Cognitive OFDM network sensing: a free probability approach

1 Cogniti v e OFDM network sensing: a free probability approach Romain Couillet ST -NXP W ireless - Supelec 505 Route des Lucioles 06560 Soph ia Anti polis, Franc e Email: romain.couillet@nxp.com M ´ erouane Debbah Alcatel-Lucent Chair , Supe lec Plateau de Moulon, 3 rue Joliot-Curie 91192 Gif sur Yvette, France Email: merouan e.debba h@supe lec.fr Abstract — In th is paper , a practical power detection scheme fo r OFDM terminals, based on recent fre e probability tools, is proposed. The objective is fo r the receiving terminal to determine the transmission power and the nu mber of the surrounding base stations in the network. Howev er , the system dimensions of the network model turn energ y detection in to an under-determined problem. The focus of this p aper is t hen twof old: (i ) discuss th e maximum amount of in for mation th at an OFDM terminal can gather from the surrounding base stations in the network, (ii) propose a practical solution for blind cell detection using th e free decon volution tool. The efficiency of this solution is measured through simulations, which show better performance than the classical p ower detection methods. I . I N T RO D U C T I O N The ever increasing deman d of hig h d ata rate has pushe d system designers to exploit the wireless channel medium to the smallest granular ity . In this respect, the ortho gonal frequ ency division multiplexing (OFDM) m odulation h as been c hosen as the next com mon standard in mo st wireless communication systems, e.g. WiMax [5], 3GPP-L TE [4 ]. OFDM converts a frequen cy selecti ve fading channel in to a set of flat fad- ing chann els [ 20], therefo re pr oviding a high flexibility in terms of p ower an d rate allocatio n. Future wireless networks therefor e tend to be based on highly load ed OFDM cells. Howe ver, in multiple cell en v ironme nts, inte r-cell in terferen ce is still the bottleneck factor whic h con siderably re duces th e network-wide capacity . Cooperation between base stations are envisioned to reach the capacity perfor mance o f the so- called br oa dcast chann el [13], but many problems (essen- tially of power allocation and user scheduling ) prevent those solutions to appe ar soon in practical standar ds. The refore, it is essential fo r mobile termin als to be ab le to d etermine which n eighbo ring cell provides the best quality of serv ice, so th at the termin al quick ly hands over this best perfor mance base station. Classically , only scarce and n arrow-band pilot sequences allow the ter minals to estimate the tra nsmission power o f the main surrou nding base stations, e.g. in 3 GPP- L TE, two sequen ces of the 0 . 7 MHz ba nd are available every 5 ms . Those synchr onization sequences are usually affected by fast channel fading and overlap data fr om other base stations; as a consequen ce numer ous occu rrences of those pilots need be accumulated to achieve a satisfying estimation of the base stations transmission p ower . The classical alterna ti ve to the pilot-aid ed ( also referred to as data-a ided ) p ower detection is to p erform a blind estimation fro m the in coming interferin g signals. This raises the fund amental cognitive radio que stion [2 3], [ 12], wh ich will be a n importan t top ic of th e presen t work: “ho w mu ch informa tion can a cogniti ve recei ver recover from the incoming signals?”. Th e response to th is q uestion answers two clas- sical concerns of eng ineers a nd system designers: (i) is the additional information brou ght by blind detection worth the computatio nal effort?, (ii) i s some gi ven blind detector solution far from providin g all the accessible informatio n?. It is clear in par ticular, from an in formatio n th eoretic v iewpoint, that the information rece i ved on the N OFDM subcarriers mu st ideally not be filtere d in order to pr ovide as much inform ation as p ossible on the problem a t hand, i. e. any filtering pro cess diminishes the a vailable inform ation in the Shannon’ s sense [1]. Theref ore, if as many as L consecutive OFDM symbols are received, the available inf ormation is c ontained in the received N × L matrix Y , with N typ ically large. As a consequen ce, since L cannot b e taken infin itely large, N / L is non tri vial. This leads to the study of large ran dom matrices problem s, which is currently a hot to pic in the wir eless commun ication co mmun ity [ 8]. This is in sharp con trast with classical p ower detection meth ods [10], [11] which are only asymptotically unb iased, i. e. these meth ods assume that on e of th e system d imensions is large with respec t to the others and this co ndition is n ecessary to en sure the c onv ergence of the u nderly ing algorith ms. Our purpo se is to r etrieve r elev an t infor mation on the base station transmission powers. It will be shown herea fter th at, depend ing on th e a priori knowledge of the receiv er , the essential part of the p ower inf ormation is, in most pr actical situations, contained in the eigenvalue distribution of the 2 matrix 1 L YY H . This naturally leads to the consideration of recent r esearch on random matrix theory (RMT) [8] and more specifically o n fre e deconv olution [1 6]. In particular, in [7], a similar stud y of terminal p ower detection in code division multiple access (CDMA) networks is derived from these to ols. Howe ver, the mo del in [7] only c onsiders flat fading ch annels and dod ges the difficulty of multi-p ath chann els; m oreover the structure of the CDMA encod ing matrix allows to easily recover the tr ansmitted signal variances, which is not the case of m ulti-cell OFDM in which m ultiple streams overlap with no dedicated code to separate them. W e p ropo se here first to discuss the o ptimal amou nt of informa tion which th e receiver can extract from the incoming data to blind ly retrieve the values of the powers transmitted b y all surround ing base stations, wh en the rece i ver’ s prior state of knowledge about the environment is very limited. From th is analysis, it will be observed th at the general problem is not very tractab le b oth in term s o f mathematical derivation an d therefor e in ter ms of practical implem entation. Seco ndly , we propo se a suboptimal but im plementable approa ch to so lve the cell detection prob lem, ba sed o n the free pr obability framework, fo r which we derive a novel sig nal detectio n algorithm for OFDM. The re mainder of the pape r is structu red as fo llows: In Section II, we in troduce the mod el of the mu ltiple cell OFDM network. I n Section III, we discuss the amount of infor mation about tran smitted p owers which can be collected by the terminal from the received matrix Y . The ob servation that all the n ecessary inform ation is contained in the eig en values of 1 L YY H leads to Section I V, in which we evaluate the classical energy detection meth ods, which assume L/ N → ∞ . In Sectio n V, we intr oduce some basic concep ts of random matrix theo ry [8], which a re n eeded to the u nderstand ing o f the subsequent sections. In Sec tion VI, we provide a novel algorithm to detect th e cell transmission powers. Simu lation results are th en provided in Section VII. A d iscussion on th e gains an d limitations of this n ovel meth od is carried out in Section VIII. Finally , in Section I X we draw our conclusion s. Notations: In the following, bo ldface lower case sym bols represent vector s, capital bold face characters den ote matrices ( I N is the size- N identity ma trix). The spaces M ( A , i, j ) and M ( A , i ) are the sets of i × j and i × i matr ices over the algebra A , respectively . The transpose and Herm itian transpose operator s are den oted ( · ) T and ( · ) H , respectively . The oper ator diag( x ) tur ns the v ector x in to a diagonal matrix. The function 1 ( · ) denotes th e indicator function . I I . D O W N L I N K M O D E L Consider a network of N B base stations and on e terminal equippe d with a single receiving antenn a. Th is scena rio is depicted in Figure 1. Assume m oreover th at the receiver is already connected to o ne serving base station , which we purpo sely exclude fr om the set of the N B surroun ding base stations since we already kn ow all informa tion abou t it. The network is assumed to be sync hronized in time and fr equency and to u se OFDM m odulation with a size N discr ete Fourier transform (DFT). Denote then M the m aximum number of P S f r a g r e p l a c e m e n t s s ( l ) 1 s ( l ) 2 s ( l ) 3 s ( l ) M Fig. 1. System Model base stations the receiver expects to de tect. Ideally N B ≤ M . In the following, for simplification, we assume that this condition is always fulfilled and we will only co nsider the parameter M , consider ing then a n etwork of M base stations, of which some co uld be of n ull power . Th e lin k between the terminal and th e base station k is modelled as a fast-fading complex f requen cy domain chann el h k = [ h k 1 . . . h kN ] T ∈ C N , coupled to a slow-fading path loss L k = 1 /P k with P k the mean receiv ed p ower origin ating from base station k . Th e terminal also receives additi ve white Gau ssian noise σ n ∈ C N with e ntries of variance σ 2 . T he base station k sends at time l the frequ ency-dom ain OFDM symb ol s ( l ) k = [ s ( l ) 1 k , . . . , s ( l ) N k ] T . Therefo re, the recei ved signal vector y ( l ) = [ y ( l ) 1 , . . . , y ( l ) N ] T at time l read s y ( l ) = M − 1 X k =0 P 1 2 k D k s ( l ) k + σ n ( l ) (1) with D k = dia g ( h k ) . This sum mation over the M cells can be rewritten y ( l ) = HP 1 2 θ ( l ) + σ n ( l ) (2) with θ ( l ) ∈ C M N the concatenated vector θ ( l ) = [ s ( l ) 1 , . . . , s ( l ) M ] T , H ∈ M ( C , N , M N ) the conc atenated m atrix of the D k , k ∈ { 1 , . . . , M } H =    h 11 · · · 0 · · · h M 1 · · · 0 . . . . . . . . . · · · . . . . . . . . . 0 · · · h 2 N · · · 0 · · · h M N    (3) and P ∈ M ( R , N ) th e diagon al matrix P =       P 1 0 · · · 0 0 P 2 . . . . . . . . . . . . . . . 0 0 · · · 0 P M       ⊗ I N (4) Now assum e th at the M ch annels ha ve a coher ence tim e of order (or more than) L times the OFDM symbol duration. The L samples y ( l ) , l = 1 , . . . , L , c an be concatenated into an N × L matrix Y = [ y ( 1 ) · · · y ( L ) ] to lead to the more general matrix expression of the r eceiv ed sign al Y Y = HP 1 2 Θ + σ N (5) where Θ ∈ M ( C , M N , L ) and N ∈ M ( C , N , L ) are the con- catenation matrices of th e L vectors θ ( l ) and n ( l ) , respectively . 3 I I I . P RO B L E M S TA T E M E N T The power detection pro blem consists in th e pre sent situa- tion to retrie ve the en tries of th e ma trix P fr om the rece iv e matrix Y . T he prior information at the receiver is no t s ufficient to find P in a straigh tforward manner . Inde ed, the chann el matrix H and the tr ansmitted signal Θ are un known and, worse, e ven th eir stochastic distribution are usually unkn own; in p articular, rec ent flexible multiple acce ss OFDM standar ds adapt their transmission rates to the channel quality so that the termina l ca nnot a priori assume to receive eithe r QPSK, 16 -QAM, 6 4 -QAM or any other type of modu lation. The a priori kn owledge I at the terminal is therefore limited to: ( i) the appr oximated back groun d covariance E[ nn H ] = σ 2 I , (ii) the fast-fading chan nel p ower E[ h H k h k ] = 1 , (iii) th e chan nel delay spread known to be lesser than th e cyclic prefix len gth, (iv) the transmitted signal covariance E[ θ H k θ k ] = I N . From this amou nt of prio r inform ation I , the most reli- able chan nel model 1 is obtain ed from the ma ximum entropy principle [14]. The latter states that the transmitted data Θ must be mode lled as a Ga ussian indep endent an d identically distributed process (i.i.d. ). As fo r the sh ort-term chann els h k , giv en a delay spread τ d (counted in integer n umber of time- domain samples), the time- domain representatio n of h k must be mo delled as a Gau ssian i.i.d. vector of length τ d ; ther efore, h k is to b e modelled as a Gaussian vector with cov ariance matrix the DFT of I τ d . Since little information about τ d is initially kn own to the receiver , the chan nels must be modelled as the marginal distribution of those Ga ussian p rocesses with τ d varying from 1 to the cyclic prefix length. Of course, this mo del migh t be very different fr om r eality and mig ht provide totally wrong results, as lon gly discussed in [1 5]. Howev er , this is the best on e can b lindly infer on the transmission scheme from the av ailable informa tion. The objective now is to determine what is the proba bility p ( P | I ) that a sequence of transmitted powers { P 1 , . . . , P M } fits the previous model kn owing I . From tho se probabilities, computed for all vectors in ( R + ) M , an estimate ˆ P of P can be designed which minimizes some erro r measure, e .g. ˆ P = E[ P | Y , I ] would be the min imum mean square erro r estimate of P . The prob ability p ( P | Y , I ) assigned to the info rmation ( P | Y , I ) can be written, thanks to Bayes’ rule p ( P | Y , I ) = p ( Y | P , I ) · p ( P | I ) p ( Y | I ) (6) in which ( P | I ) is th e a priori knowledge a bout P . It is classically assign ed a u niform d istribution over so me sub- space [0 , P max ] M for a m aximum rec eiv e power P max . As fo r p ( Y | P , I ) , it can b e expanded as p ( Y | P , I ) = Z Θ , H p ( Y | P , H , Θ , I ) p ( H | I ) p ( Θ | I )d H d Θ (7) in which all integrands are known from the max imum entropy model aforem entioned , 1 by “most reliable model”, we mean the model which satisfies the con- straints imposed by I and which is the most noncommittal regardi ng unknown system informati on. • p ( Θ | I ) is standard mu ltiv ariate i.i.d. Gaussian. • the compou nd chann el H is assigned a distribution p ( H | I ) = τ max X k =1 p ( H | τ d , I ) · p ( τ d | I ) (8) with τ d the channel delay spread, τ max the cyclic pr efix length, p ( τ d | I ) = 1 /τ max and p ( H | τ d , I ) with stan dard i.i.d. Gaussian diagonals. Howe ver, th e explicit com putation of ( 7) is very in volved and req uires advanced tools from ran dom matrix theor y . A similar calculus was p erform ed by the autho rs f or the simpler single-cell MIMO energy detector [31]. In the latter it was shown that, surprisingly , the standard i.i.d . Gau ssian model assigned to the main system pa rameters m akes the energy detection depen d only on the eig en value distribution of the receive ma trix Y . The multi-cell detection problem at han d is very similar in configu ration, a part for the chan nel marginal- ization of equa tion (8) whic h is not i.i.d. Gaussian. Since we cannot provide an optimal info rmation theoretical solu tion to our problem and since both aforem entioned pr oblems are very similar , it seems relev an t to concentrate on the close-to-o ptimal random m atrix theoretical appro ach. Some important infor mation can noneth eless be alre ady deduced from the integral form of equation (7). If the tran s- mission ch annels are extrem ely freq uency flat, i.e . for all k , h k 1 ≃ h k 2 . . . ≃ h kN , then { HP 1 2 Θ } ij = P k √ P k h k θ kj . Therefo re, even if the realizations o f N a nd Θ wer e perfectly known, o ne will have access at best to the variables √ P k h k , k = 1 , . . . , M , from which no reliable estimation of P k be drawn; in such a situation, the p osterior probab ility p ( P | Y , I ) is very b road and is m aximized on a large continuo us set of P 1 , . . . , P M . On an inf ormation theoretical viewpoint, this means th at the o ptimal in ference on P given Y and I cannot lead to any valuable information . In the random m atrix approa ch, the situatio n is even worse. If on e knew perf ectly the entries of HPH H , then nothin g at all can b e said ab out P 1 , . . . , P M . Indeed , { HPH H } ij = P k P k | h k | 2 (see Section VI fo r details) and the o nly piec e of informa tion which on e has about P 1 , . . . , P M is the sum P k P k | h k | 2 ; the l atter cannot lead to any estimate o f P when M > 1 an d the problem cannot be solved. This means that, given the limited prior informa tion of the terminal, it is impossible to come up with a reliable estimate of P when the channe ls are f requen cy flat. In the rem ainder o f this paper , we shall theref ore consider that the OFDM ch annels are very freq uency selecti ve 2 . In the following, we investigate the classical power detection techniqu es, which shall prove inefficient in this large n on-trivial matrix problem . I V . C L A S S I C A L P O W E R D E T E C T I O N Usual p ower detection conside rs the second o rder statistics of th e received signals. I n th e scalar case, i.e. y ( l ) reduces to a single value y ( l ) , it was proved [10] that the o ptimal detector with the aforemention ed state of knowledge at the terminal 2 note that this assumption ensures high effici ency of the network in terms of per -user outage capa city , which is very desirab le in the current trend for pack et-switch ed communications. 4 consists in evaluating 1 L ([ y (1) , . . . , y ( L ) ][ y (1) , . . . , y ( L ) ] H ) − σ 2 , with L > > 1 . W e show in what follows that this classical scheme can be simply extended to our network situation but that it is very in efficient for small L/ N ratios. Assuming that L/ N is very large, the expression of the normalized Gram matrix associated to Y reads YY H L − → L →∞ E  1 L ( HP 1 2 Θ + σ N )( HP 1 2 Θ + σ N ) H  − → L →∞ HP 1 2 E  ΘΘ H L  P 1 2 H H + HP 1 2 L E h ΘN H i + 1 L E h NΘ H i P 1 2 H + E  1 L NN H  − → L →∞ HPH H + σ 2 I N (9) the last line comes from th e f act that, N being finite, the N × N matrices 1 L NN H and 1 L ΘΘ H conv erge in distribution to an identity matrix, and the cross produ cts to null matrices. As a consequen ce, as will be de tailed in Section VI-B, one can estimate the values P k , k ∈ { 1 , . . . , M } from the moments { ( 1 L YY H − σ 2 I N ) k } , k ∈ { 1 , . . . , M } , wh en L is large co mpared with N . Our situa tion does not fall into this asymptotic L / N → ∞ co ntext. In present a nd f uture OFDM techn ologies, the number N of av ailable subcarr iers is large, e.g . of ord er a thousand subcarr iers, while L is limited in o ur model to the channel coh erence tim e or in general to the number of OFDM symbols the terminal is willing to memorize b efore treatin g informa tion. The refore, even if N and L ar e large, their ratio N / L is not in general close to zero. The fund amental asymptotic assumption is therefor e no long er satisfied. W e show in T able I that the non -trivial ratio N/ L im pairs significantly th e p erform ance of the classical power detection. The latter is the result of a simulation in which we applied the algo rithm that will be described in Section VI-B, based on th e sample mo ments o f 1 L ( YY H − σ 2 I N ) ( instead o f the sample mom ents of 1 L HPH H , whose cor respond ing results are shown between b rackets). Th e typical situation consid ered in this examp le is a thre e-base station scenario o f respective powers P 1 = 4 , P 2 = 2 an d P 3 = 1 an d a n oise lev el σ 2 = 0 . 1 , i.e. SNR = 10 dB , N = 2 56 and L is taken in a range from 256 to 3 2 , 7 68 . It turns out indeed that L need s to be large for this method to be satisfyin g. In this pr ecise example, this compels L/ N to be of o rder 6 4 , which is not acceptab le in our current system settings. Such problems inv olv ing large matrices with no n-trivial N /L r atios a re at the hear t o f a re cent field of research , known as rando m matrix theory (RMT), wh ich is a particular case of the more gen eral free pro bability theory intr oduced by V oiculescu [2 2]. In the subsequen t section, we pr ovide a quick introdu ction to impo rtant notions o f RMT which ar e necessary to handle the rest of the multiple cell detection study . V . R M T A N D F R E E D E C O N VO L U T I O N A. Ra ndom ma trix theory Definition 1: A ran dom matrix is a multi-variate ran - dom v ariable X = { X 11 , X 12 , . . . , X M N } f or a given N = 256 , P = { P 1 , P 2 , P 3 } = { 4 , 2 , 1 } L Estimated ˜ P [our algorithm] k P − ˜ P k 2 256 { 7 . 93 , 1 . 62 , − 2 . 5 } [ { 4 . 28 , 1 . 35 , 1 . 35 } ] 27 . 63 512 { 5 . 82 , 2 . 90 , − 1 . 7 } [ { 3 . 80 , 2 . 16 , 1 . 00 } ] 11 . 42 1024 { 4 . 26 , 3 . 60 , − 0 . 8 } [ { 3 . 62 , 2 . 37 , 0 . 94 } ] 5 . 87 2048 { 4 . 52 , 2 . 69 , − 0 . 2 } [ { 4 . 22 , 1 . 55 , 1 . 12 } ] 1 . 77 4096 { 4 . 20 , 2 . 65 , 0 . 18 } [ { 4 . 09 , 2 . 06 , 0 . 78 } ] 1 . 41 8192 { 4 . 10 , 2 . 28 , 0 . 58 } [ { 4 . 05 , 1 . 89 , 0 . 92 } ] 0 . 27 16384 { 3 . 97 , 2 . 42 , 0 . 89 } [ { 3 . 95 , 2 . 24 , 0 . 99 } ] 0 . 19 32768 { 4 . 07 , 1 . 95 , 0 . 99 } [ { 4 . 03 , 1 . 95 , 0 . 98 } ] 0 . 01 T ABL E I C L A S S I C A L M O M E N T - B A S E D M E T H O D ( M , N ) ∈ N 2 couple. As such, X is a m atrix who se entries X ij ∈ C M × N are ruled b y a joint pro bability distribution p ( X 11 , X 12 , . . . , X M N ) . Free pr obability is the study of rando m variables in non- commutative algeb ras, i.e. algebras in which the prod uct operation is non-c ommutative. The algeb ra of large He rmitian random ma trices is a particu lar case of those n on-com mutative algebras. In the f ollowing, we shall qualify fr ee any notion attached to the free prob ability (or RMT) framework while we shall q ualify c lassical any notion attached to th e cla ssical probab ility framework of commuta ti ve algebras. Similarly to the classical theory of pro bability , a free ex- pectation fun ctional φ can be d efined. For a given Her mitian random m atrix X , the free expectation read s φ ( X ) = lim N →∞ E [tr N X ] (10) and we can similarly define free moments m k , k ∈ N of a random m atrix. Those are m k = φ  X k  (11) Thanks to the trace pro perties, note that the free m oments are strongly linked to the eig en values λ i , i ∈ { 1 , N } of X , since m k can also be written m k = lim N →∞ 1 N N X i =1 λ k i (12) Indeed , denote X = Q Λ ΛQ Λ H with Λ = diag( { λ 1 , . . . , λ N } ) an d Q Λ unitary , X k = Q Λ Λ k Q Λ H . T aking the trace of X k leads to equation (12). The asy mptotic ( N , L → ∞ with N /L constant) marginal distribution o f the eigen values of X is called the empirical distribution of X and will b e de noted µ X . Its associated cumulative d istribution fun ction F X reads [8] F X ( λ ) = lim n →∞ 1 N N X i =1 1 ( λ i ≤ λ ) (13) Therefo re the free m oments are directly linked to the empirical d istribution of the matrix X , m k = lim N →∞ Z R + λ k µ X ( λ )d λ (14) which is the classical definitio n of mo ments associated to the distribution µ X . 5 0 0 . 5 1 1 . 5 2 2 . 5 3 0 0 . 2 0 . 4 0 . 6 0 . 8 1 1 . 2 x Densit y µ η c ( x ) c = 0 . 1 c = 0 . 2 c = 0 . 5 Fig. 2. Marchenk o-Pastur la w µ η c Interestingly , for mo st usua l 3 random ma trices A of large dimensions N , L → ∞ with N/L = c constant, the eigenv alu e density of X = 1 L AA H conv erges to a definite distribution. For instance, in our cu rrent p roblem, since the inpu t signals Θ and noise N ar e modelled as stand ard i. i.d. Gaussian, w e are interested in the so-called W ishart matrice s that we define hereafter Definition 2: An N × N ran dom matrix X = AA H , with A a rand om N × L matrix whose co lumns are zero mean Gaussian vectors with covariance matr ix Σ , is called a generalized Wis hart matrix of L degre es of freedom. This is denoted X ∼ W N ( L, Σ) (15) W ishar t matrices W N ( L, I N ) are known to h av e an e igen- value distribution which conv erges, whe n ( N , L ) grows to in- finity with a constant ra tio c = N /L , tow ards the Marchenko- Pastur law µ η c [8]. T he Marchenko-Pastur law is defined by µ η c =  1 − 1 c  + δ ( x ) + p ( x − a ) + ( b − x ) + 2 π cx (16) with ( a, b ) =  (1 − √ c ) 2 , (1 + √ c ) 2  . In Figure 2 we p rovide the distribution of µ η c for d ifferent values of c . Note that when c tends to 0 , i.e. when L/ N → ∞ , the Marchenko- Pastur law con verges to a sing le Dirac in 1 and we recover the classical law o f large num bers. Equiv alently to classical probab ility theory , many results of free pro bability in volve the distribution of sum , difference, produ ct and inverse of ran dom matr ices. The ch aracteristic function of a distribution, used to de riv e the distribution o f the sum of independen t co mmutative ran dom variables, has a free cou nterpar t called the R -Transform. The Mellin trans- form, used to derive the p roduct of indepen dent co mmutative random variables, also has a fre e counterp art, kn own as th e 3 by usual, we qualify matrices found in common wirel ess communication problems. S -Transform 4 . Giv en two large rand om Hermitian m atrices A an d B , whose sum is C = A + B and whose prod uct is D = AB , one can then deri ve the empirical distributions of C and D from th e empirical distributions of A and B , w hich we d enote µ C = µ A ⊞ µ B (17) µ D = µ A ⊠ µ B (18) Equation (17) is called a dditive free con volution and equation (18) is called multip licativ e free conv olution . Similarly , giv en only the d istributions of B , C and D , one can recover the distribution of A µ A = µ C ⊟ µ B (19) µ A = µ D  µ B (20) in which equation (19) is called additive free dec onv olu tion and equation (20) is called multiplicative free d econv olution. B. F ree deco n volution for informatio n plus n oise mo del Our interest is to trea t a particu lar comm unication model, known as the information plus noise model , Definition 3: Given two N × L ( N / L → c ) large r andom matrices R (standing for an informative signal) and X (stand - ing for a n oise ad ditive signal) an d a scalar σ (the standa rd deviation of the noise proc ess), the mo del given by W = 1 L ( R + σ X )( R + σ X ) H (21) is called the information plus noise model. W e shall therefore call W an informa tion plu s noise matrix. It has been re cently shown [7] that the empirical distribution of 1 L RR H in the pr evious definition can be recovered from the empirical distributions of the matrices W an d XX H when X is Gaussian with i.i.d. en tries of zero m ean and v ariance 1 /L . This is giv en by µ 1 L RR H = (( µ W  µ η c ) ⊟ δ σ 2 ) ⊠ µ η c (22) For more details about the demonstration of formula (22), refer to [ 16]. Thanks to th is RMT fr amew ork and th e f ree con volution tools, we can now address ou r multiple cell detectio n problem. V I . A P P L I C A T I O N O F F R E E D E C O N VO L U T I O N T O M U LT I P L E C E L L D E T E C T I O N A. Sig nal an d noise deconvolution In the m odel (5), th e N × N matrix 1 L YY H is an in- formation plu s noise matrix with N a Gaussian ran dom matrix. Th erefore, a ccording to eq uation (2 2), when N , L are sufficiently large, one ca n derive th e empir ical distribution 4 note that independe nce in the classical probability s ense is not enough in free probabilit y to deriv e the empiric al distribu tion of the sum , dif ference , product and in verse product of two random matri ces. This independe nce notion is ext ended to the asymptotic fr eeness concept [8] w hich is not a techni cal issue in our study . 6 µ 1 L HP 1 2 ΘΘ H P 1 2 H H from the emp irical d istribution µ 1 L YY H as follows µ 1 L HP 1 2 ΘΘ H P 1 2 H H =  ( µ 1 L YY H  µ η c ) ⊟ δ σ 2  ⊠ µ η c (23) where c = N / L , for N is of size N × L . Also, the m atrix Θ in equ ation (5) is modelled as standard i.i.d. Gaussian. Therefo re 1 L P 1 2 H H HP 1 2 ΘΘ H is a g eneralized W ishart matrix with covariance P 1 2 H H HP 1 2 . An alogou sly to (15), this can be written 1 L P 1 2 H H HP 1 2 ΘΘ H ∼ W N ( L, P 1 2 H H HP 1 2 ) (24 ) Then, the emp irical distribution of the covariance matrix P 1 2 H H HP 1 2 of the W ishart matrix 1 L P 1 2 H H HP 1 2 ΘΘ H can be recovered from the empirical distribution µ 1 L P 1 2 H H HP 1 2 ΘΘ H when the couple ( N , L ) tend s to infinity with a constant r atio c ′ = M N /L ( M is constant). Consequently , similarly to (20), µ P 1 2 H H HP 1 2 = µ 1 L P 1 2 H H HP 1 2 ΘΘ H  µ η c ′ (25) The left side of equation (23) is slightly different fr om the desired expression in the rig ht side of equation (25). Still, thanks to the trace comm utativity p roperty , we have the link [8] µ 1 L P 1 2 H H HP 1 2 ΘΘ H = 1 M µ 1 L HP 1 2 ΘΘ H P 1 2 H H +  1 − 1 M  δ 0 (26) This relation is due to the fact that the positive eigen- values o f 1 L HP 1 2 ΘΘ H P 1 2 H H are the same as those of 1 L P 1 2 H H HP 1 2 ΘΘ H (since their traces are identical and then all their m oments match). But th e rank o f both matrices differ an d then a cer tain amoun t of null eigenv alues mu st be intr oduced . Here, 1 L HP 1 2 ΘΘ H P 1 2 H H is o f full r ank ( N ) while 1 L P 1 2 H H HP 1 2 ΘΘ H is on ly o f rank N fo r a matrix size M N , h ence the M factor in eq uation ( 26). Finally , we similarly c onnect the left side of eq uation (25) to µ HPH H throug h µ P 1 2 H H HP 1 2 = 1 M µ HPH H +  1 − 1 M  δ 0 (27) The emp irical distribution of HPH H was then der iv ed from the em pirical distribution of 1 L YY H . As a con sequence, the free moments d k = E[tr N ( HPH H ) k ] can be retrieved from the free momen ts m k = E[tr N ( 1 L YY H ) k ] . Surp risingly , it is shown in [16] that fo r all aforemen tioned fre e conv o lution operation s, the set o f the first k momen ts o f the [de]co n volved distributions can be re covered fro m the set of the k first moments of the operand s. This sub stantially reduces the com- putational effort, as is describe d in the following. Let us work in d etail all the steps to derive the mo ments d k from the momen ts m k . 1) first, the noise contribution to the signal Y is dec on- volved thanks to formula (23). The m ultiplicative conv o - lution (resp. d econv o lution) µ ( out ) of a distribution µ ( in ) and the March enko-Pastur law µ η c can be comp uted from all the m oments m ( in ) k . It is shown in [ 7] tha t c · m ( out ) k can be compu ted from th e moments/cu mulants transform (resp . cumu lants/moments transfor m) [8] o f the coefficients c · m ( in ) k , with c = N /L . As fo r additive conv olu tion (resp. deco n volution) µ ( add ) of two distribu- tions µ ( a ) and µ ( b ) , the f ree cumulan ts of µ ( add ) are the sum ( resp. difference) of th e cumulants o f µ ( a ) and µ ( b ) . From (23), the mo ments m ′ k of µ 1 L HP 1 2 ΘΘ H P 1 2 H H are obtained from the mo ments m k of µ 1 L YY H ; th erefore, in m athematical terms, this reads ( m ′ 1 , . . . , m ′ M ) = S 1 ( m 1 , . . . , m M , σ 2 ) (28) with S 1 = 1 c M  c · M  C  1 c C ( cm 1 , . . . , cm M )  − C  σ 2 , . . . , σ 2 M  (29) where the functio ns M ( · ) and C ( · ) stand respectively for the cumu lants/moments transform and the mo- ments/cumulan ts tr ansform th at b oth take as argument a vecto r of size k (the fir st k mo ments or cumulants, respectively) a nd outpu t a size k vector (th e fir st k cumulants or moments, respectively). Th e inner 1 c C ( c · ) operation multiplicatively deco n volve the Marc henko- Pastur law µ η c . T hen th e next C application tu rns the resulting mo ments into cumulants. T he ad ditiv e decon - volution o f δ σ 2 is then p erform ed thro ugh the cum ulant difference and the outp ut is turned back into the mo ment space throu gh the M app lication. The outer 1 c M ( c · ) is finally perfo rmed to m ultiplicatively c onv olve the result with µ η c . 2) the mo ments m ′′ k of µ 1 L P 1 2 H H HP 1 2 ΘΘ H are then gi ven throug h eq uation (26) by a simple scaling of the mo - ments m ′ k by 1 / M . This reads ( m ′′ 1 , . . . , m ′′ M ) = 1 M ( m ′ 1 , . . . , m ′ M ) (30) 3) the Marchenko-Pastur law µ η c ′ , c ′ = M N /L , is then deconv olved fr om µ 1 L P 1 2 H H HP 1 2 ΘΘ H accordin g to equation (25). This leads then to the mo ments m ′′′ k of µ P 1 2 H H HP 1 2 , ( m ′′′ 1 , . . . , m ′′′ M ) = 1 c ′ C ( c ′ d ′ 1 , . . . , c ′ d ′ M ) (31 ) 4) finally , th e resulting momen ts m ′′′ k are scaled by M to obtain th e d k coefficients, ( d 1 , . . . , d M ) = M ( m ′′′ 1 , . . . , m ′′′ M ) (32) Figure 3 summarizes these steps. Our final interest th ough is to find the d iagonal values of P . The d istribution of the cha nnel matrix H was mo delled in Section II by a m ixture of corr elated Gaussian sub channels. It is difficult, at this p oint o f kn owledge in free probability theory , to deconv olve the ef fect of H from the random ma trix HPH H . Only classical meth ods can help in this situation . Remarkably , it tu rns out that the matrix HPH H is diagonal HPH H =    P k P k | h k 1 | 2 · · · 0 . . . . . . . . . 0 · · · P k P k | h kN | 2    (33) 7 For k ∈ { 1 , . . . , M } , c omputatio n of m k = E h tr L ( YY H ) k i Deconv o lution of the additive noise ( m ′ 1 , . . . , m ′ M ) = 1 M S 1 ( { m k } , σ 2 ) Retriev al o f th e d k ’ s ( d 1 , . . . , d M ) = M c ′ C ( c ′ m ′ 1 , . . . , c ′ m ′ M ) Fig. 3. m k to d k Block Diagram Therefo re the theoretical mome nts d k of µ HPH H are th e normalized traces of th e asympto tic dia gonal m atrices of entries, n ( HPH H ) k o ij = M X k =1 P k | h ki | 2 ! k δ j i (34) and then the p th order mom ent d p = E[tr N ( HPH H ) p ] of HPH H can then be equated for large N to d p = lim N →∞ 1 N N X j =1 M X k =1 P k | h kj | 2 ! p (35) In the f ollowing we provide a meth od to estimate the entries P k under the assump tion, which is nev er verified in practice, that the channels a re extremely frequ ency selecti ve, i.e. such that, fo r any k and any coup le j 6 = j ′ , the entries h k,j and h k,j ′ are independ ent. B. Estimation of th e p owers P k As alre ady co ncluded in Section III, if the channel delay spreads are very short, then the channel frequen cy resp onses { h k 1 , . . . , h kN } , k ∈ { 1 , . . . , M } , ar e stro ngly correlated and almost nothin g can be ded uced o n the entries of P . On the contrary , assume that the chann el delay spread is very large, and rewrite equatio n (35) as d p = ¯ d p + w p (36) = 1 N N X j =1 M X k =1 P k | h kj | 2 ! p + w p (37) with ¯ d p the p th sample order mome nt ( N < ∞ ) and w p some noise pro cess. The latter conver ges to 0 when N → ∞ and the channel frequ ency response is i.i.d. Gaussian. Howe ver, w hen N is finite or when the channel is less frequency selective, then w p is a bit more difficult to hand le. T o push th e computatio n for ward, we need first to de riv e the classical order p moment m ( h ) p of the v ariables | h kj | 2 , for any co uple ( k , j ) , in the complex ca se. This g iv es m ( h ) p = E [ | h kj | 2 p ] = 1 2 2 p p X i =0 C i p (2 i )!(2[ p − i ])! i !( p − 1)! (38) One can th en derive the g eneral expression for d p as d p = p ! 2 2 p X k 1 ,...,k M P i k i = p M Y i =1 ( k i X k =0 (2 k )!(2[ k i − k ])! ( k !) 2 ([ k i − k ]!) 2 ) P k i i (39) The complete deriv atio n of fo rmula (39) is provided in the append ix. 1) Bayesian a ppr o ach: Let K be some integer greater than or equa l to M . The no ise process w = [ w 1 , . . . , w K ] T , as already men tioned, is in gen eral difficult to analyze. W e shall consider in the following that one actually has a limited knowledge abou t w which reduc es to the covariance matrix C = E[ ww T ] gathered from p revious simulations 5 . Now , consider that th e set of prior informatio n I con tains the following elements: • the values o f P 1 , . . . , P M are real p ositiv e an d k nown not to b e larger than some value P max . • the typ ical err or variance C in the observed mom ents { d k } , k ∈ { 1 , . . . , M } , is k nown. In fact, th e exact covariance matrix C canno t be known since its computatio n req uires the exact k nowledge of P . Indeed , a few lines of calculu s of C = E h  d − ¯ d  H  d − ¯ d  i (40) with ¯ d = [ ¯ d 1 , . . . , ¯ d K ] T , lead to equ ation (41), which depends on P 1 , . . . , P M . Howe ver, let us first co nsider C known before we intro duce alternative solutions wh en C is u nknown. The objective is to infer on the set { P 1 , . . . , P M } giv en I and the ob served sample moments ¯ d . An err or measu re must be con sidered to come up with an estimate o f { P 1 , . . . , P M } . W e co nsider here th e estimate of P which minimizes th e mean quadra tic err or (MMSE). Th is MMSE estimate ˜ P is given by ˜ P = E  P | ¯ d  (42) = Z P P p ( P | ¯ d , I )d P (43) = Z P P p ( ¯ d | P , I ) p ( P | I ) R P p ( ¯ d | P , I ) p ( P | I )d P d P (44) Since the prio r information ( P | I ) is limited to the fact that all entries are upper-boun ded b y P max , p ( P | I ) sh ould be set unifor m on the space [0 , P max ] M accordin g to the max imum entropy principle. However , n ote that if { ˜ P 1 , . . . , ˜ P M } mini- mizes the M MSE then also does any p ermutation of this set. Therefo re, to have a cor rectly d efined problem (with a un ique solution), the set { P 1 , . . . , P M } must be order ed; we will then state in the following that P 1 ≤ P 2 ≤ P M . Ther efore the p rior p ( P k | I ) is taken un iformly on the set [0 , P k − 1 ] when P k − 1 is set, wh ich leads to p ( P | I ) = P max Q M − 1 i =1 P − 1 i . A lso, since only the error cov ariance matrix C in the observed sample moments ¯ d is known, the maxim um entropy princ iple requests that the p rocess w is assigned a Gau ssian distrib ution with 5 honesty woul d require tha t we actu ally deriv e the maximum ent ropy distrib ution of w but this would lead to in volv ed computation. 8 C a,b = − ¯ d a ¯ d b + X k 1 ,...,k M k ′ 1 ,...,k ′ M P i k i = a P j k j = b                  ( N − 1 ) a ! b ! N Y i,j 1 ≤ i ≤ a 1 ≤ j ≤ b P k i k i P k ′ j k ′ j m ( h ) k i m ( h ) k ′ j k i ! k ′ j ! + a ! b ! N Y i,j 1 ≤ i ≤ a 1 ≤ j ≤ b k i = k ′ j P 2 k i k i m ( h ) 2 k i ( k i !) 2 × Y i,j 1 ≤ i ≤ a 1 ≤ j ≤ b k i 6 = k ′ j P k i k i P k ′ j k ′ j m ( h ) k i m ( h ) k i k i ! k ′ j !                  (41) variance C . Theref ore, equ ation (4 4) bec omes ˜ P = Z P 1 ≤ ... ≤ P M P e − w ( P ) H C − 1 w ( P ) R P 1 ≤ ... ≤ P M e − w ( P ) H C − 1 w ( P ) d P d P (4 5) where we denoted w = w ( P ) to remind the actual dependence of w in th e p owers P 1 , . . . , P M (throu gh the expression of d in equation (39)). Unfortu nately , the in tegration space of equ ation ( 45) m akes both integrals rather in volved to com pute. A way to prac tically computed ˜ P co nsists in turning the integrals into finite sums over thin sliced versions of the integration space. Also, as previously m entioned, the covariance matrix C is obviously unknown when trying to decipher the cell powers P 1 , . . . , P M . Howe ver, iterative m ethods can be con sidered in which C is initially defined as th e cov ariance matrix C init of an hyp othetic set o f p owers, say P 1 = P 2 = . . . = P M = P max / 2 . Then, runnin g the MMSE estimator with C init returns a first set P (1) 1 , . . . , P (1) M from which a r efined version of C can be ev alu ated ( from formu la (4 1)). Note that, in runn ing k instances o f the algor ithm, the sample mo ments ˜ d p can b e accumulated a nd the covariance matrix C has to be compute d as if as many as k N subcar riers we re actually used in the free deconv olution algorith m. This proc ess can be processed in a loop for a satisfying number of iterations. 2) Altern ative estimators: Other estimator s th an MMSE, such as m aximum- likelihood (ML), migh t b e conside red which take as an estimate the set P which minimizes w ( P ) H C − 1 w ( P ) . Howe ver , the measure associate d to the ML estimator does not suit the broa d a posteriori d istribution p ( P | ¯ d , I ) as will b e shown in simulation s. Indeed , a large estimation error is a s bad as a small estimation erro r in the ML context; theref ore, when the posterior p ( P | d , I ) is not peaky , large estimatio n errors are expected. T he MMSE estimator is, in this scenario, more appro priate. A zero-fo rcing method can also b e de riv ed. Fro m equ ation (39), if no thing wer e known abou t th e noise pro cess w p , one might naively consider solvin g th e system of M e quations (39), with p = 1 , . . . , M in the M un knowns P 1 , . . . , P M , in which d p is set equ al to the ob served ˜ d p . This can be solved by tur ning this system of equ ations into the eq uiv a lent system M X k =1 P k = Q 1 ( d 1 ) M X k =1 P 2 k = Q 2 ( d 1 , d 2 ) . . . = . . . M X k =1 P M k = Q M ( d 1 , . . . , d M ) (46) with Q k ∈ R [ d 1 , . . . , d k ] . This system can be so lved using the Newton-Girar d for - mulas [18]; the solutions P 1 , . . . , P M are foun d to be th e M roots o f an M th degree po lynom ial. This solution does not require an y kno wledge on the c ovariance matrix C , which does not h av e any significatio n in this context. Howev er , it often turns out that the roots P k are not all real, which makes the solution u seless in pr actice. Also, this zero- forcing metho d is very a wkward as it stron gly suffers from the presence of no ise. Especially , the large variance E[ | w M | 2 ] is consider ed equally to the typica lly very small variance E[ | w 1 | 2 ] , and therefo re the first sample mom ent ¯ d 1 is no t more impo rtant in the final computatio n than the last sample momen t ¯ d M 6 . V I I . S I M U L A T I O N A N D R E S U LT S In the follo wing, we use th e results that wer e p reviously derived in the case of a th ree-cell network that the ter minal wishes to track . The set o f cells studied alo ng this part are of relativ e powers P 1 = 4 , P 2 = 2 , P 3 = 1 . Before perfor ming the first simulation of the com plete algorithm , we present in Figure 4 the relative erro r in the estimation o f the momen ts ν k = tr N ( HPH H ) k from the free deco n volved sample mom ents ¯ d k . It is observed that ev en for N = 256 , L = 51 2 the m ean relative erro r in the computed third moments is of o rder 1% . This suggests that the free deconv olution techniqu e is very a ccurate ev en f or non-in finite values of N and L . The bottleneck approx imation in the estimation of P 1 , . . . , P M , as will be o bserved in the coming plots, therefor e lies in th e convergence of the entries h kj tow ards a Gaussian i.i.d. process, and not in the infinite matrix size assumption. In a first simulation, we stud y th e conv ergence properties of the p roposed scheme. W e consider a large OFDM sy stem with 6 this would be here again a dishonest considerati on as one at least knows that d M is m ore uncert ain than d 1 . 9 256 512 1 , 024 2 , 048 0 0 . 2 0 . 4 0 . 6 0 . 8 1 · 10 − 2 Num b er of sub carriers N relativ e error | ¯ d 1 − ν 1 | ν 1 | ¯ d 2 − ν 2 | ν 2 | ¯ d 3 − ν 3 | ν 3 Fig. 4. Relati ve error on recovere d moments ν k = tr N ( HPH H ) k from the free decon vol ution algor ithm, L = 2 N 1 2 4 0 0 . 02 0 . 04 0 . 06 0 . 08 0 . 10 Estimated p o w ers Densit y Fig. 5. Cell power detect ion, N = 2048 , L = 4096 , MMSE estimat e, Perfect kno wledge of C 1 2 4 0 0 . 02 0 . 04 0 . 06 0 . 08 Estimated p o w ers Densit y 1 2 4 0 0 . 02 0 . 04 0 . 06 0 . 08 Estimated p o w ers Densit y Fig. 6. Cell power detecti on, N = 512 , L = 1024 , MMSE estimate (lef t) and ML estimat e (right ), Perfect knowledg e of C 1 2 4 0 0 . 05 0 . 10 0 . 15 0 . 20 Estimated p o w ers Densit y Fig. 7. Cell po wer detection, N = 512 , L = 1024 , 10 a ccumulat ions, MMSE estimate , Perfec t knowle dge of C 1 2 4 0 1 3 2 3 1 Estimated p o w ers Cum ulativ e distribution No accum ulation 10 accum ulations Optimal estimation Fig. 8. Ce ll powe r detect ion CDF , N = 512 , L = 1024 , Perfect knowl edge of C N = 2048 subcarr iers an d L = 4096 sampling per iods under ideal Gaussian i.i.d . input symbo ls, un correlated Gaussian channel freq uency respo nses and Gau ssian add itiv e wh ite noise with signal to n oise ratio SNR = 20 dB . T he covariance matrix C is the exact covariance matrix. Figure 5 provid es the results o f this simu lation for thousan d channel realisations. It is observed that the distribution o f the eigenv alues is largely spread over the expected eigenv alu es. This is explained by the slow conver gence natu re of the compu ted d p tow ards the correspo nding m oments when N → ∞ . Also, it is ob served that th e peak ce nters ar e offset from the expected p owers. This is main ly due to th e fact that the MMSE estimator is meant to minimize th e mea n quadratic error averaged over all possible sets P 1 , P 2 , P 3 and not for the particular set selected here 7 . Also, the actually n on-Gaussian prop erty of the noise p rocess 7 it was observ ed in particula r from other simulations that cells of equal po wers are generally better estimat ed. 10 1 2 4 0 0 . 05 0 . 10 0 . 15 0 . 20 Estimated p o w ers Densit y Fig. 9. Cell po wer detecti on, N = 512 , L = 1024 , MMSE estimate, Recursi ve update of C , 10 steps w as well as the inexact results fro m the free deconv olution process contribute to the offset. Figure 6 provide s a comp arison between the results g iv en by the MMSE estimator and the ML e stimator when N = 512 an d L = 1 0 24 . It tu rns out, as discussed ear lier , that th e ML estimate is mo re largely spread ar ound the expected cell powers than the MMSE estimate. In the following, we perform more r ealistic simu lations in which input signals are QPSK modulated instead of co mplex Gaussian, an d with more realistic channels of length varying from 1 to N/ 4 symbols. T o red uce the v ariance o f the estimates, we also average the sample d p over ten channel realizations, which in practice requir es aro und 1 ms o f data to proc ess. I n Figure 7, we took N = 512 , L = 1024 and a Rayleigh ch annel o f le ngth N / 8 . The cov ariance matrix C is still the exact covariance matrix. Then a hund red r ealisations of this proc ess are run. Th e SNR is still SNR = 20 dB in this secon d experiment. The cumulative distribution f unction (CDF) of the detected p owers is presented in Figure 8 and compare d to th e CDF when no accum ulation is performed . The three thresholds correspond ing to the three d etected c ell powers ca n be observed, with a slight shift fro m the expected cell powers caused both by the non-exact Gaussian assumption on w and o n the n on-exact Gau ssian i.i. d. assumption on the channel frequen cy responses. Now we propose to examine the performance of the iterativ e cell p ower recovery . W e initially set C init to the cov ariance matrix of a set of cells of powers P 1 = P 2 = P 3 = 2 . 5 . Then ten iterations of the cell-po wer detector are run, with at each step a refinement of C . Note th at, since a t each step the sample moments ¯ d p are ac cumulated , th e en tries of the covariance matrix C are comp uted accordin gly , i.e. k accumulatio ns demand that C is compu ted from an effectiv e number k N sub carriers, as presented in Section VI. Figure 9 provides the results of the recursive algorithm, which p roves to perform s very accurately , with surpr isingly little dispersion around the detected powers com pared to Figu re 7. 1 2 4 0 1 3 1 6 1 Estimated p o w ers Cum ulativ e distribution i.i.d. Gaussian L TE-ETU L TE-EV A Optimal estimation Fig. 10. Cell po wer detection CDF in L T E channe ls, N = 512 , L = 1024 , 10 accumula tions, preset C Channel T ype RMS dela y spread Channel length EV A 357 ns N / 27 ETU 991 ns N/ 13 T ABL E II 3 G P P - LT E S TA N D A R D I Z E D S H O RT D E L AY C H A N N E L S Also, we test the r obustness of our algo rithm ag ainst prac- tical shor t channels, instead of high de lay spread c hannels. This is shown in Figure 10 which prop oses a compar ison between the id eal long chann el situation and the 3GPP-L TE [4] stand ardized E xtended V ehicular A (EV A) and E xtended T ypical Urb an (ETU) c hannels with para meters g iv en in T able II. Here we considered a m obile handset with N r = 2 anten nas, 256 subc arriers p er an tenna (an d then in total an effectiv e number of N = 512 subcarr iers), L = 1024 , SNR = 20 dB . The c ovariance m atrix C of the n oise pr ocess w is an approx imated matrix ob tained from intensive simulations on short chann els. Indeed, the covariance pa ttern is very different from the m atrices used in the Gaussian i.i.d. scenario and is difficult to derive an alytically; especially we noticed that the smaller the delay sprea d, the larger the un certainty on the higher moments, whic h tu rns C into an ill-condition ed matrix. The results are averaged over ten chan nel realizations. The CDF of the detected power d istribution f or those channels is provide d in Figure 10. The latter shows a r ather good behaviour in both ETU and E V A chan nels. Nonetheless their short delay spreads lead to a larger variance in the mean power estimation. Surprisingly , it turn s out that th e distribution of the in put signals s ( l ) does not impa ct the system p erforma nce as lo ng as it is i.i.d . with zero m ean and unit variance. T his is a known result in free deconv o lution which has no t been proven yet. Therefo re in our simulation s, QPSK modu lations showed the exact same behaviour as purely Gau ssian d istributed input signals. 11 V I I I . D I S C U S S I O N A. Data , p rior and c onver gence W e previously d escribed and simulated a recursive algo- rithm m eant to con verge to an accurate estimate of th e cell powers P 1 , . . . , P M , whose conver gence we did not prove yet. Actually , a m athematical as well a s a philosoph ical reasons for this convergence can be ad vanced. First, note that a large num- ber of iteration s lead to a smaller variance of w = d − ˆ d ( P ) , therefor e to smaller entries of the noise c ovariance matrix C . As a con sequence C − 1 has large entries and the exponential terms e − 1 2 ( d − ˆ d ( P )) T C − 1 ( d − ˆ d ( P )) in the MMSE integrals (45) are rele vant on ly when th e differences d − ˆ d ( P ) a re very small; this is, when ˆ d ( P ) ≃ d . Therefo re the accuracy in the entries of C − 1 is n ot of fun damental importan ce, as lon g as they are large en ough . Th e conver gence of the itera ti ve process is ensured by the accum ulated d ata d themselves; but of cou rse this convergence is accelerated with good appr oximation s of C . This obser vation is very gen eral in the Bayesian prob ability theory con text and h as been observed and analyzed tho roug hly by Jaynes [ 15]. Bayesian pr obabilities rely on a balance between priors and d ata . If the available data is scarce, then prior infor mation is very valuable; in o ur situation, as shown by simulation, if C wer e known, then e ven a sin gle chann el realization allows to app roxima tely recover the tran smitted powers. On th e contrar y , large amounts of d ata prev ail over prior information s o that even u nfortu nate prio rs may not badly alter the a posteriori pr obability; this explains why a precise estimation of C is n ot ma ndatory when large accum ulations of d ata ar e consider ed. Note th at in this case, ML estimates and MMSE e stimates show similar perform ance, since the posterior d istribution p ( P | Y , I ) is very peaky in the correct value for P . Howe ver, it is importan t to und erline the fact th at the offset problem, observed in simulatio n, in the estimation of P 1 , . . . , P M will not be redu ced by m ere data accum ulations. Indeed , the issue co mes here from the free deco n volution process which is not a ccurate for fi nite N . Therefo re, an appro- priate trade-off between large N an d many accumulations mu st be foun d: large N entail mo re accu rate f ree deconv olution processes at the expense o f compu tationally dema nding large matrix p rodu cts, while many accumulatio ns ensures a faster conv ergence ( to offset cell p owers) at a lower computation cost. Ano ther app roach consists in compu ting higher o rder moments to strengthen the cell power estimation. Howe ver, high order sam ple m oments have a large variance for finite N . Their impact on the fina l estimation might then be very limited since they are very un reliable. For instance we o bserved in simulations th at, for N = 51 2 , typical matrices C verify C 11 ≃ 1 e − 2 and C 33 ≃ 1 e 4 , which gives a million times more credit to the first o rder sample moment than to th e th ird order samp le m oment. Bring ing in the compu tation f ourth order mom ents with N = 512 would turn ou t n ot worth the computatio n increase. B. Ap plicability As reminded in the intro duction of this paper, usual cell power detection tech niques u se scar ce an d largely interfered synchro nization sequen ces. M uch time, but low compu tation, is then requ ired to detect cells with high efficiency . From the autho rs’ experience in the co ntext of 3GPP-L TE, many problem s arise when those p ilot seq uences are syn chron ized and the emitting cells ha ve equal p owers since both signals interfere to the detrimen t of th e decode r . In the previously derived sch eme, even cells of equal power can be counted and isolated . Indeed, if M cells are expected and the CDF of the estimated powers shows a large jump of 2 / M for a g iv en power P , then we can deduce that two cells have eq ual power P . It is th erefor e n ot necessary to enhanc e th e qua lity of the estimation to sep arate tho se two cells. Howe ver, some limitations can be f ound in th e applicab ility of this cog nitive scheme . Firstly , load ed cells are req uired to ensu re reliability of th e estimates; this r equires that the underly ing standards o ptimally r euse the allocated bandwidth . Secondly , if many cells are desired to b e de tected, th en very high ord er mome nts must be computed which , as discussed earlier are on ly re liable if the num ber of subcarrie rs N and the n umber of accumulation s are very large. High order moments also require very large matrix prod ucts, which migh t be too d emandin g to the embedd ed system processors. Also , under the limited state of k nowledge of the system mod el, neighbo ring cells can only b e de tected if the propag ation channels ar e very freq uency selectiv e. The flatter th e ch annels, the more num erous the accumulations requir ed to come up with a reliable estimation . The latter limitation is obviously the m ost constrainin g factor . I X . C O N C L U S I O N In this pape r , we pr ovided a practical way to blindly detect neighbo ring cells in a distributed OFDM network. Assumin g constant tran smission in those cells on a fairly large band - width, we showed that one can d etermine the ind ividual SNR of ev ery surrou nding cell provided that th e chann el delay spread is sufficiently large. This scheme is particularly suited for the future co gnitive OFDM systems which aim to r educe the amou nt of synchr onization sequences while keepin g track of the neighb oring cells. X . A C K N O W L E D G E M E N T This work was partially supporte d by the Europ ean Com- mission in the framework of the FP7 Network of Excellence in Wireless Com municatio ns NEWCOM++. A P P E N D I X Consider Gau ssian channels with independ ent fr equency responses h ij ( i ∈ { 1 , . . . , M } , j ∈ [1 , N ] ). Th en, for a g iv en 12 j , noise taken ap art, we denote d ( j ) p = M X i =1 P i | h ij | 2 ! p = X k 1 ,k 2 ,...,k M P m k m = p C k 1 p C k 2 p − k 1 . . . C k M p − k 1 − ... − k M − 1 M Y l =1 ( P i | h ij | 2 ) k l (47) where the pr oduct of the binom ial coefficients C k 1 p C k 2 p − k 1 . . . C k M p − k 1 − ... − k M − 1 is the multinomial coefficient p ! k 1 ! k 2 ! ...k M ! . Hence the simplified expression, when averaging d ( j ) p over j ∈ [1 , N ] ( with N → ∞ to ensure 1 N P j | h ij | 2 → E j [ | h ij | 2 ] ) d p = p ! X k 1 ,k 2 ,...,k M P m k m = p E j h Q M i =1 ( P i | h ij | 2 ) k i i Q M i =1 k i ! 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