The information that can be conveyed through a wireless channel, with multiple-antenna equipped transmitter and receiver, crucially depends on the channel behavior as well as on the input structure. In this paper, we present very recent analytical results, concerning the probability density function (pdf) of the output of a single-user, multiple-antenna communication. The analysis is carried out under the assumption of an optimized input structure, and assuming Gaussian noise and block-fading. A further simplification of the output pdf expression presented in our last paper is derived, without the need for resorting to involved integration rules over unitary matrices. With respect to the former result, presented at the main track of this conference, the newly derived formula has the appealing feature of being numerically implementable with open access Matlab codes developed at MIT for the evaluation of zonal polynomial

Alfano, Giuseppa

Chiasserini, Carla Fabiana

Nordio, Alessandro

Zhou, Siyuan

English

PORTO Publications Open Repository TOrino

Toward an efficiently computable formula
for the output statistics of MIMO block-fading
channels
G. Alfano
DET
Politecnico di Torino
Torino, Italy, 10129
alfano@tlc.polito.it
C-F. Chiasserini
DET
Politecnico di Torino
Torino, Italy, 10122
chiasserini@polito.it
A. Nordio
CNR IEIIT
Torino, Italy, 10129
alessandro.nordio@ieiit.cnr.it
S. Zhou
DET
Politecnico di Torino
Torino, Italy, 10129
siyuan.zhou@polito.it
Abstract—The information that can be conveyed through a
wireless channel, with multiple-antenna equipped transmitter
and receiver, crucially depends on the channel behavior as
well as on the input structure. In this paper, we present
very recent analytical results, concerning the probability density
function (pdf) of the output of a single-user, multiple-antenna
communication. The analysis is carried out under the assumption
of an optimized input structure, and assuming Gaussian noise
and block-fading. A further simplification of the output pdf
expression presented in our last paper is derived, without the
need for resorting to involved integration rules over unitary
matrices. With respect to the former result, presented at the
main track of this conference, the newly derived formula has
the appealing feature of being numerically implementable with
open access Matlab codes developed at MIT for the evaluation
of zonal polynomials.
I. INTRODUCTION
Several communication and information theoretic perfor-
mance indices depend on the statistical characterization of the
output of a wireless channel. In the MIMO case, this is posing
challenging issues due to the need for exploiting sophisti-
cated tools from multivariate statistics and finite-dimensional
random matrix theory. This work is intended to describe
the ongoing efforts toward an efficiently computable closed
form characterization of the joint density of the entries of a
matrix variate, which represents the output of a MIMO system
affected by AWGN and block-fading. The rationale of the
work stems from the fact that, albeit compact, the expression
we recently provided for the conditional output probability
density function (pdf) of the abovementioned wireless system
was not suitable for computer evaluation. Our current aim is,
then, to provide a formula whose numerical evaluation could
be done via Matlab codes developed for the approximation
of zonal polynomials and functions of matrix arguments like
those from [1]. The result we introduce later in the paper is
the first relevant step toward our main goal.
II. NOTATION AND SYSTEM MODEL
Throughout the work, matrices are denoted by uppercase
boldface letters, vectors by lowercase boldface. The pdf of
a random matrix Z, pZ(Z), is simply denoted by p(Z). (·)†
indicates the conjugate transpose operator, |·| and Tr(·) denote,
respectively, the determinant and the trace of a square matrix,
and || · || stands for the Euclidean norm1. Γp(q), with p≤q, is
the complex multivariate Gamma function [2]
Γp(q) = π
p(p−1)
2
p￿
￿=1
(q − ￿)!
and
pF q(a1, . . . , ap; b1 . . . , bq; ·, . . . , ·) ,
with p and q non-negative integers, denotes the generalized
hypergeometric function [3]. The arguments of such a function
can be either scalars or square matrices; there is in general
no limit to the number of arguments, and hypergeometric
functions of multiple matrix arguments are defined also for
set of square matrices of different size. We denote by Im the
m×m identity matrix.
We consider a single-user multiple-antenna communication,
with nR and nT denoting the number of receive and transmit
antennas. For sake of simplicity, let nT ≤ nR, which allows
the modeling of a massive-MIMO channel in the large nR
limit2. Assuming block-fading of length nb, the channel output
can be described by the following linear relationship:
Y =
√
γHX+N. (1)
In (1), Y is the nR × nb output, X is the complex nT × nb
input matrix, and N is the nR × nb matrix of additive com-
plex circularly symmetric Gaussian noise. H is the nR×nT
complex channel matrix, whose entries represent the fading
coefficients between each transmit and each receive antenna.
Finally, γ = SNR/nT represents the normalized per-transmit
antenna Signal-to-Noise Ratio (SNR).
The input matrix X, unless otherwise stated, is assumed
to have a product structure, i.e., X = D1/2Φ, where D is a
random, nT -dimensional, diagonal matrix, which is positive
definite w.p. 1. The entries of D represent the amount of
1As applied to a matrix, we mean ||A||2 = Tr(A†A).
2The extension of the results to the case nT > nR is straightforward.
transmit power allocated to each of the nT transmit antennas,
while Φ is an nT × nb isotropic matrix. As usually done in
the literature [4], [5], we will refer to square isotropic matrices
as Haar and to rectangular isotropic matrices as Stiefel. We
stress that the above structure of the input matrix X allows to
achieve the capacity limit in absence of CSI at both the ends
of the communication link [5, Thm2].
III. STATISTICAL CHARACTERIZATION OF THE CHANNEL
OUTPUT
Theorem 1. Given a channel as in (1), the pdf of its matrix-
variate output, conditionally on the nT -dimensional (diagonal)
input power allocation matrix D and on the nT -dimensional
matrix of the non-zero squared singular values of the channel,
Σ, can be expressed as
p(Y|D,Σ) = e
−||Y||2￿vπnR[nT−nb]+γnRnT |Σ|−nR|D|nR
0F 0(ΣD−1,DΣ−1,Y†Y) , (2)
with ￿v = 2[nT−nb]+π[nT−nb]+2/Γ[nT−nb]+([nT − nb]+) the
volume of the unitary group of dimension [nT − nb]+3. The
hypergeometric function of three matrix arguments in (2) can
be expanded in zonal polynomials [2] Cκ(·), following [6,
Appendix B] as
0F 0(ΣD−1,DΣ−1,Y†Y) = (3)
(2π)2q−nR−nT
+∞￿
k=0
￿
#κ=k
Cκ(ΣD−1)Cκ(DΣ−1)Cκ(Y†Y)
k!C2κ(Iq)
,
with q = max{nT , nR, nb}and #κ the cardinality of κ, which
is a partition of the integer k.
Proof. Due to the Gaussianity of both the channel and the
noise, the conditional output pdf of the MIMO channel (1)
can be expressed as follows:
p(Y|X,H) = e
−Tr
￿
(Y−√γHX)(Y−√γHX)†
￿
πnRnb
.
Notice that, expanding the product in the exponent, decom-
posing in its singular values/vectors the channel matrix H =
UΣ1/2V†, and recalling that X = √γD1/2Φ, one obtains,
term by term, e−||Y||
2
, which is independent of H and X;
exp
￿−γTr ￿HXX†H†￿￿ = exp￿−γTr ￿ΣV†DV￿￿ ,
which depends only on V, and finally
exp{√γTr ￿YX†H† +HXY†￿} ,
which depends on both V and U. In turn, the conditional pdf
we are interested in can be expressed as
p(Y|D,Σ) =
￿
p(Y|Φ,U,V)p(Φ)p(U)p(V)dΦdUdV ,
(4)
3Here, we assume [nT −nb]+ = nT −nb if nT ≥ nb, and [nT −nb]+ =
nb − nT otherwise.
with the integral being over the appropriate matrix spaces,
which can be performed as described below.
The integration over V can be carried out by exploiting [7,
Integral B.I], that over U follows by the splitting formula for
the hypergeometric functions of matrix argument [2, Formula
(92)], while the last one, over Φ, can be performed by the
help of [8, Formula (54)].
IV. DISCUSSION AND FUTURE WORK
The main expression we obtained for the output law is
conditioned to the channel eigenvalues and to the input power
allocation, while we were able to average over the input and
the channel eigenvectors distributions.
Notice that, tough no determinant representation is known
yet for hypergeometric functions of more than two matrix
arguments, still the zonal polynomials the output pdf can
be expanded in can be represented as ratio of determinants.
Moreover, the zonal polynomial of a matrix and this of its
inverse are strongly related (see e.g. [2]), and this yields to
further simplification of (3). The procedure adopted in [9,
Appendix] to perform the average over some of the matrices
appearing in the argument of the zonal polynomials may not be
fully generalized to our case, however the obtained expression
(2) can be already numerically evaluated, as opposite to its
counterpart in [10], by exploiting tools in [1].
REFERENCES
[1] Plamen Koev and Alan Edelman, “The efficient evaluation of the hy-
pergeometric function of a matrix argument,” Math. Comp. Vol. 75, pp.
833–846, 2006.
[2] A. T. James, “Distribution of Matrix Variates and Latent Roots Derived
from Normal Samples,” The Annals of Mathematical Statistics, Vol. 35,
No. 2, pp. 474–501, June 1964.
[3] M. Abramowitz, and I. A. Stegun, ”Handbook of Mathematical Func-
tions”, New York: Dover Publications, 1972.
[4] L. Zheng, D. Tse, “Communicating on the Grassmann Manifold: A Geo-
metric Approach to the Non-coherent Multiple Antenna Channel,” IEEE
Trans. on Information Theory, Vol. 48, No. 2, pp. 359–383, Feb. 2002.
[5] T. L. Marzetta, B. M. Hochwald, “Capacity of a mobile multiple-antenna
communication link in Rayleigh flat fading,” IEEE Trans. on Information
Theory, Vol. 45, No. 1, pp. 139–157, Jan. 1999.
[6] L. Hanlen, A. Grant, “Capacity Analysis of Correlated MIMO Chan-
nels,” IEEE Trans. Information Theory, Vol. 58, No. 11, pp. 6773–6787,
Nov. 2012.
[7] G. Taricco, “Asymptotic Mutual Information Statistics of Separately
Correlated Rician Fading MIMO Channels,” IEEE Trans. Information
Theory, Vol. 54, No. 8, pp. 3490–3504, Aug. 2008.
[8] W. Yang, G. Durisi, E. Riegler, “On the capacity of large-MIMO block-
fading channels,” IEEE J. Sel. Areas Commun., Feb. 2013, to appear.
[9] M. R. McKay, A. J. Grant, I.B. Collings, “Performance Analysis of
MIMO-MRC in Double-Correlated Rayleigh Environments,” IEEE Trans.
on Communications, Vol. 55, No. 3, pp. 497–507, Mar. 2007.
[10] G. Alfano, C.-F.Chiasserini, A. Nordio, S. Zhou, “Output Statistics of
MIMO Channels with General Input Distribution,” Proc. of ISIT 2013,
Istanbul, July 7-13, 2013.


Asymptotic Mutual Information Statistics of Separately Correlated Rician Fading

Capacity Analysis of Correlated

Capacity of a mobile multiple-antenna communication link in Rayleigh ﬂat fading,”

Communicating on the Grassmann Manifold: A Geometric Approach to the Non-coherent Multiple Antenna Channel,”

Distribution of Matrix Variates and Latent Roots Derived from Normal Samples,” The Annals of Mathematical Statistics,

Handbook of Mathematical Functions”,

On the capacity of large-MIMO blockfading channels,”

The efﬁcient evaluation of the hypergeometric function of a matrix argument,”

Toward an efficiently computable formula for the output statistics of MIMO block-fading channels

The information that can be conveyed through a wireless channel, with multiple-antenna equipped transmitter and receiver, crucially depends on the channel behavior as well as on the input structure. In this paper, we present very recent analytical results, concerning the probability density function (pdf) of the output of a single-user, multiple-antenna communication. The analysis is carried out under the assumption of an optimized input structure, and assuming Gaussian noise and block-fading. A further simplification of the output pdf expression presented in our last paper is derived, without the need for resorting to involved integration rules over unitary matrices. With respect to the former result, presented at the main track of this conference, the newly derived formula has the appealing feature of being numerically implementable with open access Matlab codes developed at MIT for the evaluation of zonal polynomials

ALFANO, GIUSEPPA

CHIASSERINI, Carla Fabiana

NORDIO, ALESSANDRO

ZHOU, SIYUAN

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

04 August 2020POLITECNICO DI TORINORepository ISTITUZIONALEToward an efficiently computable formula for the output statistics of MIMO block-fading channels / Alfano G.; ChiasseriniC.F.; Nordio A.; Zhou S.. - ELETTRONICO. - (2013). ((Intervento presentato al convegno ISIT 2013 tenutosi a Istanbul(Turkey) nel July 2013.OriginalToward an efficiently computable formula for the output statistics of MIMO block-fading channelsieeePublisher:PublishedDOI:Terms of use:openAccessPublisher copyrightcopyright 20xx IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all otheruses, in any current or future media, including reprinting/republishing this material for advertising or promotionalpurposes, creating .(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: 11583/2507801 since: 2016-01-20T10:49:38ZIEEE / Institute of Electrical and Electronics Engineers Incorporated:445 Hoes Lane:Piscataway, NJ 08854:Toward an efficiently computable formulafor the output statistics of MIMO block-fadingchannelsG. AlfanoDETPolitecnico di TorinoTorino, Italy, 10129alfano@tlc.polito.itC-F. ChiasseriniDETPolitecnico di TorinoTorino, Italy, 10122chiasserini@polito.itA. NordioCNR IEIITTorino, Italy, 10129alessandro.nordio@ieiit.cnr.itS. ZhouDETPolitecnico di TorinoTorino, Italy, 10129siyuan.zhou@polito.itAbstract—The information that can be conveyed through awireless channel, with multiple-antenna equipped transmitterand receiver, crucially depends on the channel behavior aswell as on the input structure. In this paper, we presentvery recent analytical results, concerning the probability densityfunction (pdf) of the output of a single-user, multiple-antennacommunication. The analysis is carried out under the assumptionof an optimized input structure, and assuming Gaussian noiseand block-fading. A further simplification of the output pdfexpression presented in our last paper is derived, without theneed for resorting to involved integration rules over unitarymatrices. With respect to the former result, presented at themain track of this conference, the newly derived formula hasthe appealing feature of being numerically implementable withopen access Matlab codes developed at MIT for the evaluationof zonal polynomials.I. INTRODUCTIONSeveral communication and information theoretic perfor-mance indices depend on the statistical characterization of theoutput of a wireless channel. In the MIMO case, this is posingchallenging issues due to the need for exploiting sophisti-cated tools from multivariate statistics and finite-dimensionalrandom matrix theory. This work is intended to describethe ongoing efforts toward an efficiently computable closedform characterization of the joint density of the entries of amatrix variate, which represents the output of a MIMO systemaffected by AWGN and block-fading. The rationale of thework stems from the fact that, albeit compact, the expressionwe recently provided for the conditional output probabilitydensity function (pdf) of the abovementioned wireless systemwas not suitable for computer evaluation. Our current aim is,then, to provide a formula whose numerical evaluation couldbe done via Matlab codes developed for the approximationof zonal polynomials and functions of matrix arguments likethose from [1]. The result we introduce later in the paper isthe first relevant step toward our main goal.II. NOTATION AND SYSTEM MODELThroughout the work, matrices are denoted by uppercaseboldface letters, vectors by lowercase boldface. The pdf ofa random matrix Z, pZ(Z), is simply denoted by p(Z). (·)†indicates the conjugate transpose operator, |·| and Tr(·) denote,respectively, the determinant and the trace of a square matrix,and || · || stands for the Euclidean norm1. Γp(q), with p≤q, isthe complex multivariate Gamma function [2]Γp(q) = πp(p−1)2p￿￿=1(q − ￿)!andpF q(a1, . . . , ap; b1 . . . , bq; ·, . . . , ·) ,with p and q non-negative integers, denotes the generalizedhypergeometric function [3]. The arguments of such a functioncan be either scalars or square matrices; there is in generalno limit to the number of arguments, and hypergeometricfunctions of multiple matrix arguments are defined also forset of square matrices of different size. We denote by Im them×m identity matrix.We consider a single-user multiple-antenna communication,with nR and nT denoting the number of receive and transmitantennas. For sake of simplicity, let nT ≤ nR, which allowsthe modeling of a massive-MIMO channel in the large nRlimit2. Assuming block-fading of length nb, the channel outputcan be described by the following linear relationship:Y =√γHX+N. (1)In (1), Y is the nR × nb output, X is the complex nT × nbinput matrix, and N is the nR × nb matrix of additive com-plex circularly symmetric Gaussian noise. H is the nR×nTcomplex channel matrix, whose entries represent the fadingcoefficients between each transmit and each receive antenna.Finally, γ = SNR/nT represents the normalized per-transmitantenna Signal-to-Noise Ratio (SNR).The input matrix X, unless otherwise stated, is assumedto have a product structure, i.e., X = D1/2Φ, where D is arandom, nT -dimensional, diagonal matrix, which is positivedefinite w.p. 1. The entries of D represent the amount of1As applied to a matrix, we mean ||A||2 = Tr(A†A).2The extension of the results to the case nT > nR is straightforward.transmit power allocated to each of the nT transmit antennas,while Φ is an nT × nb isotropic matrix. As usually done inthe literature [4], [5], we will refer to square isotropic matricesas Haar and to rectangular isotropic matrices as Stiefel. Westress that the above structure of the input matrix X allows toachieve the capacity limit in absence of CSI at both the endsof the communication link [5, Thm2].III. STATISTICAL CHARACTERIZATION OF THE CHANNELOUTPUTTheorem 1. Given a channel as in (1), the pdf of its matrix-variate output, conditionally on the nT -dimensional (diagonal)input power allocation matrix D and on the nT -dimensionalmatrix of the non-zero squared singular values of the channel,Σ, can be expressed asp(Y|D,Σ) = e−||Y||2￿vπnR[nT−nb]+γnRnT |Σ|−nR|D|nR0F 0(ΣD−1,DΣ−1,Y†Y) , (2)with ￿v = 2[nT−nb]+π[nT−nb]+2/Γ[nT−nb]+([nT − nb]+) thevolume of the unitary group of dimension [nT − nb]+3. Thehypergeometric function of three matrix arguments in (2) canbe expanded in zonal polynomials [2] Cκ(·), following [6,Appendix B] as0F 0(ΣD−1,DΣ−1,Y†Y) = (3)(2π)2q−nR−nT+∞￿k=0￿#κ=kCκ(ΣD−1)Cκ(DΣ−1)Cκ(Y†Y)k!C2κ(Iq),with q = max{nT , nR, nb}and #κ the cardinality of κ, whichis a partition of the integer k.Proof. Due to the Gaussianity of both the channel and thenoise, the conditional output pdf of the MIMO channel (1)can be expressed as follows:p(Y|X,H) = e−Tr￿(Y−√γHX)(Y−√γHX)†￿πnRnb.Notice that, expanding the product in the exponent, decom-posing in its singular values/vectors the channel matrix H =UΣ1/2V†, and recalling that X = √γD1/2Φ, one obtains,term by term, e−||Y||2, which is independent of H and X;exp￿−γTr ￿HXX†H†￿￿ = exp￿−γTr ￿ΣV†DV￿￿ ,which depends only on V, and finallyexp{√γTr ￿YX†H† +HXY†￿} ,which depends on both V and U. In turn, the conditional pdfwe are interested in can be expressed asp(Y|D,Σ) =￿p(Y|Φ,U,V)p(Φ)p(U)p(V)dΦdUdV ,(4)3Here, we assume [nT −nb]+ = nT −nb if nT ≥ nb, and [nT −nb]+ =nb − nT otherwise.with the integral being over the appropriate matrix spaces,which can be performed as described below.The integration over V can be carried out by exploiting [7,Integral B.I], that over U follows by the splitting formula forthe hypergeometric functions of matrix argument [2, Formula(92)], while the last one, over Φ, can be performed by thehelp of [8, Formula (54)].IV. DISCUSSION AND FUTURE WORKThe main expression we obtained for the output law isconditioned to the channel eigenvalues and to the input powerallocation, while we were able to average over the input andthe channel eigenvectors distributions.Notice that, tough no determinant representation is knownyet for hypergeometric functions of more than two matrixarguments, still the zonal polynomials the output pdf canbe expanded in can be represented as ratio of determinants.Moreover, the zonal polynomial of a matrix and this of itsinverse are strongly related (see e.g. [2]), and this yields tofurther simplification of (3). The procedure adopted in [9,Appendix] to perform the average over some of the matricesappearing in the argument of the zonal polynomials may not befully generalized to our case, however the obtained expression(2) can be already numerically evaluated, as opposite to itscounterpart in [10], by exploiting tools in [1].REFERENCES[1] Plamen Koev and Alan Edelman, “The efficient evaluation of the hy-pergeometric function of a matrix argument,” Math. Comp. Vol. 75, pp.833–846, 2006.[2] A. T. James, “Distribution of Matrix Variates and Latent Roots Derivedfrom Normal Samples,” The Annals of Mathematical Statistics, Vol. 35,No. 2, pp. 474–501, June 1964.[3] M. Abramowitz, and I. A. Stegun, ”Handbook of Mathematical Func-tions”, New York: Dover Publications, 1972.[4] L. Zheng, D. Tse, “Communicating on the Grassmann Manifold: A Geo-metric Approach to the Non-coherent Multiple Antenna Channel,” IEEETrans. on Information Theory, Vol. 48, No. 2, pp. 359–383, Feb. 2002.[5] T. L. Marzetta, B. M. Hochwald, “Capacity of a mobile multiple-antennacommunication link in Rayleigh flat fading,” IEEE Trans. on InformationTheory, Vol. 45, No. 1, pp. 139–157, Jan. 1999.[6] L. Hanlen, A. Grant, “Capacity Analysis of Correlated MIMO Chan-nels,” IEEE Trans. Information Theory, Vol. 58, No. 11, pp. 6773–6787,Nov. 2012.[7] G. Taricco, “Asymptotic Mutual Information Statistics of SeparatelyCorrelated Rician Fading MIMO Channels,” IEEE Trans. InformationTheory, Vol. 54, No. 8, pp. 3490–3504, Aug. 2008.[8] W. Yang, G. Durisi, E. Riegler, “On the capacity of large-MIMO block-fading channels,” IEEE J. Sel. Areas Commun., Feb. 2013, to appear.[9] M. R. McKay, A. J. Grant, I.B. Collings, “Performance Analysis ofMIMO-MRC in Double-Correlated Rayleigh Environments,” IEEE Trans.on Communications, Vol. 55, No. 3, pp. 497–507, Mar. 2007.[10] G. Alfano, C.-F.Chiasserini, A. Nordio, S. Zhou, “Output Statistics ofMIMO Channels with General Input Distribution,” Proc. of ISIT 2013,Istanbul, July 7-13, 2013.

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Toward an efficiently computable formula for the output statistics of MIMO block-fading channels

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