On the Spectrum of Large Random Hermitian Finite-Band Matrices

On the Spectrum of Lar ge Rando m Hermiti an Finite-Band Matrice s Oren Somekh ∗ , Osvaldo Simeone † , Benjamin M. Za idel ‡ , H. V incent Poor ∗ , and Shlomo Sh amai (Shitz) § ∗ Departmen t of Electrical Engine ering, Princeton Uni versity , Princeton, NJ 085 44, USA † CWCSPR, Dep artment of Electric al and Computer Engineer ing, NJIT , Newark, NJ 07 102, USA ‡ Departmen t of E lectronics and T elecommunicatio ns, N TNU, Trondheim 749 1, Norway § Departmen t of E lectrical Engin eering, T echnion , Haifa 320 00, Isr ael Abstract — The open p roblem of calculating the limiting sp ec- trum (or its Sh annon transform) of in creasingly large rand om Hermitian finite-band matrices is described. In general, these matrices includ e a finite number of non-zero diagonals around their main diagonal regardless of their si ze. T wo d ifferent com- munication setups which may be modeled using such matrices are presented: a simple cellular up link channel, and a time varying inter -symbol interference ch annel. Selected recent in form ation- theoretic works dealing directly with such channels ar e rev iewed. Finally , sev eral characteristics of the still unknown limiting spectrum of such matrices are listed, and some reflections are touched upon. I . P R O B L E M D E S C R I P T I O N Consider a linear channe l of th e form y = H N x + z , (1) where x is the N K × 1 z ero-mean complex Gaussian inp ut vector x ∼ C N (0 , P K I N K ) 1 , y is the N × 1 ou tput vecto r , and z d enotes the N × 1 z ero-mean comp lex Gaussian additi ve noise vector z ∼ C N (0 , I N K ) , which is inde penden t of x and H N . Acco rdingly ρ = P K is th e tran smitted signal-to - noise ratio ( SNR). In addition, the N × N K chann els tran sfer matrix H N is defined by H N = 0 B B B B B B B B B @ a 1 β c 1 0 · · · 0 0 α b 2 a 2 β c 2 0 · · · 0 0 α b 3 a 3 β c 3 . . . . . . . . . 0 α b 4 . . . . . . 0 0 . . . . . . . . . a N − 1 β c N − 1 0 0 · · · 0 α b N a N 1 C C C C C C C C C A , (2) where { a i , b i , c i } are statistically in depend ent 1 × K ran dom row vectors w ith in depend ent identically d istributed (i.i.d .) entries a i,j ∼ π a , b i,j ∼ π b , and c i,j ∼ π c . For simplicity , we assume that the p ower mo ments o f the entries for a ny finite order are b ound ed. Finally , α, β ∈ [0 , 1] are con stants. The norm alized input-ou tput m utual infor mation of (1) condition ed on H N (also known as the Sh annon tran sform) 1 An N × N identity matrix is denoted by I N . is 2 1 N I ( x ; y | H N ) = 1 N log det “ I N + ρ H N H † N ” = 1 N N X i =1 log “ 1 + ρλ i ( H N H † N ) ” = Z ∞ 0 log(1 + ρx ) d F H N H † N ( x ) , (3) where λ i ( H N H † N ) d enotes the i th eigenv alue of the Hermi- tian fi ve-diagon al matrix H N H † N . Furtherm ore, den oting the indicator f unction by 1 {·} , F H N H † N ( x ) = 1 N N X i =1 1 { λ i ( H N H † N ) ≤ x } (4) is the emp irical cumulative d istribution fu nction o f the eigen- values (also refer red to as the spectru m or em pirical distribu- tion) of H N H † N . Fixing K and a ssuming that F H N H † N ( x ) conv erges almost surely (a.s.) to a u nique limiting spectru m F H N H † N ( x ) a . s . − → N →∞ F ( x ) , it can b e shown that the expectation of (3) with respec t to (w .r .t.) the d istribution of H N conv erges as well. T his is since (3) is unifo rmly in tegrable due to the Hadamard inequa lity and the b ounded power moment assump- tion, and h ence the a.s. conv ergence implies co n vergence in expectation [1]. In Sectio n II it will be realized that if the channel H N is known at th e receiver an d its variation over time is station ary and ergodic, then the expectation o f (3) w .r .t. the distribution of H N is the per-cell sum-ra te capac ity of a certain cellu lar uplink chann el model. In a nother setting (see Section II) , the same expectation m ay be interpreted as the capacity of a certain time variant inter-symbol interf erence (I SI) c hannel, assuming aga in tha t th e ch annel is k nown at th e rece iv er . A. Analytical Difficulty Many r ecent studies have analyzed the asymptotic rates of various vector chann els u sing r esults from th e theo ry of (large) ra ndom m atrix (see [2] for a recent revie w). In tho se cases, the numbe r of random variables inv olved is of the order of the numb er of elem ents in the matrix H N , and self-averaging is strong en ough to ensur e convergence of the 2 Unless explici tly denoted otherwise a natural base logarithm is used throughout this presenta tion. empirical measure o f eigenv alues, a nd to de riv e eq uations for the limiting spectrum (o r its Stieltjes transform) . In particular, this is the case if the normalized continu ous power profile of H N , which is defined with r, t ∈ [0 , 1] as P N ( r, t ) , E ( | [ H N ] i,j | 2 ) i − 1 N ≤ r < i N , j − 1 N K ≤ t < j N K , (5) conv erges uniformly to a boun ded, piecewise continuous func - tion as N → ∞ , see e.g. [2, Theo rem 2.5 0]. In th e case under consideratio n her e, it is easy to verify that fo r K fixed, P M ( r , t ) does no t converge unifo rmly , a nd other techn iques are re quired. Remark: It is noted that the setting o f ( 1) can be extended in m any ways such as increasing the number of non-zer o block diago nals, or replacing each K -d imensional random row vector with an n × m r andom matr ix. Such settings result in H N H † N which includes m ore than fiv e non- zero d iagonals and are refer red to as Hermitian finite-b and rand om matr ices; the resu lting ma trices con tain o nly z ero en tries ou tside a finite band (finite nu mber of non-zer o diag onals) ar ound their main diagona l regardless of N . T o the best of th e authors’ knowledge, n either the limiting spectrum of Herm itian finite- band ran dom ma trices, nor th e expectation of the no rmalized input-o utput co nditional mutual informa tion (3), is known in gen eral except for a few special cases (see Section III). Mor eover , even the high- SNR r egime characterizatio n (d efined in [3][4]) of th e latter is known only for a few special cases (see Sectio n II I) and remains an o pen problem in g eneral. I I . M O T I V AT I O N In this section we present two d ifferent multi-acce ss com - munication channels whose chann el tr ansfer matrices are finite-band . a) Cellular uplink: Motiv ated b y the fact tha t a mobile user in a cellu lar sy stem effectiv ely “ sees” only a finite number of base- stations, a simplified cellu lar mode l family has been intr oduced by W yner in [5] ( see also [6 ] for an indepen dent earlier w ork wh ich deals with similar setup s). According to the or iginal lin ear variant setup p resented in [5], the K homogeno us users of each c ell are colloc ated at the cell’ s center and “see” their local b ase-station anten na and the antennas of the two adjacent base-station s only . Wh ile the signals travel to the local an tenna with no path- loss, the path-losses to the adjacent cell antenna on th e left and to the o ne on the right, are characterized by two parameters α, β ∈ [0 , 1 ] , respectively . W yner assumed that the users cannot cooperate in any way and that all the base-stations are connected to a central receiver via an id eal error-free infinite capacity b ackhau l network. With optimal jo int pro cessing of all th e received signals, the chan nel can b e considered as a multiple-acce ss cha nnel wh ose vector r epresentation is given by (1). Th e non-fading setup of [ 5] was extended to includ e flat fadin g chann els in [7][ 8]. Conside ring an infinite nu mber of c ells and assuming that th e channe l state informa tion is known by the cen tral receiver , th e per-cell sum-rate capacity of the W yner model is giv en by setting π a = π b = π c = π , and averaging the mu tual inform ation of (3) over the e ntries of H N . I t is noted th at th e b asic m odel can be extended to cases where each mobile “sees” any finite numb er of cell-site antennas an d the resulting H N H † N is a finite-ban d matrix. Remark: Using the uplink-d ownlink duality (e.g. [9]), the per- cell sum-rate capacity of th e W yner uplink chan nel is also an achiev able per-cell sum -rate (a lower bou nd of the per-cell capacity) o f th e recip rocal W yner downlink channel, assum ing the joint mu lticell transmitter has full channel state inform ation (CSI) wh ile each m obile is aware o f its own CSI o nly . Since its intr oduction in [5], the W yner model family has provided a powerful fra mew ork for research assessing the perfor mance o f v ariou s j oin t multicell processing schemes (see [10] and [ 11] for recent surveys). Overcoming th e analy tical difficulties relating to these mo dels an d calcu lating the spectra (or their tran sforms) of the resu lting finite-ban d matr ices, would greatly enh ance our understan ding and insight into the theoretical performa nce of future cellular (and wireless) systems. b) T ime varying ISI chann els: Here we consider K homog enous users commu nicating with a r eceiver over an L - tap time varying ISI channel. Assuming that the cha nnel taps are i.i. d. between different user s and also i. i.d. in the time index it is ea sily verified that the received signal is given by (1). Assuming tha t L = 3 , the sum -rate of this multiple access chann el is given b y averaging th e mutu al info rmation of (3) over th e entries o f H N . This setup may describ e a “fast” mu ltipath fading chan nel wher e the ch annel taps are indepen dent over the time index. As with th e previous setu p for any finite L the resulting H N H † N is a finite- band matrix . In co ntrast to the pr evious mo del where the entries of the received signal are in the spatial domain, the entries of the received signa l here are in the time dom ain. I I I . S E L E C T E D P R I O R W O R K In this section we briefly review selected previous works dealing with the spec trum o f finite- band m atrices, its Shan non transform , and related issues. The reader is r eferred to [1 0] and [11] for detailed surveys o f relevant in formatio n-theor etic works. The non-fading (or deterministic) c ase w as analyze d by W yner in [ 5] for th e special case of β = α . Setting a i,j = b i,j = c i,j = 1 we get that 1 K H N H † N becomes a five- diagona l T oeplitz matrix with no n-zero entries ( α 2 , 2 α, 1 + 2 α 2 , 2 α, α 2 ) . Using well known results regarding th e limiting spectrum of large T oeplitz matrices (Szeg ¨ o’ s Theo rem [12]), W yner sh owed th at the p er-cell sum-rate cap acity a pproach es as N → ∞ to C = Z 1 0 log ` 1 + P (1 + 2 α cos(2 π f ) ) 2 ´ d f . (6) It is noted that the result is independ ent o f K as long as the total transmit power per-cell P is fixed. The reader is referred to [13] fo r a deriv ation of the Stieltjes transform o f the spec trum for similar five-diagonal T oeplitz m atrices. The infinite lin ear W yner model in the presence o f flat fading cha nnels is conside red in [8]. For the special case of β = α , π a = π b = π c = π and K = 1 it is shown that the unor dered eigenv alue distribution E ( F H N H † N ) conv erges weakly to a uniq ue distribution. It is co njectured that using similar methods the spec trum can be proved to converge a. s. to a un ique lim it as well. In ad dition, using a standard weighted paths sum mation over a restricted grid, th e limiting values of the first several mo ments o f this distribution were calcu lated for the special case in which the amplitude of an individual fading coefficient is statistically indepen dent of its uniform ly distributed p hase (e.g. Ra yleigh fading π = C N (0 , 1 ) ). For example, listed below are the first thre e limiting mo ments: M 1 = m 2 + 2 m 2 α 2 M 2 = m 4 + 8 m 2 2 α 2 + (4 m 2 2 + 2 m 4 ) α 4 M 3 = m 6 + (6 m 3 2 + 12 m 2 m 4 ) α 2 + (36 m 3 2 + 12 m 2 m 4 ) α 4 + (6 m 3 2 + 12 m 2 m 4 + 2 m 6 ) α 6 , (7) where m i is the i -th p ower momen ts of the amplitude of an individual f adin g coefficient. It is noted th at this pro cedure can be extended in p rinciple, although in a tedio us manner, for any finite K or also for H N to include more than thr ee non - zero block diagonals. Since the l imiting momen ts of in creasing order are func tions of in creasing order s of the mome nts of the fading co efficients, it is c onjectured that the limitin g distributions (and also the spectra) of finite- band matrices depend o n th e actua l fading d istribution and not ju st on its few first momen ts. Focusing o n th e case in wh ich K is large while P is kept constan t, and apply ing the stro ng law o f large number s (SLLN) , the en tries o f 1 K H N H † N consolidate a.s. to their me an values and the latter becom es a T oep litz matrix . By app lying Szeg ¨ o’ s T heorem fo r N → ∞ it is shown in [ 8] that the p er-cell sum-rate capacity is given b y C = Z 1 0 log (1 + P [ σ 2 (1 + 2 α 2 ) + | m 1 | 2 (1 + 2 α cos(2 π θ )) 2 i dθ , (8) where σ 2 = m 2 − | m 1 | 2 is th e variance of an in dividual fading coefficient. An alternati ve app roach wh ich replaces the role of the eigen- values of H N H † N with the diagonal elemen ts o f its Cholesky decomp osition, is p resented by Narula [1 4]. With α = 1 , β = 0 , π a = π b = π , an d K = 1 , the resulting H N H † N is a three- diagona l matrix (a lso known as Jacobi m atrix). Originally , Narula has stud ied the capac ity of a “fast” tim e varying two- tap ISI channe l, wh ere the c hannel coefficients ar e i.i.d . zero- mean complex Gaussian (i.e. π = C N (0 , 1) ). Following [14], the diago nal entries of the Chole sky deco mposition applied to the cov arian ce matrix “ I N + P H N H † N ” = L N D N U N , ar e giv en b y d n = 1+ P | a n | 2 + P | b n | 2 „ 1 − P | a n − 1 | 2 d n − 1 « , n = 2 , . . . , N , (9) with an initial cond ition d 1 = 1 + P | a 1 | 2 + P | b 1 | 2 . Thus, the diago nal entries { d m } for m a discr ete-time continu ous space Mar kov chain . Remarkably , Narula manag ed to prove that this M arkov chain possesses a unique ergodic stationary distribution, given by f d ( x ) = log( x ) e − x ¯ P Ei ` 1 ¯ P ´ ¯ P ; x ≥ 1 , (10) where Ei ( x ) = R ∞ x exp( − t ) t dt is the exponential integral function . Further, it is proven in [14] that the SLLN holds f or the sequence { log d n } as N → ∞ , and the chann el capacity is C = Z ∞ 1 (log( x ) ) 2 e − x ¯ P Ei ` 1 ¯ P ´ ¯ P dx . (11) It is noted that Nar ula’ s approach is clo sely match ed to the above s etting and any attempt so far to ch ange a key parameter in this setting (such as the entries’ d istribution, th e num ber of users per-cell, and the number of n on-zer o d iagonals) leads to an an alytically intractable d eriv ation. This is p robab ly related to the unique prop erties o f Jacobi matrices wh ich does not apply to finite-band matrice s in gen eral. F or example, the determinan t of a Jacobi matrix is equ al to a weigh ted sum of the determ inants of its two largest principa l sub -matrices. In additio n, Narula’ s analysis provide s additional evidence to support the conjectu re tha t th e limiting spectrum of finite- band ran dom m atrices is depen dent on the d istribution of their entries. On this note, in [15] an equi valent cellular u plink setup but with un iform ph ase fading ( | a i,j | 2 = 1 and θ i,j = ∡ a i,j ∼ U [0 , 2 π ] ) kn own at the joint receiver is considered , and the per-cell su m-rate cap acity is shown to coin cide with the n on- fading setup f or N → ∞ . I t is worth mentio ning that the latter result h olds o nly f or the trid iagonal ca se. As an alternative to deriving exact analy tical results, som e works focus on extrac ting parameter s that characterize the channel cap acity unde r extreme SNR scen arios (see [3] - [4] fo r more details on the extreme SNR ch aracterization ). The low-SNR regime is character ized thro ugh the minimum transmit E b / N 0 that enables reliable commun ications, i.e., E b / N 0 min , an d the low-SNR spectral efficiency slope S 0 . As- suming full r eceiv er CSI and no user cooperatio n, it is shown in [16] that the d eriv ation of the low-SNR parameter s r educes to the calculation of tr “ E ( H † N H N ) ” and tr „ E “ H † N H N ” 2 « . For example, the low-SNR param eters for the capacity o f the W yn er setu p ar e given for N → ∞ b y [17] E b N 0 min = log 2 m 2 (1 + 2 α 2 ) S 0 = 2 K (1 + 2 α 2 ) 2 K + K − 1 + 4(1 + K ) α 2 + 2( K + 2 K ) α 4 , (12) where the kurtosis of an in dividual fadin g coef ficient is defined as K = m 4 / ( m 2 ) 2 . Th is re sult can b e extend ed in a straight- forward yet tediou s mann er to gen eral finite-ba nd matrice s. The high -SNR regime is chara cterized throug h the high- SNR slope S ∞ (also referr ed to as the “mu ltiplexing gain”) and the high -SNR power offset L ∞ . Recently [1], the per- cell capacity high -SNR parameter s for a two diagonal H N ( K = 1 , α = 1 , and β = 0 ) were calculated for N → ∞ and rather genera l fading distributions: S ∞ = 1 ; L ∞ = − 2 max ( E π a log 2 | x | , E π b log 2 | x | ) . (13) The main idea is to lin k the spectral pr operties of H N H † N with the expo nential growth of the elements of its eig en vectors. Since H N H † N in this case is an Hermitian Jacobi matrix , and hence is tridiago nal, its eigenv ector s can be consider ed to be sequences with second or der linear recu rrence. Th erefor e, the problem reduce s to the study of the expo nential g rowth of produ cts of two by two matrices. Th is is closely related to the ev aluation of the to p Lyapunov expon ent of the p rodu ct; The explicit link between the Shanno n transfor m (3) an d the top L yap unov exponen t is the Tho uless formula [18]. Moreover, for arb itrary finite K , it is shown in [1] that S ∞ = 1 while th e power offset is b ound ed by a seq uence o f explicit uppe r- and lower -bou nds; the gap between the lower an d the u pper bounds decreases with the boun ds’ ord er an d co mplexity . It is n oted that calculating exact expressions for the high-SNR parameters of chann els with general fading distribution an d arbitrary finite K rem ains an ope n prob lem even for the tridiagonal case. In addition, (1 3) also fu rther suppo rts the conjecture made r egarding the depend ency of the lim iting spectr um o f finite-band matrices o n th eir en tries’ d istribution. Recently [ 13], the limiting spectru m of 1 1+2 α 2 H N H † N for the W yner setup an d c omplex Gaussian vector s, has b een loosely shown by fr ee pr o bability tools to be ap prox imated by the Mar ˘ oenko-Pastur distribution with par ameter K . The approx imation, is shown to fairly well match th e sp ectrum by Monte-Carlo simu lations on ly for r elativ ely large values of α . It shou ld b e emp hasized that such a match is n ot gu aranteed for o ther fading distributions excluding the complex Gaussian distribution (i.e. Rayleigh fading ). A possible reasoning fo r the approxim ation inaccu racy in th e low α regime is that in the extreme case o f α = 0 , the eigenv alues are evidently exponentially d istributed, with no finite suppo rt (in contrast to the M ar ˘ oenko-Pastur distribution). I V . C O N C L U D I N G R E M A R K S The limiting spectru m (or its Shannon tr ansform) of certa in large finite-band Hermitian random matrices is known for a few limited cases and remains an open problem in general. Moreover , even the high-SNR c haracterizatio n of their Shan- non tran sforms is still u nsolved. Due to their sp ecial p ower profile, standard tools from the theory of rand om matrice s cannot be used for th is prob lem. It is co njectured that u nlike “full” rand om matrices, the limiting spectra of finite-band random matrices depend on the actual distribution of their entries. It seems that uncon vention al m ethods such as the method used by Na rula, replacing the role o f eigenv alues with the diag onal elements of the Cho lesky d ecompo sition, are requir ed to shed light on this problem . Nevertheless, it is noted th at the tri- diagon al (Jacobi matrices) case is un ique and these techniques may not apply t o general finite-band matrices. Finally , we note that solving the p roblem w ould facilitate analytical treatment, which in turn gains much insight into the effect of key system parameters on the perf ormanc e of ce rtain cellular uplink ch annels and time varying ISI ch annels. A C K N O W L E D G M E N T The researc h was supported in par t by a M arie Curie Outg o- ing Intern ational Fellowship and th e NEWCOM++ network of excellence both within the 6th Eu ropean Community Fr ame- work Programme , by the U.S. Natio nal Science Foundation under Grants CNS-06-256 37 and CNS-06- 2661 1, an d also by the REMON Con sortium. R E F E R E N C E S [1] N. Levy , O. S omekh, S. Shamai, and O. Z eitouni , “On certain large random hermit ian jacobi matric es with appl ications to wirele ss commu- nicat ions. ” Submitted to the IEE E T rans. Inform. Theory , Oct. 2007. [2] A. M. Tulino and S. V erd ´ u, “Random m atrix theory and wireless communicat ions, ” in F oundati ons and T rends in Communicati ons and Informatio n Theory , v ol. 1, (Han over , MA, USA), Now Publishers, 2004. [3] S. Shamai (Shitz) and S. V erd ´ u, “The impact of frequency-fla t fading on the spectral ef ficienc y of CDMA, ” IEEE T rans. Inform. Theory , vol. 47, pp. 1302–1327, May 2001. [4] A. 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