 1 e − ˜ C ( L ) ˜ k ( L ) P L max L = L min det( ˜ M ( L ) ˜ Q L ) − 1 e − ˜ C ( L ) (19) in which we u pdated o ur p revious notations to in corpor ate th e depend ence on L ,                      ˜ M ( L ) = ˜ Q − 1 L + 1 σ 2 P H P = ˜ Q − 1 L ( I N + 1 σ 2 ˜ Q L P H P ) ˜ k ( L ) = 1 σ 2 ( I N + 1 σ 2 ˜ Q L P H P ) − 1 ˜ Q L P H Ph ′ ˜ C ( L ) = 1 σ 2 h ′ H P H Ph ′ − ˜ k H ( ˜ Q − 1 + 1 σ 2 P H P ) ˜ k = 1 σ 2 h ′ H P H Ph ′ − 1 σ 4 h ( I N + 1 σ 2 ˜ Q L P H P ) − 1 ˜ Q L P H Ph ′ i H P H Ph ′ = h ′ H  ( I N + 1 σ 2 ˜ Q L P H P ) − 1  H P H P σ 2 h ′ (20) The d eterminan t det( ˜ M ( L ) ) ca n be further developed as det( ˜ M ( L ) ) = det( I N + 1 σ 2 ˜ Q L P H P ) · det( ˜ Q L ) − 1 (21) which entails det( ˜ M ( L ) ˜ Q L ) = det( I N + 1 σ 2 ˜ Q L P H P ) (22) No inv ersion of ˜ Q k matrices is then n ecessary so that th e limiting pro cess is now straig htforward ˆ h = L max X L = L min det  ( I N + 1 σ 2 Q L P H P ) − 1  e − C ( L ) k ( L ) L max X L = L min det  ( I N + 1 σ 2 Q L P H P ) − 1  e − C ( L ) (23) in which k ( L ) and C ( L ) are th e limits o f ˜ k ( L ) and ˜ C ( L ) respectively , ( k ( L ) =  I N + 1 σ 2 Q L P H P  − 1 1 σ 2 Q L P H Ph ′ C ( L ) = h ′ H  ( I N + 1 σ 2 Q L P H P ) − 1  H P H P σ 2 h ′ (24) Since C ( L ) comprises the quadra tic term h ′ H P H Ph ′ , the MMSE estimation o f h is not linear in h ′ , ther efore the LMMSE estimate in the scenario when L is unkn own do es n ot coincide with th e MMSE estimate. W e also no te th at formu la ( 23) is no m ore than a weig hted fun ction o f the individual LMMSE estimates for different hy pothetical values of L . The weigh ting coefficients allow to en hance the estimates that r ather fit the correct L h ypothesis and to discard tho se estimates that do n ot concor d with th e h ′ observation. C. Using time co rr elation When the cha nnel co her ence time , defined as the typ ical duration for which th e channel realization s are co rrelated [11], is of the same order or larger than a few OFDM symbo ls, then past (and future) receiv ed d ata c arry important inf ormation on the present c hannel. Th is in formation mu st b e taken into accoun t. Classically , channel time cor relation is described through Jakes’ model [1 2]. For a Do ppler spr ead f d (prop ortional to the vehicular speed ), the cor relation figure is mo deled as E[ ν t + T ,p ν ∗ t,p ] = 1 L · J 0 (2 π f d T ) (25) in which p is one of th e paths o f the multipath chan nel ν and J 0 is the Bessel functio n of the first kind . This model actually makes two assumptions th at, un der th e p roper in formation setting, can be turned into the output of a maximum entropy p rocess. Tho se assumption s [18] are • the signal scatterers are unc orrelated in the sense that two rays arriving a t different a ngles to the receiver face indep endent attenuation proper ties. Under n o kn owledge on th e environmental scatterers, this has to be the logical assumptio n. • the an gles of arrival (i.e. the angle betwe en the a ntenna body and the inco ming wa ve) are u niformly distributed. Again, this is what the max imum entropy p rinciple would state if no pa rticular knowledge on the po sitions of the scatterers, transmitter and receiver is a p riori given. For those reasons, Jakes’ model is r easonable wh en no geometrical in formatio n on the channel is given. Practically speak ing, it will be difficult for the receiver to be aware of the exact D oppler fr equency f d . In the f ollowing the oretical derivations and in the forthco ming simulations, we shall co nsider that the receiver exactly kn ows the expected value of equa tion (25). It is o f course possible, either to find estimate s fo r E[ ν p,t + T ν ∗ p,t ] or to complete the subsequ ent study by integrating out the possible Doppler frequen cies given a p rior distribution for f d . Let us consider the simple scenario in which only the pr esent and last past p ilot symbo ls are con sidered by the term inal. Those corr espond to two time instants t 1 and t 2 , respec ti vely . W e also con sider first that L is known. For n otational simplicity , we shall denote h k = h t k . The MMSE estimator for h 2 under this state o f knowledge is then ˆ h 2 = E[ h 2 | h ′ 1 h ′ 2 ] (26) = Z h 2 h 2 P ( h ′ 1 h ′ 2 | h 2 ) P ( h 2 ) P ( h ′ 1 h ′ 2 ) d h 2 (27) = Z h 2 h 2 P ( h 2 ) P ( h ′ 2 | h ′ 1 h 2 ) P ( h ′ 1 | h 2 ) P ( h ′ 1 h ′ 2 ) d h 2 (28) = Z h 2 h 2 P ( h 2 ) P ( h ′ 2 | h 2 ) P ( h ′ 1 | h 2 ) P ( h ′ 1 h ′ 2 ) d h 2 (29) = Z h 2 h 2 P ( h 2 ) P ( h ′ 2 | h 2 ) R h 1 P ( h ′ 1 | h 2 h 1 ) P ( h 1 | h 2 ) d h 1 P ( h ′ 1 h ′ 2 ) d h 2 (30) = Z h 2 h 2 P ( h 2 ) P ( h ′ 2 | h 2 ) R h 1 P ( h ′ 1 | h 1 ) P ( h 1 | h 2 ) d h 1 P ( h ′ 1 h ′ 2 ) d h 2 (31) in which eq uations (29) and (31) ar e verified sin ce h 1 and h 2 do n ot br ing any ad ditional info rmation to ( h ′ 2 | h 2 ) and ( h ′ 1 | h 1 ) respectively . At this point, we re cognize tha t, apart from the new ter m P ( h 1 | h 2 ) , all the pro babilities to be deriv ed h ere hav e already been produ ced in the previous sections. Now , our knowledge on ( h 1 | h 2 ) is limited to equ ation (25). Burg’ s theorem [1 7] states then that th e maxim um entro py distribution for ( ν 1 | ν 2 ) is an L -multivariate Gaussian distribution of mean λ ν 2 and variance 1 L (1 − λ 2 ) I L with λ = J 0 (2 π f d T ) . Therefo re, thank s to the same linearity argu ment as ab ove, the distribution of ( h 1 | h 2 ) is giv en by P ( h 1 | h 2 ) = lim ˜ Φ → Φ 1 π N det( ˜ Φ ) e − ( h 1 − λ h 2 ) H ˜ Φ − 1 ( h 1 − λ h 2 ) (32) with Φ ( T ) = (1 − λ 2 ) Q (33) Consider first the inner integral in equ ation (31). Similarly to above, we can expre ss P ( h 1 | h 2 ) P ( h ′ 1 | h 1 ) = lim ˜ Φ → Φ 1 π M 1 + N σ 2 M 1 det( ˜ Φ ) e − ( h 1 − ˜ k 1 ) H ˜ M 1 ( h 1 − ˜ k 1 ) − ˜ C 1 (34) with                  ˜ M 1 = ˜ Φ − 1 + 1 σ 2 P H 1 P 1 ˜ k 1 = ˜ M − 1 1 ( λ ˜ Φ − 1 h 2 + 1 σ 2 P H 1 P 1 h ′ 1 ) = ( I N + 1 σ 2 ˜ ΦP H 1 P 1 ) − 1 ( λ h 2 + 1 σ 2 ˜ ΦP H 1 P 1 ) ˜ C 1 = λ 2 h H 2 ˜ Φ − 1 h 2 + 1 σ 2 h ′ 1 H P H 1 P 1 h ′ 1 − ˜ k H 1 ˜ M 1 ˜ k 1 = λ 2 h H 2 ˜ Φ − 1 h 2 + 1 σ 2 h ′ 1 H P H 1 P 1 h ′ 1 − ( λ h 2 + 1 σ 2 ˜ ΦP H 1 P 1 h ′ 1 ) H   I N + 1 σ 2 ˜ ΦP H 1 P 1  − 1  H ˜ Φ − 1 ( λ h 2 + 1 σ 2 ˜ ΦP H 1 P 1 h ′ 1 ) (35) and M 1 is th e n umber of p ilot po sitions in th e first pilot seq uence. Now , the integration o f the part dep endent on h 1 giv es 1 π M 1 + N σ 2 M 1 det( ˜ Φ ) Z h 1 e − ( h 1 − ˜ k 1 ) H ˜ M 1 ( h 1 − ˜ k 1 ) d h 1 = det( ˜ M 1 ) − 1 π M 1 σ 2 M 1 det( ˜ Φ ) (36) = det( I N + 1 σ 2 ˜ ΦP H 1 P 1 ) π M 1 σ 2 M 1 (37) which leads to Z h 1 P ( h 1 | h 2 ) P ( h ′ 1 | h 1 ) d h 1 = det( I N + 1 σ 2 ΦP H 1 P 1 ) − 1 π M 1 σ 2 M 1 e − C 1 (38) with, like pr e viously , Φ an d C 1 the re spectiv e limits of ˜ Φ and ˜ C 1 in the limiting pr ocess ˜ Q → Q . W e n ow need to co nsider the ou ter integra l, that we sha ll similar ly develop (not forgetting C 1 that depen ds on h 2 ) as 1 π M 2 + N σ 2 M 2 det( ˜ Q ) Z h 2 e − ( h 2 − ˜ k 2 ) H ˜ M 2 ( h 2 − ˜ k 2 ) − ˜ C 2 d h 2 (39) with                                  ˜ M 2 = ˜ Q − 1 + λ 2 ˜ Φ − 1 − λ 2  ˜ Φ + 1 σ 2 ˜ ΦP H 1 P 1 ˜ Φ  − 1 + 1 σ 2 P H 2 P 2 = ˜ Q − 1  (1 + λ 2 1 − λ 2 ) I N − λ 2 1 − λ 2 ( I N + 1 σ 2 ˜ ΦP H 1 P 1 ) − 1 + 1 σ 2 ˜ QP H 2 P 2  ˜ k 2 = ˜ M − 1 2  1 σ 2 P H 2 P 2 h ′ 2 +  ˜ Φ + 1 σ 2 ˜ ΦP H 1 P 1 ˜ Φ  − 1 λ σ 2 ˜ ΦP H 1 P 1 h ′ 1  =  (1 + λ 2 1 − λ 2 ) I N − λ 2 1 − λ 2 ( I N + 1 σ 2 ˜ ΦP H 1 P 1 ) − 1 + 1 σ 2 ˜ QP H 2 P 2  − 1 ˜ Q ×  1 σ 2 P H 2 P 2 h ′ 2 + ( I N + 1 σ 2 P H 1 P 1 ˜ Φ ) − 1 λ σ 2 P H 1 P 1 h ′ 1  ˜ C 2 = 1 σ 2 h ′ 2 H P H 2 P 2 h ′ 2 − ˜ k H 2 ˜ M 2 ˜ k 2 + 1 σ 2 h ′ 1 H P H 1 P 1 h ′ 1 − h ′ 1 H   I N + 1 σ 2 ˜ ΦP H 1 P 1  − 1  H 1 σ 2 P H 1 P 1 h ′ 1 with M 2 the n umber o f pilo ts in the seco nd pilot sequ ence. In the expression of C 2 , it is readily seen that expandin g k H 2 M 2 k 2 leads to invert M 2 . The ˜ Q − 1 factor cancels then out (from the d ev elopment of k 2 ). Th en M 2 and M − 1 2 cancel out as well (in the development of M 2 k 2 ). Th erefore, no prob lem of m atrix inversion is f ound in tho se expr essions. W e can then take the limit to finally have ˆ h = lim ˜ Q → Q det( ˜ M 2 ) − 1 det( ˜ M 1 ) − 1 π M 2 + M 1 σ 2( M 2 + M 1 ) det( ˜ Q ) det ( ˜ Φ ) e − ˜ C 2 · ˜ k 2 · det( ˜ M 2 ) − 1 det( ˜ M 1 ) − 1 π M 2 + M 1 σ 2( M 2 + M 1 ) det( ˜ Q ) det ( ˜ Φ ) e − ˜ C 2 ! − 1 (40) = lim ˜ Q → Q ˜ k 2 (41) =  (1 + λ 2 1 − λ 2 ) I N − λ 2 1 − λ 2 ( I N + 1 − λ 2 σ 2 QP H 1 P 1 ) − 1 + 1 σ 2 QP H 2 P 2  − 1 Q ×  1 σ 2 P H 2 P 2 h ′ 2 + ( I N + 1 − λ 2 σ 2 P H 1 P 1 Q ) − 1 λ σ 2 P H 1 P 1 h ′ 1  (42) (43) This formula stands only wh en the chan nel leng th L is known. Then, with the same notation s as in previous sections, if the channel len gth were o nly k nown to belong to an interval { L min , . . . , L max } , then ˆ h 2 = L max X L = L min det( A L ) − 1 det( B L ) − 1 e − C 2 ( L ) ! − 1 L max X L min det( A L ) − 1 det( B L ) − 1 e − C 2 ( L ) k ( L ) 2 (44) with        A L = I N + 1 − λ 2 σ 2 Q L P H 1 P 1 B L =  1 + λ 2 1 − λ 2  I N − λ 2 1 − λ 2  I N + 1 σ 2 QP H 1 P 1  C 2 = 1 σ 2 h ′ 2 H P H 2 P 2 h ′ 2 − k H 2 M 2 k 2 + 1 σ 2 h ′ 1 H P H 1 P 1 h ′ 1 − h ′ 1 H  I N + 1 − λ 2 σ 2 QP H 1 P 1  − 1 1 σ 2 P H 1 P 1 h ′ 1 (45) The final form ulas ( 42) and (45) are interesting in th e sense th at they do not direc tly carry any intuitive prope rties. I ndeed, if we were to find an a d-hoc techniq ue th at is to po nder the relative importan ce of our p rior info rmation on h 2 , of the p ilot data h ′ 2 and o f the past (or f uture) pilot data h ′ 1 , we would sugg est a lin ear com bination o f those c onstraints. O ur result is n ot linear in those constraints. Howe ver it c arries the expected intuition in the limits, • when λ = 0 , then the past an d present ch annels are completely uncorrelated so that no in formation carried by the past pilots sho uld b e of any use. This is what is o bserved since then, equa tion (42) reduce s to LMMSE solution (15). • when λ → 1 , then ˆ h 2 =  I N + 1 σ 2 QP H 1 P 1 + 1 σ 2 QP H 2 P 2  − 1 Q  1 σ 2 P H 2 P 2 h ′ 2 + 1 σ 2 P H 1 P 1 h ′ 1  (46) which is a gain th e sam e equ ation as (42) but n ow the past and present pilots h ′ 1 and h ′ 2 can be co mpiled into a sing le pilot seq uence h ′′ 2 with the projector R H 2 R 2 = P H 1 P 1 + P H 2 P 2 , ˆ h 2 =  I N + 1 σ 2 QR H 2 R 2  − 1 Q  1 σ 2 R H 2 R 2 h ′′ 2  (47) Note also that (42) is linear in the variables h ′ 1 and h ′ 2 , so that th e final MMSE so lution when L is kn own is also th e LMMSE so lution. D. T ime-fr eq uency Channe l Estimation Now , instead o f merely estimating channels at times when pilot sequences are fou nd, we can extend our scheme to estimate channels at any time position. For th is, we shall in the following consider a c hannel h 12 that we want to estimate, given the knowledge of p ilot signals h ′ 1 and h ′ 2 found at positions o f the r espectiv e h 1 and h 2 channels. W e are also aware of λ 1 and λ 2 , the r espectiv e time-cor relation coefficients between the coup les ( h 1 , h 12 ) an d ( h 2 , h 12 ) . W e assume for br evity h ere that the channel length L is known (this will avoid the heavy compu tation o f some c oefficients). Using th e same deriv ations as in the previous sections, the MMSE estimation for h 12 is given by ˆ h 12 = Z h 12 P ( h 12 ) ·  R h 2 P ( h ′ 2 | h 2 ) P ( h 2 | h 12 ) d h 2  ·  R h 1 P ( h ′ 1 | h 1 ) P ( h 1 | h 12 ) d h 1  P ( h ′ 1 h ′ 2 ) d h 12 (48) W e d o no t provid e th e complete d eriv ation, which is identical in spirit as all the previous der i vations to fin ally ob tain, ˆ h 12 =  (1 + λ 2 1 1 − λ 2 1 + λ 2 2 1 − λ 2 2 ) I N − λ 2 1 1 − λ 2 1 ( I N + 1 − λ 2 1 σ 2 QP H 1 P 1 ) − 1 − λ 2 2 1 − λ 2 2 ( I N + 1 − λ 2 2 σ 2 QP H 2 P 2 ) − 1  − 1 Q ×  λ 1 ( I N + 1 − λ 2 1 σ 2 QP H 1 P 1 ) − 1 1 σ 2 P H 1 P 1 h ′ 1 + λ 2 ( I N + 1 − λ 2 2 σ 2 QP H 2 P 2 ) − 1 1 σ 2 P H 2 P 2 h ′ 2  (49) which generalizes eq uation (4 2). This is further gener alized fo r a giv en nu mber K of pilo t sign als sequences h ′ k , k ∈ { 1 , . . . , K } , ( sent throug h chann el h k ) and a chann el h ( with in verse Fourier transfo rm ν ) wh ich satisfies ∀ ( i, k ) ∈ { 1 , . . . , L } × { 1 , . . . , K } , E[ ν i,t ν ∗ i,t + k ] = λ k L (50) The m aximum e ntropy prin ciple in this situation gives the M MSE estimator for h , ˆ h = (1 + K X k =1 λ 2 k 1 − λ 2 k ) I N − K X k =1 λ 2 k 1 − λ 2 k ( I N + 1 − λ 2 k σ 2 QP H k P k ) − 1 ! − 1 Q × K X k =1 λ k ( I N + 1 − λ 2 k σ 2 QP H k P k ) − 1 1 σ 2 P H k P k h ′ k ! (51) E. Un known correlation factor λ The r eader migh t object at this p oint th at the prior knowledge either of the vehic ular speed or of the mean correlation factor λ might only be accessible th rough yet ano ther estimatio n pr ocess. Mo reover λ expresses as an expectation so th at possibly some tim e is re quired to hav e an accur ate estimation . This of course goes against our idea of fast chann el estimation. In the same trend as we did p reviously with the possibly un known parameter L , we can equ ally integer out the parameter λ f rom our formu las. In general th is m ight b e a n un easy task, since the n the estimated ch annel ˆ h would re ad, ˆ h = Z h h P ( h | h ′ 1 , . . . , h ′ K )d h (52) = Z h  Z λ h P ( h | h ′ 1 , . . . , h ′ K , λ ) P ( λ ) dλ  d h (53) in which P ( λ ) is o ur p rior knowledge on the parame ter λ . But, going furth er in the compu tation, this last integration is rather in volved. It cou ld then be well app roximated b y the finite summ ation, ˆ h ≃ Z h λ max X λ = λ min h P ( h | h ′ 1 , . . . , h ′ K , λ ) P ( λ ) ! d h (54) = λ max X λ = λ min P ( λ ) Z h h P ( h | h ′ 1 , . . . , h ′ K , λ ) d h (55) As shall be illustrated in Section IV, it actually makes almo st no difference to assume that λ is or is not k nown. T his suggests tha t ou r estimators are ab le themselves to co pe with the lack of inform ation concernin g λ , just by in ductive inf erence on λ from the data h ′ 1 , . . . , h ′ K . F . Non -homogeneous S N R In a ll p revious section s, we consider ed an homog eneous noise power σ 2 over the fr equency bandwidth. This situation is usually far from re al ( and of ten far from the actual k nowledge the r eceiv er has on the noise correlation matrix ). T ypically , in the presence of stro ng interf erers working on selective fr equencies, the no ise correlation matrix C n = E[ nn H ] is n ot prop ortional to an identity matrix (and has even n o particular tendency to be diago nal). If the information about the noise is updated to consider C n , then all the previous equation s are to be upd ated by replacin g all terms 1 σ 2 P H P by the corrected terms P H C n P . 2 8 14 20 26 32 0 0 . 2 0 . 4 0 . 6 0 . 8 1 1 . 2 Sub c a rriers Channel energy Real c hannel Ba y e sia n estimate LMMSE estimate Pilot p ositions Fig. 2. Channel estimated ener gy - N = 32 , L = 3 , L max = 6 , SNR = 20 dB I V . S I M U L A T I O N S A N D R E S U LT S In this sectio n, we p ropose simula tions of the p reviously derived f ormulas. In or der to pr oduce insigh tful p lots, we con sider a shor t discrete Fourier transform of size N = 3 2 . In a first simulation, we assume the chann el length L = 3 is unkn own to the rec ei ver that o nly knows L ∈ { 1 , . . . , 6 } . The pilot symbols ar e separated by 6 subcarr iers a s in the 3 GPP-L TE stand ard [2] and de picted in Figu re 1. The SNR is set to SNR = 20 dB . The results are propo sed in Fig ure 2 tha t comp ares the novel Bayesian MM SE solution to th e classical LMMSE solution assuming m aximum ch annel leng th ˜ L = L max = 6 . It is observed that o ur solution has b etter results th an LMMSE in th is particu lar c ase, du e to the prio r error in ˜ L th at does n ot occur in th is novel scheme. A co mparison betwee n the perfo rmance of our Bayesian MMSE e stimator when th e cha nnel leng th L is k nown or is unknown is then prop osed in Figure 3. W e take here a chann el length L = 5 that is known to belon g to the ran ge { 1 , . . . , 1 0 } . Interestingly , the perfo rmance de cay due to the a bsence of k nowledge in L is not large. If we co nsider the p revious situation , i.e. L = 3 for a ran ge { 1 , . . . , 6 } , we even visu ally ob serve no p erform ance difference. This raises a very in teresting featu re o f the indu cti ve reasoning framew ork since, by tr ying to infer on the ch annel k nowledge giv en the re ceiv ed signal ( h | y ) , th e Bayesian f ramework also enco mpasses infer ence on L . Ind eed, P ( L | y ) = P ( L ) P ( y | L ) P ( y ) (56) = P ( L ) P ( y ) Z h P ( y | h ) P ( h | L )d h (57) in which the integral is the same has in Section II I-A and P ( L ) is the unifo rm prior distribution for L . The indu cti ve infer ence on e very hypothe sis L = 1 , . . . , L max can then be com pared thanks to the evidence fun ction defined by Jaynes [ 13] which reads e ( L | y ) = log 10 P ( L | y ) P l 6 = L P ( l | y ) ! (58) The r esults, for different SNR are pr oposed in Figur e 5. Howe ver , since th e evidence for any hy pothesis on L is no t large, we instead draw the odd s fun ction O ( L | y ) = P ( L | y ) P l 6 = L P ( l | y ) ! (59) which is ten to the p ower of the evidenc e function . W e observe, as pred icted, that evidence for L = 5 is raised hig her than th e other h ypothesis and that th is behaviour is especially noticeab le when the SNR is hig h. This p roduces an u pdated posterior distribution P ( L | y ) tha t a lmost discards 2 8 14 20 26 32 0 0 . 2 0 . 4 0 . 6 0 . 8 1 1 . 2 1 . 4 Sub c a rriers Channel energy Real c hannel L unkno wn L kno wn Pilot p ositions Fig. 3. Channel estimated ener gy - N = 32 , L = 5 , L max = 10 , SNR = 20 dB 0 2 4 6 8 10 12 14 16 18 20 10 − 3 10 − 2 10 − 1 SNR [dB] Mean Square Error L tested = { 6 } L tested = { 1 , . . . , 9 } L tested = { 2 , 5 , 8 } Fig. 4. Mean square error of channel estimati on when L is unkno wn - SNR = 20 dB , N = 128 , L = 6 all wrong hy pothesis. T herefor e, at hig h SNR, the impact of th e hypo thesis L 6 = 5 in the final equatio ns is n egligible. Th is automatic infere nce on the chan nel leng th is a direct co nsequenc e of the pr oposed MMSE form ula that would no t be possible throug h classical orthod ox proba bility approach es. W e want in the following to observe first the effects of using time correlation p roperties bef ore dealing with performa nce figures. In Figures 7 and 8 , we propo se th e situation of two pilot sequences c orrespon ding to two corre lated chann els with correlation factor λ = 0 . 99 . W e estimate here one of the two ch annels either using both p ilot seque nces. It is ob served that the high co rrelation λ allows to perfor m the estimation of long chann els. Indeed , the h igh den sity of p ilots in the time-f requency grid allows to better a pproach the genuine ch annel. Th is is shown in Figur e 7 on a rand om chan nel, for N = 32 , L = 6 , SNR = 20 dB , which clear ly illustrates that using bo th pilo t signals incr eases the accu racy of th e estimator . Another go od effect is o bserved wh en th e noise power σ 2 is strong (or equ i valently wh en the SNR is lo w). Then , using twice as ma ny sam ples for chan nel estimatio n leads effectiv ely to twice as accu rate channel estimation pr ovided that b oth 1 2 3 4 5 6 7 8 9 10 0 0 . 2 0 . 4 0 . 6 0 . 8 1 1 . 2 1 . 4 1 . 6 1 . 8 2 Hyp othetic c hannel lengths L hyp Evidence O ( L hyp ) for L hyp SNR = 20 dB SNR = 10 dB SNR = 0 dB Fig. 5. Channel length inferenc e O ( L ) = P ( L | y ) P l 6 = L P ( l | y ) for differe nt SNR - N = 32 , L = 5 , L max = 10 channels are strongly correlated . This is demon strated in Figure 8 in which we used SNR = 10 dB . The c orrespon ding nu merical perf ormance are p roposed in Figu res 9 and 10 respectively . In the f ormer, the ch annel length is L = 5 for a DFT -size N = 64 . A single train of pilo ts is the n en ough to estimate the chan nel. Howe ver , as p reviously discussed, in the presence of highly time-cor related channels, the estimation noise can be redu ced using both train s of pilo ts. Essentially , it is obser ved here that, since the numb er o f pilots for both sequences is almost equal, the channel estimation based on b oth tr ains of pilo ts is realized on twice as many pilot position s. T herefore , when the time-co rrelation λ is high, for low SNR, u p to 3 dB gain can be obser ved. As f or lo ng chan nels, it is clear in Figu re 10 that high cor relation be tween chann els in time is dem anded so to perfor m accurate channel e stimates. Indeed, w hile not using tim e-adjacent channels p rove dev astating ( over 6 dB o f mean square e rror), high time-cor relations allows to sign ificantly reduce th e mean square er ror . Finally , in Figure 1 1, we p ropose to compa re the p erform ance of a ch annel estimator using two p ilot sequence s correla ted in time, p rovided that the process kn ows or d oes not know the exact time-correlatio n coefficient λ , set h ere to λ = 93 % . The channel len gth L = 15 is known and the DFT size is kept to N = 64 . Surprisin gly , a poor prior knowledge o n λ does not stro ngly imp act the fin al mean quadr atic error o f the estimation. At least, wh en the cor relation is kn own to be mor e than 1 / 2 , the perfo rmance in terms of mean squ are error is similar to th at when λ is perfectly known. T his suggests, as alread y mentioned , that the Bay esian mac hinery is ab le to efficiently infer on λ whatever th e SNR lev el. The reader must be wary that this last senten ce do es not imply at all that th e time-corr elation p arameter do es not intervene in th e system per forman ce. What we stated above is just that p rior k nowledge abou t th is parameter is not m andatory since the posterio r distribution for λ (given h ′ 1 , h ′ 2 ) is alread y very peaky a round the co rrect value for λ . V . D I S C U S S I O N In th e following, we d iscuss th e advantages of the gene ral framework w hich encom passes the previous chan nel estimators. Some limitations, co ncerning co mplexity mainly , are also con sidered. Finally we discuss the po tential dr awbacks in using alternative ch annel estimation techn iques. Those channel estimatio n algorith ms were p roposed on the sole basis of Jaynes’ probab ility theo ry [13] that m ainly encomp asses the Bayesian ru le and the maximu m en tropy p rinciple 1 . Those rules can b e applied to larger problems than the m ere scope of chan nel estimation. For in stance, o ptimal Bayesian signal detection is p roposed in [20]. Also, maximum entropy ch annel m odeling are derived in [21]. All tho se stud ies lead to the gene ral idea o f cogn itive r eceivers . Indeed , a few years after the in troductio n of the concept of cognitive rad ios [22], some attempts have been propo sed to clearly define the fundam ental b asis of a cognitive radio [2 3] but still no cor rect definition has been derived. Th is has the rath er unp leasant consequen ce to see m any contr ibutions on cognitive tech niques, each based on very d ifferent fundame ntal assumptions. In o ur 1 the Bayesia n rule has actua lly been prov en to be a partic ular case of the genera lized ME principl e [14] 0 5 10 15 20 25 30 10 1 10 2 10 3 SNR [dB] O (6) / max k 6 =6 O ( k ) Sim ulation P oin ts Linear Fitting Fig. 6. Mean ev idence gap of channel length reco ver y - SNR = 20 dB , N = 128 , L = 6 , L tested ∈ { 1 , . . . , 9 } minds, pr obability theor y as extend ed logic is an in teresting cand idate in the info rmation theo retic definition of a co gnitive receiver . By the latter, we mean a re cei ving termina l which, gi ven any am ount of information I is able to optimally infer on any system par ameter . This way the d e vice would be cap able of learning. Having done th at, it will then have to take decision s, i.e. what informatio n to send ba ck with which acc uracy , wh at additiona l informa tion to req uest etc., so to m aximize some utility fu nction. Howe ver , this second require ment fo r a cogn iti ve re ceiv er goes beyo nd the mere scope of Jayne s’ pro bability theory an d in volves decision theoretic d iscussions as well as ep istemological considera tions such as th e brilliant inquiry theo ry from Cox [15] and Kn uth [ 16]. This being said , th e reader will th en raise the objection that our current work obviou sly did n ot consider all the info rmation av ailable to the receiver . Ind eed, as previously mentioned , if pilots we re designed to help c hannel estimation th en infor mativ e data 2 also carr y inform ation o n the chan nel they face. W e co uld mentally envision a joint chann el e stimator and sign al decoder device tha t would help recover a better estima te on th e chann el and on the sent signal. But of course, at this poin t, we would face stro ng implem entation difficulties as well pro bably as inv olved mathematical p roblems. Still, throug h this study we suggest that in g eneral ad-ho c solution s ar e not the directio n to head for when one wishes to have both g ood performan ce and non-inv olved algorithm s. I f on e wishes to reduce th e comp utational load of an algo rithm, it is prefer able to develop the full Bayesian solu tion and only then start to simplify the mathe matical expressions. For instanc e, in our pr evious examples, when the channel length L is un known but the com plexity of summin g up hundr eds of poten tials can didates for L is to o large, a solution that o nly considers ten sample d v alues for L co uld be envisioned. This is the cor rect way to keep a gr asp on what simplification we perform ed; in classical techniq ues, f rom the very b eginning, stro ng assum ptions and approx imations are made which effects are often invisible in terms o f perfo rmance (an d mig ht only be o bservable throu gh extensive simulations). One of tho se classical tech niques in the chann el estimation realm is propo sed, among others, in [1 9]. The basic idea is very insightful since it consists in estimating present chann els based on the k nowledge of the estimation d one on a p ast chann el a nd the time- correlation coefficient that link b oth chann els. Such solution s might seem interesting in the fact that, by recur sion, the previous estimate “car ries the in formation on all past pilots signals” but this is actually very deluding. Indeed , this estimate does n ot actu ally contain th e whole inform ation o n th e previous pilot sign als. It merely c onsists in some post-filtering result of th ose pilots sig nals. W e migh t then raise a few ob jections to using pr e vious estimates, • if we were to “select” information 3 , then we would better want to conside r some o f the past (an d possibly f uture) pilo t signals than previous estimates • if on ly the previous estimates are av ailable, then it would seem disho nest not to mentio n to the Bayesian machine ry that those actu ally ar e estimates of a ch annel. This suggests th at, when using th ose previous estimates, it mu st be somehow mentioned in th e equ ations th at they are MMSE or least-squa re [5] estimates of the chan nel, th at orig inated from pilots 2 by informat i ve, we suggest here data dedicate d to communication purposes and not synchronizati on purposes 3 which is dishone st according to Jaynes’ theory fundament al desiderata [13] 1 4 7 10 13 16 19 22 25 28 31 0 0 . 2 0 . 4 0 . 6 0 . 8 1 1 . 2 Sub c a rriers Channel energy Real c hannel Time correlation LMMSE lo cal Pilots 1 Pilots 2 Fig. 7. Channel estimated ener gy with time correlati on - 99% correlatio n, N = 32 , L = 6 , SNR = 20 dB 1 4 7 10 13 16 19 22 25 28 31 0 0 . 2 0 . 4 0 . 6 0 . 8 1 1 . 2 1 . 4 Sub c a rriers Channel energy Real c h a nn el Time correlation LMMSE lo cal Pilots 1 Pilots 2 Fig. 8. Channel estimated ener gy with time correlati on - 99% correlatio n, N = 32 , L = 3 , SNR = 10 dB sequences pr esent (if this is also known) at particula r p ositions and so on an d so fo rth. This would allow for the Bay esian process to provide indu ctiv e in ference on the actu al pilot sig nals and then make th e similar derivations as tho se we have propo sed along th is p aper . Any oth er usag e of previous estimates cou ld n ot be claimed op timal . V I . C O N C L U S I O N In th is work, a novel view on cha nnel estimation for OFDM systems is prop osed. Optimal f ormulas for the mea n square error criterion are der iv ed tha t r e-demon strate known classical solutions while new f ormulas are a lso p roposed for scena rios based on different levels of knowledge. T he who le work can b e su mmarized as a uniqu e novel framework that allows to in tegrate any info rmation th e receiver is aw are of to perform optimal chann el infer ence b ased on this k nowledge. Also, some h ints on the long- term introduc tion of f oundatio ns f or co gnitive receivers are suggested that would encomp ass the pr esent work. 0 5 10 15 20 25 30 − 35 − 30 − 25 − 20 − 15 − 10 − 5 SNR [dB] Mean Square Error [dB] λ = 1 − 10 − 2 λ = 1 − 10 − 4 λ = 1 − 10 − 7 λ = 0 Fig. 9. Estimation Mean Square Error - Short channel - N = 64 , L = 5 0 5 10 15 20 25 30 − 30 − 25 − 20 − 15 − 10 − 5 0 5 SNR [dB] Mean Square Error [dB] λ = 1 − 10 − 2 λ = 1 − 10 − 4 λ = 1 − 10 − 7 λ = 0 Fig. 10. Es timati on Mean Square E rror - Long channel - N = 64 , L = 15 R E F E R E N C E S [1] http://wirel essman.org/ [2] S. Sesia, I. T oufik and M. Baker , “L T E, The UMTS Long T erm Evol ution: From T heory to Practice” , W ile y & S ons, 2008. [3] J. A. Bingham, “Multicar rier m odulati on for data transmission: An idea whose time has come, ” IEEE Commun. Mag., vol. 28, pp. 5-14, May 1990. [4] S. Coleri, M. E rgen, A. Puri, and A. 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