Alfano, Giuseppa

Chiasserini C.-F.

Nordio, Alessandro

Zhou, Siyuan

English

PORTO Publications Open Repository TOrino

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Handbook of Mathematical Functions”,

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Orlov “New solvable matrix integrals,”

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Information densities for block-fading MIMO channels

G. Alfano

C.-F. Chiasserini

A. Nordio

S. Zhou

Crossref

ALFANO, GIUSEPPA

CHIASSERINI, Carla Fabiana

NORDIO, ALESSANDRO

ZHOU, SIYUAN

PORTO@iris (Publications Open Repository TOrino - Politecnico di Torino)

04 August 2020POLITECNICO DI TORINORepository ISTITUZIONALEInformation densities for block-fading MIMO channels / Alfano G.; Chiasserini C.-F.; Nordio A.; Zhou S.. - STAMPA. -(2013), pp. 1374-1377. ((Intervento presentato al convegno IEEE-APS Topical Conference on Antennas andPropagation in Wireless Communications (IEEE APWC 2013) tenutosi a Turin (Italy) nel September 2013.OriginalInformation densities for block-fading MIMO channelsPublisher:PublishedDOI:10.1109/APWC.2013.6624940Terms of use:openAccessPublisher copyright(Article begins on next page)This article is made available under terms and conditions as specified in the  corresponding bibliographic description inthe repositoryAvailability:This version is available at: 11583/2507386 since:IEEE / Institute of Electrical and Electronics Engineers Incorporated:445 Hoes Lane:Piscataway, NJ 08854:Information densities for block-fadingMIMO channelsG. Alfano∗ C.-F.Chiasserini† A. Nordio‡ S. Zhou§Abstract — This paper provides a characterizationof the output and of the transition probability den-sity function (pdf) of a MIMO block-fading channel,paving the way to its information-theoretic analysis.The information density, whose average in ergodicconditions gives the mutual information conveyedover the channel, is needed to characterize the chan-nels with arbitrary behavior, as one in this case canneither resort to ergodicity nor to consequences ofany law of large numbers. The information densityexpression is then provided under several assump-tions on the channel model and also in the presenceof multiple-antenna equipped interferers.1 INTRODUCTIONInformation density methods allow for the study ofarbitrary channels, as they do not take recourse toergodic results like, e.g. the Asymptotic Equiparti-tion Property (AEP). Relying on the survey in [1],in this work we aim at characterizing the informa-tion density for some MIMO channel of interest,characterized by block-independent fading. Thecrucial steps toward the derivation of a formula forthe information density are the characterization ofthe probability density function (pdf) of the chan-nel output, as well as of the transition probability,i.e. the conditional law of the channel output giventhe input. A recent analysis in [2], whose authorswere mainly interested in proposing a computation-ally efficient way for evaluating the mutual infor-mation conveyed on a MIMO block fading channelwith no Channel State Information (CSI) at eitherends of the link, paved the way to our derivation.Indeed, the strategy in [2] for evaluating the outputpdf can be extended from block-Rayleigh MIMOchannels to channels with different fading laws, ex-ploiting some results of finite-dimensional randommatrix theory. The transition probability follows∗Dipartimento di Elettronica, Politecnico di Torino,C.so Duca degli Abruzzi 24, 10129 Torino, Italy, e-mail:alfano@tlc.polito.it, tel.: +39 011 5644030, fax: +39 0115644099.†Dipartimento di Elettronica, Politecnico di Torino,C.so Duca degli Abruzzi 24, 10129 Torino, Italy, e-mail:chiasserini@polito.it, tel.: +39 011 5644183, fax: +39011 5644099.‡CNR-IEIIT, C.so Duca degli Abruzzi 24, 10129 Torino,Italy, e-mail: alessandro.nordio@ieiit.cnr.it§Dipartimento di Elettronica, Politecnico di Torino,C.so Duca degli Abruzzi 24, 10129 Torino, Italy, e-mail:siyuan.zhou@polito.it, tel.: +39 011 5644194, fax: +39011 5644099.in many cases from simple observations on the fad-ing law. Our focus is on MIMO channel fed byi.i.d. Gaussian inputs, which exhibit the nice fea-ture of being invariant (in law) with respect to leftand/or right multiplication times a unitary matrix,and serve as a reference with respect to otherwiseoptimized inputs.The paper is organized as follows: next Sectiondescribes the System Model, while in Section 3 theanalytical derivation of the information density foreach channel model of interest is reported.Throughout the paper, matrices are denoted byuppercase boldface letters, vectors by lowercaseboldface. The pdf of a random matrix Z, pZ(Z),is simply denoted by p(Z). The complex, matrix-variate Gaussian law with mean matrix ∆ and co-variance matrix Ξ is denoted by CN (∆,Ξ). Thecomplex matrix-variate F law with p and n degreesof freedom is denoted by Fm(p, n), with m the sizeof the matrix and, in order to avoid rank deficien-cies, p ≥ m and n ≥ m. (·)† indicates the con-jugate transpose operator, | · | and Tr(·) denote,respectively, the determinant and the trace of asquare matrix, and || · || stands for the Euclideannorm1. Γp(q), with p ≤ q, is the complex multi-variate Gamma function [3]Γp(q) = pip(p−1)2p∏`=1(q − `)!We denote by Im the m ×m identity matrix and,letting A be an n × n Hermitian matrix with or-dered eigenvalues2 λ1, . . . , λn, we denote by V(A)the Vandermonde determinant of A [4], i.e.,V(A) =∏1≤i<`≤n(λi − λ`).2 SYSTEM MODELWe consider a single-user multiple-antenna commu-nication, with nr and nt denoting the number ofreceive and, respectively, of transmit antennas. As-suming block-fading with block length nb, the chan-nel output can be described by the following linear1As applied to a matrix, we mean ||A||2 = Tr(A†A)2In the following, all the joint eigenvalue distributionsassume the eigenvalues to be ordered.relationship:Y =√γH X + N . (1)In (1), Y is the nr×nb output, X is the i.i.d. zero-mean, unit variance complex Gaussian nt × nb in-put matrix, and N is the nr×nb matrix of additivecomplex circularly symmetric Gaussian noise. H isthe nr × nt complex channel matrix, whose entriesrepresent the fading coefficients between each trans-mit and each receive antenna, and whose singularvalue decomposition reads as3 H = UΣ1/2V†. Fi-nally, γ = SNR/nt represents the normalized per-transmit antenna signal-to-noise ratio (SNR).We apply the well known definition of the infor-mation density to the channel in (1), and writeiX,Y(X,Y) = logp(Y|X)p(Y). (2)As largely discussed in [5, and references therein],this quantity plays an essential role in the in-formation theoretic analysis of finite block-lengthcommunications. Moreover, information densitymethods are crucial in the characterization of non-ergodic channels at large. In the remaining of thepaper, we will then provide analytical results forthe output and the transition pdf of a block-fadingchannel, since they constitute the building blocksof (2).3 ANALYTICAL RESULTSThe characterization we propose works for arbi-trary number of transmit and receive antennas andarbitrary fading duration, however the analyticaldetails of the derivation depend on the relative val-ues of nt and nr (as in [2, and references therein]).Due to the lack of space, we focus on the case ofnt ≥ nr.Given a channel as in (1), with i.i.d. Gaussian X,the output pdf when nt ≥ nr can be convenientlywritten [2, Formulae (40) and (41)] asp(Y) = T∫Σ>0p(Σ)V(Σ) |E|nr∏`=1(1 + γσ`)nr−nb−1 dΣ ,(3)withT =∏nr`=1(`− 1)!γnr(nr−1)/2pinrnbe−||Y||2V(Y†Y) ,3Herein, U and V are unitary matrices of size nr, and,respectively, nt, and Σ1/2 has non-negative entries. In thefollowing, by Σ we’ll mean the square matrix of size s =min{nr, nt} of the non-zero squared singular values of H.Due to the further assumptions of next Section, s = nrnecessarily.σi the i-th eigenvalue of Σ, y2j the j-th non-zeroeigenvalue of YY† and (E)i,j = ey2i γσj/(1+γσj). Byparticularizing this integral to the channel modelin force, in the following we will evaluate and listthe output pdf in closed form for some scenarios ofinterest.3.1 Ricean channelIn this case, the channel is traditionally modeled asa superposition of a scattered plus a Line-Of-Sight(LOS) component, i.e.H =√κκ+ 1H¯ +√1κ+ 1Hw (4)where κ is the Ricean factor, representing the ra-tio between the average power of the unfaded andfaded channel components, H¯ is deterministic in-dependent zero-mean unit-variance complex Gaus-sian. The joint density of the unordered squaredsingular values of the channel matrix reads in thiscase as [6, Formula (34)]p(Σ) = (1+κ)nr(nt−nr+1)e−nrntκnr!V(H¯H¯†)·∏nr−1`=1 (nr − `)`(nt − `)`∏nr`=1(nr − `)!(nt − `)!·e−(1+κ)Tr(Σ)|F||Σ|nt−nrV(Σ) , (5)with (F)i,j = 0F 1(nt − nr + 1, (1 + κ)2µiσj), µibeing the i-th squared singular value of H¯. As an-ticipated earlier in this section, the output pdf canbe evaluated by replacing (5) in (3) and performingthe average over Σ, by exploiting [7, Corollary I].This way, we obtain for a MIMO channel affectedby block-Ricean fading:p(Y) =∏nr`=1(`− 1)!γnr(nr−1)/2pinrnbe−||Y||2V(Y†Y) ·· (1 + κ)nr(nt−nr+1)e−nrntκnr!V(H¯H¯†)·∏nr−1`=1 (nr − `)`(nt − `)`∏nr`=1(nr − `)!(nt − `)!|Z| , (6)where(Z)i,j =∫ +∞0ey2iσγ1+σγ−(1+κ)σ·σnt−nr0F 1(nt − nr + 1, (1 + κ)2µjσ)(1 + γσ)nb−nr+1dσ ,and 0F 1(·, ·) is the Bessel hypergeometric function[8]. Notice that, as already stressed in [9], theLOS matrix H¯H¯† is assumed to be full-rank inthis derivation. If this assumption is broken, thenone needs to resort to the limiting procedure in [9,Corollary I, and references therein].As for the transition probability of the channel,in this case Y|X ∼ CN(√γκκ+1 H¯X, I +γκ+1XX†).Gathering together this pdf and (6), (2) can be con-veniently evaluated by taking the logarithm.3.2 Interference-limitedIn this case the AWGN is neglected, and the actualimpairment is generated by the co-channel interfer-ers, each seen from the direct link receiver underits own random channel, which we assume to be af-fected by Rayleigh fading with block-length nb, thesame as for the useful signal. Assuming to have Lactive interferers in the network, following [10, Sec.IV], it turns out that channel matrix Gramian canbe modeled as an Fnr (nt, Lnt)4-distributed matrixvariate, whose unordered joint eigenvalues law canbe written as [3, Formula (98)]p(Σ) =|Ω|−ntV(−Ω−1)nr∏`=1((L+ 1)n− `)!(Lnt − `)!(nt − `)!(nr − `)!·nr−1∏`=1`nr−`(nt(L+ 1) + `)nr−`|Σ|nt−nrV(Σ)|F˜| ,(7)with Ω the compound receive spatial correlationmatrix5. By replacing (7) into (3), it turns outthat the output is distributed according top(Y) =T|Ω|−ntV(−Ω−1)nr∏`=1((L+ 1)nt − `)!(Lnt − `)!(nt − `)!(nr − `)!·nr−1∏`=1`nr−`(nt(L+ 1) + `)nr−`|Z| ,where(Z)i,j =∫ +∞0exp(y2iσγ1 + σγ)· σnt−nr(1 + γσ)nb−nr+1(1 + σ/ωj)nt+nb−nr+1dσ.4The parameters here exploited for the definition of theF matrix descend from our assumption nr < nt, and referto a scenario where the number of transmit antennas nt isthe same at each transmitter.5comprehensive of both the correlation among the receiveantennas as well as of the correlation arising from interferer’stransmission, for a detailed explanation on how this correla-tion structure arises the interested reader is referred to [10,and refs. teherein].If Ω = I, then p(Σ) strongly simplifies, namelyp(Σ) =nr∏`=1((L+ 1)nt − `)!(Lnt − `)!(nt − `)!(nr − `)!· |Σ|nt−nr|I + Σ|(L+1)nt V2(Σ) ,so that in this casep(Y) = Tnr∏`=1((L+ 1)nt − `)!(Lnt − `)!(nt − `)!(nr − `)! |Z| ,where(Z)i,j =∫ +∞0ey2iσγ1+σγσnt−nr+j−1(1 + γσ)nb−nr+1dσ .Notice that, as we are neglecting the thermal noise,the law of Y|X can be characterized through itscharacteristic function, exploiting tools in [11, andreferences therein].3.3 Land Mobile Satellite communicationThe Land Mobile Satellite (LMS) MIMO channelcan be viewed as a non-central channel with ran-dom mean, so that the channel matrix model isakin to that in (4), but with random H¯. Thenthe Gramian H¯†H¯ follows a matrix-variate Γ(α,Φ)distribution [13, 12]. Here, α plays the role of ashape parameter (indeed, for integer values α coin-cides with the number of degrees of freedom of theΓ-distributed matrix-variate), while Φ is a scale pa-rameter. From a physical point of view, they bothrefer to the average power of the random LOS com-ponent, as shown in detail in [12]. For this scenario,the channel eigenvalues law reads asp(Σ) =∏nr−1`=1[(α−nr+`)`(nt−nr+`)]`−nrV((I + Φ)−1) |F|V(Σ)·e−Tr(Σ)|Σ|nt−nrnr∏`=1φα` (1 + φ`)−α(nt − `)!(nr − `)!with (F)i,j = 1F 1(α− nr + 1;nt − nr + 1; σj1+φi),and φi the i-th ordered eigenvalue of Φ. Recallhere that 1F1 (a; b;x) is the confluent hypergeomet-ric function of scalar argument [8, Ch. 13].Following the lines of the previous Subsections,it turns out thatp(Y) = T∏nr−1`=1[α−nr+``(nt−nr+`)]`−nrV((I + Φ)−1)·|Z|nr∏`=1φα` (1 + φ`)−α(nt − `)!(nr − `)! ,where(Z)i,j =∫ +∞0ey2iσγ1+σγ−σ σnt−nr(1 + γσ)nb−nr+1·1F 1(α− nr + 1;nt − nr + 1; σ1 + φj)dσ.Also in this case, when6 Φ = I the channel eigen-values pdf simplifies, and reduces indeed top(Σ) =e−Tr(Σ)|Σ|nt−nrΓnr (nt)Γnr (nr)|F˜|V(Σ) ,with (F˜)i,j = σnr−ji 1F 1 (α− j + 1;nt − j + 1;σi),so that the final result isp(Y) = T|Z˜|Γnr (nt)Γnr (nr),where(Z)i,j =∫ +∞0ey2iσγ1+σγ−σ σnt−nr(1 + γσ)nb−nr+1·1F 1 (α− j + 1;nt − j + 1;σ) dσ.In the LMS MIMO case, the conditional law ofY|X can be evaluated at high-SNR resorting tothe Cholesky decomposition of the channel matrixGramian, instance of which is reported in [12, Sec.II].4 ConclusionThe output pdf and the conditional law of the out-put given the i.i.d. Gaussian input has been eval-uated for block-fading channels. The results areexploited to characterize the information density ofthe analyzed channel models, which is needed in theinformation-theoretic characterization of arbitrarynon-ergodic channels. Extensions of the results toother input/channel structures are currently underinvestigation.5 AcknowledgementThe work of G. Alfano has received funding fromthe European Union Seventh Framework Pro-gramme (FP7/2007-2013) under grant agreementn. 257740 (Network of Excellence TREND). Shewas also supported by Newcom]. S. Zhou andC. Chiasserini were supported by NPRP grant]5 − 782 − 2 − 322 from the Qatar National Re-search Fund (a member of Qatar Foundation). Thestatements made herein are solely the responsibilityof the authors.6This case is of particular interest because under this as-sumption the channel turns out to be unitarily invariant ;for the definition and the practical impact of this statisti-cal feature on the signalling strategy see e.g. [14, and refs.therein].References[1] K.P. Jagannathan, “Information DensityMethods and the Characterization of Ar-bitrary Channels,” MIT student project re-port, http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.92.2318, 2009.[2] F. Rusek, A. E. Lozano, N. Jindal “Mutual Infor-mation of IID Complex Gaussian Signals on BlockRayleigh-Faded Channels”, IEEE Trans. on Inf.Theory Vol. 58, No. 1, pp. 331–340 Jan. 2012.[3] A. T. James, “Distribution of Matrix Variates andLatent Roots Derived from Normal Samples,” TheAnnals of Math. Stat., Vol. 35, No. 2, pp. 474–501,June 1964.[4] R. A. Horn, C.R. Johnson, Matrix Analysis, Cam-bridge University Press, 1985.[5] Y. Polyanskiy, H. V. Poor, and S. 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Information densities for block-fading MIMO channels

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