cs.IT 2010-07-28 0

MIMO Networks: the Effects of Interference

Multiple-input/multiple-output (MIMO) systems promise enormous capacity increase and are being considered as one of the key technologies for future wireless networks. However, the decrease in capacity due to the presence of interferers in MIMO networ…

Authors: Marco Chiani, Moe Z. Win, Hyundong Shin

MIMO Networks: the Effects of Interference
MIMO Netw orks: the Ef fe cts of Interference Marco Chiani, Senior Member , IEEE , Moe Z. W in, F ellow , IEEE , and Hyundong Shin, Member , IEEE . Corresponding Address: Marco Chiani DEIS, Uni v ersity of Bologna V .le Risor gimento 2, 40136 Bologna, IT AL Y T el: +39-0512093084 Fax: +39-0512093540 e-mail: marco.ch iani@unibo .it This research was supported, in part, by the European Commission in the scope of the FP7 project CoExisting Short Range Radio by Adv anced Ul tra-W ideBand Radio T echnology (E UWB), the National Science F oundation under Grants E CCS-0636519 and ECCS-0901034, the Of fice of Nav al Research Presidential Early Career A ward for Scientists and Engineers (PECASE) N00014-09 -1-0435, the MIT Instit ute for Soldier Nanotechnologies, the Ko rea Science and Engineering Found ation (KOSEF) grant funded by the K orea gov ernment (MOST) (No. R01-2007-000-112 02-0), and the Basic S cience R esearch Program through the National Research Founda tion of Korea ( NRF) funded by the Ministry of Education, Science and T echnology . M. Chiani is with WiLab/DEIS, Uni v ersity of Bologna, V .le Risorgimento 2 , 40136 Bologna, IT AL Y (e-mail: marco.chiani@ unibo.it ). M. Z. Win is with the Laboratory for Information and Decision Systems (LIDS ), Massachusetts Institute of T echnology , Room 32-D666, 77 Massachu setts A ven ue, Cambridge, MA 0213 9 US A (e-mail: m oewin@mit.edu ). H. Shin is with t he School of Electronics and Information Ky ung Hee Univ ersity Y ongin, Ky ungki 446-701, K orea (e-mail: hshin@khu.ac. kr ). CHIANI, WIN, SHIN: MIMO NETWORKS. 1 Abstract MIMO system s are b eing c onsidered as one o f the key enablin g technolo gies for futu re wireless networks. Howe ver , the decrea se in capac ity due to the presen ce of interfe rers in MI MO networks is not well un derstood. In this pap er , we develop an analytical framework to chara cterize the capacity of MIMO commun ication sy stems in th e p resence o f m ultiple MIMO co-chann el interf erers and no ise. W e co nsider th e situation in wh ich transmitters ha ve n o channel state informatio n and all links undergo Rayleigh fading. W e first generalize th e d eterminant repr esentation o f hypergeo metric f unctions with matrix argum ents to the case when the argu ment matrices have eigen values of arbitrary multiplicity . This enables th e derivation of the distribution o f the eige n values of Gau ssian quadratic f orms and Wishart matrices w ith arbitrary co rrelation, with ap plication to both sing le-user and mu ltiuser MIMO systems. In particular, we der i ve the ergodic mu tual info rmation for MIMO systems in the pr esence of mu ltiple MIMO interfe rers. Ou r analysis is valid fo r any numbe r of interferer s, each with arbitrary nu mber of antennas having possibly u nequal power levels . This fra mew ork, therefor e, accommo dates th e stud y of distributed MI MO system s and accounts for different spatial p ositions of the MIMO interfe rers. Index T erms Eigenv alues distribution, Gaussian quadratic f orms, Hypergeo metric fun ctions of m atrix argum ents, Interfer ence, MIMO, W ishart matrices. I . I N T R O D U C T I O N The use of m ultiple transmitt ing and receiving antennas can provide high spectral efficienc y and link reliability for point-to-point communication in f ading en vironments [1], [2]. Th e analysis of capacity for MIMO channels in [3] suggested prac tical receiv er structures to obtain such spectral ef ficiency . Since t hen, many studies hav e been dev oted to the analysis o f MIMO sys tems, starting from the er godic [4] and out age [5] capacity for uncorrelated fading to the case where correlation is present at one of the two sides (either at the transm itter or at the recei ver) or at both si des [6]–[8]. The effect of time correlation is studied in [9]. Only a few papers, by usi ng simulatio n or approximations, have studied the capacity of MIMO systems in the presence of cochannel interference. In parti cular , a s imulation st udy is presented in [10] for cell ular systems , assuming up to 3 transm it and 3 receiv e antennas. The sim ulations showed that cochannel interference can seriousl y d egrade the overall capacity when MIMO li nks are used in cellular networks. In [11], [12] it is studied whether , i n a MIMO multiu ser scenario, DRAFT 2 SUBM. TO IEEE T RANS. ON INF . TH. it is alw ays con venient to use all transm itting antennas. It was found t hat for some values of SNR and SIR, allocating all power into a single transm itting antenna, rather than dividing the power equally amon g independent st reams from the different antennas, would l ead to a hig her overall system mu tual information . The stud y in [11 ], [12] adopts simulation to e valuate the capacity of MIMO sy stems in the presence of cochannel int erference, and the difficulties in t he ev aluations limited the result s to a scenario wit h two MIMO users emp loying at most two antenna elements. In [13] the replica method is used to obtain approxim ate moments of the capacity for MIMO systems wit h large number of antenna elements in cluding the presence of interference. The approximation requires iterativ e numerical methods to solve a system of non-linear equation s, and i ts accuracy has to be verified by com puter simul ations. A mul tiuser MIMO sys tem with specific receiver st ructures is analyzed for the interference-limited case in [14], [15 ]. The MIMO capacity at high and low SNR for i nterference-limited scenarios i s addressed in [16], [17]. A worst-case anal ysis for M IMO capacity wit h CSI at the t ransmitter and at the recei ver , conditi oned on the channel matrix, can b e found in [18]. Asymptoti c results for the Rician channel in the p resence of interference can be found in [19]. In t his paper , we dev elop an analytical frame work to analyze the ergodic capacity of MIMO systems in the presence of m ultiple MIMO cochannel interferers and A WGN. Throughout the paper we consider ri ch scattering environments in which transmitters have no CSI, t he receiv er has perfect CSI, and all lin ks undergo frequency flat Rayleigh fading. The key contributions o f the paper are as foll o ws: • Generalizatio n of the determi nant representation of hyp er geometric functions with m atrix ar guments to t he case where matrices in the arguments have eigen v alues with arbitrary multipli city . • Deriva tion, using the generalized representation, of the joint p.d.f. of the eigen v alues of complex Gaussi an quadratic forms and W ishart m atrices, with arbitrary m ultipliciti es for the eigen va lues of the associated covariance matri x. • Deriva tion of th e ergodic capacity of singl e-user MIM O system s that accounts for arbitrary power lev els and arbitrary correlatio n across the transmitti ng antenna elements, or arbitrary correlation at the recei ver si de. • Deriva tion of capacity expressions for M IMO syst ems in the presence of multip le MIMO interferers, valid for any number of interferers, each wi th arbitrary num ber of antenn as DRAFT CHIANI, WIN, SHIN: MIMO NETWORKS. 3 having po ssibly unequal power leve ls. The paper i s organized as follows: in Section II we int roduce the sy stem model for mul tiuser MIMO setting, relati ng the er godic capacity of MIMO systems in the presence of multiple MIMO interferers to that of single-user MIMO syst ems with no int erference. General results on hypergeometric functions of m atrix arguments are giv en in Section III. The joint p .d.f. of eigen values for Gaussian quadrati c forms and W i shart matrices with arbit rary correlation is given in Section IV. In Section V we give a u nified expression for the capacity of single-user MIMO systems that accounts for arbitrary correlation matrix at one side. Numerical resul ts for MIMO relay networks and multiuser MIMO are presented in Section VI, and conclusions are giv en in Section V II. Throughout the paper vectors and matrices are ind icated by bol d, | A | and det A denot e the determinant of matrix A , and a i,j is th e ( i, j ) th element of A . Expectation operator is denoted by E {·} , and in particular E X {·} denotes expectation wi th respect to the random variable X . The superscript † d enotes conjug ation and transposition , I i s the identity matrix (in particular I n refers to the ( n × n ) identity m atrix), t r { A } is the trace of A and ⊕ is us ed for the direct sum of m atrices defined as A ⊕ B = diag ( A , B ) [20]. I I . S Y S T E M M O D E L S W e consider a network scenario as s hown in Fig. 1, where a MIMO- ( N T 0 , N R ) lin k, with N T 0 and N R denoting the num bers of transm itting and receiving antennas, respectiv ely , is su bject t o N I MIMO co-channel in terferers from o ther li nks, each with arbitrary num ber of antennas. The N R -dimensional equiv alent lowpass s ignal y , after matched filterin g and s ampling, at the out put of t he receiving antennas can be written as y = H 0 x 0 + N I X k =1 H k x k + n (1) where x 0 , x 1 , . . . , x N I denote the complex transmitted v ectors wit h dim ensions N T 0 , N T 1 , . . . , N T N I , respectiv ely . Subscript 0 is used for the desired signal, whi le subscript s 1 , . . . , N I are for the interferers. The additiv e noise n i s an N R -dimensional random ve ctor with zero-mean i.i .d. circularly symmetric comp lex Gaussian entries, each wit h independent real and imaginary parts having variance σ 2 / 2 , so that E  nn †  = σ 2 I . The power transm itted from the k th user i s E n x † k x k o = P k . DRAFT 4 SUBM. TO IEEE T RANS. ON INF . TH. N T 0 ✚ ✚ ❩ ❩ ✚ ✚ ✄ ✂  ✁ N R ✚ ✚ ✚ ✚ ❩ ❩ ❩ ❩ ✄ ✂  ✁ P 0 , x 0 H 0 H 1 H N I ✲ ✸ ✸ ✼ ✣ y N T N I N T 1 ✚ ✚ ✚ ✚ ❩ ❩ ❩ ❩ ✚ ✚ ✚ ✚ ✄ ✂  ✁ ✄ ✂  ✁ P N I , x N I P 1 , x 1 ♣ t t t ❢ ❢ ✻ ✻ n ✲ Fig. 1. MIMO Network. The matrices H k in (1) d enote t he channel m atrices of size ( N R × N T k ) with com plex elements h ( k ) i,j describing the gain of the radio channel between the j th transmittin g antenna of the k th MIMO interferers and the i th recei ving antenna o f the desired l ink. In particul ar , H 0 is the matrix describi ng the channel of the desired lin k (see Fig. 1). When consi dering stati stical v ariations of th e channel, the channel gain s must be described as r .v .. In particular , we assume uncorrelated MIMO Rayleigh fading channels for which the entries of H k are i.i .d. circularly symmetric com plex Gaussian r .v . with zero-mean and variance one, i.e., E n | h ( k ) i,j | 2 o = 1 . W i th thi s normalization, P k represents the short-term avera ge recei ved power per antenna element from user k , which depends on t he transmit power , path-loss , and shadowing between t ransmitter k and the (i nterfered) receiv er . Thus , the P k are in general diffe rent. Conditioned to th e channel matrices { H k } N I k =0 , the mutual information between the receiv ed vector , y , and the desired transmitt ed vector , x 0 , is [21] I  x 0 ; y | { H k } N I k =0  = H  y | { H k } N I k =0  − H  y | x 0 , { H k } N I k =0  (2) DRAFT CHIANI, WIN, SHIN: MIMO NETWORKS. 5 where H ( · ) denotes differential entropy . Here we con sider the s cenario in whi ch the receiver has perfect CSI, and all the transmit ters hav e no CSI. Note t hat t he term CSI in cludes t he information about the channels associated with all other M IMO interfering users. In this case, since the users do not know what is the interference seen at the receiv er (if any), a rea sonable strategy is that each user transmit s circularly symmetric Gaussian vector sig nals with zero m ean and i.i.d. element s. Thus, the transmi t power per antenna element of the k th user is P k /N T k . Note that this model includes t he case in which t he power lev els of the individual antennas are differ ent: i t suffic es to decompose a transmitter int o v irtual sub-transmitt ers, each with t he proper power l e vel. Hence, conditioned on all channel matrices { H k } N I k =0 in (1), bo th y and y | x 0 are circularly symmetric Gaussian. Since the differe ntial entropy of a Gaussian vec tor is proportio nal to the logarit hm of the determinant of its covariance matrix , we obtain th e conditional mutu al information C MU  { H k } N I k =0  = lo g det K y det K y | x 0 (3) where K y and K y | x 0 respectiv ely deno te the covariance matrices of y and y | x 0 , condi tioned on the channel gains { H k } N I k =0 . By expanding the covariance matrices us ing (1), the condit ional mutual information of a MIMO link in the presence of mu ltiple MIMO interferers with CSI only at the recei ver is th en given by: C MU  { H k } N I k =0  = lo g det  I N R + ˜ H ˜ Ψ ˜ H †  det ( I N R + HΨH † ) (4) where the N R × ( P N I i =1 N T i ) m atrix H is H = [ H 1 | H 2 | · · · | H N I ] the N R × ( P N I i =0 N T i ) m atrix ˜ H is ˜ H = [ H 0 | H ] the cov ariance mat rices Ψ , ˜ Ψ are Ψ =  1 I N T 1 ⊕  2 I N T 2 ⊕ · · · ⊕  N I I N T N I (5) and ˜ Ψ =  0 I N T 0 ⊕ Ψ (6) DRAFT 6 SUBM. TO IEEE T RANS. ON INF . TH. with  i = P i N T i σ 2 . (7) W it h random channel m atrices th e mut ual i nformation i n (4) is the d if ference b etween random var iables of the form log det  I + HΦH †  where t he elem ents of H are i.i.d. complex Gaussian and Φ is a covar iance matrix. The statist ics of such random variables has been i n vestigated in [6]–[8], assuming that the eigen values of Φ were distinct. Howev er , in the scenario under analysis these results cannot be used directly , since in (4) each eigen va lue  i of Ψ and ˜ Ψ has multipli city N T i . W e consider the ergodic mu tual information as a performance measure: taking t he expectation of (4) with respect to the distribution of { H k } N I k =0 , we get C MU , E n C MU  { H k } N I k =0 o = C SU N I X i =0 N T i , N R , ˜ Ψ ! − C SU N I X i =1 N T i , N R , Ψ ! (8) where C SU ( n T , n R , Φ ) , E H  log det  I n R + HΦH †  denotes t he er godic mutual informatio n of a si ngle-user MIMO- ( n T , n R ) Rayleigh fading channel with unit noise variance per receiving antenna and channel cov ariance matrix Φ at the transmitter . Note that the “building block” E H  log det  I + HΦH †  is simple to ev aluate when the cov ariance matrix Φ is proportional t o an identity matrix, which corresponds to a t ypical interference-free case with equal transmit power among all trans mitting antennas (see, e.g., [4]). In contrast, in the presence of interference, the cov ariance m atrix is of the type indicated in (5) and (6), where the power l e vels of the different users are in general differe nt. Note that e ven when the power for the i th user is equally spread over the N T i antennas, the m atrices in (5) and (6) are in general n ot propo rtional to identity matri ces and their eig en values hav e multipli cities greater than one. Therefore, studying MIMO syst ems i n the presence of mu ltiple MIMO cochann el i nterferers requires the characterization of C SU ( n T , n R , Φ ) in a general s etting in which the cov ariance matrix Φ has eigen values of arbitrary multipli cities. T o thi s aim, we derive in the next sections simp le expressions for the hypergeometric functions of matrix arguments with not necessarily distinct ei gen values; t hen, we obt ain the joint p .d.f. of the ei gen values o f central W ishart m atrices as well as that of Gaussian quadratic forms with arbitrary covariance matri x. DRAFT CHIANI, WIN, SHIN: MIMO NETWORKS. 7 I I I . H Y P E R G E O M E T R I C F U N C T I O N S W I T H M A T R I X A R G U M E N T S H A V I N G A R B I T R A RY E I G E N V A L U E S Hypergeometric funct ions with matrix arguments [22] hav e been used extensiv ely in mul ti- var iate stati stical analysi s, esp ecially in problem s related to the distribution of random matrices [23]. These functi ons are defined i n terms of a series of zonal polynomials , and, as such, they are functions only of th e eigen values (or latent roo ts) of the argument m atrices [22], [23]. Definition 1: The hypergeometric functions of two Hermitian m × m matri ces Λ and W are defined by [22] p ˜ F q ( a 1 , . . . , a p ; b 1 , . . . , b q ; Λ , W ) , ∞ X k =0 X κ ( a 1 ) κ · · · ( a p ) κ ( b 1 ) κ · · · ( b q ) κ C κ ( Λ ) C κ ( W ) k ! C κ ( I m ) (9) where C κ ( · ) is a s ymmetric homogeneou s polyn omial o f degree k i n the eigen va lues of i ts ar gument, called zo nal polynomial , th e sum P κ is over all partition s of k , i.e., κ = ( k 1 , . . . , k m ) with k 1 ≥ k 2 ≥ · · · ≥ k m ≥ 0 , k 1 + k 2 + · · · + k m = k , and the generalized hypergeometric coef ficient ( a ) κ is giv en by ( a ) κ = Q m i =1  a − 1 2 ( i − 1)  k i with ( a ) k = a ( a + 1) · · · ( a + k − 1) , ( a ) 0 = 1 . W e remark t hat zon al pol ynomials are symm etric po lynomials in the eigen v alues of the matrix ar gument. Therefore, hypergeometric functions are only functions of the eigenv alu es o f their matrix arguments. In ot her words, without lo ss of generality we can replace Λ and W wi th the diagonal matrices diag ( λ 1 , . . . λ m ) and diag ( w 1 , . . . w m ) , where λ i and w j are the eigen values of Λ and W , respective ly . Clearly the order of Λ and W is unimportant. It is quite evident that these functions expressed as a series of zon al po lynomials are in general very di f ficult t o manage and the form of (9) is no t tractable for further analysis. Fortunately , when the ei gen values of Λ and W are all dis tinct, a simpler expression in terms of determinants of matrices whose elements are hypergeometric functio ns of scalar arguments can be obtained as fol lows [24, Lemma 3]: Lemma 1: ([Khatri, 1970]) Let Λ = di ag ( λ 1 , . . . λ m ) and W = diag ( w 1 , . . . w m ) with λ 1 > · · · > λ m and w 1 > · · · > w m . Then we ha ve p ˜ F q ( a 1 , . . . , a p ; b 1 , . . . , b q ; Λ , W ) = Γ ( m ) ( m ) ψ ( m ) q ( b ) ψ ( m ) p ( a ) | G | Q i w 2 · · · > w m , and the functions f i ( w ) hav e deriva tive s f ( n ) i ( w ) = d n f i ( w ) dw n of orders at l east m − 1 throughout neighborhoods of the points w 1 , . . . , w m . Then, the contin uous extension ˘ P ( w 1 , w 2 , . . . , w m ) of t he function P ( w 1 , w 2 , . . . , w m ) to those point s in R m with L coincident arguments w K = w K +1 = · · · w K + L − 1 is obtained by DRAFT CHIANI, WIN, SHIN: MIMO NETWORKS. 9 removing the zero fa ctors from the denom inator i n (14), replacing the columns of the matrix in (14) correspond ing to the coi ncident ar guments with the successive deriv ativ es f ( L − l ) i ( w K ) , l = 1 , . . . , L , and dividing by a scalin g factor Γ ( L ) ( L ) = Q L − 1 i =1 i ! . For example, for w 1 = w 2 = · · · w L , this procedure gives ˘ P ( w 1 , w 2 , . . . , w m ) = 1 Q i · · · > λ m and w 1 > · · · > w k = w k +1 = · · · = w k + L − 1 > w k + L > · · · > w m . Then we ha ve 1 0 ˜ F 0 ( Λ , W ) = Γ ( m ) ( m ) Γ ( L ) ( L ) | G | Q i 1 give s ri se to L columns λ L − 1 i e λ i w , λ L − 2 i e λ i w , . . . , λ 2 i e λ i w , λ i e λ i w , e λ i w in the matrix G of (16), w ith the proper scaling factor Γ ( L ) ( L ) . Using Lemma 3 with k = m − L + 1 and w k = 0 results in the fol lowing corollary , for the case where som e eigen v alues are equal to zero. Cor ol lary 1: Let Λ = diag ( λ 1 , . . . λ m ) and W = diag ( w 1 , . . . w m ) with λ 1 > · · · > λ m and w 1 > · · · > w m − L +1 = w m − L +2 = · · · = w m = 0 . Then we h a ve 0 ˜ F 0 ( Λ , W ) = Γ ( m ) ( m ) Γ ( L ) ( L ) | G | Q i · · · > λ m and w 1 > · · · > w k = w k +1 = · · · = w k + L − 1 > w k + L > · · · > w m . Then we ha ve 1 ˜ F 0 ( r ; Λ , W ) = Γ ( m ) ( m ) Γ ( L ) ( L ) ( − 1) ( L − 1) L/ 2 ψ ( m ) 1 ( r ) · γ L − 1 ( γ − 1) L − 2 · · · ( γ − L + 2) | A | Q i 1 g i ves ris e to L columns λ L − 1 i (1 − λ i w ) γ − ( L − 1) , . . . , λ 2 i (1 − λ i w ) γ − 2 , λ i (1 − λ i w ) γ − 1 , (1 − λ i w ) γ in the matrix A of (20), and to a factor ( − 1) ( L − 1) L/ 2 γ L − 1 · · · ( γ − L + 2) / Γ ( L ) ( L ) . DRAFT CHIANI, WIN, SHIN: MIMO NETWORKS. 11 Using Lem ma 4 wi th k = m − L + 1 and w k = 0 results in the following corol lary . Cor ol lary 2: Let Λ = diag ( λ 1 , . . . λ m ) and W = diag ( w 1 , . . . w m ) with λ 1 > · · · > λ m and w 1 > · · · > w m − L +1 = w m − L +2 = · · · = w m = 0 . Then we h a ve that (20) hol ds, with a i,j =    λ m − j i j = m − L + 1 , . . . , m (1 − λ i w j ) γ elsewhere . (22) In other words, the matrix A has in this case the last L colum ns with elements λ L − 1 i , λ L − 2 i , . . . , λ i , 1 . Finally , we giv e the result for the p ˜ F q ( · ) . Lemma 5: Let Λ = d iag ( λ 1 , . . . λ m ) and W = diag ( w 1 , . . . w m ) with λ 1 > · · · > λ m and w 1 > · · · > w k = w k +1 = · · · = w k + L − 1 > w k + L > · · · > w m . Then we ha ve p ˜ F q ( a 1 , . . . , a p ; b 1 , . . . , b q ; Λ , W ) = Ξ | C | Q i µ (2) . . . > µ ( L ) are the L distinct eigen values of Φ − 1 , with corresponding multiplicities m 1 , . . . , m L such that P L i =1 m i = n . The ( n × n ) mat rix ˜ G ( x , µ ) has elements ˜ g i,j =    ( − x j ) d i e − µ ( e i ) x j j = 1 , . . . , n min [ n − j ] d i µ n − j − d i ( e i ) j = n min + 1 , . . . , n (27) where [ a ] k = a ( a − 1) · · · ( a − k + 1) , [ a ] 0 = 1 , e i denotes the unique integer such that m 1 + . . . + m e i − 1 < i ≤ m 1 + . . . + m e i and d i = e i X k =1 m k − i. Pr oof: See Appendix I. Note that Lemma 6 gives, i n a compact form, the general joi nt distribution for the eigen values of a central W ishart ( p ≥ n ), and central pseudo-W ishart or quadratic form ( n ≥ p ), wi th arbitrary on e-sided correlation matrix with not-necessarily distinct eig en values. In fact, Lemm a 6 can be us ed for both p ≥ n and n ≥ p ; i n particular , for n ≥ p we ha ve Q n min i =1 x p − n min i = 1 in (25), while for p ≥ n the second r ow in (27) dis appears and ( − 1) p ( n − n min ) = 1 in (26). Moreover , using Lemma 6 and the results in [32], [33] we can also deri ve the marginal distribution of individual eigen v alues or of an arbitrary subset of the eig en values. V . E R G O D I C M U T U A L I N F O R M A T I O N O F A S I N G L E - U S E R M I M O S Y S T E M In this section we provide a unified analysi s of the ergodic mu tual informati on of a s ingle-user MIMO sy stem with arbitrary power levels/correlation among the transm itting antenna elements or arbitrary correlation at the receive r , admitting correlation matri ces with not-necessarily dist inct eigen values. Let us consider the functi on C SU ( n, p, Φ ) = E H  log det  I p + HΦH †  (28) DRAFT CHIANI, WIN, SHIN: MIMO NETWORKS. 13 where Φ is a generic ( n × n ) posi tiv e definite m atrix and H is a ( p × n ) random matrix with zero-mean, uni t variance complex Gaussian i .i.d. entries. Now , consider a sin gle-user MIMO- ( n T , n R ) Rayleigh fading channel with Ψ T , Ψ R denoting the ( n T × n T ) transmit and ( n R × n R ) recei ve correlation matri ces, respectiv ely , having di agonal elements equal t o one. Assume the transmit vector x is zero-mean complex Gaussian, with arbitrary (but fixed) ( n T × n T ) covariance matrix Q = E  xx †  so th at tr { Q } = P . Then, the function (28) can be used to express the er godic mutual information in the following cases [6]–[8]: 1) the MIM O- ( n T , n R ) chann el with no correlation at the recei ver ( Ψ R = I ), co variance matrix at t he transm itter side Ψ T , and transmit cov ariance m atrix Q . In t his case the m utual in formation is C SU ( n T , n R , Φ ) with Φ = (1 /σ 2 ) Ψ T Q . If also Ψ T = I , we ha ve Φ = (1 / σ 2 ) Q and therefore tr { Φ } = P / σ 2 . 2) the M IMO- ( n T , n R ) channel with no correlation at the transmi tter ( Ψ T = I ), covariance matrix at the receiv er side Ψ R , and equal power all ocation Q = P / n T I . In this case the capacity is C SU ( n R , n T , Φ ) with Φ = ( P /n T σ 2 ) Ψ R , giving tr { Φ } = ( P /σ 2 )( n R /n T ) , in accordance to [6, Theorem 1 ]. In both cases P /σ 2 represents the SNR per receiving antenna. By indicating with n min = min( n, p ) and wi th f λ ( · , . . . , · ) the joint p.d.f. of the (real) ordered non-zero eigen v alues λ 1 ≥ λ 2 ≥ . . . ≥ λ n min > 0 o f th e ( p × p ) random matrix W = HΦH † , we can write: C SU ( n, p, Φ ) = E ( n min X i =1 log (1 + λ i ) ) = Z · · · Z D ord f λ ( x 1 , . . . , x n min ) n min X i =1 log (1 + x i ) d x (29) where the m ultiple int egral is ov er the d omain D ord = {∞ > x 1 ≥ x 2 ≥ . . . ≥ x n min > 0 } and d x = dx 1 dx 2 · · · dx n min . The nested integral in (29) can be e valuated u sing t he results from pre vious sections and Appendix II, leading to the following Theorem. Theor em 1: The ergodic m utual information of a MIMO Rayleigh fading channel with CSI at the recei ver only and one-sid ed correlation matrix Φ having eigen va lues of arbitrary multipliciti es DRAFT 14 SUBM. TO IEEE T RANS. ON INF . TH. is given by C SU ( n, p, Φ ) = K n min X k =1 det  R ( k )  . (30) In t he previous equation n min = min( n, p ) , the matrix R ( k ) has elements r ( k ) i,j =            ( − 1) d i R ∞ 0 x p − n min + j − 1+ d i e − x µ ( e i ) dx j = 1 , . . . , n min , j 6 = k ( − 1) d i R ∞ 0 x p − n min + j − 1+ d i e − x µ ( e i ) log (1 + x ) dx j = 1 , . . . , n min , j = k [ n − j ] d i µ n − d i − j ( e i ) j = n min + 1 , . . . , n (31) and [ a ] k , e i , d i , K are defined as in Lemma 6, where µ (1) > µ (2) . . . > µ ( L ) are the L disti nct eigen values of Φ − 1 , wi th corresponding multiplicit ies m 1 , . . . , m L . Pr oof: In Section IV i t is shown th at the j oint p.d.f. of t he ordered eigen v alues of W can b e writt en as (25), where t he el ements of V ( x ) , ˜ G ( x , µ ) are real function s of x 1 , . . . , x n min . Thus, by using Appendix II, the multip le in tegral in (29) reduces to (30). Note that the integral in (31) can be ev aluated easily with st andard numerical techniques; howe ver , the integral can be further sim plified, using the identiti es R ∞ 0 x m e − xµ dx = m ! /µ m +1 , and R ∞ 0 x m e − xµ ln(1 + x ) dx = m ! e µ P m i =0 Γ( i − m, µ ) /µ i +1 , where Γ( · , · ) is the incomplete Gamma functio n. Theorem 1 gives, in a uni fied way , the exact mutual information for MIM O sy stems, encom- passing the cases of n R ≥ n T and n T ≥ n R with arbitrary correlation at t he transmitter or at the receiv er , av oiding the need for Monte Carlo ev aluation. The applicatio n of t he results in Sections III-V enables a un ified analysis for MIMO system s, which allow the generalization for ergodic and outage capacity [6]–[8], [29], for optimum combi ning multi ple antenna systems [26], [27], for MIMO-MM SE sy stems [28], for MIMO relay networks [34], [35], as well as fo r multiuser MIMO system s and for dis tributed M IMO systems , accounting arbitrary covariance matrices. For example, after the first deriv ation of the hypergeometric functions of m atrices with non-distinct eigen v alues in [36], other applicati ons to mu ltiple ant enna systems hav e appeared in [32], [37]–[40]. V I . N U M E R I C A L R E S U LT S DRAFT CHIANI, WIN, SHIN: MIMO NETWORKS. 15 Let us first apply Theorem 1 to the analysis of a single-user MIMO system with unequal power levels among the transmitti ng antennas. Figure 2 s hows the ergodic mutual inform ation 2 of a MIMO- (6 , 3) Rayleigh channel, where the relative t ransmitted power levels are { 1 + ∆ , 1 + ∆ , 1 + ∆ , 1 − ∆ , 1 − ∆ , 1 − ∆ } . The particular cases ∆ = 0 and ∆ = 1 are equiv alent to the equal po wer l e vels over 6 and 3 transmit ting antennas, respectiv ely . This figure shows h o w the capacity decreases as ∆ increases from 0 to 1 , with a beha vior in accordance to analysis based on majori zation theory [41]. As another example o f appli cation, we ev aluate the performance of MIMO relay networks in Rayleigh fading [34], [35]. For such networks the network capacity i s u pper bounded by [35, eq. (5)], which can be easily put in the fo rm C u = 1 2 E H  log det  I + HΦH †  , and e valuated in closed form by Theorem 1. In Fig. 3 we report the exact C u as obtained from Theorem 1, compared with the Jensen’ s inequalit y [35, Theorem 1 ]. The figure has been obtained for a source node with 4 ant ennas, 5 relays each equipped wit h 2 antennas, as a function of the total equiv alent SNR here defined as SNR = tr { Φ } . W e assume, for the 5 relays, that the recei ved power is di stributed proportionally to the w eights { 1 , 2 , 5 , 1 0 , 20 } . It can observed that the result s based o n the Jensen’ s inequality can be overly opti mistic. As a third example of application we ev aluate, usi ng (8) together with Theorem 1, the exact expression of the ergodic mu tual information of MIM O systems in the presence of mu ltiple MIMO i nterferers in Rayleigh fading. In particular , the eigen v alues to be used in Theorem 1 are giv en by µ ( i ) = 1 / i = σ 2 N T i /P i , all o wing an easy analysi s for sev eral scenarios . W e define the avera ge SNR per receiving antenna as SNR = P 0 /σ 2 giving  0 = SNR /N T 0 , and th e SIR as SIR = P 0 / P i ≥ 1 P i . 3 Fig. 4 sh o ws the ergodic mutual informati on for a MIMO- (6 , 6) system as a function of the SIR, in t he presence of one MIMO cochannel int erferer ha ving N T 1 equal power transmitti ng antennas. It can be no ted that the capacity decreases w ith the increase in t he number of int erfering ant enna elements, tendin g to the curve obtained b y using th e Gaus sian approximation. 4 Despite the fact that the receiv ed vector y in (1) is Gaussi an condit ioned on 2 For the numerical results we use t he base 2 of logarithm in all formulas, gi ving a mutual information in bits/s/Hz. 3 W e recall that, with our normalization on the channel gains, the mean receive d power from user i is P i , and our definition of SIR account for the total interference power . 4 W ith G aussian approximation the performance is ev aluated as if interference were absent, exc ept the ov erall noise po wer is set to σ 2 + P i ≥ 1 P i , giv ing a signal-to-interference-plus-noise ratio SINR = ` 1 SNR + 1 SIR ´ − 1 . DRAFT 16 SUBM. TO IEEE T RANS. ON INF . TH. the channel matrices, and t hat t he elements of H k are Gaussian, approximati ng the cumul ativ e interference as a spat ially white complex Gaussian vector is pessimist ic for analyzing MIMO systems in the presence of interference, unless t he num ber o f transm itting antenna of the interferer is lar ge compared with that of the desired user . This is because the Gaussian approximati on implicitl y assumes that t he recei ver does not exploit th e CSI of the int erferers (single-user recei ver), whereas the exact capacity accounts for the knowledge of all CSI at the recei ver . In the same figure we also report, using circles, the c apacity of a si ngle-user MIMO- ( N T 0 , N R − N T 1 ) for N R > N T 1 . It can be observed th at the capacity o f the MIMO- ( N T 0 , N R ) i n t he presence of N T 1 interfering antenna elements approaches asymptoti cally , for large interference power , to a floor given by the capacity of a si ngle-user M IMO- ( N T 0 , N R − N T 1 ) system . Th is behavior can be thought of as using N T 1 DoF at the receiv er to null the interference in a small SIR regime. On the other hand, when N R ≤ N T 1 the capacity approaches to zero for small SIR. T his is due to the limi ted DoF at the recei ver (related to t he n umber N R of recei ving antenna elements) that pre vents miti gating all interfering signals (one from each antenna elements) while, at the same time, processing the N T 0 useful parallel streams, as previously ob served for mu ltiple antenna systems with optimu m com bining [2], [26], [27]. Finally , in Fig. 5 we consi der a MIMO- ( N T 0 , 6) system in t he presence of one and two MIMO interferers i n the network, each equi pped with the s ame number of antenn as as for the desired user . W e clearly see here two diff erent regions: for small SIR the interference effect is dominant, and it is better for all users to employ the m inimum number of transm itting antennas (i.e., MIMO - (3 , 6) for all users), so as to allow the receiv er to m itigate the int erfering signals. On the contrary , for large SIR the channel tends to th at of a singl e-user MIMO system and it is better to emp loy the m aximum n umber of transmitt ing antennas. In the same figure we also report the capacity for i nterference-fre e channels, which represents the asymptotes of the four curves, as w ell as the Gaussian approxi mation, which incorrectly indi cates that i t i s always better to use the largest pos sible num ber of transmitting antennas. It can be also verified that, in a network where all nodes are using the s ame MIM O- ( n, n ) systems, lar ger values of n achieve hig her mut ual i nformation, for all va lues of SIR and SNR. Note, howe ver , that when increasing the number o f antennas and u sers, correlation may arise in the channel matrices. DRAFT CHIANI, WIN, SHIN: MIMO NETWORKS. 17 V I I . C O N C L U S I O N W e h a ve studi ed MIMO comm unication system s in the presence of multiple MIM O i nterferers and noise. T o this aim, we first generalized the determinant representations for hypergeometric functions with m atrix arguments to the case where t he eigenv alu es of th e argument matrices have arbitrary multipli cities. Then, we derived a u nified form ula for the joi nt p.d.f. of the eigen v alues for central W i shart matrices and Gaussian quadratic forms, allowing arbitrary mult iplicities for the cov ariance matrix eigen va lues. These ne w results enable th e analys is of m any scenarios in v olving MIMO s ystems. For example, we deriv ed a uni fied expression for the ergodic mutu al information of MIMO Rayleigh fading channels, which applies to transmit or receive correlation matrices wit h eigen v alues of arbitrary mul tiplicities. W e have s hown how to apply the n e w expressions to MIMO networks, deriving in closed form the er godic mutual i nformation o f MIMO sy stems in the presence of multipl e MIM O interferers. A P P E N D I X I P RO O F S A. Pr oof of Lemma 2 For ease of notation and wit hout lo ss of g enerality we cons ider t he case K = 1 , where th e application of the lemm a l eads to (15). For the proo f we p roceed by induction. First, the result in (15) i s obvious for L = 1 , since in t his case (15) coincides with (14 ). Then, we must show that if (15) i s t rue for any L then i t i s also true for L + 1 . So, assum ing that (15) hold s for L , we m ust find lim w L +1 → w L ˘ P ( w 1 , . . . , w m ) . In this regard note that, with w 1 = w 2 = · · · = w L the product Q i

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