Asymptotic Behaviour of Random Vandermonde Matrices with Entries on the Unit Circle

IEEE TRANSACTIONS ON INF ORMA TION THEOR Y , V OL. 1, NO. 1, J ANU AR Y 2009 1 Asymptotic Beha viour of Random V andermonde Matric es with Entries o n the Unit Circle Øyvind Ryan, Member , I EEE and M ´ erouane Debbah, Senior Member , IEEE Abstract —Analytical methods for ﬁnding moments of random V an dermonde matrices with entries on the u nit circle are d ev el- oped. V andermonde Matrices play an important role i n signal processing and wireless applications su ch as direction of arriv al estimation, precoding, and sp arse sampling theory , just to name a few . Within thi s framework, we extend classical freeness results on random matrices with in dependent, identically distributed (i.i.d.) entries and show that V andermonde structured matrices can b e treated in the same vein with different tools. W e fo cus on v arious types o f matrices, such a s V andermonde matrices with and without uniform phase d istributions, as well as gener- alized V andermonde matrices. In each case, we pro vide explicit expressions of the moments of the associated Gram matrix, as well as more advanced models in volving the V andermonde matrix. Compar isons with classical i.i. d. random matrix theory are provided, and decon v olution r esults ar e discussed. W e re view some applications of the results to the ﬁeld s of signal p rocessing and wireless communications. Index T erms —V andermonde matrices, R andom Matrices, de- con v olution, limiting eigen v alue d istribution, MIMO. I . I N T RO D U C T I O N V andermo nde matrices have for a lo ng time had a central position in signal processing du e to their c onnectio ns with importan t tools in the ﬁeld such as the FFT [1] or Ha damard [2] transforms, to n ame a few . V an dermo nde matrices o ccur frequen tly in many applications, such as ﬁnance [3], signal ar- ray processing [4], [5], [ 6], [7], [8], ARMA pro cesses [9], cog- nitiv e radio [10], security [1 1], wireless commun ications [12], and biology [13], and have been much studied. The applied research h as b een som ewhat temper ed by the fact th at very few theore tical results have been av ailable. A V and ermond e matrix with en tries on the u nit circle has the following fo rm: V = 1 √ N      1 · · · 1 e − j ω 1 · · · e − j ω L . . . . . . . . . e − j ( N − 1) ω 1 · · · e − j ( N − 1) ω L      (1) W e will co nsider the case where ω 1 ,..., ω L are i. i.d., tak ing values in [0 , 2 π ) . Throu ghou t the paper, the ω i will be This project is parti ally sponsored by the projec t BIONET (INRIA) This work wa s supporte d by Alcatel-Luc ent within the Alc atel -Lucent Chai r on ﬂexi ble radio at SUPELEC as well as the ANR project SESAME This paper was presented in part at the 1st W orkshop on Physics-Inspired Parad igms in W irel ess Communications and Networ ks, 2008, Berlin, Germany Øyvind Ryan is with the Centr e of Mathemat ics for Applications, Uni- versi ty of Oslo, P .O. Box 1053 Blindern, NO-0316 Oslo, NOR W A Y , oyvi n- dry@iﬁ.uio.no M ´ erouane Debbah is with SUPELE C, Gif- sur-Yv ette , France, mer- ouane.deb bah@supel ec.fr called phase distributions . V will be u sed only to deno te V andermo nde m atrices with a given p hase distribution, and the dimension s of the V an dermon de matr ices will always be N × L . Known results on V an dermon de matrices are related to the distribution of the d eterminan t [14]. The large major- ity of k nown results on the eig en values o f the associated Gram matr ix conc ern Gaussian matrices [15] or matrices with indepen dent entr ies. V ery few resu lts are available in the literature on matrices whose structure is strong ly related to the V andermo nde case [16], [17]. Known resu lts depen d hea vily on the distribution o f the entr ies, and do not give any hint on the asymp totic behaviour as the m atrices b ecome large. In th e realm of wireless chann el modelin g, [18] has provid ed some insight o n the beh aviour of the eige n values of rand om V andermo nde matrices for a speciﬁc case, without a ny fo rmal proof . In many application s, N and L are q uite large, and we m ay be interested in studyin g th e case where bo th go to ∞ at a giv en r atio, L N → c . Results in the liter ature say very little on the asymptotic behaviour of (1) under this growth con dition. The results, howev er , are well known for other models. Th e factor 1 √ N , as w ell as th e assump tion that the V an dermo nde entries e − j ω i lie on the unit circle, are included in (1) t o ensure that the an alysis will gi ve limiting asymptotic beha viour . W ith out this assumption, the prob lem at hand is more in volved, since the rows of the V ande rmond e matrix w ith th e high est powers would do minate in the ca lculations of the moments fo r large matrices, a nd also grow faster to inﬁnity than the 1 √ N factor in ( 1), mak ing asymp totic analysis difﬁcult. In gen eral, often the mom ents, n ot the mo ments o f the d eterminants, are the qu antities we seek. Results in the literature say also very little on the momen ts of V an dermo nde matrices (however , see [16]), and also o n the mixed mo ments of V an dermo nde matrices an d matrices in depend ent fro m them. This is in contrast to Gau ssian ma trices, where exact exp ressions [19] and their asymptotic behaviour [20] are known throug h the concept o f fr eeness [2 0], which is central for d escribing the mixed mo ments. The fr amework an d resu lts pr esented in this paper are reminiscent of similar results conce rning i.i.d. random matrices [21] which have shed lig ht on the design of m any impor- tant wireless co mmun ication problem s such as CDMA [22], MIMO [23], or OFDM [2 4]. This contribution aim s to do the same. W e will show that, asymptotically , the momen ts of the V and ermon de m atrices d epend on ly o n the ratio c and the ph ase distribution, and have explicit expressions. The IEEE TRANSACTIONS ON INFORMA TION THEOR Y , VOL. 1, NO. 1, JANU ARY 2009 2 expressions are more in volved than what was claimed in [18]. Moments are useful for perfo rming d econv olution. Decon - volution for our purposes will mean retrieving ”mom ents” tr L (( D ( N )) i ) , ..., tr L (( D ( N )) i ) (wh ere D ( N ) are unk nown matrices), from ”mixed moments” of D ( N ) and matrices on the form (1), with L and N large. W e are only able to perform such d econv olution when the D ( N ) are square diagona l matrices independ ent from V . W e will see th at such deconv olution can be very useful in many applications, since the retrieved moments can giv e usefu l informatio n abou t the system under study . Deconv olution has previously been handled in cases where V is replac ed with a Gaussian ma- trix [25], [26], [19], [27]. As will be seen, th e way th e phase distribution inﬂuences these mome nts can b e split into several cases. Uniform phase d istribution play s a cen tral role in that it minimizes the momen ts. When the phase distribution has a ”nice” density (for our pu rposes th is m eans that the density of th e phase distribution is co ntinuo us), a n ice con nection with the mo ments for u niform phase d istribution can be gi ven. When the d ensity o f the phase distribution has singular ities, for instance when it has point masses, it tu rns out that the asymptotics of the mo ments chang e dr astically . W e will also extend our results to gene ralized V anderm onde matrices, i.e. matr ices where the colu mns do not consist of unifor mly distrib uted powers. Such matr ices are important for ap plications to ﬁnance [3]. Th e tools used for stan dard V andermo nde matr ices in this p aper will allow us to ﬁnd the asym ptotic behaviour of m any generalized V ander monde matrices as well. While we pr ovide the f ull computatio n o f lower order moments, we also describe h ow the highe r o rder moments can be computed. T edious evaluation of many integrals is needed for this. It turn s out tha t th e ﬁrst three limit mom ents coincide with th ose of the Mar ˘ chenko Pas tur law [20], [2 8]. For h igher ord er m oments this is not the case, alth ough we state an interesting inequ ality in volving the V andermond e limit moments and the m oments of the classical Poisson distribution and the Mar ˘ chenko Pastur law . The p aper is organized as follows. Section I I provides backgr ound ess entials on random matrix theory needed to state the main results. Sectio n III states the main results of the paper . It starts with a gener al result for the mixed mo ments of V andermo nde matrices and matrices indep endent fro m them. Results for the unifor m p hase distribution are stated next, bo th fo r the asym ptotic moments, and the lower ord er moments. Af ter this, the nice con nection b etween u niform phase distribution an d o ther phase distributions is stated. The case where th e density of ω has singu larities is then h andled. The section ends with r esults on generalized V an dermo nde matrices, and m ixed moments of (mor e than one) inde penden t V andermo nde matrices. Section IV discusses our results a nd puts them in a g eneral deconv o lution persp ectiv e, comp aring with oth er deconv olution results, such as those for Gaussian deconv olution. Section V presents some simulations and usefu l applications showing the implication s of th e presented results in v arious applied ﬁelds, and discusses the validity of the asymptotic claims in the ﬁnite regime. First we ap ply th e presented V anderm onde deco n volution fr amew ork to wireless systems, wh ere we estimate the number of paths, th e tra nsmis- sions powers of the users, the n umber of sources, and what is com monly referr ed to as wav elength. Finally we ap ply the results o n V and ermon de matrices to th e very active ﬁeld o f sparse signal reco nstruction . I I . R A N D O M M AT R I X BA C K G RO U N D E S S E N T I A L S In the fo llowing, upper (lower boldface) symbols will be used for matrices (column vectors), wher eas lo wer symbols will repr esent scalar values, ( . ) T will deno te transpo se opera- tor , ( . ) ⋆ conjuga tion, and ( . ) H =  ( . ) T  ⋆ hermitian transpo se. I L will r epresent the L × L identity matr ix. W e let T r b e the (non- normalized ) trace fo r square matrice s, deﬁned by , T r ( A ) = L X i =1 a ii , where a ii are the d iagonal elements of the L × L matrix A . W e also let tr L be the n ormalized trace, d eﬁned by tr L ( A ) = 1 L T r ( A ) . Results in random matrix th eory o ften refer to the em pirical eigenv a lue distribution o f matric es: Deﬁnition 1: W ith th e empirical eige n value d istribution of an L × L h ermitian random matrix T we mean the ( random ) function F L T ( λ ) = # { i | λ i ≤ λ } L , (2) where λ i are the (r andom) eigenv a lues of T . In the follo wing, D r ( N ) , 1 ≤ r ≤ n will denote no n- random d iagonal L × L matrices, wher e we im plicitly assume that L N → c . W e will assume that the D r ( N ) ha ve a joint limit distribution as N → ∞ in the following sense: Deﬁnition 2: W e will say that the { D r ( N ) } 1 ≤ r ≤ n have a joint limit distribution as N → ∞ if the limit D i 1 ,...,i s = lim N →∞ tr L ( D i 1 ( N ) · · · D i s ( N )) (3) exists f or all cho ices of i 1 , ..., i s ∈ { 1 , .., n } . The matrices D i ( N ) are assumed to be non -rando m throug hout th e pap er . Howev er , all presented fo rmulas extend naturally to the case when D i ( N ) are random and independen t from the V an dermo nde m atrices. The difference when the D i ( N ) ar e random is th at expectations of prod ucts of traces also co me into play , in the sense that, sim ilarly to expr essions on the form ( 3), expressions of the form lim N →∞ E [ tr L ( D i 1 ( N ) · · · D i s ( N )) · · · tr L ( D j 1 ( N ) · · · D j r ( N )) ] also enter the pictu re. Our f ramework can a lso b e extended naturally to com pute the covariance o f traces, deﬁne d in the following way: Deﬁnition 3: By the covariance C i,j of two traces tr L ( A i ) and tr L ( A j ) of an L × L random m atrix A , we mean the quantity C i,j ( A ) = E  tr L  A i  tr L  A j  − E  tr L  A i  E  tr L  A j  . (4) IEEE TRANSACTIONS ON INFORMA TION THEOR Y , VOL. 1, NO. 1, JANU ARY 2009 3 When A is replaced with an ensemble of ma trices, A L , the limits lim L →∞ LC i,j ( A L ) are also called secon d order moments. The nor malizing factor L is include d in order to o btain a limit. It will be explained later why this is the cor- rect n ormalizing factor for the matrices we conside r . The term seco nd ord er momen t is taken from [29], where dif- ferent matrix ensembles w ere considere d. For the se m a- trices, th e second order m oments were instead deﬁne d as lim L →∞ L 2 C i,j ( A L ) (i.e. a hig her orde r normalizin g factor was used), since these matrices displayed faster conv ergence to a limit. W e will p resent expressions fo r the secon d o rder moments lim N →∞ LC i,j ( D ( N ) V H V ) . Most th eorems in this paper will present expr essions for various mixed mome nts, de ﬁned in the following way: Deﬁnition 4: By a mixed mome nt we mean th e limit M n = lim N →∞ E [ tr L ( D 1 ( N ) V H VD 2 ( N ) V H V · · · × D n ( N ) V H V )] , (5) whenever th is exists. A join t limit d istribution of { D r ( N ) } 1 ≤ r ≤ n is always assumed in the presented results o n mixed moments. Note that when D 1 ( N ) = · · · = D n ( N ) = I L , th e M n compute to the asymptotic moments of the V andermonde matrices themselves, deﬁned by V n = lim N →∞ E h tr L   V H V  n i = lim N →∞ E  Z λ n dF L V H V ( λ )  . Similarly , whe n D 1 ( N ) = · · · = D n ( N ) = D ( N ) , we will also write D n = lim N →∞ tr L ( D ( N ) n ) . (6) Note th at this is in conﬂict with the notatio n D i 1 ,...,i s , but the name of the index will resolve such con ﬂicts. T o p rove the results of th is paper, the r andom matrix concepts p resented up to now need to be extended using concepts from partition theory . W e denote by P ( n ) the set o f all partitions of { 1 , ..., n } , and u se ρ as notation for a partition in P ( n ) . Also, we will write ρ = { W 1 , ..., W k } , wher e W j will be used repeatedly to denote the block s of ρ , | ρ | = k will denote the num ber of b locks in ρ , and | W j | will den ote the number o f elemen ts in a given block . Deﬁnition 2 can now be extended as follows. Deﬁnition 5: For ρ = { W 1 , ..., W k } , with W i = { w i 1 , ..., w i | W i | } , we deﬁn e D W i = D i w i 1 ,...,i w i | W i | (7) D ρ = k Y i =1 D W i . (8) T o b etter understand the presented expre ssions fo r mixed moments, the n otion of f ree cumulants will be helpfu l. They are deﬁned in terms of n oncro ssing pa rtitions [ 30]. Deﬁnition 6: A partition ρ is called non crossing if, wh en- ev er we hav e i < j < k < l with i ∼ k , j ∼ l ( ∼ meanin g belongin g to the same block), we also have i ∼ j ∼ k ∼ l (i.e. i, j, k , l are all in th e same block) . Th e set o f non crossing partitions of { 1 , , , ., n } is denoted N C ( n ) . The noncrossing partitions ha ve alread y sho wn th eir use- fulness in expressing what is called the free ness relation in a particularly nice way [30]. Deﬁnition 7: Assume that A 1 , ..., A n are L × L -rando m matrices. By the free cu mulants of A 1 , ..., A n we mean the unique set of multilinear func tionals κ r ( r ≥ 1 ) which satisfy E [ tr L ( A i 1 · · · A i n )] = X ρ ∈ N C ( n ) κ ρ [ A i 1 , ..., A i n ] (9) for all cho ices of i 1 , ..., i n , where κ ρ [ A i 1 , ..., A i n ] = k Y j =1 κ W j [ A i 1 , ..., A i n ] κ W i [ A i 1 , ..., A i n ] = κ | W i | [ A i w i 1 , ..., A i w i | W i | ] , where ρ = { W 1 , ..., W k } , with W i = { w i 1 , ..., w i | W i | } . By the classical cum ulants o f A 1 , ..., A n we mean the unique set of multilinear f unctiona ls which satisfy (9) with N C ( n ) replaced by the set of all p artitions P ( n ) . W e ha ve restricted our deﬁnition of cumula nts to r andom matrices, althou gh th eir gener al de ﬁnition is in term s of more general probability space s (Lec ture 11 of [30]). (9) is also called th e (fr ee or classical) m oment-cu mulant formula. The importan ce o f the fr ee mom ent-cum ulant for mula comes f rom the fact that, had we replaced V andermond e matrices with Gaussian m atrices, it cou ld h elp us per form dec onv o lution. For this, the cumulan ts of the Gaussian m atrices are n eeded, which asymptotica lly ha ve a very nice form . F or V andermo nde matrices, it is not known what a usefu l deﬁnition of cumulants would be. Ho wev er , fro m the calculations in Appe ndix A, it will turn ou t that the following quantities are helpf ul. Deﬁnition 8: For ρ ∈ P ( n ) , deﬁne K ρ,ω , N = 1 N n +1 −| ρ | × R (0 , 2 π ) | ρ | Q n k =1 1 − e jN ( ω b ( k − 1) − ω b ( k ) ) 1 − e j ( ω b ( k − 1) − ω b ( k ) ) dω 1 · · · dω | ρ | , (10) where ω W 1 , ..., ω W | ρ | are i.i. d. (ind exed by the blocks o f ρ ), all with th e sam e d istribution as ω , and where b ( k ) is the block of ρ which co ntains k (notation is cyclic, i.e. b (0) = b ( n ) ). If the limit K ρ,ω = lim N →∞ K ρ,ω , N exists, then it is called a V andermonde mixed m oment expan- sion coefﬁcient . These quantities do not b ehave exactly as cu mulants, but rather as weights which tell us how a p artition in the mo ment formu la we present sh ould be weighted. In this respect our formu las for the moments are different fro m classical or free m oment-c umulant formula s, since these do not perf orm this weigh ting. The limits K ρ,ω may not always exist, and necessary and sufﬁcient cond itions for their existence seem to be hard to ﬁnd . Howe ver , it is easy to p rove from their deﬁnition that they do n ot exist if the density o f ω has singularities (f or instanc e wh en the density has po int masses). On the oth er h and, T heorem 3 will show that they exist wh en the same den sity is continuo us. IEEE TRANSACTIONS ON INFORMA TION THEOR Y , VOL. 1, NO. 1, JANU ARY 2009 4 P ( n ) is equ ipped with the reﬁn ement order ≤ [3 0], i.e. ρ 1 ≤ ρ 2 if and only if any block of ρ 1 is contained within a block of ρ 2 . The partition with n blo cks, denoted 0 n , is the smallest partition within this ord er , while the partition with 1 block, deno ted 1 n , is the largest par tition within this ord er . In the following sections, we will en counter the com plementatio n map of Kreweras (p. 14 7 of [30]), whic h is an order-reversing isomorph ism of N C ( n ) onto itself. T o deﬁne th is we need the circular rep resentation of a p artition: W e mark n equidistan t points 1 , ..., n (n umber ed clockwise) on the circle, and f orm the conv ex hull of points lying in the same block of the partition. This gi ves u s a number o f con vex sets H i , equally many as there are block s in the partition, which do not intersect if an d only if the par tition is noncro ssing. Put names ¯ 1 , ..., ¯ n on the midpo ints of the 1 , ..., n (so th at ¯ i is the mid point of the segment fr om i to i + 1 ). The com plement of the set ∪ i H i is again a u nion of disjoint conve x sets ˜ H i . Deﬁnition 9: The Kre weras co mplemen t of ρ , den oted K ( ρ ) , is deﬁn ed as the partition on { ¯ 1 , ..., ¯ n } determin ed b y i ∼ j in K ( ρ ) ⇐ ⇒ ¯ i, ¯ j b elong to the same con ve x set ˜ H k . An importan t prop erty of the Kreweras comp lement is that (p. 148 o f [30]) | ρ | + | K ( ρ ) | = n + 1 . (11) I I I . S TA T E M E N T O F M A I N R E S U LT S W e ﬁrst state the main result of th e paper, which applies to V andermo nde matrices with any p hase distribution. I t restricts to the case when the expan sion coefﬁcients K ρ,ω exist. Differ - ent versions of it adapted to d ifferent V ander monde ma trices will be stated in succeeding sections. Theor em 1 : Assume that the { D r ( N ) } 1 ≤ r ≤ n have a joint limit distribution as N → ∞ . Assume also that all V a nder- monde mixed mom ent expansion coefﬁcients K ρ,ω exist. Then the limit M n = lim N →∞ E [ tr L ( D 1 ( N ) V H VD 2 ( N ) V H V · · · × D n ( N ) V H V )] (12) also exists when L N → c , and equ als X ρ ∈ P ( n ) K ρ,ω c | ρ |− 1 D ρ . (13) The p roof o f Th eorem 1 can be found in Appendix A. Theorem 1 explains ho w ”conv olution” with V andermond e matrices can be perfo rmed, and also provides us with an extension o f the concep t of fr ee conv o lution to V an dermo nde matrices. It also gives us mean s for perfo rming deco nv o lution. Indeed , sup pose D 1 ( N ) = · · · = D n ( N ) = D ( N ) , and that one knows all the moments M n . One can then in fer on the moments D n by insp ecting (13) for increasing values of n . For in stance, the ﬁrst two equations can also be written D 1 1 = M 1 K 1 1 ,ω D 1 2 = M 2 − cK 0 2 ,ω D 0 2 K 1 2 ,ω , where we have used (8), that the one-block partition 0 1 = 1 1 is the o nly partition of length 1 , and that the two-block partition 0 2 and the o ne-block partition 1 2 are the on ly partitio ns o f length 2 . This gives us the ﬁrst mo ments D 1 and D 2 deﬁned by (6), since D 1 1 = D 1 , D 0 2 = D 2 1 , and D 1 2 = D 2 . A. Uniformly distributed ω For th e case of V andermon de matrices with uniform phase distribution, it turns o ut th at the noncr ossing par titions play a central role. The role is some what dif ferent th an the relation for fre eness. L et u denote the unifor m distribution on [0 , 2 π ) . Pr op osition 1: The V an dermo nde mixed moment expansion coefﬁcient K ρ,u = lim N →∞ K ρ,u,N exists for all ρ . Mo reover , 0 < K ρ,u ≤ 1 , th e K ρ,u are ration al number s for all ρ , and K ρ,u = 1 if and only if ρ is noncrossing. The proo f of Pr oposition 1 can b e fou nd in Appendix B. The same r esult is proved in [16], wh ere the K ρ,u are giv en an equiv alent descrip tion. The pr oof in the appendix o nly translates th e result in [16] to the current no tation. Due to Proposition 1, Theo rem 1 guar antees that the mixed momen ts (12) exist in the limit for the u niform phase distrib ution, and are given by (13). T he K ρ,u are in general ha rd to co mpute for higher order ρ with cr ossings. It tu rns out th at the following computatio ns sufﬁce to obtain the 7 ﬁrst mo ments. Pr op osition 2: The following holds: K {{ 1 , 3 } , { 2 , 4 }} ,u = 2 3 K {{ 1 , 4 } , { 2 , 5 } , { 3 , 6 }} ,u = 1 2 K {{ 1 , 4 } , { 2 , 6 } , { 3 , 5 }} ,u = 1 2 K {{ 1 , 3 , 5 } , { 2 , 4 , 6 }} ,u = 11 20 K {{ 1 , 5 } , { 3 , 7 } , { 2 , 4 , 6 }} ,u = 9 20 K {{ 1 , 6 } , { 2 , 4 } , { 3 , 5 , 7 }} ,u = 9 20 . The proof o f Proposition 2 is given in Appendix C . Com- bining Pro position 1 and Proposition 2 one can prove the following: Pr op osition 3: Assume D 1 ( N ) = · · · = D n ( N ) = D ( N ) , and that the lim its m n = ( cM ) n = c lim N →∞ E h tr L   D ( N ) V H V  n i (14) d n = ( cD ) n = c lim N →∞ tr L ( D n ( N )) . (15) IEEE TRANSACTIONS ON INFORMA TION THEOR Y , VOL. 1, NO. 1, JANU ARY 2009 5 exist. When ω = u , we h ave that m 1 = d 1 m 2 = d 2 + d 2 1 m 3 = d 3 + 3 d 2 d 1 + d 3 1 m 4 = d 4 + 4 d 3 d 1 + 8 3 d 2 2 + 6 d 2 d 2 1 + d 4 1 m 5 = d 5 + 5 d 4 d 1 + 25 3 d 3 d 2 + 10 d 3 d 2 1 + 40 3 d 2 2 d 1 + 10 d 2 d 3 1 + d 5 1 m 6 = d 6 + 6 d 5 d 1 + 12 d 4 d 2 + 15 d 4 d 2 1 + 151 20 d 2 3 + 50 d 3 d 2 d 1 + 20 d 3 d 3 1 + 11 d 3 2 + 40 d 2 2 d 2 1 + 15 d 2 d 4 1 + d 6 1 m 7 = d 7 + 7 d 6 d 1 + 49 3 d 5 d 2 + 21 d 5 d 2 1 + 497 20 d 4 d 3 + 84 d 4 d 2 d 1 + 35 d 4 d 3 1 + 1057 20 d 2 3 d 1 + 693 10 d 3 d 2 2 + 17 5 d 3 d 2 d 2 1 + 35 d 3 d 4 1 + 77 d 3 2 d 1 + 280 3 d 2 2 d 3 1 + 21 d 2 d 5 1 + d 7 1 . Proposition 3 is proved in Appe ndix D. Sev eral of th e following theorems will also be stated in term s o f th e scale d moments (1 4)-(15), rather th an M n , D n . The reason for this is that th e depen dency o n the m atrix aspect ratio c can b e absorbed in m n , d n , so that the result itself can be expressed indepen dently of c , as in the equations of Pro position 3. The same usag e of scaled mo ments has b een applied fo r large W ish art m atrices [27]. Similar com putation s to those in the proof o f Pro position 3 are p erform ed in [ 16], although the computatio ns th ere do not go up as high as th e ﬁrst seven mixed mo ments. T o compu te hig her order m oments, K ρ,u must be comp uted for partitions of higher ord er also. The computatio ns perfor med in Appen dix C and D should con vince the read er th at this can b e done, but that it is very ted ious. Follo wing the pr oof of Proposition 1, we can also obtain formu las fo r the second order mo ments of V anderm onde matrices. Since it is easily seen that C 1 ,n ( D ( N ) V H V ) = C n, 1 ( D ( N ) V H V ) = 0 , the ﬁrst n ontrivial second ord er moment is th e following: Pr op osition 4: Assume that V has un iform phase distribu- tion, let d n be as in ( 15), and deﬁne m i,j = c lim L →∞ LC i,j  D ( N ) V H V )  . (16) Then we h av e that m 2 , 2 = d 4 + 4 d 3 d 1 4 3 d 2 2 + 4 d 2 d 2 1 . (17) Proposition 4 is proved in App endix E, and r elies on the same type o f calculations as tho se in Appen dix C. Following the pro of of Pro position 1 again, we c an also obtain exact expressions for moments of lower orde r rand om V anderm onde matrices with u niform pha se distribution, no t only the limit. W e state these on ly fo r the ﬁrst f our moments. Theor em 2 : Assume D 1 ( N ) = D 2 ( N ) = · · · = D n ( N ) , set c = L N , and d eﬁne m ( N ,L ) n = cE h tr L   D ( N ) V H V  n i (18) d ( N ,L ) n = ctr L ( D n ( N )) . (19) When ω = u we h av e th at m ( N ,L ) 1 = d ( N ,L ) 1 m ( N ,L ) 2 =  1 − N − 1  d ( N ,L ) 2 + ( d ( N ,L ) 1 ) 2 m ( N ,L ) 3 =  1 − 3 N − 1 + 2 N − 2  d ( N ,L ) 3 +3  1 − N − 1  d ( N ,L ) 1 d ( N ,L ) 2 + ( d ( N ,L ) 1 ) 3 m ( N ,L ) 4 =  1 − 20 3 N − 1 + 12 N − 2 − 19 3 N − 3  d ( N ,L ) 4 +  4 − 12 N − 1 + 8 N − 2  d ( N ,L ) 3 d ( N ,L ) 1 +  8 3 − 6 N − 1 + 10 3 N − 2  ( d ( N ,L ) 2 ) 2 +6  1 − N − 1  d ( N ,L ) 2 ( d ( N ,L ) 1 ) 2 + ( d ( N ,L ) 1 ) 4 . Theorem 2 is pr oved in Ap pendix F. Exa ct formulas for the higher ord er mom ents also exist, but they becom e increasing ly complex, as higher order ter ms N − k also en ter the picture. These for mulas are a lso ha rder to prove for highe r orde r moments. In many cases, exact expressions are not what we need: ﬁrst order appr oximation s (i.e. expressions where only the N − 1 -terms are included ) can sufﬁce for m any purp oses. In Appendix F, w e explain how the simpler case of these ﬁrst order approx imations can be compu ted. It seems much har der to prove a similar result when the phase distribution is not unifor m. An im portant r esult building on the results we present is the following, which provid es a major difference f rom the limit eigenv a lue distributions of Gaussian matrices. Pr op osition 5: The asymptotic mean eigenv alue distrib u- tion of a V a ndermo nde matrix with u niform phase distribution has unb ound ed support. Proposition 5 is p roved in App endix G . B. ω with con tinuou s density The following re sult tells us that the limit K ρ,ω exists for many ω , and also gives a useful expr ession for th em in terms of K ρ,u and the d ensity o f ω . Theor em 3 : The V a ndermo nde mixed mome nt expan sion coefﬁcients K ρ,ω = lim N →∞ K ρ,ω , N exist when ev er the density p ω of ω is contin uous on [0 , 2 π ) . If this is f ulﬁlled, then K ρ,ω = K ρ,u (2 π ) | ρ |− 1  Z 2 π 0 p ω ( x ) | ρ | dx  . (20) The proof is given in Append ix H. Althoug h the proo f assumes a con tinuous density , we rem ark that it can be generalized to cases where the density contains a ﬁnite set of jump discon tinuities also. In Section V, se vera l exam ples are provided where the in tegrals ( 20) are c omputed . An importa nt consequen ce of Theorem 3 is the fo llowing, which giv es the unifor m phase distribution an impo rtant role. IEEE TRANSACTIONS ON INFORMA TION THEOR Y , VOL. 1, NO. 1, JANU ARY 2009 6 Pr op osition 6: Let V ω denote a V ander monde matrix with phase distribution ω , and set V ω ,n = lim N →∞ E h tr L   V H ω V ω  n i . Then we h av e that V u,n ≤ V ω ,n . The proof is gi ven in Appen dix I. An immed iate conse- quence of this and Propo sition 5 is tha t all phase distributions, not on ly unif orm phase distribution, giv e V andermo nde m atri- ces with unbou nded mean eig en value distributions in the limit. Besides providing us with a deconv olution m ethod f or ﬁnd ing the mixed mom ents of the { D r ( N ) } 1 ≤ r ≤ n , The orem 3 also provides us w ith a way o f inspecting th e p hase distribution ω , by ﬁrst ﬁnding the mo ments of the den sity , i.e. R 2 π 0 p ω ( x ) k dx . Howe ver , note that we can n ot expec t to ﬁnd the density o f ω itself, o nly the density of the density of ω . This follows immediately by no ting that R 2 π 0 p ω ( x ) k dx r emains unchan ged when the phase distribution ω is cyclically shif ted. C. ω with density singularities The asympto tics of V an dermo nde matrices are different when the density of ω h as singular ities, and depen ds o n the density g rowth rates n ear the singu lar po ints. It will be clear from the followi ng r esults th at one can n ot perform deconv olution fo r such ω to obtain the higher or der moments of th e { D r ( N ) } 1 ≤ r ≤ n , as only their ﬁrst moment can be obtained. The asymptotics are ﬁrst described for ω with atomic density sin gularities, as this is the simplest case to p rove. After this, den sities with polyn omic growth rate s n ear the singularities are add ressed. Theor em 4 : Assume that p ω = P r i =1 p i δ α i is ato mic (where δ α i ( x ) is dirac measu re (point m ass) at α i ), and denote by p ( n ) = P r i =1 p n i . Then lim N →∞ E [ T r ( D 1 ( N ) 1 N V H VD 2 ( N ) 1 N V H V · · · × D n ( N ) 1 N V H V )] = c n − 1 p ( n ) lim N →∞ n Y i =1 tr L ( D i ( N )) . Note here that th e non-n ormalized trace is u sed. The proof can be found in Append ix J . In p articular, Theorem 4 states that th e asym ptotic momen ts of 1 N V H V can be co mputed fro m p ( n ) . The theore m is of g reat im portance for the estimation of the po int masses p i . In blind seismic and telecomm unication app lications, on e would like to de tect the lo cations α i . Un fortun ately , Theorem 4 tells us that th is is impossible with our d econv olution framework, since th e p ( n ) , which a re the q uantities we can ﬁn d thr ough deco n volution, have no depend ency to them. This parallels Theorem 3, since also th ere we could not recover the density p ω itself. Having found th e p ( n ) throug h d econv olution, o ne can ﬁnd th e poin t masses p i , by solving for p 1 , p 2 , ... in th e V an dermo nde equation    p 1 p 2 · · · p r p 2 1 p 2 2 · · · p 2 r . . . . . . . . . . . .       1 1 . . .    =    p (1) p (2) . . .    . The case wh en the density has non-a tomic singularities is more complicated. W e provide only the following result, which addresses the case when the density has polynomic gro wth rate near the singular ities. Theor em 5 : Assume that lim x → α i | x − α i | s p ω ( x ) = p i for some 0 < s < 1 for a set of points α 1 , ..., α r , with p ω continuo us f or ω 6 = α 1 , ..., α r . Then lim N →∞ E [ T r ( D 1 ( N ) 1 N s V H VD 2 ( N ) 1 N s V H V · · · × D n ( N ) 1 N s V H V )] = c n − 1 q ( n ) lim N →∞ n Y i =1 tr L ( D i ( N )) where q ( n ) =  2Γ(1 − s ) cos  (1 − s ) π 2  n p ( n ) × R [0 , 1] n Q n k =1 1 | x k − 1 − x k | 1 − s dx 1 · · · dx n , (21) and p ( n ) = P i p n i . Note here that the non -norm alized trace is used. The proof can be fo und in Appendix K. Also in this case it is only the poin t masses p i which can be fou nd through deconv olution, not the locations α i . Note that the integral in (21) can also b e w ritten as an m -f old co n volution. Similarly , the deﬁnition of K ρ,ω , N giv en by (1 0) c an also be viewed as a 2 -fold con v olution when ρ has two bloc ks, and as a 3 -fo ld conv o lution when ρ has th ree blocks (but not for ρ with mo re than 3 block s). A useful app lication of Th eorem 5 we will return to is th e case when ω = k sin( θ ) f or some constant k ( see (39)), with θ unifor mly distributed on some interval. This case is simulated in Section V -A. It is apparent fro m (40) that the density goes to inﬁnity near ω = ± k , with rate x − 1 / 2 . Th eorem 5 thus ap plies with s = 1 / 2 . For this case, howev er , the ”ed ges” a t ± π / 2 are never reached in practice. Indeed, in a rray p rocessing [3 1], th e antenna array is a sector anten na which scans an a ngle inter val which never inclu des the ed ges. W e can therefor e restrict ω in our analysis to clusters of in tervals [ α i , β i ] not co ntaining ± 1 , for which th e results of Sectio n III-B sufﬁce. In this way , we also av oid the compu tation of the cu mbersom e integral (21). D. Generalized V andermon de matrices W e will co nsider g eneralized V andermo nde matrices o n the form V = 1 √ N      e − j ⌊ N f (0) ⌋ ω 1 · · · e − j ⌊ N f (0) ⌋ ω L e − j ⌊ N f ( 1 N ) ⌋ ω 1 · · · e − j ⌊ N f ( 1 N ) ⌋ ω L . . . . . . . . . e − j ⌊ N f ( N − 1 N ) ⌋ ω 1 · · · e − j ⌊ N f ( N − 1 N ) ⌋ ω L      , (22) IEEE TRANSACTIONS ON INFORMA TION THEOR Y , VOL. 1, NO. 1, JANU ARY 2009 7 where f is called th e power distribution , and is a fun ction from [0 , 1) to [0 , 1) . W e will also consider the m ore general case when f is replaced with a random variable λ , i. e. V = 1 √ N      e − j N λ 1 ω 1 · · · e − j N λ 1 ω L e − j N λ 2 ω 1 · · · e − j N λ 2 ω L . . . . . . . . . e − j N λ N ω 1 · · · e − j N λ N ω L      , (23) with the λ i i.i.d. and distributed as λ , deﬁned and taking v a lues in [0 , 1) , an d also indepen dent fro m the ω j . W e will deﬁne mixed m oment expa nsion coefﬁcients for generalized V ander monde matrices also. The difference i s that, while we in Deﬁnition 8 simpliﬁed using the geometric sum formu la, we can not do th is now since we do n ot assume unifor m power distribution a nymore. T o d eﬁne expansio n coefﬁcients for g eneralized V and ermon de matrices of the for m (22), deﬁn e ﬁrst integer functio ns f N from [0 , N − 1 ] to [0 , N − 1 ] by f N ( r ) = ⌊ N f  r N  ⌋ . Let p f N be th e correspon d- ing density f or f N . The procedu re is similar for m atrices of the for m ( 23). The following d eﬁnition captures bo th cases: Deﬁnition 10: For (22) and (23 ), deﬁne K ρ,ω , f ,N = 1 N 1 −| ρ | × R (0 , 2 π ) | ρ | Q n k =1  P N − 1 r =0 p f N ( r ) e j r ( ω b ( k − 1) − ω b ( k ) )  dω 1 · · · dω | ρ | K ρ,ω , λ,N = 1 N 1 −| ρ | × R (0 , 2 π ) | ρ | Q n k =1  R 1 0 N e j N λ ( ω b ( k − 1) − ω b ( k ) ) dλ  dω 1 · · · dω | ρ | , (24) where ω W 1 , ..., ω W | ρ | are as in Deﬁn ition 8. I f the limits K ρ,ω , f = lim N →∞ K ρ,ω , f ,N K ρ,ω , λ = lim N →∞ K ρ,ω , λ,N , exist, the n they are called V a ndermond e mixed moment expan- sion coefﬁcients . Note that (1) corr esponds to (22) with f ( x ) = x . The following result ho lds: Theor em 6 : Theor em 1 holds also with V andermon de m a- trices (1 ) r eplaced with g eneralized V and ermon de matrices on either form (22) or (23), and with K ρ,ω replaced with either K ρ,ω , f or K ρ,ω , λ . The proof follows the same lines as those in Appen dix A, and is therefore only explained brieﬂy at the end of that append ix. As for matrices o f the form (1 ), it is the case of uniform phase distribution which is most easily described how to c ompute for generalized V andermon de matrices also. Append ix B sho ws how the computation of K ρ,u boils down to comp uting ce rtain integrals. The sam e comm ents are v alid for matrices of th e form (22) o r (23) in ord er to co mpute K ρ,ω , f and K ρ,ω , λ . This is fur ther commented at the end of that app endix. W e will not consid er gen eralized V an dermon de matrices with den sity singularities. E. The joint distribution of independent V a ndermond e matri- ces When many independent random V a ndermo nde matrices are in volv ed, the following holds: Theor em 7 : Assume that the { D r ( N ) } 1 ≤ r ≤ n have a joint limit distribution as N → ∞ . Assum e also that V 1 , V 2 , ... are ind epend ent V an dermo nde m atrices with the sam e ph ase distribution ω , and that the density of ω is con tinuous. Th en the limit lim N →∞ E [ tr L ( D 1 ( N ) V H i 1 V i 2 D 2 ( N ) V H i 2 V i 3 · · · × D n ( N ) V H i n V i 1 )] also exists wh en L N → c . The limit is 0 when n is odd, and equals X ρ ≤ σ ∈ P ( n ) K ρ,ω c | ρ |− 1 D ρ , (25) where σ = { σ 1 , σ 2 } = { { 1 , 3 , 5 , ..., } , { 2 , 4 , 6 , ... }} is the partition wh ere th e two blo cks ar e the even num bers, and the odd numbe rs. The proof of Theorem 7 can be foun d in Ap pendix L. That append ix also contains so me remarks on the case when the matrices D i ( N ) are placed at different position s relative to the V andermo nde matrices. From Theo rem 7 , the following corollary is immed iate: Cor ollary 1: T he ﬁrst th ree mixed m oments V (2) n = lim N →∞ E h tr L   V H 1 V 2 V H 2 V 1  n i of indepe ndent V an dermon de matrices V 1 , V 2 are given by V (2) 1 = I 2 V (2) 2 = 2 3 I 2 + 2 I 3 + I 4 V (2) 3 = 11 20 I 2 + 4 I 3 + 9 I 4 + 6 I 5 + I 6 , where I k = (2 π ) k − 1  Z 2 π 0 p ω ( x ) k dx  . In particular, when the p hase distribution is uniform, th e ﬁrst three mixed moments are given by V (2) 1 = 1 V (2) 2 = 11 3 V (2) 3 = 411 20 The results her e can also be extended to the case with indepen dent V andermond e matrice s with different phase dis- tributions: Theor em 8 : Assume that { V i } 1 ≤ i ≤ s are indepen dent V an- dermon de m atrices, whe re V i has c ontinuo us phase distribu- tion ω i . Denote by p ω i the density of ω i . Th en Eq uation ( 25) still holds, with K ρ,ω replaced by K ρ,u (2 π ) | ρ |− 1 Z 2 π 0 s Y i =1 p ω i ( x ) | ρ i | dx, where ρ i consists of all numbers k such that i k = i . The proof is o mitted, as it is a straightfo rward extension of the proof s of Theorem 3 and T heorem 7. IEEE TRANSACTIONS ON INFORMA TION THEOR Y , VOL. 1, NO. 1, JANU ARY 2009 8 I V . D I S C U S S I O N In th e recen t work [1 6], the V andermond e m odel (1) is encoun tered in recon struction of multidimension al signals in wireless sensor networks. The autho rs also recogn ize a similar expression f or the V andermond e mixed mom ent expansion coefﬁcient as in Deﬁnitio n 8. They also state th at, for the case of uniform p hase d istribution, closed form expressions fo r th e moments can be f ound, building on an analysis of partitions and calculation of volumes of convex poly topes d escribed by certain co nstraints. This is very similar to wh at is do ne in th is paper . W e will in th e following discu ss some differences and similarities between Gau ssian and V anderm onde matrices. A. Conver gence rates In [ 19], alm ost sure co n vergence of Gaussian matrices was shown by p roving exact formulas for the d istribution of lower order Gaussian matrices. These de viated from their limits b y terms of order 1 / N 2 . In Theorem 2, we see th at terms of orde r 1 / N are inv o lved. This slower rate of conver gence may n ot b e enoug h to ma ke a statemen t on whether we have alm ost sure conv ergence f or V andermo nde matrices. Howe ver , [32] shows some almo st sure co n vergence properties for certain Hankel and T o eplitz matrices. These matrices are seen in that pap er to have similar combinator ial descriptions for the moments, when compare d to V and ermon de matrices in this paper . Therefore, it may be the case th at the tech niques in [32] can b e gener alized to ad dress almo st sure conver gence of V andermo nde m atrices also. Figu re 1 shows the speed of conv ergence of the moments of V an dermo nde matrices (with un iform ph ase d istribution) tow ards the asym ptotic mom ents as the matrix dimension s grow , and as the number of samples g row . The differences between th e a symptotic moments and the exact moments are also shown. T o be m ore p recise, the MSE values in Figure 1 are compu ted as follo ws: 1) K samples V i are indepen dently g enerated using (1). 2) The 4 ﬁrst sample moments ˆ v j i = 1 L tr n   V H i V i  j  ( 1 ≤ j ≤ 4 ) are co mputed from the samples. 3) The 4 ﬁrst estimated moments ˆ V j are computed as the mean of the sample mo ments, i.e. ˆ V j = 1 K P K i =1 ˆ m j i . 4) The 4 ﬁrst exact m oments E j are com puted using Theorem 2. 5) The 4 ﬁrst asymp totic moments A j are comp uted usin g Proposition 3. 6) The mean squar ed error (MSE) of the ﬁrst 4 esti- mated moments fr om the exact moments is computed as P 4 j =1  ˆ V j − E j  2 . 7) The MSE of the ﬁrst 4 exact mo ments f rom the asymp - totic mom ents is computed as P 4 j =1 ( E j − A j ) 2 . Figure 1 is in sharp co ntrast with Gau ssian matrices, as shown in Figure 2. First of all, it is seen that the asymptotic moments can be used just as well in stead of the exact m oments (for which expressions can be f ound in [33]), due to the O (1 / N 2 ) conv ergence of th e m oments. Secondly , it is seen that only 5 samples were need ed to g et a r eliable estimate for th e moments. 50 100 150 200 250 300 350 400 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 N MSE MSE between exact and asymptotic moments MSE between estimated and exact moments (a) 80 samples 50 100 150 200 250 300 350 400 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 N MSE MSE between exact and asymptotic moments MSE between estimated and exact moments (b) 320 samples Fig. 1. MSE of the ﬁrst 4 estimate d moments from the exac t m oments for 80 and 320 sample s for va rying matrix size s, with N = L . Matrice s are on the form V H V with V a V andermonde matri x with uniform phase distri but ion. The MSE of the ﬁrst 4 e xact moments from the asymptotic moments is al so sho wn. B. Inequ alities between momen ts of V andermonde matrices and moments of kn own distributions W e will state an in equality inv olving the moments of V an- dermon de matrices, an d the mo ments of known distributions. The classical Poisson d istribution with rate λ an d ju mp size α is d eﬁned as the limit of  1 − λ n  δ 0 + λ n δ α  ∗ n as n → ∞ [3 0], where ∗ d enotes classical ( additive) conv o- lution, an d ∗ n deno tes n -fo ld conv olution with itself. For ou r analysis, we will on ly need the classical Poisson distribution with rate c and jum p size 1 , denoted ν c . Th e free Poisson distribution with r ate λ and jump size α is d eﬁned similarly IEEE TRANSACTIONS ON INFORMA TION THEOR Y , VOL. 1, NO. 1, JANU ARY 2009 9 50 100 150 200 250 300 350 400 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 N MSE MSE between exact and asymptotic moments MSE between estimated and exact moments Fig. 2. MSE of the ﬁrst 4 estimated moments from the exact moments for 5 samples for vary ing matrix sizes, with N = L . Matrices are on the form 1 N XX H with X a complex stand ard Gaussian m atrix. The MSE of the ﬁrst 4 exac t moments from the asympto tic moments is also sho wn. as the limit of  1 − λ n  δ 0 + λ n δ α  ⊞ n as n → ∞ , whe re ⊞ is the free pr obability counter part of ∗ [30], [ 20], and where ⊞ n denotes n -fold free conv o lution with itself. F or our analysis, we will only need the free Poisson distribution with rate 1 c and jum p size c , d enoted µ c . µ c is th e same as the better known Mar ˘ chenko Pastur law , i.e. it has the density [20] f µ c ( x ) = (1 − 1 c ) + δ 0 ( x ) + p ( x − a ) + ( b − x ) + 2 π cx , (26) where ( z ) + = max (0 , z ) , a = (1 − √ c ) 2 , b = (1 + √ c ) 2 . Since the classical (free) cumulants of th e c lassical (fr ee) Poisson distribution are λα n [30], we see that th e (classical) c umulants of ν c are c, c, c, c, ... , and that the (f ree) cu mulants of µ c are 1 , c, c 2 , c 3 , ... . In other words, if a 1 has the distribution µ c , then φ ( a n 1 ) = P ρ ∈ N C ( n ) c n −| ρ | = P ρ ∈ N C ( n ) c | K ( ρ ) |− 1 = P ρ ∈ N C ( n ) c | ρ |− 1 . (27) Here we have used the Kreweras co mplementatio n map and (11), with φ d enoting th e expectation in a non- commuta ti ve probab ility spa ce [20]. Also, if a 2 has the d istribution ν c , then E ( a n 2 ) = X ρ ∈ P ( n ) c | ρ | . (28) W e immediately reco gnize the c | ρ |− 1 -entry o f Th eorem 1 in (27) an d (28) (with an addition al power of c in (2 8)). Co mbin- ing Propo sition 1 with D 1 ( N ) = · · · = D n ( N ) = I L , (27), and (28), we thus get the f ollowing cor ollary to Proposition 1: Cor ollary 2: Assume that V has u niform p hase distribu- tion. Then th e limit momen t V n = lim N →∞ E h tr L   V H V  n i satisﬁes the inequ ality φ ( a n 1 ) ≤ V n ≤ 1 c E ( a n 2 ) , where a 1 has the d istribution µ c of the Mar ˘ chenko Pastur law , and a 2 has th e Po isson distribution ν c . I n particular , equa lity occurs f or m = 1 , 2 , 3 a nd c = 1 (since all partitions are noncro ssing for m = 1 , 2 , 3 ). Corollary 2 thus states th at the momen ts of V ander monde matrices with unifor m phase distribution are bound ed above and below b y the momen ts of the classical and free Poisson distributions, respectiv ely . Th e left pa rt of th e inequ ality in Corollary 2 was also ob served in Section VI in [1 6]. The dif- ferent Poisson d istributions enter here becau se their (free and classical) cumulants resem ble the c | ρ |− 1 -entry in Th eorem 1, where we also can u se that K ρ,u = 1 if and only if ρ is noncro ssing to get a con nection with the Mar ˘ chenko Pastur law . T o see how close the asymptotic V an dermon de mo ments are to these upper and lower bo unds, the following corollar y to Proposition 3 co ntains the ﬁrst mo ments: Cor ollary 3: Wh en c = 1 , the limit m oments V n = lim N →∞ E h tr L   V H V  n i , the moments f p n of the Mar ˘ chenko Pastur law µ 1 , and the moments p n of the Poisson d istribution ν 1 satisfy f p 4 = 14 ≤ V 4 = 44 3 ≈ 14 . 67 ≤ p 4 = 15 f p 5 = 42 ≤ V 5 = 146 3 ≈ 48 . 67 ≤ p 5 = 52 f p 6 = 132 ≤ V 6 = 3571 20 ≈ 178 . 55 ≤ p 6 = 203 f p 7 = 429 ≤ V 7 = 2141 3 ≈ 713 . 67 ≤ p 7 = 877 . The ﬁrst th ree momen ts co incide for th e three distributions, and are 1 , 2 , an d 5 , respectively . The numb ers f p n and p n are simply the n umber of partitions in N C ( n ) and P ( n ) , respe ctiv ely . Th e n umber of partitions in N C ( n ) equals the Catalan number C n = 1 n +1  2 n n  [30], and are easily comp uted. T he num ber of partitions of P ( n ) ar e also known as the Bell numbers B n [30]. They can ea sily be co mputed from the recu rrence relation B n +1 = n X k =0 B k  n k  . In Figure 3, the mean e igenv alu e distribution of 6 40 samples of a 16 00 × 1200 (i.e. c = 0 . 75 ) V and ermon de matrix with uniform phase distrib ution is sho wn. While the Poisson distribution ν 1 is p urely atomic and ha s masses at 0 , 1 , 2 , a nd 3 which are e − 1 , e − 1 , e − 1 / 2 , and e − 1 / 6 (the ato ms consist of all in teger m ultiples), the V an dermo nde histogra m shows a more co ntinuou s eigen value distrib ution, with the peaks wh ich the Poisson d istribution has at integer multiples clear ly visible, although not as sharp. W e rem ark that the supp ort of V H V for a ﬁxed N goes all the way u p to N , but lies within [0 , N ] . It is unk nown wh ether the peaks a t integer mu ltiples in the V an dermon de h istogram grow to inﬁnity as we let N → ∞ . From th e histogr am, o nly the p eak at 0 seems to be of atom ic n ature. The effect of dec reasing c amoun ts to stretching the eigenv alue density vertically , and compressing it horizontally , just as the case for the different M ar ˘ chenko Pastur laws. An eigenv alue histogra m fo r Gau ssian matrices IEEE TRANSACTIONS ON INFORMA TION THEOR Y , VOL. 1, NO. 1, JANU ARY 2009 10 0 1 2 3 4 5 6 7 8 9 10 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 λ Density Fig. 3. Histogram of the mean eigen val ue distri but ion of 640 samples of V H V , with V a 1600 × 1200 V andermonde matr ix with uniform phase distrib ution . 0 1 2 3 4 5 6 7 8 9 10 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 λ Density Fig. 4. Histogram of the mean eige n value distribut ion of 20 samples of 1 N XX H , with X an L × N = 1200 × 1600 complex, standard, Gaussian matrix. which in the limit gi ve the correspon ding (in the sense of Corollary 2) Mar ˘ chenko Pastur law f or Figure 3 (i.e. µ 0 . 75 ) is shown in Figu re 4. Figure 5 shows an eig env alu e histogram in th e case of a non-u niform phase d istribution. Here we have taken 64 0 samp les of a 1 600 × 1200 V ander mond e matrix with phase distribution with density (40), with λ = 2 d, α = π 4 . Th is density , also shown in Figu re 6, is used in the applications o f Section V -A. Expe riments show that the eigenv alue histogram tends to ﬂatten when the phase distrib ution beco mes ”less unifor m”, with a higher co ncentra tion of larger eigenv alues. It is unkn own whe ther the inequalities fo r the m oments ca n be exten ded to inequalities fo r the associated ca pacity . If X is an N × N stand ard, complex, Gaussian matrix, then an explicit 0 1 2 3 4 5 6 7 8 9 10 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 λ Density Fig. 5. Histogram of the mean eigen val ue distributi on of 640 samples of V H V , with V a 1600 × 1200 V andermonde matrix with phase distrib utio n p ω deﬁned in (40) with λ = 2 d, α = π 4 . −3 −2 −1 0 1 2 3 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 ω Density Fig. 6. The densit y p ω ( x ) gi ve n by (40), with λ = 2 d, α = π 4 . expression f or the asymp totic cap acity exists [28]: lim N →∞ 1 N log 2 det  I N + ρ  1 N XX H  = 2 log 2  1 + ρ − 1 4  √ 4 ρ + 1 − 1  2  − log 2 e 4 ρ  √ 4 ρ + 1 − 1  2 . (29) In Figure 7(a ), sev eral realizations of th e capac ity are com- puted for Gaussian matrix samples of size 36 × 36 . T he asymptotic capacity (29) is also shown. In Figure 7(b), se veral realizations o f the capacity are computed fo r V and ermond e matrix samp les of the same size, for the ca se of un iform pha se distribution. It is seen that the variance of the V anderm onde capacities is h igher tha n fo r the Gau ssian cou nterparts. This should come as n o surprise, due to the slo wer co n vergence to the asymptotic lim its for V andermo nde matrices. Although the capac ities o f V ande rmond e matrices with un iform phase IEEE TRANSACTIONS ON INFORMA TION THEOR Y , VOL. 1, NO. 1, JANU ARY 2009 11 0 1 2 3 4 5 6 7 8 9 10 0 0.5 1 1.5 2 2.5 3 ρ Capacity Asymptotic capacity sample capacity (a) Realiza tions of 1 N log 2 det ` I N + ρ 1 N XX H ´ when X is standard, comple x, Gaussian. The asymptoti c capacity (29) is also sho wn. 0 1 2 3 4 5 6 7 8 9 10 0 0.5 1 1.5 2 2.5 3 ρ Capacity (b) Realiza tions of 1 N log 2 det ` I N + ρ VV H ´ when ω has uniform phase distrib ution . Fig. 7. Realiza tions of the capacity for Gaussian and V andermond e matrices of size 36 × 36 . distribution and Gaussian ma trices seem to b e close, we have no proof th at the capacities of V an dermon de matrices are e ven ﬁnite due to th e unb ounde dness o f its sup port. C. Deconvolution Deconv olution with V ander monde matrices (as stated in (13) in Theor em 1) d iffers from the Ga ussian dec onv o lution counterp art [30 ] in the sense that there i s no multiplicativ e [30] structure in volved, since K ρ,ω is not multip licativ e in ρ . The Gaussian eq uiv alent of Proposition 3 (i. e. V H V replaced with 1 N XX H , with X an L × N com plex, stand ard, Gaussian matrix) is m 1 = d 1 (30) m 2 = d 2 + d 2 1 (31) m 3 = d 3 + 3 d 2 d 1 + d 3 1 (32) m 4 = d 4 + 4 d 3 d 1 + 2 d 2 2 + 6 d 2 d 2 1 + d 4 1 (33) m 5 = d 5 + 5 d 4 d 1 + 5 d 3 d 2 + 10 d 3 d 2 1 + 10 d 2 2 d 1 + 10 d 2 d 3 1 + d 5 1 (34) m 6 = d 6 + 6 d 5 d 1 + 6 d 4 d 2 + 15 d 4 d 2 1 + 3 d 2 3 + 30 d 3 d 2 d 1 + 20 d 3 d 3 1 + 5 d 3 2 + 10 d 2 2 d 2 1 + 15 d 2 d 4 1 + d 6 1 (35) m 7 = d 7 + 7 d 6 d 1 + 7 d 5 d 2 + 21 d 5 d 2 1 + 7 d 4 d 3 + 42 d 4 d 2 d 1 + 35 d 4 d 3 1 + 21 d 2 3 d 1 + 21 d 3 d 2 2 + 10 5 d 3 d 2 d 2 1 + 35 d 3 d 4 1 + 35 d 3 2 d 1 + 70 d 2 2 d 3 1 + 21 d 2 d 5 1 + d 7 1 , (36) where the m i and the d i are computed as in (14)-(15). This fol- lows immediately fr om asymptotic fre eness [2 0], an d from the fact tha t 1 N XX H conv erges to the Mar ˘ chenko Pastur law µ c . In p articular, when all D i ( N ) = I L and c = 1 , we obtain the limit moments 1 , 2 , 5 , 14 , 42 , 13 2 , 429 , which also were listed in Cor ollary 3. One can also write down Gaussian equiv alents to the seco nd ord er mom ents of V and ermond e matric es (17) using techniques from [29]. Howe ver the fo rmulas look quite different, and the asymptotic b ehaviour is different. W e h av e for instance lim L →∞ L 2 D 1 , 1  D ( N ) 1 N XX H  = cd 2 , (37) where it is not n eeded that the matrices D ( N ) are diago nal. Similarly , o ne can write d own an equ iv alent to Theorem 2 for the exact momen ts. For the ﬁrst three moments (the fou rth moment is drop ped, sinc e this is m ore inv o lved), these ar e m 1 = d 1 m 2 = d 2 + d 2 1 m 3 =  1 + N − 2  d 3 + 3 d 1 d 2 + d 3 1 . This follows from a car eful c ount of all po ssibilities af ter the matrices hav e been multiplied together (see also [33], where one can see that the restrictio n th at the matrices D i ( N ) are diagona l can be dropped in the Gaussian case). It is seen, contrary to Theorem 2 for V andermo nde matrices, that the second exact mo ment equa ls the second asymp totic mome nt (31), and also that the co nv ergence is faster ( i.e. O ( N − 2 ) ) for the thir d moment (th is will also b e the case for higher moments). The two typ es o f (de) conv olution also differ in h ow they can be com puted in practice. In [27], an algor ithm for free conv o lution with the Mar ˘ chenko Pastur la w was sketched. A similar algorithm may not exist for V ander mond e con volu- tion. Ho we ver , V an dermo nde con volution can b e sub ject to numerical approxima tion: T o see t his, note ﬁrst that Theo rem 3 splits the numeric s into two parts: The approxim ation of the integrals R p ω ( x ) | ρ | dx , an d th e app roxima tion of the K ρ,u . A IEEE TRANSACTIONS ON INFORMA TION THEOR Y , VOL. 1, NO. 1, JANU ARY 2009 12 strategy for obtaining the latter quantities cou ld be to randomly generate many numb ers b etween 0 and 1 and estimate the volume as the ratio of the solu tions wh ich satisfy (63) in Append ix B. Imp lementation s of the various V and ermon de conv o lution variants giv en in this p aper can b e fo und in [34]. In practice, one often has a random matrix model where indepen dent Ga ussian and V anderm onde matrices are both present. I n such cases, it is p ossible to comb ine the in dividual results for b oth o f them . In Section V, exam ples on h ow this can be d one are presen ted. V . A P P L I C AT I O N S The ap plications p resented here all use the deco n volution framework fo r V anderm onde ma trices. Since additive, white, Gaussian n oise also is taken into account, V ander mond e d e- conv o lution is comb ined with Gaussian deconv olution. Matlab code for running the different simulations can be found in [34]. In the eige n value h istograms fo r V an dermo nde matrices shown in ﬁgures 3 a nd 5, large matrices were u sed in o rder to obtain so mething close to th e asympto tic limit. I n practical scenarios, an d in the applications we present, N and L are much smaller than what was used in these ﬁgures, which partially explains the uncertain ty in some of the simulations. In particular, the un certainty for n on-un iform phase distributions such as those in Section V -A is high, sin ce exact expr essions for the lower order moments are not k nown, contrary to the case of uniform phase distribution. In all the following, d is the distance b etween the antennas whereas λ is the wa velength. The ratio d λ is a ﬁgure of th e resolution with wh ich the system will be ab le to separate (an d therefo re estimate the position of) users in spa ce. A. Detection of the number of sour ces Let us consider a basestation eq uipped with N r eceiving antennas, an d with L mobiles (each with a sing le antenna) in the cell. The re ceiv ed sign al at the ba se station is g iv en b y r i = VP 1 2 s i + n i . (38) Here r i is th e N × 1 recei ved vector, s i is th e L × 1 transmit vector by the L users wh ich is a ssumed to satisfy E  s i s H i  = I L , n i is N × 1 additive, white, Gaussian noise of variance σ √ N (all c ompon ents in s i and n i are assumed indepen dent). In the case of a line of sight between the user s and the base station, and consid ering a Unifor m Linear Array (ULA), the matrix V has th e following fo rm: V = 1 √ N      1 · · · 1 e − j 2 π d λ sin( θ 1 ) · · · e − j 2 π d λ sin( θ L ) . . . . . . . . . e − j 2 π ( N − 1) d λ sin( θ 1 ) · · · e − j 2 π d λ sin( θ L )      (39) Here, θ i is the angle of the user in th e cell and is suppo sed to be un iformly d istributed over [ − α, α ] . P 1 2 is an L × L diagona l power matrix due to the different distances from which the user s emit. In o ther words, we assume th at the phase distribution h as the form 2 π d λ sin( θ ) with θ u niform ly distributed on [ − α, α ] . The fact that the phase h as the fo rm 2 π d λ sin( θ ) is a w ell kn own resu lt in array pr ocessing [3 1]. The user’ s distribution can be k nown (in the case of these simulations, the un iform distribution has been accoun ted for without lo ss of generality) thro ugh measureme nts in wir eless systems up to some param eters (here, α typically ). This is usually done to have a better und erstanding of the user ’ s behaviour . It is easily seen, b y taking inverse f unction s, th at the density is, whe n 2 d sin α λ < 1 , p ω ( x ) = 1 2 α q 4 π 2 d 2 λ 2 − x 2 (40) on [ − 2 π d sin α λ , 2 π d sin α λ ] , and 0 elsewhere (see Figu re 6) . Throu ghout the paper we will assume, as in Figu re 5, that λ = 2 d, α = π 4 when model (39) is used . W ith this assumption, 2 d sin α λ < 1 is always fulﬁlled. The goal is to detect th e number of sources L and th eir respective power b ased on th e sample covariance m atrix supposing that we hav e K observations, of the same order as N . When the num ber of observation is qu ite high er th an N (and the n oise variance is k nown), classical subspace methods [35] provid e tools to detect the n umber of sou rces. Ind eed, let R be the tru e covariance matrix given by VPV H + σ 2 I N , where σ 2 is the n oise variance. This matrix has N − L eigenv a lues eq ual to σ 2 and L eigen v alues strictly superior to σ 2 . One can therefo re determin e the number of sou rce by counting the number of eigenvalues different from σ 2 . Howe ver , in pr actice, one has only access to th e sample covariance m atrix given by W = 1 K YY H , with Y = [ r 1 , ... r K ] = VP 1 2 [ s 1 , ..., s K ] + [ n 1 , ..., n K ] . (41) If one has only the sample covariance matrix W , we hav e three indepen dent parts which mu st be d ealt with in or der to g et an estimate of P : the Gaussian ma trices S = [ s 1 , ..., s K ] and N = [ n 1 , ..., n K ] , an d the V a ndermo nde matrix V . It should thus be possible to comb ine Gaussian deco n volution [33] and V and ermond e deconv olution by pe rformin g the f ollowing steps: 1) Estimate the m oments of 1 K VP 1 2 SS H P 1 2 V H using multiplicative free con volution as describ ed in [27]. T his is the den oising part. 2) Estimate the mom ents of PV H V , again u sing multi- plicative fr ee deco n volution. 3) Estimate th e mo ments of P using V andermo nde d econ- volution as d escribed in this p aper . Putting these steps together , we will prove the following: Pr op osition 7: Deﬁne I n = (2 π ) n − 1 Z 2 π 0 p ω ( x ) n dx, (42) IEEE TRANSACTIONS ON INFORMA TION THEOR Y , VOL. 1, NO. 1, JANU ARY 2009 13 and denote the m oments of P and th e sample cov ariance matrix, respectively , by P i = tr L ( P i ) W i = tr N ( W i ) . Then the equ ations W 1 = c 2 P 1 + σ 2 W 2 = c 2 P 2 + ( c 2 2 I 2 + c 2 c 3 )( P 1 ) 2 +2 σ 2 ( c 2 + c 3 ) P 1 + σ 4 (1 + c 1 ) W 3 = c 2 P 3 + (3 c 2 2 I 2 + 3 c 2 c 3 ) P 1 P 2 +  c 3 2 I 3 + 3 c 2 2 c 3 I 2 + c 2 c 2 3  ( P 1 ) 3 +3 σ 2 (1 + c 1 ) c 2 P 2 +3 σ 2 ((1 + c 1 ) c 2 2 I 2 + c 3 ( c 3 + 2 c 2 ))( P 1 ) 2 +3 σ 4 ( c 2 1 + 3 c 1 + 1) c 2 P 1 + σ 6 ( c 2 1 + 3 c 1 + 1) provide an asympto tically u nbiased estimator fo r the mo- ments P i from the m oments of W i (or vice versa) when lim N →∞ N K = c 1 , lim N →∞ L N = c 2 , lim N →∞ L K = c 3 . The proof of this can be found in Appendix M. Note that c 3 = c 1 c 2 , so th at the deﬁnition of c 3 is really n ot necessary . W e still include it h owe ver , sin ce c 1 , c 2 and c 3 are matrix aspect ratios which re present different d econv olution stages, so that they all are used when th ese stages are imp lemented and co mbined serially . I n the simulations, Pro position 7 is pu t to the test when P has th ree sets o f powers, 0.5, 1, and 1 .5, with e qual p robability , with phase distribution gi ven by (3 9). Both th e n umber of sour ces an d the powers are estimated. For the phase distribution (39), the integrals I 2 and I 3 can be com puted exactly (f or general pha se d istributions th ey ar e computed numeric ally), and ar e [36] I 2 = λ 4 dα 2 ln  1 + sin α 1 − sin α  I 3 = λ 2 tan α 4 d 2 α 3 . Under the a ssumptions λ = 2 d, α = π 4 used th rough out th is paper, the integrals above take the values I 2 = 40 π 2 ln 2 + √ 2 2 − √ 2 ! I 3 = 1600 π 3 . For estimation of the powers, kn owing that we have only three sets of powers with equal proba bility , it sufﬁces to estimate the th ree lowest mo ments in o rder to get an estimate of the powers (which are the three distinct eigenv alu es of P ). Therefo re, in the following simulation s, Propo sition 7 is ﬁr st used to get an estima te of the mo ments o f P . Then these ar e used to obtain an estimate of the th ree d istinct eigenv alues o f P u sing th e Newton-Girard formulas [ 37]. Th ese shou ld then lie close to th e three powers of P . Power estimation for the model (39) is shown in th e ﬁrst plot of Figur e 1 0. In the plot, K = L = N = 144 , and σ = √ 0 . 1 . Experim ents sh ow that when the phase distribution becomes ”less” u niform , larger matrix sizes are needed in order for accurate power estimation using th is m ethod. T his will also be seen when we per form power estimation u sing unif orm p hase distribution in the next section. For estimation of the number of user s L , we assume that the power distribution of P is known, but n ot L itself. Since L is un known, in the simulation s we enter different c andidate values of it in to the fo llowing proce dure: 1) Computin g the m oments P i = tr L ( P i ) of P . 2) The moments tr L ( P i ) are fed into the form ulas of Proposition 7, an d we th us obtain c andidate moments W i of the samp le covariance m atrix W . 3) Compute the sum of the square errors between these can- didate moments, and th e momen ts ˆ W i of the observed sample covariance ma trix ˆ W , i.e. com pute P 3 i =1 | W i − ˆ W i | 2 . The estimate L f or the number of users is chosen as the one which g iv es the minimum value for the sum o f squ are errors after these steps. In Figure 8, we have set σ = √ 0 . 1 , N = 100 , and L = 36 . W e tried the procedure d escribed a bove for 1 all the way up to 1 00 o bservations. It is seen that only a small number of observations are need ed in ord er to get an ac curate estimate of L . When K = 1 , it is seen that mor e observations are needed to g et an accu rate estimate of L , when compar ed to K = 10 . B. Estimation of the number of paths In many chan nel modeling applica tions, one needs to deter- mine the number of paths in the channel [38 ]. For this pur pose, consider a multi- path channel of the fo rm: h ( τ ) = L X i =1 s i δ ( τ − τ i ) Here, s i are i.d . Ga ussian ra ndom variables with power P i and τ i are uniformly distrib uted delays over [0 , T ] . The s i represent the attenua tion factors due to th e different reﬂections. L is the total numb er of paths. In the fr equency domain, the cha nnel is given by H ( f ) = L X i =1 s i G ( f ) e − j 2 π f τ i . Sampling the continuo us frequen cy signal at f i = i W N where W is the b andwidth , the model be comes (fo r a g iv en chann el realization) H = VP 1 2 s where V = 1 √ N       1 · · · 1 e − j 2 π W τ 1 N · · · e − j 2 π W τ L N . . . . . . . . . e − j 2 π ( N − 1) W τ 1 N · · · e − j 2 π ( N − 1) W τ L N       , (43) W e will here set W = T = 1 , which m eans that the ω i of (1) ar e u niform ly d istributed over [0 , 2 π ) . The cor respond ing IEEE TRANSACTIONS ON INFORMA TION THEOR Y , VOL. 1, NO. 1, JANU ARY 2009 14 0 10 20 30 40 50 60 70 80 90 100 0 10 20 30 40 50 60 70 Number of observations L Estimate of L Actual value of L (a) K = 1 0 10 20 30 40 50 60 70 80 90 100 0 10 20 30 40 50 60 70 Number of observations L Estimate of L Actual value of L (b) K = 10 Fig. 8. Estimate for the number of users. Actual val ue of L is 36 . Also, σ = √ 0 . 1 , N = 100 . The po wers were 0 . 5 , 1 , and 1 . 5 , with equal probability . eigenv a lue histog ram was sho wn in Figure 3. When additiv e noise ( n ) again is taken into con sideration, our model aga in becomes that of ( 38), the on ly difference being that th e phase distribution of the V andermo nde matrix now is uniform. L now is the number of paths, N the num ber of fr equency samples, and P is the unknown L × L diagonal power matrix. T aking K observations we arrive at the same fo rm as in (41). In this case with uniform ph ase distribution, we can do ev en better than Proposition 7, in that on e can wr ite down estimators for the moments which are unbiased fo r any number of ob servations and frequ ency samples: Pr op osition 8: Assume that V has un iform phase distribu- tion, a nd le t P i be the m oments of P , and W i = tr N ( W i ) the momen ts of th e sample covariance m atrix. Deﬁne also c 1 = N K , c 2 = L N , and c 3 = L K . Then E [ W 1 ] = c 2 P 1 + σ 2 E [ W 2 ] = c 2  1 − 1 N  P 2 + c 2 ( c 2 + c 3 )( P 1 ) 2 +2 σ 2 ( c 2 + c 3 ) P 1 + σ 4 (1 + c 1 ) E [ W 3 ] = c 2  1 + 1 K 2   1 − 3 N + 2 N 2  P 3 +  1 − 1 N   3 c 2 2  1 + 1 K 2  + 3 c 2 c 3  P 1 P 2 +  c 3 2  1 + 1 K 2  + 3 c 2 2 c 3 + c 2 c 2 3  ( P 1 ) 3 +3 σ 2  (1 + c 1 ) c 2 + c 1 c 2 2 K L   1 − 1 N  P 2 +3 σ 2  c 1 c 3 2 K L + c 2 2 + c 2 3 + 3 c 2 c 3  ( P 1 ) 2 +3 σ 4  c 2 1 + 3 c 1 + 1 + 1 K 2  c 2 P 1 + σ 6  c 2 1 + 3 c 1 + 1 + 1 K 2  Just as Prop osition 7, this is proved in Appendix M. In the following, this result is used in order to d etermine the number of paths as well as the p ower of each path. The different conv ergence rates of the ap proxim ations are clearly seen in the plots. In Figure 9, the num ber of p aths is estimated b ased on the proced ure sketched above. W e have set σ = √ 0 . 1 , N = 100 , and L = 36 . The proced ure is tried for 1 all the way up to 100 ob servations. Th e plo t is very similar to Figure 8, in that only a small numb er of obser vations are n eeded in ord er to get an accura te estimate of L . Wh en K = 1 , it is seen that more ob servations are needed to get an accura te estimate of L , when com pared to K = 10 . For th e estimation of p owers simulation , we ha ve set K = N = L = 144 , an d σ = √ 0 . 1 , following th e proced ure a lso described ab ove, up to 100 0 observations. The second plot in Figure 10 shows the results wh ich conﬁrm s the usefu lness of the appro ach. C. Estimation of wavelen gth In the ﬁeld of MI MO co gnitive sensing [39], [40], term inals must decide on the band on wh ich to transmit and in particu lar sense which b and is o ccupied . One way of do ing so is to ﬁnd the wavelength λ in (39), based on some realizations o f the sample cov ariance ma trix. In o ur simulation , we h av e set K = 10 , L = 3 6 , N = 100 , and σ = √ 0 . 1 , in addition to λ = 2 , d = 1 , α = π 4 . W e have tried values between 0 an d 5 as cand idate wa velen gths (to be more prec ise, the values 0 . 05 , 0 . 1 , 0 . 15 , ..., 5 are tried), an d chosen th e o ne wh ich gi ves the smallest d eviation (in the same sense as above, i.e. the sum of the squared erro rs of th e ﬁrst three mom ents are taken ) from a different numbe r of realizations of sample covariance matrices. The resulting plot is shown in Figur e 1 1, and shows that th e V an dermo nde deconv o lution method can also be used for wav elength estimation. IEEE TRANSACTIONS ON INFORMA TION THEOR Y , VOL. 1, NO. 1, JANU ARY 2009 15 0 10 20 30 40 50 60 70 80 90 100 0 10 20 30 40 50 60 70 Number of observations L Estimate of L Actual value of L (a) K = 1 0 10 20 30 40 50 60 70 80 90 100 0 10 20 30 40 50 60 70 Number of observations L Estimate of L Actual value of L (b) K = 10 Fig. 9. Estimate for the number of paths. Actual valu e of L is 36 . Also, σ = √ 0 . 1 , N = 100 . D. Signal r econ struction and estimation of th e samp ling d is- tribution For signal r econstructio n, one can pr ovide a gen eral frame- work where o nly the sampling distribution matters asymp tot- ically . Th e samp ling d istribution can be estimated with the help of the p resented resu lts. Several works have investigated how irr egular samp ling affects th e perf ormance o f signal reconstruc tion in the p resence of n oise in different ﬁeld s, namely sensor networks [41], [42], im age p rocessing [43], [44], geophysics [4 5], an d c ompressive sampling [46]. The usual Nyquist The orem states that for a sign al with maximum frequen cy f max , one nee ds to sample the signal at a rate which is a t least twice th is numb er . Howe ver, in many cases, this can no t be performed , or one has an observation of a signal at only a subset o f the f requen cies. Mo reover , one feels that if the signal h as a spar se sp ectrum, one can take fewer samples and still hav e the same information on the original signal. One of the centr al motiv ations of sparse samplin g is 0 100 200 300 400 500 600 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Number of observations Powers First eigenvalue Second eigenvalue Third eigenvalue (a) The model (39 ) of Secti on V -A. 0 100 200 300 400 500 600 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Number of observations Powers First eigenvalue Second eigenvalue Third eigenvalue (b) The model (43) of Section V -B . Fig. 10. Esti mation of po wers for the tw o models (39) and (43), for variou s number of observat ions. K = N = L = 144 , and σ = √ 0 . 1 . The actual po wers were 0 . 5 , 1 , and 1 . 5 , with equal probabili ty . exactly to understand under which condition one can still have less samples and recover the original signal up to an error of ǫ [47]. Let us co nsider the signal of interest as a superpo sition of its frequen cy compo nents (th is is also th e case for a un idimension al ban dlimited ph ysical signal), i.e. r ( t ) = 1 √ N N − 1 X k =0 s k e − j 2 π kt N and su ppose that the signal is samp led at various instants [ t 1 , ..., t L ] with t i ∈ [0 , 1] . This can be identically written as r ( ω ) = 1 √ N N − 1 X k =0 s k e − j kω , or r = V T s . In th e presence of no ise, one can write r = V T s + n , (44) IEEE TRANSACTIONS ON INFORMA TION THEOR Y , VOL. 1, NO. 1, JANU ARY 2009 16 0 20 40 60 80 100 120 140 160 180 200 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Number of observations λ Estimate of λ Actual value of λ Fig. 11. Estimation of wa v elengt h. Decon volution was performed for v aryin g number of observ ati ons, assuming diffe rent wav elen gths, In the true model (39), λ = 2 , d = 1 , α = π 4 , K = 10 , L = 36 , N = 100 , and σ = √ 0 . 1 . where r = [ r ( ω 1 ) , ...r ( ω L )] T , s and n are as in ( 38), and with V on the form (1). A similar analysis f or such cases can b e found in [16]. In the following, we suppose that o ne has K observations of the re ceiv ed samp led vector r : Y = [ r 1 , ... r K ] = V T [ s 1 , ..., s K ] + [ n 1 , ..., n K ] (45) The vecto r r is the d iscrete ou tput of the sampled continuo us signal r ( w ) for which the distribution is unknown ( howe ver , c is k nown). This case happens wh en one has a n ob servation without the knowledge of the samp ling rate for example. The difference in ( 45) from th e model (41) lies in that the adjoint of a V andermon de matrix is used, and in that there is no additional diag onal matrix P includ ed. The following result can n ow be stated and proved similarly to Pro position 7 and 8: Pr op osition 9: E [ tr n ( W )] = 1 + σ 2 (46) E  tr n  W 2  = c 2 I 2 + (1 + c 3 )(1 + σ 2 ) 2 (47) E  tr n  W 3  = 1 + 3 c 2 (1 + c 3 ) I 2 3 c 3 + c 2 3 + c 2 2 I 3 3 σ 2 (1 + 3 c 3 + c 2 3 + c 2 (1 + c 3 ) I 2 ) 3 σ 4 c 2 ( c 2 3 + 3 c 3 + 1) σ 6 ( c 2 3 + 3 c 3 + 1) , (48) where lim N →∞ N K = c 1 , lim N →∞ L N = c 2 , lim N →∞ L K = c 3 , I n is deﬁned as in Proposition 7, and W = 1 K YY H . The proof o f Pr oposition 9 is commented in Append ix M. W e have tested (46)-(48) by taking a phase distribution ω which is unif orm on [0 , α ] , and 0 elsewhere. T he density is thus 2 π α on [0 , α ] , and 0 elsewhere. In this case we can com pute 0 100 200 300 400 500 600 0 0.2 0.4 0.6 0.8 1 1.2 Number of observations α Estimated α Actual α Fig. 12. Estimated va lues of α using (46 )-(48), for vari ous number of observ ati ons, and for K = 10 , L = 36 , N = 100 , σ = √ 0 . 1 . The actual v alue of α w as π 4 . 0 100 200 300 400 500 600 0 20 40 60 80 100 120 Number of observations I n Estimated I 2 Actual I 2 Estimated I 3 Actual I 3 Fig. 13. Estimated value s of I 2 and I 3 using (46 )-(48), for var ious number of observ ati ons, and for K = 10 , L = 36 , N = 100 , σ = √ 0 . 1 . The actua l v alue of α w as π 4 . that I 2 = 2 π α I 3 =  2 π α  2 . The ﬁrst of these eq uations, co mbined with (46)- (48), ena bles us to estimate α . Th is is tested in Figur e 12 for various number of observations. In Figure 1 3 we ha ve also tested estimation of I 2 , I 3 from the observations using the same equations. When on e ha s a d istribution which is not unifor m, the in tegrals I 3 , I 4 , ... would also be n eeded in ﬁnd ing the characteristics of the und erlying pha se distribution. Figu re 1 3 shows that th e estimation of I 2 requires far fewer o bserva- tions th an the estimatio n of I 3 . In both ﬁgures, the v alues IEEE TRANSACTIONS ON INFORMA TION THEOR Y , VOL. 1, NO. 1, JANU ARY 2009 17 K = 10 , L = 36 , N = 100 , and σ = √ 0 . 1 were used and α was π 4 . It is seen th at the estimation of I 3 is a bit off even for higher number of observations. This is to be expected, since an asymptotic resu lt is applied. V I . C O N C L U S I O N A N D F U RT H E R D I R E C T I O N S W e have sh own how asy mptotic m oments o f ran dom V an- dermon de matrices with entries on th e u nit circle can be computed analytically , and treated many different cases. V an- dermon de matrices with un iform p hase distribution pr oved to be the easiest case, and it was sho wn h ow the case with more ge neral phases cou ld be expressed in terms of th is. The case where the ph ase d istribution has singularities was also handled , as this case displayed different asymptotic behaviour . Also, mixed momen ts of indepen dent V and ermond e ma trices were investigated, as well as the moments o f genera lized V andermo nde matrices. In addition to the general asymp totic expressions stated, exact expression s for the ﬁrst moments of V andermo nde matrices with un iform phase distribution were also stated. W e h av e also provided some useful applications o f random V ander mond e matrices. The app lications c oncentra ted on d econv olution and signal sampling analysis. As shown, many useful system mod els use independent V ander mond e matrices and Gaussian matrices combine d in som e way . T he presented examples sh ow h ow rando m V andermo nde m atrices in such sy stems can be han dled in pr actice to o btain estimates on quantities such as the numb er of paths in channel modeling, the transmission p owers of the users in wireless transmission , or the sampling distrib ution fo r signal recovery . T he paper has only touched upon a limited numbe r o f applications, b ut the results alread y provide benchma rk ﬁg ures in th e non- asymptotic regime. From a theo retical p erspective, it would a lso be interesting to ﬁnd meth ods for obtaining the generalized expansion co- efﬁcients K ρ,ω , λ from K ρ,u,u , similar to how we fou nd the expansion coefﬁcients K ρ,ω from K ρ,u . This co uld also she d some lig ht on wheth er u niform ph ase- and power distribution also minimizes momen ts of gen eralized V andermon de matri- ces, similarly to how we sho wed that it m inimizes mome nts in the no n-gene ralized case. Throu ghout the paper, we a ssumed that only d iagonal matrices were in volved in mixed mo ments of V an dermo nde matrices. The case of n on-d iagonal matrices is harder, a nd should be addressed in future research. The analysis of the maximum and min imum eigen v alue is also of impor tance. The method s pre sented in this paper can not be used directly to obtain explicit expressions for the p.d.f. o f the asy mptotic mean eigen v alue distribution, so this is also a case fo r fu- ture research. A way of attacking this problem co uld b e to develop fo r V an dermon de matr ices analy tic coun terparts to what one has in fr ee pro bability , suc h as the R -, S - , and the Stieltjes transform [20]. I nterestingly , ce rtain m atrices similar to V and ermond e m atrices, have analytical expression s for th e moments: in [17], an alytical expressions for the moments of matrices with en tries of the form A i,j = F ( ω i − ω j ) ar e foun d. This is inter esting for the V and ermon de matrices we co nsider, since  1 N V H V  i,j = sin  N 2 ( ω i − ω j )  N sin  1 2 ( ω i − ω j )  . Unfortu nately , the function F N ( x ) = sin ( N 2 x ) N sin ( 1 2 x ) depend s on the m atrix dim ension N , so tha t we can no t ﬁnd a func tion F which ﬁts the r esult from [17]. Finally , another ca se f or f uture resear ch is th e asymptotic behaviour of V andermo nde m atrices whe n the matrix entries lie outside the un it circle. A P P E N D I X A T H E P RO O F O F T H E O R E M 1 W e can write E  tr L  D 1 ( N ) V H VD 2 ( N ) V H V · · · D n ( N ) V H V  (49) as L − 1 P i 1 ,...,i n j 1 ,...,j n E ( D 1 ( N )( j 1 , j 1 ) V H ( j 1 , i 2 ) V ( i 2 , j 2 ) D 2 ( N )( j 2 , j 2 ) V H ( j 2 , i 3 ) V ( i 3 , j 3 ) . . . D n ( N )( j n , j n ) V H ( j n , i 1 ) V ( i 1 , j 1 )) (50) The ( j 1 , ..., j n ) uniquely identiﬁes a partition ρ of { 1 , ..., n } , where each b lock W j of ρ consists of the position s of the indices which equ al j , i.e. W j = { k | j k = j } . W e will also say that ( j 1 , ..., j n ) give rise to ρ . Write W j = { w j 1 , w j 2 , ..., w j | W j | } . When ( j 1 , ..., j n ) give rise to ρ , w e see that since j w j 1 = j w j 2 = · · · = j w j | W j | , we also have that ω j w j 1 = ω j w j 2 = · · · = ω j w j | W j | , and we will d enote the ir com mon value by ω W j as in Deﬁni- tion 8. With this in m ind, it is straightfor ward to verify that (50) can be wr itten as X ρ ∈ P ( n ) X ( i 1 ,...,i n ) X ( j 1 , ..., j n ) giving rise to ρ N − n L − 1 × | ρ | Y k =1 E  e j “ P k ∈ W j i k − 1 − P k ∈ W j i k ” ω W k  × D 1 ( N )( j 1 , j 1 ) × · · · × D n ( N )( j n , j n ) , ( 51) IEEE TRANSACTIONS ON INFORMA TION THEOR Y , VOL. 1, NO. 1, JANU ARY 2009 18 where i 1 , ..., i n takes values b etween 0 and N − 1 . W e will in the following switch between th e form (5 1) a nd the f orm X ρ ∈ P ( n ) X ( j 1 , ..., j n ) giving rise to ρ X ( i 1 ,...,i n ) N | ρ |− n − 1 c | ρ |− 1 L −| ρ | × E n Y k =1  e j ( ω b ( k − 1) − ω b ( k ) ) i k  ! × D 1 ( N )( j 1 , j 1 ) × · · · × D n ( N )( j n , j n ) , (52 ) where we also have reorganized the powers of N and L in (51), an d changed the order of sum mation (i.e. sum med over the different i 1 , ..., i n ﬁrst). Noting that X ( i 1 ,...,i n ) N | ρ |− n − 1 E n Y k =1 e j ( ω b ( k − 1) − ω b ( k ) ) i k ! (53) = N | ρ |− n − 1 E   X ( i 1 ,...,i n ) n Y k =1 e j ( ω b ( k − 1) − ω b ( k ) ) i k   (54) = N | ρ |− n − 1 E n Y k =1 N − 1 X i k =0 e j ( ω b ( k − 1) − ω b ( k ) ) i k !! (55) = N | ρ |− n − 1 E n Y k =1 1 − e j N ( ω b ( k − 1) − ω b ( k ) ) 1 − e j ( ω b ( k − 1) − ω b ( k ) ) ! (56) = N | ρ |− n − 1 × Z (0 , 2 π ) | ρ | n Y k =1 1 − e j N ( ω b ( k − 1) − ω b ( k ) ) 1 − e j ( ω b ( k − 1) − ω b ( k ) ) dω 1 · · · dω | ρ | (57) = K ρ,ω , N , (58) Deﬁnition 8 o f the V ander mond e mixed moment expansion coefﬁcients com es into play , so that (52) can also be written X ρ ∈ P ( n ) X ( j 1 , ..., j n ) giving rise to ρ c | ρ |− 1 L −| ρ | K ρ,ω , N × D 1 ( N )( j 1 , j 1 ) · · · × × D n ( N )( j n , j n ) . (59) The notation fo r a joint limit distribution simpliﬁes (52). Indeed , add to ( 52) fo r each ρ the term s X ρ ′ ∈ P ( n ) ,ρ ′ >ρ X ( j 1 , ..., j n ) giving rise to ρ ′ c | ρ |− 1 L −| ρ | K ρ,ω , N × D 1 ( N )( j 1 , j 1 ) · · · × D n ( N )( j n , j n ) . (60) These go to 0 as N → ∞ , since they are boun ded by c | ρ |− 1 L −| ρ | K ρ,ω , N L | ρ ′ | = K ρ,ω , N c | ρ |− 1 L | ρ ′ |−| ρ | = O ( L − 1 ) . After this add ition, the limit of ( 59) can b e written X ρ ∈ P ( n ) c | ρ |− 1 K ρ,ω D ρ , (61) which is what we had to sho w . W e also need to co mment on the statem ent o f Theo- rem 6, where generalized V and ermon de ma trices are con- sidered. In this case, the deriv ations after (52) are different since the p ower distribution is no t u niform. For the case of (22), we can in (55) repla ce P n i k =1 e j ( ω b ( k − 1) − ω b ( k ) ) i k with P N − 1 r =0 N p f N ( r ) e j r ( ω b ( k − 1) − ω b ( k ) ) , since the n umber of occurre nces of the power e j r ( ω b ( k − 1) − ω b ( k ) ) is N p f N ( r ) . The rest of the pr oof of Theo rem 6 follows by ca nceling n powers of N after this replacem ent. Th e d etails are similar fo r the case (23), wher e the law of large number s is app lied to arrive at the second f ormula in (24). A P P E N D I X B T H E P R O O F O F P RO P O S I T I O N 1 Note that fo r each block W j , E  e j “ P k ∈ W j i k − 1 − P k ∈ W j i k ” ω W j  = 0 when X k ∈ W j i k − 1 6 = X k ∈ W j i k , and 1 if X k ∈ W j i k − 1 = X k ∈ W j i k . (62) If we deno te b y S ρ,N the set o f all n -tuples ( i 1 , ..., i n ) ( 0 ≤ i k ≤ N − 1 , 1 ≤ k ≤ n ) which solve (6 2), and de ﬁne | S ρ,N | to be the card inality of S ρ,N , it is clear that K ρ,u = lim N →∞ K ρ,u,N = lim N →∞ 1 N n +1 −| ρ | | S ρ,N | . It is straightforward to show that the so lution set of (62 ) h as n + 1 − | ρ | free variables. After dividing the equatio ns (62) b y N and letting N go to inﬁnity , K ρ,u can thus altern ativ ely b e expressed as the volume in R n +1 −| ρ | of the solu tion set of X k ∈ W j x k − 1 = X k ∈ W j x k , (63) with 0 ≤ x k ≤ 1 . It is clear that th e volume of th is solutio n set computes to a rational numb er . It is the form (63) which will be used in the other appendices to compute K ρ,u for certain lower order ρ . Ap pendix D of [16] states the same equation s for ﬁnding quantities equi valent to V andermond e mixed moment expansion co efﬁcients fo r the u niform phase distribution. T he fact that K ρ,u ≤ 1 follows directly f rom Appen dix D of [16]. The same applies for the fact that K ρ,u = 1 if an d only if ρ is noncr ossing. For any ρ , we can deﬁne a partition of { 1 , ..., n } into n + 1 − | ρ | block s, wher e two elements are deﬁned to b e in the same block if and on ly if th e cor respond ing variables in solutions to (63) are linearly depend ent. When ρ is noncro ssing, it is straightfor ward to show that two such variables are d epende nt if and only if they are equ al, an d also that th is partition is the Kreweras complem ent K ( ρ ) of ρ . This fact is u sed elsewhere in this paper . W e will also b rieﬂy explain why the co mputation s in this append ix are useful fo r generalized V and ermond e matrices IEEE TRANSACTIONS ON INFORMA TION THEOR Y , VOL. 1, NO. 1, JANU ARY 2009 19 with un iform phase d istribution. For ( 22), the numb er o f solutions i 1 , ..., i k to (62) nee ds to be multip lied by N p f N ( i 1 ) · · · N p f N ( i k ) , since each i j now may occur N p f N ( i j ) times. This mean s that K ρ,ω , f can b e compu ted as the integrals in this ap pendix , but that we also need to multiply with the d ensity p f for each variable. The comp utations of these new integrals beco me rather inv olved when f is n ot unif orm, and are theref ore dropp ed. A P P E N D I X C T H E P R O O F F O R P RO P O S I T I O N 2 W e will in the following compute the volume of the solution set of (63), as a volume in [0 , 1] n +1 −| ρ | ⊂ R n +1 −| ρ | , as explained in the proo f of Proposition 1. These in tegrals are very ted ious to compu te, and many of the deta ils ar e skipped . The form ula r ! s ! ( r + s + 1)! = Z 1 0 x r (1 − x ) s dx can be used to simplify som e of the calculations for high er values of n . A. Computatio n o f K {{ 1 , 3 } , { 2 , 4 }} ,u This is equiv alent to ﬁnding the volume of the solution set of x 1 + x 3 = x 2 + x 4 in R 3 . Since this me ans that x 4 = x 1 + x 3 − x 2 lies between 0 and 1 , we can set up the following integral bounds: When x 1 + x 3 ≤ 1 , we must have that 0 ≤ x 2 ≤ x 1 + x 3 , so that we g et the contribution Z 1 0 Z 1 − x 1 0 Z x 1 + x 3 0 dx 2 dx 3 dx 1 , which computes to 1 3 . When 1 ≤ x 1 + x 3 , we must have that x 1 + x 3 − 1 ≤ x 2 ≤ 1 , so that we g et the con tribution Z 1 0 Z 1 1 − x 1 Z 1 x 1 + x 3 − 1 dx 2 dx 3 dx 1 , which also computes to 1 3 . Ad ding the contributions to gether we get 2 3 , which is the stated v alue for K {{ 1 , 3 } , { 2 , 4 }} ,u . It turns out that when the blocks of ρ are cyclic shifts of each other, the computatio n of K ρ,u can b e simpliﬁed. Examp les of such ρ are {{ 1 , 3 } , { 2 , 4 }} (fo r which we just co mputed K ρ,u ), { { 1 , 3 , 5 } , { 2 , 4 , 6 }} , and { { 1 , 4 } , { 2 , 5 } , { 3 , 6 }} . W e will in the following d escribe this simpliﬁed compu tation. Let a ( m ) l ( x ) b e the polynomial which gives the v olume in R m − 1 of the solutions set to x 1 + · · · + x m = x (co nstrained to 0 ≤ x i ≤ 1 ) for l ≤ x ≤ l + 1 . It is clear that these satisfy the integral equ ations a ( m +1) l ( x ) = Z l x − 1 a ( m ) l − 1 ( t ) dt + Z x l a ( m ) l ( t ) dt, (64) which can be used to compute the a m l ( x ) recursively . Note ﬁrst that a (1) 0 ( x ) = 1 . For m = 2 we have a (2) 0 ( x ) = Z x 0 a (1) 0 ( t ) dt = x a (2) 1 ( x ) = Z 1 x − 1 a (1) 0 ( t ) dt = 2 − x. For m = 3 we h ave a (3) 0 ( x ) = Z x 0 a (2) 0 ( t ) dt = 1 2 x 2 a (3) 1 ( x ) = Z 1 x − 1 a (2) 0 ( t ) dt + Z x 1 a (2) 1 ( t ) dt = 1 − 1 2 ( x − 1) 2 − 1 2 (2 − x ) 2 a (3) 2 ( x ) = Z 2 x − 1 a (2) 1 ( t ) dt = 1 2 (3 − x ) 2 . B. Computatio n of K {{ 1 , 3 , 5 } , { 2 , 4 , 6 }} ,u For m = 3 , integratio n gives Z 1 0 ( a (3) 0 ) 2 ( t ) dt + Z 2 1 ( a (3) 1 ) 2 ( t ) dt + Z 3 2 ( a (3) 2 ) 2 ( t ) dt, which compu tes to 11 20 . This is the stated expression fo r K {{ 1 , 3 , 5 } , { 2 , 4 , 6 }} ,u . C. Computation of K {{ 1 , 4 } , { 2 , 5 } , { 3 , 6 }} ,u This is equiv alent to ﬁnding the volume of the solution set of x 1 + x 4 = x 2 + x 5 = x 3 + x 6 in R 4 , which is co mputed as Z 1 0 ( a (2) 0 ) 3 ( t ) dt + Z 2 1 ( a (2) 1 ) 3 ( t ) dt, which com putes to 1 2 . Th is is the stated expression for K {{ 1 , 4 } , { 2 , 5 } , { 3 , 6 }} ,u . D. Computatio n of K {{ 1 , 4 } , { 2 , 6 } , { 3 , 5 }} ,u This is equiv alent to ﬁnding the volume of the solution set of x 1 + x 4 = x 2 + x 5 x 2 + x 6 = x 3 + x 1 in R 4 . Since this me ans that x 5 = x 1 − x 2 + x 4 lies between 0 and 1 , x 6 = x 1 − x 2 + x 3 lies between 0 and 1 , we can set up the following integral bound s: For x 2 ≥ x 1 we must have x 2 − x 1 ≤ x 3 , x 4 ≤ 1 , so that we get the contribution Z 1 0 Z 1 x 1 Z 1 x 2 − x 1 Z 1 x 2 − x 1 dx 4 dx 3 dx 2 dx 1 , which co mputes to 1 4 . It is clear that for x 1 ≥ x 2 we get the same result by symmetry , so that the total contribution is 1 4 + 1 4 = 1 2 , which pr oves the claim. IEEE TRANSACTIONS ON INFORMA TION THEOR Y , VOL. 1, NO. 1, JANU ARY 2009 20 E. Computatio n o f K {{ 1 , 5 } , { 3 , 7 } , { 2 , 4 , 6 }} ,u This is equiv alent to ﬁnding the volume of the solution set of x 1 + x 5 = x 2 + x 6 x 3 + x 7 = x 4 + x 1 in R 5 , or x 6 = x 5 + x 1 − x 2 lies between 0 and 1 , x 7 = x 4 + x 1 − x 3 lies between 0 and 1 . (65) This can be split into the following volumes: 1) x 1 ≤ x 2 ≤ x 3 , 2) x 1 ≤ x 3 ≤ x 2 , 3) x 3 ≤ x 2 ≤ x 1 , 4) x 2 ≤ x 3 ≤ x 1 , 5) x 2 ≤ x 1 ≤ x 3 , 6) x 3 ≤ x 1 ≤ x 2 . Each o f these volumes can be com puted by setting up an in- tegral with c orrespo nding bound s. Computin g the se in tegrals, we get the v alues 1 15 , 1 15 , 1 15 , 1 15 , 11 120 , 11 120 , respectiv ely . Adding these con tributions together, we get 4 15 + 11 60 = 27 60 = 9 20 , which proves the claim . F . The co mputation of K {{ 1 , 6 } , { 2 , 4 } , { 3 , 5 , 7 }} ,u This is equiv alent to ﬁnding the volume of the solution set of x 1 + x 6 = x 2 + x 7 x 2 + x 4 = x 3 + x 5 in R 5 , or x 6 = x 7 + x 2 − x 1 lies between 0 and 1 , x 5 = x 4 + x 2 − x 3 lies between 0 and 1 , . This can be obtained from (65) by a per mutation of th e variables, so the contribution from K {{ 1 , 6 } , { 2 , 4 } , { 3 , 5 , 7 }} ,u must also be 9 20 , which pr oves the claim. A P P E N D I X D T H E P R O O F F O R P RO P O S I T I O N 3 Note ﬁrst that multiplying both of sides of (13) with c gives cM n = X ρ ∈ P ( n ) K ρ,ω ( cD ) ρ , (66) where we now can substitute the scaled mom ents (1 4)-(15). W ith D 1 ( N ) = D 2 ( N ) = · · · = D n ( N ) = D ( N ) , D ρ as d eﬁned in Deﬁnition 2 does only d epend o n the block cardinalities | W j | , so that we can group toge ther the K ρ,ω for ρ with equ al block cardinalities. I f w e group th e b locks of ρ so that the ir cardin alities are in descend ing or der, an d set P ( n ) r 1 ,r 2 ,...,r k = { ρ = { W 1 , ..., W k } ∈ P ( n ) || W i | = r i ∀ i } , where r 1 ≥ r 2 ≥ · · · ≥ r k , and also write K r 1 ,r 2 ,...,r k = X ρ ∈ P ( n ) r 1 ,r 2 ,...,r k K ρ,ω , (67) (66) can be wr itten m n = X r 1 ,...,r k r 1 + ··· + r k = n K r 1 ,r 2 ,...,r k k Y j =1 d r j . (68) For th e ﬁrst 5 mo ments this beco mes m 1 = K 1 d 1 (69) m 2 = K 2 d 2 + K 1 , 1 d 2 1 (70) m 3 = K 3 d 3 + K 2 , 1 d 2 d 2 1 + K 1 , 1 , 1 d 3 1 (71) m 4 = K 4 d 4 + K 3 , 1 d 3 d 1 + K 2 , 2 d 2 2 + K 2 , 1 , 1 d 2 d 2 1 + K 1 , 1 , 1 , 1 d 4 1 (72) m 5 = K 5 d 5 + K 4 , 1 d 4 d 1 + + K 3 , 2 d 3 d 2 + K 3 , 1 , 1 d 3 d 2 1 + K 2 , 2 , 1 d 2 2 d 1 + K 2 , 1 , 1 , 1 d 2 d 3 1 + K 1 , 1 , 1 , 1 , 1 d 5 1 . (73) Thus, to prove Prop osition 3, w e h av e to compute the K r 1 ,r 2 ,...,r k by g oing throu gh all partition s. W e will h av e use for the following result, taken from [30]: Lemma 1: The numb er of noncro ssing partitions in N C ( n ) with r 1 blocks of len gth 1 , r 2 blocks of len gth 2 and so on (so that r 1 + 2 r 2 + 3 r 3 + · · · nr n = n ) is n ! r 1 ! r 2 ! · · · r n !( n + 1 − r 1 − r 2 · · · r n )! . Using th is and a similar f ormula for the n umber of p ar- titions with prescribe d block sizes, we obtain cardinalities for n oncrossing partition s and th e set of all partitions with a giv en block structure . T hese numbers are th e used in th e following calculations. F or the proof of Pro position 3, we need to co mpute (67) for all possible b lock car dinalities ( r 1 , ..., r k ) , and insert these in (69)-(73). Th e f ormulas for the three ﬁrst moments are obvio us, since all partitions of length ≤ 3 are noncro ssing. For the remainin g computation s, the following two observations sav e a lot of work: • If ρ 1 ∈ P ( n 1 ) , ρ 2 ∈ P ( n 2 ) with n 1 < n 2 , and ρ 1 can be obtained f rom ρ 2 by o mitting eleme nts k in { 1 , ..., n 2 } such that k an d k + 1 are in the same b lock, then we must have that K ρ 1 ,u = K ρ 2 ,u . This is straightf orward to pr ove since it follows from the proo f of Pro position 1 that i k +1 can be ch osen arbitrarily between 0 and N − 1 in such a case. • K ρ 1 ,u = K ρ 2 ,u if the set of equation s ( 63) for ρ 1 can be obtained by a permu tation o f th e v ariables in th e set of equatio ns for ρ 2 . Since the rank of the matrix for (63) eq uals the numbe r of equations − 1 , we actu ally need o nly hav e that | ρ 1 | − 1 of the | ρ 1 | equations can be obtained fr om permutation o f | ρ 2 | − 1 e quations of the | ρ 2 | equatio ns in the equ ation system for ρ 2 . IEEE TRANSACTIONS ON INFORMA TION THEOR Y , VOL. 1, NO. 1, JANU ARY 2009 21 A. The mom ent o f fourth order The result is here obvious except fo r th e case for the three partitions with b lock car dinalities (2 , 2 ) (fo r all other block cardinalities, all partitio ns are nonc rossing, so that K r 1 ,r 2 ,...,r k is simply the numb er of no ncrossing partition s with bloc k cardinalities ( r 1 , ..., r k ) . this number can be computed from Lemma 1 ). T wo of the p artitions with blo cks of cardina lity (2 , 2) are noncro ssing, the thir d one is not. W e see from Proposition 2 th at the total con tribution is K 2 , 2 = 2 + K {{ 1 , 3 } , { 2 , 4 }} ,u = 2 + 2 3 = 8 3 . The form ula fo r the four th mo ment follows. B. The mom ent o f ﬁfth order Here two cases requ ire extra attentio n: 1) ρ = { W 1 , W 2 } with | W 1 | = 3 , | W 2 | = 2 : There are 1 0 such partition s, an d 5 of them have crossings and contribute with K {{ 1 , 3 } , { 2 , 4 }} ,u . The total contribution is therefor e 5 + 5 × K {{ 1 , 3 } , { 2 , 4 }} ,u = 5 + 5 × 2 3 = 25 3 . 2) ρ = { W 1 , W 2 , W 3 } with | W 1 | = | W 2 | = 2 , | W 3 | = 1 : There a re 15 such pa rtitions, of which 5 ha ve c rossings. T he total con tribution is therefo re 10 + 5 × K {{ 1 , 3 } , { 2 , 4 }} ,u = 10 + 5 × 2 3 = 40 3 . The c omputatio ns f or th e sixth an d seventh order m oments are similar, but th e de tails ar e skip ped. T hese ar e more tedio us in the sense th at one has to c ount the nu mber of par titions with a given block structure, an d identify each partitio n with one of the co efﬁcients listed in Pro position 2. A P P E N D I X E T H E P R O O F O F P R O P O S I T I O N 4 C i,j ( D ( N ) V H V ) is comp uted as in Append ix A. Since some terms in E  tr L  A i  tr L  A j  cancel those in E  tr L  A i  E  tr L  A j  , we can r estrict to summing over partitions of 1 , 2 , ..., i + j where at least one block co ntains elements from both [1 , ..., i ] and [ i + 1 , ..., i + j ] . W e denote this set by P ( i, j ) , an d set n = i + j . In o ur new calculations,(5 2) now instead takes the form L X ρ ∈ P ( i,j ) X ( j 1 , ..., j n ) giving rise to ρ X ( i 1 ,...,i n ) N | ρ |− i − j − 1 L − 1 c | ρ |− 1 L −| ρ | × E n Y k =1  e j ( ω b ( k − 1) − ω b ( k ) ) i k  ! × D 1 ( N )( j 1 , j 1 ) × · · · × D n ( N )( j n , j n ) , (74 ) where the n ormalizing factor L f rom De ﬁnition 4 ha s bee n included. Simplif ying this as in Ap pendix A, and restricting to unif orm ph ase distribution, we obtain lim L →∞ LC i,j ( D ( N ) V H V ) = X ρ ∈ P ( i,j ) c | ρ |− 1 K 2 ,ρ,u D ρ , where K 2 ,ρ,u is the volume of the solu tion set of X k ∈ W j x σ − 1 ( k − 1) = X k ∈ W j x k , (75) where σ is the pe rmutation which shifts [1 , i ] and [ i + 1 , ..., i + j ] to th e right cyclically so that the result is contain ed within the same interval. Thus, when the normalizin g factor L is included, we see that the second order momen ts exist. C 2 , 2 ( D ( N ) V H V ) in (17) is compu ted by notin g that K 2 , {{ 1 , 3 } , { 2 , 4 }} ,u and K 2 , {{ 1 , 4 } , { 2 , 3 }} ,u both equal 2 3 , and th at there are 9 other partitions in P (2 , 2) , an d K 2 ,π ,u = 1 for all these π (all these values ar e compu ted as in Ap pendix C). By adding up for th e different blo ck cardinalities we get that c lim L →∞ LC 2 , 2 ( D ( N ) V H V ) = d 4 + 4 d 3 d 1 4 3 d 2 2 + 4 d 2 d 2 1 , and using the substitution ( 16) we arrive at the desired r esult. A P P E N D I X F T H E P RO O F O F T H E O R E M 2 In ord er to get th e exact expressions in Theor em 2 , we now need to keep track of th e K ρ,u,N deﬁned by (1 0), not only the limits K ρ,u (if we had no t assumed ω = u , the calculation s for K ρ,ω , N would b e much m ore cumbersom e). When ρ is a partition of { 1 , ..., n } and n ≤ 4 , we h av e that K ρ,u,N = K ρ,u = 1 when ρ 6 = {{ 1 , 3 } , { 2 , 4 } } . W e also have that K {{ 1 , 3 } , { 2 , 4 }} ,u,N = 2 3 + 1 3 N 2 , (76) where we have u sed that P N i =1 i 2 = N 3 ( N + 1)( N + 1 2 ) [ 36]. W e also n eed the exact expression for the q uantity T ρ = X ( j 1 ,...,j n ) giving rise to ρ L −| ρ | D 1 ( N )( j 1 , j 1 ) ×· · ·× D n ( N )( j n , j n ) from (59) (i.e. we can not add (60) to obtain the appr oximatio n (61) here). Setting D ( N ,L ) n = tr L ( D n ( N )) , and D ( N ,L ) ρ = Q k i =1 D ( N ,L ) W i , we see that T ρ = D ( N ,L ) ρ − X ρ ′ >ρ L | ρ ′ |−| ρ | T ρ ′ , (77) IEEE TRANSACTIONS ON INFORMA TION THEOR Y , VOL. 1, NO. 1, JANU ARY 2009 22 which can be used recursively to expr ess the T ρ in terms of the D ( N ,L ) ρ . W e obtain the f ollowing for mulas f or n = 4 : T {{ 1 , 2 , 3 , 4 }} = D ( N ,L ) 4 (78) T {{ 1 , 2 , 3 } , { 4 }} = D ( N ,L ) 3 D ( N ,L ) 1 − L − 1 D ( N ,L ) 4 (79) T {{ 1 , 2 } , { 3 , 4 }} = ( D ( N ,L ) 2 ) 2 − L − 1 D ( N ,L ) 4 (80) T {{ 1 , 2 } , { 3 } , { 4 }} = D ( N ,L ) 2 ( D ( N ,L ) 1 ) 2 − 2 L − 1 ( D ( N ,L ) 3 D ( N ,L ) 1 − L − 1 D ( N ,L ) 4 ) − L − 1  ( D ( N ,L ) 2 ) 2 − L − 1 D ( N ,L ) 4  − L − 2 D ( N ,L ) 4 = D ( N ,L ) 2 ( D ( N ,L ) 1 ) 2 − L − 1 ( D ( N ,L ) 2 ) 2 − 2 L − 1 D ( N ,L ) 3 D ( N ,L ) 1 +2 L − 2 D ( N ,L ) 4 (81) T {{ 1 } , { 2 } , { 3 } , { 4 }} = ( D ( N ,L ) 1 ) 4 − 6 L − 1 ( D ( N ,L ) 2 ( D ( N ,L ) 1 ) 2 − L − 1 ( D ( N ,L ) 2 ) 2 − 2 L − 1 D ( N ,L ) 3 D ( N ,L ) 1 +2 L − 2 D ( N ,L ) 4 ) − 3 L − 2 ( D ( N ,L ) 2 ) 2 + 3 L − 3 D ( N ,L ) 4 − 4 L − 2 D ( N ,L ) 3 D ( N ,L ) 1 +4 L − 3 D ( N ,L ) 4 − L − 3 D ( N ,L ) 4 = − 6 L − 3 D ( N ,L ) 4 + L − 2 (8 D ( N ,L ) 3 D ( N ,L ) 1 +3( D ( N ,L ) 2 ) 2 ) − 6 L − 1 D ( N ,L ) 2 ( D ( N ,L ) 1 ) 2 + ( D ( N ,L ) 1 ) 4 . (82) For n = 3 and n = 2 the formulas a re T {{ 1 , 2 , 3 }} = D ( N ,L ) 3 (83) T {{ 1 , 2 } , { 3 }} = D ( N ,L ) 1 D ( N ,L ) 2 − L − 1 D ( N ,L ) 3 (84) T {{ 1 } , { 2 } , { 3 }} = ( D ( N ,L ) 1 ) 3 − 3 L − 1 D ( N ,L ) 1 D ( N ,L ) 2 +2 L − 2 D ( N ,L ) 3 (85) T {{ 1 , 2 }} = D ( N ,L ) 2 (86) T {{ 1 } , { 2 }} = ( D ( N ,L ) 1 ) 2 − L − 1 D ( N ,L ) 2 . (87) It is clear that (78)-(82) a nd (83)-(87) cover all possibilities when it c omes to partition blo ck sizes. Using (14)-(15), and putting (7 6), (78)-(8 2), and (83)-(87) in to (5 9) we g et the expressions in Theorem 2 after some c alculations. If we are only interested in ﬁrst ord er appr oximation s rather than exact expressions, ( 77) gives us T ρ ≈ D ρ − X ρ ′ >ρ | ρ |−| ρ ′ | =1 L − 1 D ρ ′ , which is easier to comp ute. Also, we need only ﬁrst ord er approx imations to K ρ,u,N , which is much easier to compu te than the exact expression. For ( 76), K {{ 1 , 3 } , { 2 , 4 }} ,u,N ≈ 2 3 is already a ﬁrst order appro ximation. Inserting the approxima- tions in (59) giv es a ﬁrst order approx imation of the mom ents. A P P E N D I X G T H E P R O O F O F P RO P O S I T I O N 5 W e only state th e pr oof for the case c = 1 . I n [32] it is stated that the asymp totic 2 n -moment ( m 2 n ) of certain Hankel and T oeplitz matrices can be expr essed in terms of volumes o f solution sets o f equatio ns o n the f orm (63), with ρ restricted to partitions with a ll blo cks of len gth 2 . Rep hrased in our language of V a ndermo nde m ixed moment e xpansion coefﬁcients, this mean s that m 2 n = X ρ ∈ P (2 n ) ρ has two elements in each block K ρ,u (88) In th e language o f [32], the formula is not stated exactly like this, but rather in terms of v olumes of solution sets o f equations of the fo rm (63). This tran slates to (88), since we in Append ix B interpr eted K ρ,u as such volumes. In Prop osition A.1 in [32], unbound ed su pport was proved by sho wing that ( m 2 n ) 1 /n → ∞ . Again denotin g the asympto tic m oments of V andermo nde matrices with unifo rm phase distribution by V n , we ha ve that m 2 n ≤ V 2 n , since we sum over a greater class of partition s than in (88) wh en comp uting the V ande rmond e moments. Th is means tha t ( V 2 n ) 1 /n → ∞ also, so that the asymptotic mean eigen value distribution of the V ander monde matrices have unb ounde d su pport also. A P P E N D I X H T H E P RO O F O F T H E O R E M 3 W e will use the fact that K ρ,u,N = 1 (2 π ) | ρ | N n +1 −| ρ | × R (0 , 2 π ) | ρ | Q n k =1 1 − e jN ( x b ( k − 1) − x b ( k ) ) 1 − e j ( x b ( k − 1) − x b ( k ) ) dx 1 · · · dx | ρ | , (89) where integration is w .r .t. Leb esgue measure. For ρ = 1 n Theorem 3 is tri vial. W e will thu s assume that ρ 6 = 1 n in the following. W e ﬁrst prove that lim N →∞ K ρ,ω , N exists whenever p ω is continuo us. T o simplif y notatio n, deﬁne F ( ω ) = n Y k =1 1 − e j N ( ω b ( k − 1) − ω b ( k ) ) 1 − e j ( ω b ( k − 1) − ω b ( k ) ) = n Y k =1 sin  N ( ω b ( k − 1) − ω b ( k ) ) / 2  sin  ( ω b ( k − 1) − ω b ( k ) ) / 2  , and set ω = ( ω 1 , ..., ω | ρ | ) and dω = dω 1 · · · dω | ρ | . Sin ce ω is continuo us, th ere exists a p max such that p ω ( ω i ) ≤ p max for all ω i . Then we ha ve that | K ρ,ω , N | ≤ p | ρ | max N m +1 −| ρ | × R [0 , 2 π ) | ρ | Q n k =1     sin ( N ( x b ( k − 1) − x b ( k ) ) / 2 ) sin ( ( x b ( k ) − x b ( k +1) ) / 2 )     dx, IEEE TRANSACTIONS ON INFORMA TION THEOR Y , VOL. 1, NO. 1, JANU ARY 2009 23 where we have co n verted to Lebesgue measure, an d wh ere we have also written dx = dx 1 · · · dx | ρ | . Consider ﬁrst the set U = { ω || x b ( k − 1) − x b ( k ) | ≤ π ∀ k } . When 2 π N ≤ | ω b ( k − 1) − ω b ( k ) | ≤ π , it is clear that      sin  N ( x b ( k − 1) − x b ( k ) ) / 2  sin  ( x b ( k − 1) − x b ( k ) ) / 2       ≤     4 x b ( k − 1) − x b ( k )     , (90) since   sin  N ( x b ( k − 1) − x b ( k ) ) / 2    ≤ 1 , and since | s in( x ) | ≥ | x 2 | when | x | ≤ π 2 . When | x b ( k − 1) − x b ( k ) | ≤ 2 π N we have that      sin  N ( x b ( k − 1) − x b ( k ) ) / 2  sin  ( x b ( k − 1) − x b ( k ) ) / 2       ≤ N . (91) Let k 1 , ..., k | ρ | ∈ Z , and assume that k | ρ | = 0 . By using the triangle inequ ality , it is clear that on the set D k 1 ,...,k | ρ |− 1 = { ω |     x i − 2 k i π N     ≤ π N ∀ 1 ≤ i ≤ | ρ |} , when | k r − k s | ≥ 2 for all r, s , the i ’th factor in F ( x ) is bound ed by 4 N ( | k b ( r − 1) − k b ( r ) |− 1 ) π due to (9 0). Also, when | k r − k s | < 2 for some r, s , th e correspond ing factors in F ( x ) are bound ed by N o n D k 1 ,...,k | ρ | due to (91). Note also that the volume of D k 1 ,...,k | ρ |− 1 is (2 π ) | ρ |− 1 N 1 −| ρ | . By adding some more terms (to compensate for the dif ferent beha viour for | k r − k s | ≥ 2 and | k r − k s | < 2 ), we ha ve that we can ﬁnd a constant D that 1 N n +1 −| ρ | R U | F ( x ) | dx ≤ 1 N n +1 −| ρ | N n × P 0 ≤ k 1 ,...,k | ρ |− 1 1 0 fo r | x | ≤ 1 (94) The assump tion th at f ( x ) = 0 in a neighbourh ood of zero is due to the fact that the k i are all different. No te that | f ( x ) | ≤ 1 | x | 1 − ǫ for any 0 < ǫ < 1 . Also, the n − 2 -fold conv olution (we wait with the n − 1 ’th conv olution till th e end) of 1 | x | 1 − ǫ with itself exist outside 0 when ever 0 < ( n − 2) ǫ < 1 , and is on th e form r 1 | x | 1 − ( n − 2) ǫ for some constant r [36]. Th erefor e, (93) is bo unded by Z | x | > 1 r 1 | x | 1 − ( n − 2) ǫ 1 | x | dx = Z | x | > 1 r 1 | x | 2 − ( n − 2) ǫ dx = 2 r ( n − 2 ) ǫ − 1 . This proves that th e entire sum (93) is boun ded, and thus also the statement on the existence of the limit K ( ρ, ω ) in Theorem 3 when th e density is co ntinuo us. For th e re st of the proof of Theor em 3 , we ﬁrst reco rd the following result: Lemma 2: For any ǫ > 0 , lim N →∞ 1 N n +1 −| ρ | Z B ǫ,r F ( ω ) dω = 0 , (95) where B ǫ,r = { ( ω 1 , ..., ω | ρ | ) || ω b ( r − 1) − ω b ( r ) | > ǫ } . Proof: The set B ǫ,r correspo nds to tho se k 1 , ..., k | ρ | in ( 93) for which | k b ( r − 1) − k b ( r ) | > N 2 π ǫ . T hus, for large N , we sum over k 1 , ..., k | ρ | in (9 3) fo r wh ich | k b ( r − 1) − k b ( r ) | is arb itrarily large. By the con vergenc e of the F ourier in tegral o f 1 | x | , it is clear that this c onv erges to ze ro. Deﬁne B ǫ = { ( ω 1 , ..., ω | ρ | ) || ω i − ω j | > ǫ for some i, j } . If ω ∈ B ǫ , ther e mu st exist an r so that | ω b ( r − 1) − ω b ( r ) | > 2 ǫ n , so that ω ∈ B r, 2 ǫ/n . This m eans that B ǫ ⊂ ∪ r B r, 2 ǫ/n , so that by Lem ma 2 also lim N →∞ 1 N n +1 −| ρ | Z B ǫ F ( ω ) dω = 0 . This means that in the integral for K ρ,ω , N , we need only integrate over the ω wh ich are arbitrarily close to th e diago nal, (where ω 1 = · · · = ω | ρ | ). W e thus have K ρ,ω = lim N →∞ 1 N n +1 −| ρ | R [0 , 2 π ) | ρ | F ( x ) Q | ρ | r =1 p ω ( x r ) dx = lim N →∞ 1 N n +1 −| ρ | R [0 , 2 π ) | ρ | F ( x ) p ω ( x | ρ | ) | ρ | dx = lim N →∞ 1 N n +1 −| ρ | R 2 π 0 p ω ( x | ρ | ) | ρ |  R [0 , 2 π ) | ρ |− 1 F ( x ) dx 1 · · · dx | ρ |− 1  dx | ρ | . W e used here that the density is co ntinuo us. Using th at lim N →∞ 1 N n +1 −| ρ | R [0 , 2 π ) | ρ |− 1 F ( x ) dx 1 · · · dx | ρ |− 1 = (2 π ) | ρ |− 1 K ρ,u (96) when x | ρ | is kept ﬁxed at an arb itrary v alue (th is is straight- forward by using the metho ds fro m the pro of of Pro position 1 and (89)), we get that the above equals K ρ,u (2 π ) | ρ |− 1 Z 2 π 0 p ω ( x | ρ | ) | ρ | dx | ρ | , which is what we had to sho w . IEEE TRANSACTIONS ON INFORMA TION THEOR Y , VOL. 1, NO. 1, JANU ARY 2009 24 A P P E N D I X I T H E P R O O F O F P R O P O S I T I O N 6 Proposition 6 w ill fo llow directly if we can p rove the following result: Lemma 3: Let ω k ( 1 ≤ k ≤ n ) b e the uniform distribution on [ 2 π ( k − 1) n , 2 π k n ] and deﬁne ω λ 1 ,...,λ n ( 0 ≤ λ i ≤ 1 , λ 1 + · · · + λ n = 1 ) as the ph ase distribution with density p ω λ 1 ,...,λ n = λ 1 p ω 1 + · · · + λ n p ω n . Then K ρ,ω 1 n ,..., 1 n ≤ K ρ,ω λ 1 ,...,λ n . Proof: This follo ws imm ediately by noting th at K ρ,ω λ 1 ,...,λ n = K ρ,u (2 π ) | ρ |− 1  Z 2 π 0 p ω λ 1 ,...,λ n ( x ) | ρ | dx  = K ρ,u (2 π ) | ρ |− 1 × Z 2 π 0 ( λ 1 p ω 1 ( x ) + · · · + λ n p ω n ( x )) | ρ | dx = K ρ,u (2 π ) | ρ |− 1 × (( λ 1 ) | ρ | Z 2 π 0 p ω 1 ( x ) | ρ | dx + · · · +( λ n ) | ρ | Z 2 π 0 p ω n ( x ) | ρ | dx ) = K ρ,u (2 π ) | ρ |− 1 × (( λ 1 ) | ρ | Z 2 π 0 p ω 1 ( x ) | ρ | dx + · · · +( λ n ) | ρ | Z 2 π 0 p ω 1 ( x ) | ρ | dx ) = K ρ,u (2 π ) | ρ |− 1  ( λ 1 ) | ρ | + · · · + ( λ n ) | ρ |  × Z 2 π 0 p ω 1 ( x ) dx ≥ K ρ,u (2 π ) | ρ |− 1  1 n  | ρ | + · · · +  1 n  | ρ | ! × Z 2 π 0 p ω 1 ( x ) | ρ | dx = K ρ,ω 1 n ,..., 1 n , where we h ave used that x | ρ | 1 + · · · x | ρ | n constrained to x 1 + · · · + x n = 1 achie ves its minimum for x 1 = · · · = x n = 1 n . A P P E N D I X J T H E P R O O F O F T H E O R E M 4 The co ntribution in the integral K ρ,ω , N comes on ly from when the ω i coincide w ith the atoms of p . Actually , we ev a luate 1 − e jN ω 1 − e jω in points on the form ω = α i − α j . This ev a luates to N n p n i when all ω i are cho sen equal to the same atom α j . Since lim N →∞ 1 − e jN ω N (1 − e jω ) = 0 for any ﬁxed ω 6 = 0 , lim N →∞ K ρ,ω , N N − n = 0 when ω is cho sen from noneq ual atoms. (52) (with additional 1 / N - factors) thus b ecomes P ρ ∈ P ( n ) P ( j 1 ,...,j n ) giving rise to ρ P ( i 1 ,...,i n ) N | ρ |− 2 n − 1 c | ρ |− 1 L −| ρ | ( P i N n p n i + a ρ,N N n )) D 1 ( N )( j 1 , j 1 ) D 2 ( N )( j 2 , j 2 ) · · · × D n ( N )( j n , j n ) , (97) where lim N →∞ a ρ,N = 0 . M ultiplying bo th sides with N an d letting N go to inﬁnity giv es lim N →∞ X ρ ∈ P ( n ) N | ρ |− n c | ρ |− 1 X i p n i + a ρ,N ! D ρ . It is clear that this conv erges to 0 when ρ 6 = 0 n (since | ρ | < n in this case), so that th e limit is c n − 1 X i p n i ! α 0 n = c n − 1 p ( n ) lim N →∞ n Y i =1 tr L ( D i ( N )) , which proves the claim A P P E N D I X K T H E P RO O F O F T H E O R E M 5 W e need th e following identity [36]: Z ∞ 0 x − s e j nx dx = Γ(1 − s ) | n | 1 − s e jsg n ( n )(1 − s ) π 2 , where sg n ( x ) = 1 if x > 0 , sg n ( x ) = − 1 if x < 0 , and 0 otherwise. From this it follows that R ∞ −∞ p i | x − α i | − s e j nx dx = 2 p i e j nα i Γ(1 − s ) | n | 1 − s cos  (1 − s ) π 2  . (98) Note that the measure with density p , has the same asympto tics near α i as the measure with density p i | x − α i | − s on −  1 − s 2 p i  1 1 − s ,  1 − s 2 p i  1 1 − s ! . As in the proof in Appen dix J, the in tegral for the expan sion coefﬁcients is d ominated by the behaviour n ear the points ( α i , ..., α i ) . T o see this, note that the behaviour near the singular points on the diagonal is O ( s ( | ρ | − n ) − 1) when polyno mic g rowth of order s o f the density near th e singu lar points is assumed. This is very much r elated to (93) in Append ix H, since K ρ,ω here in a similar way can be boun ded by (taking into acco unt new powers of N ) C 1 N n + ns +1 −| ρ | N n N −| ρ | N | ρ | s × P 0 ≤ k 1 ,...,k | ρ | ǫ x − s e j nx dx = 0 for all ǫ > 0 , and since the contributions from lar ge n domin ate in (101) below ( since P n | n | − s div erges), it is clear that we can restrict to an interval around ω i when co mputing the limit also (since p ω is continuo us outside the singularity points, th is follows from Theore m 3 , and due to the add itional 1 N s -factor added to (1)). After restricting to 0 n , mu ltiplying both sides with N , summing over all singularity po ints, and using (98), we obtain the a pprox imation X ( i 1 ,...,i n ) X a N − ns c n − 1 ×  2 p a Γ(1 − s ) cos  (1 − s ) π 2  n × n Y k =1 e j ( i k − 1 − i k ) α a | i k − 1 − i k | 1 − s × tr L ( D 1 ( N )) × · · · × tr L ( D n ( N )) (101 ) to (52). Since Q n k =1 e j ( i k − 1 − i k ) α a = 1 , we reco gnize q ( n,N ) =  2Γ(1 − s ) cos  (1 − s ) π 2  n ( P a p n a ) × P ( i 1 ,...,i n ) N − ns Q n k =1 1 | i k − 1 − i k | 1 − s , as a factor in (101) such that the limit o f (101) as N → ∞ can be written c n − 1 lim N →∞ q ( n,N ) lim N →∞ n Y i =1 tr L ( D i ( N )) . It th erefore suf ﬁces to prove tha t lim N →∞ q ( n,N ) = q ( n ) . T o see this, write N − s | i k − 1 − i k | 1 − s = 1 N 1  1 N  1 − s | i k − 1 − i k | 1 − s = 1 N 1    i k − 1 N − i k N    1 − s . Summing over all 1 ≤ i 1 , ..., i n ≤ N , it is clear f rom th is that q ( n,N ) can be viewed as a Riemann sum which con verges to q ( n ) as N → ∞ . A P P E N D I X L T H E P R O O F O F T H E O R E M 7 A N D C O R O L L A RY 1 Proof of Theor em 7: we deﬁne S j to be the block s of σ , i.e. S j = { k | i k = j } . Note that The orem 3 gu arantees that the limit K ρ,ω = lim N →∞ K ρ,ω , N exists. The par tition ρ simply is a gro uping of rando m variables into independ ent gro ups. I t is th erefore impossible fo r a blo ck in ρ to contain elements f rom b oth S 1 and S 2 , so that any b lock is con tained in either S 1 or S 2 . As a conseq uence, ρ ≤ σ . Until n ow , we have no t trea ted mixed m oments of the form D 1 ( N ) V i 2 V H i 2 D 2 ( N ) V i 3 V H i 3 · · · × D n ( N ) V i 1 V H i 1 , which are the same as the mixed momen ts of Theo rem 7 except for the po sition of the D i ( N ) . W e will not go into depths on this, but only remark that th is case can be treated in the same vein as generalized V anderm onde matrices by replacing the density p f (or p λ in case of con tinuou s gen er- alized V an dermo nde matrices) with fun ctions p D i ( x ) deﬁned by p D i ( x ) = D i ( N )( ⌊ L x ⌋ , ⌊ Lx ⌋ ) f or 0 ≤ x ≤ 1 . This also covers the case of m ixed moments of independe nt, generalized V andermo nde matrice s (an d, in fact, th ere are n o restrictions on the ho rizontal and vertical phase densities p ω i and p λ j for each matrix. They m ay all be different). The p roof f or this is straightfor ward. Proof o f Corollary 1 : this follows in the same way as Proposition 3 is p roved from Propo sition 2, by only consid- ering ρ which are less than σ , and also by using Theo rem 3. σ ar e fo r the listed moments {{ 1 } , { 2 } } , {{ 1 , 3 } , { 2 , 4 }} , and {{ 1 , 3 , 5 } , { 2 , 4 , 6 }} , respectively . A P P E N D I X M T H E P RO O F S O F P RO P O S I T I O N 7 A N D 8 The momen ts E  tr n  W i  will be related to the mo ments P i throug h three convolution stag es: 1) relating the mo ments o f W with the mom ents of Γ = VP 1 2  1 K SS H  P 1 2 V H , (102 ) from which we easily get the moments o f ˜ S =  1 K SS H  P 1 2 V H VP 1 2 , (103) 2) relating the mo ments o f S with th e mom ents of T = PV H V , (104) 3) relating the mo ments o f T with the mo ments of P . IEEE TRANSACTIONS ON INFORMA TION THEOR Y , VOL. 1, NO. 1, JANU ARY 2009 26 For the ﬁrst stage, the mome nts of ˆ W and Γ re late throug h the for mulas E [ tr n ( W )] = E [ tr N ( Γ )] + σ 2 (105) E  tr n  W 2  = E  tr N  Γ 2  +2 σ 2 (1 + c 1 ) E [ tr N ( Γ )] + σ 4 (1 + c 1 ) (106) E  tr n  W 3  = E  tr N  Γ 3  +3 σ 2 (1 + c 1 ) E  tr N  Γ 2  +3 σ 2 c 1 E h ( tr N ( Γ )) 2 i +3 σ 4  c 2 1 + 3 c 1 + 1 + 1 K 2  E [ tr N ( Γ )] + σ 6  c 2 1 + 3 c 1 + 1 + 1 K 2  , , (107) which a re obtained by r eplacing R in [33] by VP 1 2 S , with c = c 1 = N K . For the secon d p art of the ﬁrst stage, note that E  tr N  Γ k  = c 2 E h tr L  ˜ S k i (108) E h ( tr N ( Γ )) k i = c k 2 E   tr L  ˜ S  k  , (109) where c 2 = L N . W e can now apply Theo rem 2 to o btain c 3 E h tr L  ˜ S i = c 3 E [ tr L ( T )] (110) c 3 E h tr L  ˜ S 2 i = c 3 E  tr L  T 2  + c 2 3 E h ( tr L ( T )) 2 i (111) c 3 E h tr L  ˜ S 3 i =  1 + K − 2  c 3 E  tr L  T 3  +3 c 2 3 E  ( tr L T ) tr L  T 2  + c 3 3 E h ( tr L ( T )) 3 i (112) E   tr L  ˜ S  2  = E h ( tr L ( T )) 2 i + 1 K L E  tr L  T 2  , (113) where c 3 = L K , and T = PV H V . (105)-(10 7 ), (1 08)-(109), and (110)-(1 13) can b e comb ined to E [ tr n ( W )] = c 2 E [ tr L ( T )] + σ 2 (114) E  tr n  W 2  = c 2 E  tr L  T 2  + c 2 c 3 E h ( tr L ( T )) 2 i +2 σ 2 ( c 2 + c 3 ) E [ tr L ( T )] + σ 4 (1 + c 1 ) (115) E  tr n  W 3  = c 2  1 + 1 K 2  E  tr L  T 3  +3 c 2 c 3 E  ( tr L ( T ))  tr L  T 2  + c 2 c 2 3 E h ( tr L ( T )) 3 i +3 σ 2  (1 + c 1 ) c 2 + c 1 c 2 2 K L  E  tr L  T 2  +3 σ 2 c 3 ( c 3 + 2 c 2 ) E h ( tr L ( T )) 2 i +3 σ 4  c 2 1 + 3 c 1 + 1 + 1 K 2  c 2 E [ tr L ( T )] + σ 6  c 2 1 + 3 c 1 + 1 + 1 K 2  . (116) Up to now , all for mulas ha ve provide d exact expressions for the expectations. For the next step, exact expressions for th e expectations are o nly kn own whe n the phase d istributions ar e unifor m, in whic h case the fo rmulas are given by T heorem 2: c 2 E [ tr L ( T )] = c 2 tr L ( P ) (117) c 2 E  tr L  T 2  =  1 − N − 1  c 2 tr L ( P 2 ) + c 2 2 ( tr L ( P )) 2 (118) c 2 E  tr L  T 3  =  1 − 3 N − 1 + 2 N − 2  c 2 tr L ( P 3 ) +3  1 − N − 1  c 2 2 tr L ( P ) tr L ( P 2 ) + c 3 2 ( tr L ( P )) 3 (119) E h ( tr L ( T )) 2 i = tr L ( P ) 2 (120) E h ( tr L ( T )) 3 i = tr L ( P ) 3 (121) E  ( tr L ( T ))  tr L  T 2  =  1 − N − 1  tr L ( P ) tr L ( P 2 ) + c 2 ( tr L ( P )) 3 . (122) If the phase distribution ω is n ot u niform , Theorem 1 and Theorem 3 gives the f ollowing approx imation: c 2 E [ tr L ( T )] = c 2 tr L ( P ) (123) c 2 E  tr L  T 2  ≈ c 2 tr L ( P 2 ) + c 2 2 I 2 ( tr L ( P )) 2 (124) c 2 E  tr L  T 3  ≈ c 2 tr L ( P 3 ) + 3 c 2 2 I 2 tr L ( P ) tr L ( P 2 ) + c 3 2 I 3 ( tr L ( P )) 3 (125) E h ( tr L ( T )) 2 i = ( tr L P ) 2 (126) E h ( tr L ( T )) 3 i = ( tr L P ) 3 (127) E  ( tr L ( T ))  tr L  T 2  ≈ tr L ( P ) tr L ( P 2 ) + c 2 I 2 ( tr L ( P )) 3 , (128) where the approx imation is O ( N − 1 ) , and wh ere I k is deﬁned by (42). Proposition 8 is proved by combin ing (114)-(11 6) with (117)-(122), while Prop osition 7 is proved by c ombinin g (114)-(116) with (123)-(12 8). Prop osition 9 is p roved by ﬁrst IEEE TRANSACTIONS ON INFORMA TION THEOR Y , VOL. 1, NO. 1, JANU ARY 2009 27 observing that the roles of L and N are interchan ged, since the V anderm onde matrix is replaced by its tran spose. This means th at we obtain the formu las (114)-(1 16), with c 1 and c 3 interchang ed, and c 2 replaced with 1 c 2 . The matrix T is now instead VV H , and these can be scale d to o btain the moments of V H V . Finally the integrals I n or the ang le α can be estimated from these momen ts, using (12 3)-(128) with the momen ts of P rep laced with 1 (since no additio nal power matrix is includ ed in the mod el). Matlab cod e for implementin g the steps (105)-(1 07), (110)- (113), and ( 117)-(122) can be fo und in [34]. A C K N O W L E D G M E N T The authors would like to than k the an onymou s revie wers for their insig htful and valuable c omments, which hav e help ed improve the q uality of the paper . Th ey would also like to th ank the Associate Editor Pro f. G. T aricc o for a very professional processing of the manuscript. R E F E R E N C E S [1] C. S. Burrus and T . W . Parks, DFT/FFT and Conv oluti on Algorithms . Ne w Y ork: John Wil ey , 1985. [2] G. H. Golub and C. F . V . 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He studied mathe- matics at th e University of Oslo, where he received the M. Sc and the Ph.D. degrees in 19 93 and 1997 , re spectiv ely . From 1997 to 2 004, he worked a s a con sultant a nd pro duct developer in various information tec hnolog y projects. From 2004 to 2 007, he was a postdoctor al fellow at the In stitute of Inform atics at the University of Oslo. He is cur rently employed as a researcher at th e Centre of Mathematics f or Applicatio ns at the University of Oslo. H is research in terests are applica- tions of free p robab ility theor y and rand om matr ices to the ﬁelds o f wireless co mmun ication, ﬁnance, and info rmation theory . Merouane Debb ah was born in Mad rid, Spain. He entered the Ecole Normale Supr ieure d e Cachan ( France) in 199 6 wher e he recei ved the M.Sc and the Ph.D. degrees respe ctiv ely in 1999 and 20 02. From 1999 to 2002 , he worked f or Mo torola Labs on W ir e- less Local Area Networks and prospective fourth generatio n systems. Fro m 2 002 until 2003 , he was appo inted Senior Researcher at the V ienna Research Center for T elecommu- nications (ftw .), V ienna, Austria working on MIMO wireless channel modeling issues. From 200 3 until 2007, he joined the Mobile Communicatio ns departm ent of the I nstitute Eu recom (Sophia Antipolis, France) a s an Assistant Professor . He is presently a Professor at Sup elec (Gif-sur-Yvette, Fran ce), holder o f the Alcatel-Lucen t Chair on ﬂexible radio. His research interests are in info rmation theo ry , sign al processing and wireless com municatio ns.

Asymptotic Behaviour of Random Vandermonde Matrices with Entries on the Unit Circle

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