Computer Algebra based Analysis and Holonomic …park.itc.u-tokyo.ac.jp/atstat/conferences/201503/...Computer Algebra based Analysis and Holonomic Gradient Method based Evaluation

Computer Algebra based Analysis andHolonomic Gradient Method based Evaluationof MIMO Wireless Communications Systems

Costi Siriteanu, Osaka UniversityAkimichi Takemura, University of Tokyo

Satoshi Kuriki, Institute of Statistical MathematicsChristoph Koutschan, Austrian Academy of Sciences

Workshop on Computational and Algebraic Methods in StatisticsUniversity of Tokyo, March 3–5, 2015

Costi Siriteanu ([email protected]) Osaka University Comp. Alg. & HGM in MIMO 1 / 51

Acronyms

MIMO = multiple-input/multiple-output (communications system)ZF = zero forcing (detection method that cancels interference)SNR = signal-to-noise ratio (determines performance)H.G.M. = holonomic gradient methodm.g.f. = moment generating functionp.d.f. = probability density function


Outline

1 Multiple-Input/Multiple-Output (MIMO) Communications Systems

2 MIMO Zero-Forcing Detection (ZF) Analysis

3 Holonomic Gradient Method (HGM)-based ZF Evaluation

4 Computer Algebra Representation of Performance Measures

5 Other MIMO Analysis and Evaluation Applications


References I

[1] “MIMO Diagram,” http://en.wikipedia.org/wiki/MIMO, accessed: 2013-09-07.

[2] “PSK diagram,” http://en.wikipedia.org/wiki/Phase-shift_keying, accessed:2013-09-07.

[3] C. Siriteanu, S. D. Blostein, A. Takemura, H. Shin, S. Yousefi, and S. Kuriki,“Exact MIMO zero-forcing detection analysis for transmit-correlated Ricianfading,” IEEE Transactions on Wireless Communications, vol. 13, no. 3, pp.1514–1527, March 2014.

[4] C. Siriteanu, A. Takemura, S. Kuriki, D. Richards, T. Onoye, and H. Shin,“Performance analysis of MIMO zero-forcing for full-Rician fading,” IEEETransactions on Wireless Communications, to be submitted, 2015.

[5] C. Siriteanu, A. Takemura, S. Kuriki, and H. Shin, “MIMO zero-forcingperformance evaluation using the holonomic gradient method,” IEEETransactions on Wireless Communications, accepted, December 2014. [Online].Available: http://arxiv.org/abs/1403.3788

[6] T. Hibi, Ed., Grobner Bases. Statistics and Software Systems. Tokyo, Japan:Springer, 2013.


http://en.wikipedia.org/wiki/MIMO

http://en.wikipedia.org/wiki/Phase-shift_keying

http://arxiv.org/abs/1403.3788

References II[7] C. Koutschan, “Advanced applications of the holonomic systems approach,”

Ph.D. dissertation, Research Institute for Symbolic Computation (RISC),Johannes Kepler University, Linz, Austria, 2009. [Online]. Available:http://www.risc.jku.at/research/combinat/software/HolonomicFunctions/

[8] M. R. Bhatnagar, “Beamforming and combining in hybrid satellite-terrestrialcooperative systems,” IEEE Communications Letters, vol. 18, no. 3, pp. 483–486,2014.

[9] G. A. Ropokis, A. A. Rontogiannis, P. T. Mathiopoulos, and K. Berberidis, “Anexact performance analysis of MRC/OSTBC over generalized fading channels,”IEEE Transactions on Communications, vol. 58, no. 9, pp. 2486–2492, 2010.

[10] D. Morales-Jimenez, F. J. Lopez-Martinez, E. Martos-Naya, J. F. Paris, andA. Lozano, “Connections between the generalized Marcum-function and a classof hypergeometric functions,” IEEE Transactions on Information Theory, vol. 60,no. 2, pp. 1077–1082, 2014.

[11] M. Di Renzo and P. Guan, “A mathematical framework to the computation of theerror probability of downlink mimo cellular networks by using stochasticgeometry,” IEEE Transactions on Communications, vol. 62, no. 8, pp. 2860–2879,Aug 2014.


http://www.risc.jku.at/research/combinat/software/HolonomicFunctions/

References III

[12] M. Kang and M.-S. Alouini, “Largest eigenvalue of complex Wishart matrices andperformance analysis of MIMO MRC systems,” IEEE Journal on Selected Areasin Communications, vol. 21, no. 3, pp. 418–426, 2003.

[13] M. Chiani, “Distribution of the largest eigenvalue for real Wishart and Gaussianrandom matrices and a simple approximation for the Tracy–Widom distribution,”Journal of Multivariate Analysis, vol. 129, pp. 69–81, 2014.


SISO-to-MIMO Evolution of Wireless Systems [1]


Signal Transmitted from Each Antenna

Phase-Shift Keying and Quadrature Amplitude Modulation [2]:

110

101000

011

Q

I

010

111

100

001

Q

I

0000 0100 1100 1000

0001 0101 1101 1001

0011 0111 1111 1011

0010 0110 1110 1010


NT × NR MIMO Signal, Fading, Noise ModelsIf we transmit x = [x1 x2 . . . xNT ]T then we receive

r = H x + n = (Hd + Hr) x + n, (1)

where H = Hd + Hr is complex-Gaussian NR × NT matrix, so that:If [Hd]i,j = 0⇒ |[H]i,j | Rayleigh-distributedIf [Hd]i,j 6= 0⇒ |[H]i,j | Rician-distributed,

and n is complex-valued, zero-mean, Gaussian receiver noise.

Assumptions, for analysis tractability:Initially [3], we assumed ‘Rician–Rayleigh’ fading, i.e.,

Hd = (hd,1 Hd,2) = (hd,1 0). (2)

Recently [4], we allowed for Hd,2 6= 0, restricted ranks of Hd, Hd,2.


Rician K -Factor

If we write the channel matrix as

H = Hd + Hr, (3)

then we usually define the Rician-K factor as the power ratio

K =‖Hd‖2

E{‖Hr‖2}, (4)

whose value in practice is around K = 5, or, in decibels,

KdB = 10 log10 5 = 7 dB. (5)

Goal: evaluate MIMO performance for realistic fading type, parameters.


Outline







MIMO Zero-Forcing Detection (ZF)

For NR × 1 received-signal vector

r = H x + n, n ∼ CN (0, INR), (6)

ZF determines constellation symbols closest to each element of

[HHH

]−1 HH r = x +[HHH

]−1 HH n. (7)

where we denoted the transposed complex-conjugate of H as HH.

Then, ZF signal-to-noise ratio (SNR) for Stream 1 is

γ =1[

(HHH)−1]

1,1

, (8)

where matrix HHH is noncentral Wishart-distributed.


ZF SNR M.G.F. for Rician–Rayleigh Fading [3]Assuming

Hd = (hd,1 Hd,2) = (hd,1 0), (9)

we have shown that SNR moment generating function (m.g.f.) is [3]

M(s,a) =1

(1− s)N 1F1

(N; NR; a

s1− s

),

where

N = NR − (NT − 1) = number of degrees of freedoma ∝ K NRNT, (e.g.,a ≈ 188, for 6× 6 system,K = 5),

and confluent hypergeometric function has infinite-series expression

1F1(N; NR;σ) =∞∑

n=0

(N)n(NR)n

σn

n!=∞∑

n=0

An(σ).


ZF SNR M.G.F. and P.D.F.

Thus, SNR m.g.f. can be written as

M(s,a) =1F1

(N; NR; a s

1−s

)

(1− s)N =∞∑

n=0

An(a)n∑

m=0

(nm

)(−1)m

(1− s)N+n−m ,

so that the SNR p.d.f. is given by the infinite series

p(t ,a) =∞∑

n=0

An(a)n∑

m=0

(nm

)(−1)mt(N+n−m)−1e−t

[(N + n −m)− 1]! N+n−m . (10)

Goal is, for realistic K , i.e., large a ∝ K NRNT:Compute p(t ,a).Compute MIMO performance measures.


MIMO Performance Measures

Outage probability, for threshold SNR γth:

Po(γth,a) = Probability {γ ≤ γth} =

∫ γth

0p(t ,a) dt . (11)

Ergodic (i.e., average) capacity:

C(a) = Eγ{C(γ)} =

∫ ∞

0log2(1 + t)p(t ,a) dt . (12)

Infinite-series expression of p(t ,a) yields infinite-series expressions forPo(γth,a) and C(a).

Problem: we cannot compute (accurately or even at all) the infiniteseries for p(t ,a), Po(γth,a), and C(a) for realistic K (i.e., large a).


MATLAB Results for SNR P.D.F. for a = 0,188

10 20 30 40 50 60 700

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

t

pγ1(t)

NT=2,NR=6,K=7dB,AS=51◦,Γ s=5dB,nmax=149,ZF

Ray -Ray, se r i e sRay -Ray, s imRi c e -Ray, se r i e sRi c e -Ray, s im


Outline







1F1(N;NR;σ) from Differential Equation, with HGM

Since 1F1(N; NR;σ) satisfies

σ 1F(2)1 (N; NR;σ) + (NR − σ) 1F

(1)1 (N; NR;σ)− N 1F1(N; NR;σ) = 0,

or, equivalently,

∂σ

(1F1(N; NR;σ)

1F(1)1 (N; NR;σ)

)=

(0 1Nσ 1− NR

σ

)(1F1(N; NR;σ)

1F(1)1 (N; NR;σ)

). (13)

it can be computed with the holonomic gradient method (HGM):Compute infinite series 1F1(N; NR;σ0) for σ0 small.Numerically solve system (13) from σ0 to desired σ � 1.


Similar Approach for p(t ,a)? Yes [5]

In [3] we expressed the SNR m.g.f. as

M(s,a) =

∫ ∞

0estp(t ,a) dt =

1

(1− s)N 1F1

(N; NR; a

s1− s

). (14)

Then, we can try to obtain p(t ,a) at large values of a ∝ K as follows:Deduce differential equations for M(s,a) based on

σ 1F(2)1 (N; NR;σ) + (NR − σ) 1F

(1)1 (N; NR;σ)− N 1F1(N; NR;σ) = 0

Invert Laplace to obtain differential equations for p(t ,a).Use HGM to numerically compute p(t ,a).

Integrate numerically to evaluate MIMO performance measures.


Differential Equation for M(s,a) w.r.t. aFor σ = a s

1−s , the differential equation for 1F1(N; NR;σ) is

a s1− s 1F

(2)1

(N; NR;

a s1− s

)+

(NR −

a s1− s

)1F

(1)1

(N; NR;

a s1− s

)

−N 1F1

(N; NR;

a s1− s

)= 0, (15)

where we can substitute 1F(k)1 from

∂ka M(s,a) =

sk1F

(k)1

(N; NR; a s

1−s

)

(1− s)N+k (16)

to obtain(

a (1− s)∂2a + [NR(1− s)− a s] ∂a − N

)•M(s,a) = 0. (17)


Differential Equation for M(s,a) w.r.t. s and aOn the other hand, differentiating w.r.t. s

M(s,a) =1

(1− s)N 1F1

(N; NR; a

s1− s

)(18)

yields

∂sM(s,a) = N(1−s)N+1 1F1

(N; NR; a s

1−s

)+ a

(1−s)N+2 1F(1)1

(N; NR; a s

1−s

),

where again substitute 1F(k)1 from

1F(k)1

(N; NR; a s

1−s

)= (1−s)N+k

sk ∂ka M(s,a), (19)

to obtain[∂a − s(1−s)

a ∂s + N sa

]•M(s,a) = 0. (20)


Differential Equation for M(s,a) w.r.t. sRewriting (20) as

∂aM(s,a) =

[s (1− s)

a∂s − N

sa

]•M(s,a),

differentiating again w.r.t. a, and substituting into(

a (1− s)∂2a + [NR(1− s)− a s] ∂a − N

)•M(s,a) = 0,

yields(

s(1− s)2∂2s − [2(N + 1)s(1− s)− (1− s)NR + a s] ∂s

+N [(N + 1)s − NR − a]

)•M(s,a) = 0. (21)

Goal: invert Laplace to get differential equation for p(t ,a) w.r.t. t .


Switching Order of s and ∂sProperty (from [6, Th. 6.1.2 (Liebniz Formula), p. 282])

sl∂ks =

min(l,k)∑

m=0

(−1)m(l −m + 1)m(k −m + 1)m

m!∂k−m

s sl−m (22)

yields the following particular rules

s∂s = ∂ss − 1,s∂2

s = ∂2s s − 2∂s,

s2∂s = ∂ss2 − 2s,s2∂2

s = ∂2s s2 − 4∂ss + 2,

s3∂2s = ∂2

s s3 − 6∂ss2 + 6s,

which produce the following annihilator of M(s,a) w.r.t. s:

∂2s s3 − 2∂2

s s2 + ∂2s s + (2N − 4)∂ss2 + (6− 2N − NR − a) ∂ss

+(NR − 2)∂s + (N − 1)(N − 2)s + (N − 1)(2− NR − a). (23)


Taking the Inverse-Laplace Transform

It can be shown that∫∞

0 est [tkp(l)(t ,a)]

dt is given by

(−1)l∂ks [slM(s,a)] +

l∑

m=k+1

(−1)m p(l−m)(0+,a)(m − 1)!

(m − k − 1)!sm−k−1.

Applied to the terms in (23), it yields the following Laplace pairs:

∂2s s3M(s,a) + 2! p(0+,a) ↔ −t2p(3)(t ,a)

· · ·(N − 1)(2− NR − a)M(s,a) ↔ (N − 1)(2− NR − a) p(t ,a).

We have found that the left-hand-side terms sum to 0, ∀N ≥ 1, so thatthe right-hand-side terms also sum to 0.


Differential Equation for p(t ,a) w.r.t. tThis yields the following differential equation w.r.t. t for p(t ,a):

p(3)(t ,a) =(NR − 2)t + (N − 1)(2− NR − a)

t2 p(t ,a)

− t2 + (6− 2N − NR − a)t + (N − 1)(N − 2)

t2 p(1)(t ,a)

−2t2 − (2N − 4)tt2 p(2)(t ,a).

If we define the function vector

p(t ,a) =(

p(t ,a) p(1)(t ,a) p(2)(t ,a))T

,

then the above can be written as the system of differential equations

∂tp(t ,a) = P(t ,a) p(t ,a).


Differential Equation for p(t ,a) w.r.t. a

From (20) we deduce

a∂aM(s,a) =[s∂s − s2∂s − Ns

]•M(s,a). (24)

Switching s and ∂s order, and Laplace-transforming yields

a∂ap(t ,a) = −p(t ,a) + (N − t − 2) p(1)(t ,a)− t p(2)(t ,a).

Differentiating it twice w.r.t. t yields the system of differential equations

∂ap(t ,a) =1a

Q(t ,a) p(t ,a). (25)


MATLAB Results for p(t ,a) [5]

10 20 30 40 50 600

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

t

pγ

1(t)

NT=2,NR=6,K=7 dB, AS=51◦,Γ s=5 dB, ZF

HGM, Method 3Simulat i onSe r i e s


MATLAB Results for∫ t

0 p(y ,a)dy [5]

10 20 30 40 50 600

0.2

0.4

0.6

0.8

1

t

Pγ1(t)

NT=2,NR=6,K=7 dB, AS=51◦,Γ s=5 dB, ZF

HGM, Method 3Simulat i onSe r i e s


MIMO ZF Performance Measures

Recall:Outage probability, for threshold SNR γ1,th:

Po(γ1,th,a) = Probability {γ1 ≤ γ1,th} =

∫ γ1,th

0p(t ,a) dt . (26)

Ergodic (i.e., average) capacity:

C(a) = Eγ1{C(γ1)} =

∫ ∞

0log2(1 + t)p(t ,a) dt . (27)


MATLAB Results for Outage Probability [5]

0 1 2 3 4 5

10−4

10−3

10−2

10−1

Γb [dB]

Po

NT=2,NR=6,K=7dB,AS=51◦,γ1 , th=8. 2dB, ZF

Ray -Ray, Se r i e sRay -Ray, Simul at i onRi c e -Ray, HGMRic e -Ray, Simul at i on


MATLAB Results for Ergodic Capacity [5]

0 1 2 3 4 50

1

2

3

4

5

6

7

8

9

10

Γb [dB]

C[b

pcu]

NT=2, NR=6, K=7 dB, AS=51◦; θc=5◦, dn=1; ZF

Ri c e -Ray, HGMRic e -Ray, Simul at i onRay -Ray, Se r i e sRay -Ray, Simul at i on


Outline







Christoph Koutschan’s Mathematica Tool [7]

Properties of the HolonomicFunctions package:Deals with multivariate holonomic functions and sequences.Computes annihilating ideal, i.e., set of linear differentialequations, linear recurrences, q-difference equations, and mixedlinear equations that a given function satisfies.Executes closure properties (addition, multiplication, substitutions)for such functions.Implements creative telescoping for the summation andintegration of multivariate holonomic functions.Subtasks:

I Computations in Ore algebras (noncommutative polynomialarithmetic with mixed difference-differential operators)

I Noncommutative Gröbner basesI Solving coupled linear systems of differential or difference

equations.


Using Christoph Koutschan’s Mathematica Tool [7]

For our m.g.f. expression

M(s,a) =1

(1− s)N 1F1

(N; NR; a

s1− s

), (28)

the HolonomicFunctions command

Annihilator

(1F1(N;NR ;

as1−s )

(1−s)N , {Der(s),Der(a)}),

yields the annihilators:a∂a + ∂s

(s2 − s

)+ Ns,

∂a (as + sNR − NR) + ∂2a (as − a) + Ns.


Differential Equation w.r.t. a for P.D.F.

From

M(s,a) =1

(1− s)N 1F1

(N; NR; a

s1− s

), (29)

a few HolonomicFunctions commands produced the followingdifferential equation, whose by-hand derivation had been intractable:

a2∂3ap(t ,a) + (2a2 + 2aNR)∂2

ap(t ,a)

+

(a2 + aN + 2aNR + NR

2 − NR − at)∂ap(t ,a)

+

(aN − N + NNR − (NR − 1)t

)p(t ,a) = 0. (30)


Differential Equations for Performance MeasuresCreative telescoping [6]: deducing differential equations satisfied bythe integral of a function from the differential equation satisfied by thefunction. HolonomicFunctions yielded for Po(a), C(a):

a2∂4aPo(a) =

−(−NR + a + aN + NR + NNR)∂aPo(a)

−(−a + 3a + +a2 + aN + NR + 2aNR + NR2)∂2

aPo(a)

−2a(1 + a + NR)∂3aPo(a).

a2∂5aC(a) =

−2a(NR + a + 2)∂4aC(a)

−(2 + a + 7a + NR2 + a2 + 3NR + Na + 2NRa)∂3

aC(a)

−(3 + NR + 3a + N + 3NR + NNR + Na + 1)∂2aC(a)

−(N + 1)∂aC(a).


Outline







Our More Recent ZF Work for Full-Rician Fading [4]

Partitioning again (corresponding to intended/interfering streams) as

Hd = (hd,1 Hd,2), (31)

allowing for Hd,2 6= 0, but assuming, for tractability, that

rank(Hd) = rank(Hd,2) = 1, (32)

we derived the ZF SNR m.g.f. for Stream 1 as

Mγ1(s) =e−y

(1− s)N

∞∑

n=0

yn

n!1F1

(N; n + NR;

s1− s

x),

where x , y depend on noncentrality Hd.


Our More Recent ZF Work for Full-Rician Fading [4]

We have analyzed MIMO ZF for full-Rician fading. Partitioning as

Hd = (hd,1 Hd,2), (33)

and assuming, for tractability, that

rank(Hd) = rank(Hd,2) = 2, (34)

we derived the ZF SNR m.g.f. for Stream 1 as

Mγ1(s) =e−(x12+x22)

(1− s)N

∞∑

n12=0

∞∑

n22=0

xn1212 xn22

22n12!n22!

×2F2(N,NR − 1 + n22; NR − 1,NR + n12 + n22; δ11(s)),

where δ11(s) = s1−s‖hd,1‖2, and x12, x22 depend on noncentrality Hd.


SIMO Optimum Combining for Rician Fading [8]

For SIMO system, when the received signal vector is

r = hx + n, (35)

optimum combining means

r =hH

‖h‖r = ‖h‖x +hH

‖h‖n, (36)

and yields symbol-detection SNR ∝ ‖h‖2 =∑NR

i=1 |hi |2.

Satellite channel is described by shadowed-Rician fading.

Then, p.d.f. of ‖h‖2 is known in terms of 1F1(·; ·; ·) [8, Eq. (6)].


MIMO Space–Time Coding for General Fading [9]Output SNR Y =

∑ni=1 X 2

i m.g.f. is MY (s) =∏n

i=1 MX 2i

(s), whereper-branch SNR m.g.f. for various fading types is:


MIMO Space–Time Coding for General Fading [9]

MIMO space–time coding output SNR Y =∑n

i=1 X 2i m.g.f. is

MY (s) =n∏

i=1

exp(δi

ai s1−ai s

)(1− bis)ki

(1− ais)pi, (37)

which can be written as infinite series (β controls convergence)

MY (s) = A∞∑

r=0

cr

1− sβρ+r , (38)

so that p.d.f. of SNR is

fY (y) = A∞∑

r=0

cryρ+r−1e−y/β

βρ+r Γ(ρ+ r). (39)


MIMO ML Detection for Rician Fading [10]

MIMO maximum-likelihood (ML) receiver performance is determinedby λmin(HHH). For Wishart matrix with rank-1 noncentrality [10,Eq. (16)]

pλmin(λ) ∝ Φ3(b, c; w , z) =∞∑

k=0

∞∑

m=0

(b)k

(c)k+m

wkzm

k !m!. (40)

Because confluent hypergeometric function Φ3 is difficult to evaluate, itwas written as a finite series [10, Eq. (12)] in terms of the generalizedMarcum-Q function, which can be expressed accurately, i.e.,

Qm(a,b) =

∫ ∞

b

xm

am−1 exp(−a2 + x2

2

)Im−1(ax)dx , (41)

where Im is the mth order modified Bessel function of the first kind.


MIMO ML under Interference [11]Study MIMO network performance by assuming Poisson point processbase-station locations (e.g., with λ = 0.008) and characterizingother-cell interference using stochastic geometry.ML performance measures expressed in terms of hypergeometricfunctions, integrated from 0 to∞. Authors notice difficulties at largeargument values and employ approximations [11, Eqs. (19), (20)].

2862 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 62, NO. 8, AUGUST 2014

error probability, which is averaged with respect to the fadingdistribution and to all BSs deployments. From the averagepairwise error probability, the average frame error probabilityis obtained by using the Nearest Neighbor (NN) approximation[32, p. 138]; and3) simplifiedbutasymptoticallyaccuratemathe-matical frameworks are proposed, which are shown to provideinsight on the achievable error performance as a function of im-portant system parameters. The proposed frameworks are use-ful for better understanding and for simplifying the analysis ofcellular networks, since they do not require explicit generationand simulation of BSs locations using Monte Carlo simulations.

The remainder of the present paper is organized as follows.In Section II, the system model is introduced. In Section III,a general mathematical framework to the computation of theerror probability of downlink SISO cellular networks is pro-posed, which is applicable to arbitrary fading distributions. InSection IV, the mathematical analysis is extended to downlinkspatial multiplexing MIMO cellular networks by assuming aRayleigh fading channel model for the interfering BSs. InSection V, the performance trends of downlink SISO andMIMO cellular networks are studied, as a function of systemand channel parameters. In Section VI, numerical examplessubstantiating the mathematical findings are shown. Finally,Section VII concludes the present paper.

Notation: The following notation is used throughout thepresent paper. z∗, |z| and arg{z} denote conjugate, modulusand phase of a complex number z, respectively. C denotes thefield of complex numbers. x ∈ SN×1 denotes a N × 1 column-vector with entries belonging to the set S. The lth entry isdenoted by x(l). X ∈ SN×M denotes a N × M matrix withentries belonging to the set S. The (l, m)th entry is denotedby X(l,m). card{S} denotes the cardinality of the set S. ‖x‖denotes the norm of vector x. ‖X‖ denotes the Frobenius normof matrix X. j denotes the imaginary unit. CN (μ, σ2) andN (μ, σ2) denote a complex and a real Gaussian distributionwith mean μ and variance σ2, respectively. U(a, b) denotes auniform distribution in (a, b). (·)k denotes the Pochhammersymbol. (·)! denotes the factorial operator. Re{·} and Im{·}denote real and imaginary part operators, respectively. E{·}denotes the expectation operator. fX(·) and FX(·) denotethe Probability Density Function (PDF) and the CumulativeDistribution Function (CDF) of a Random Variable (RV) X ,respectively. ΦX(ω) = E{exp{j(ω1X

(re) + ω2X(im))}} de-

notes the CF of the complex RV X = Re{X} + jIm{X} =X(re) + jX(im), where ω = (ω1, ω2). For notational simplic-ity, the short-hand ΦX(ω) = E{exp{jωX}} is used. Pr{·}denotes probability. X

d= Y denotes that the RVs X and Y are

equal in distribution. Γ(·) denotes the Gamma function [33,Eq. 6.1.1]. erf(·) denotes the error function [34, Eq. 7.1.1].pFq(a1, . . . , ap; b1, . . . , bq; ·) denotes the generalized hyperge-

ometric function [35, Ch. IV]. Gm,np,q

((·)∣∣∣∣(ap)(bq)

)denotes the

Meijer G-function [35, Sec. 2.24].

II. SYSTEM MODEL

We consider a bi-dimensional downlink cellular networkdeployment, where a probe/typical multi-antenna MT is located

Fig. 1. Illustration of a PPP-based cellular network deployment. A normalizedbi-dimensional area of 100 × 100 is shown, where the MT is located at theorigin and the serving BS is denoted by BS0. The red dots denote the BSs. Thedensity of BSs is λ = 8 × 10−3. The blue lines denote the coverage regions,which form a Voronoi tessellation and are computed based on the shortestdistance cell association criterion.

at the origin and the multi-antenna BSs are modeled as pointsof a homogeneous PPP (Ψ) of density λ. A diagrammaticrepresentation of a PPP-based cellular network deployment isillustrated in Fig. 1. The number of antennas at each BS and atthe MT is denoted by Nt and Nr, respectively. Based on theproperties of homogeneous PPPs, there is no loss of generalityin assuming the MT to be located at the origin [8]. The distancefrom the ith BS to the MT is denoted by ri for i ∈ Ψ. The MT isassumed to be tagged to the nearest BS, i.e., a shortest distancecell association criterion is assumed. The serving BS is denotedby BS0, and its distance from the MT is denoted by r0. Owingto the random nature of the PPP-based abstraction model, r0 isa RV with PDF equal to fr0

(ξ) = 2πλξ exp(−πλξ2) [1]. Ac-cording to the Slivnyak-Mecke’s theorem [8, Vol. 1, Th. 1.4.5],the set of interfering BSs i ∈ Ψ \ {BS0} is still a homogeneousPPP outside the ball centered at the origin and of radius r0,and it is denoted by Ψ(\0). By definition of shortest distancecell association, ri > r0 for i ∈ Ψ(\0). A full frequency reusescheme is assumed, i.e., all interfering BSs transmit in thesame frequency band as BS0. Upon completion of the cellassociation, it is assumed that the interfering BSs transmitpackets, in each channel use, with equal probability 0 ≤ p ≤1. These probabilities represent independent activity factorsof the interfering BSs. This model finds application to theanalysis of, e.g., slotted-ALOHA cellular networks [36] and itis particularly suited in the context of a PPP-based abstractionmodeling of cellular networks, since, due to the independentthinning property of the PPPs [8, Proposition 1.3.5], the set ofinterfering BSs Ψ(\0) is still a PPP of density pλ.

In the depicted downlink MIMO cellular network model, thesignal received at the MT can be formulated as follows:

y =√

E/Ntr−b0 H0s0︸︷︷︸

x

+√

E/Nt

∑

i∈Ψ(\0)

r−bi Hisi

︸︷︷︸iagg(r0)

+n (1)


MIMO Beamforming for Rician Fading [12]

Assuming knowledge of H the transmitter one can transmit symbolx into a certain ‘direction’ wT, i.e., beamforming, so that receivedsignal model is:

r = H wTx + n = (Hd + Hr) wTx + n. (42)

Optimum receive processing: combine r with wR ∝ H wT; then,detection SNR is proportional with

(H wT)H(H wT) = wHT HHH wT (43)

Thus, if HHH = UΛΛΛUH =∑λiuiuHi and λ1 = λmax = max{λi},

then the optimum beamformer is wT ∝ u1.Finally, SNR ∝ λmax, i.e., we need its distribution. It was found forcomplex-valued noncentral Wishart in terms of hypergeometricfunction in [12].


Dominant-Eigenvalue Statistics, for Real Case [13]

C.d.f. of λmax of real-valued zero-mean Wishart matrix is known interms of [13, Eq. (4)]

I(a,b; x) =1

Γ(a)

∫ x

0ta−1e−tP(b, t)dt , (44)

where

P(a, x) =1

Γ(a)

∫ x

0ta−1e−tdt . (45)

The latter be computed recursively with

P(a + n, x) = P(a, x)− e−xn−1∑

k=0

xa+k

Γ(a + k + 1). (46)

C.d.f. computation requires 70 seconds for 500× 500 matrices.


Dominant-Eigenvalue Statistics, for ComplexCase [13]

For larger matrices1, use the Tracy–Widom (TW) limiting distribution,i.e., the c.d.f. of λmax

F2(x) = exp{−∫ ∞

x(y − x)q2(y)dy

}, (47)

where q(y) satisfies Painlevé II differential equation

q′′(y) = yq(y) + 2q3(y), (48)

which is solved numerically in [13] for c.d.f. evaluation.

1Massive MIMO envisions thousands of antennas.Costi Siriteanu ([email protected]) Osaka University Comp. Alg. & HGM in MIMO 47 / 51

Outline







References I

[1] “MIMO Diagram,” http://en.wikipedia.org/wiki/MIMO, accessed: 2013-09-07.

[2] “PSK diagram,” http://en.wikipedia.org/wiki/Phase-shift_keying, accessed:2013-09-07.

[3] C. Siriteanu, S. D. Blostein, A. Takemura, H. Shin, S. Yousefi, and S. Kuriki,“Exact MIMO zero-forcing detection analysis for transmit-correlated Ricianfading,” IEEE Transactions on Wireless Communications, vol. 13, no. 3, pp.1514–1527, March 2014.

[4] C. Siriteanu, A. Takemura, S. Kuriki, D. Richards, T. Onoye, and H. Shin,“Performance analysis of MIMO zero-forcing for full-Rician fading,” IEEETransactions on Wireless Communications, to be submitted, 2015.

[5] C. Siriteanu, A. Takemura, S. Kuriki, and H. Shin, “MIMO zero-forcingperformance evaluation using the holonomic gradient method,” IEEETransactions on Wireless Communications, accepted, December 2014. [Online].Available: http://arxiv.org/abs/1403.3788

[6] T. Hibi, Ed., Grobner Bases. Statistics and Software Systems. Tokyo, Japan:Springer, 2013.


http://en.wikipedia.org/wiki/MIMO

http://en.wikipedia.org/wiki/Phase-shift_keying

http://arxiv.org/abs/1403.3788

References II[7] C. Koutschan, “Advanced applications of the holonomic systems approach,”

Ph.D. dissertation, Research Institute for Symbolic Computation (RISC),Johannes Kepler University, Linz, Austria, 2009. [Online]. Available:http://www.risc.jku.at/research/combinat/software/HolonomicFunctions/

[8] M. R. Bhatnagar, “Beamforming and combining in hybrid satellite-terrestrialcooperative systems,” IEEE Communications Letters, vol. 18, no. 3, pp. 483–486,2014.

[9] G. A. Ropokis, A. A. Rontogiannis, P. T. Mathiopoulos, and K. Berberidis, “Anexact performance analysis of MRC/OSTBC over generalized fading channels,”IEEE Transactions on Communications, vol. 58, no. 9, pp. 2486–2492, 2010.

[10] D. Morales-Jimenez, F. J. Lopez-Martinez, E. Martos-Naya, J. F. Paris, andA. Lozano, “Connections between the generalized Marcum-function and a classof hypergeometric functions,” IEEE Transactions on Information Theory, vol. 60,no. 2, pp. 1077–1082, 2014.

[11] M. Di Renzo and P. Guan, “A mathematical framework to the computation of theerror probability of downlink mimo cellular networks by using stochasticgeometry,” IEEE Transactions on Communications, vol. 62, no. 8, pp. 2860–2879,Aug 2014.


http://www.risc.jku.at/research/combinat/software/HolonomicFunctions/

References III

[12] M. Kang and M.-S. Alouini, “Largest eigenvalue of complex Wishart matrices andperformance analysis of MIMO MRC systems,” IEEE Journal on Selected Areasin Communications, vol. 21, no. 3, pp. 418–426, 2003.

[13] M. Chiani, “Distribution of the largest eigenvalue for real Wishart and Gaussianrandom matrices and a simple approximation for the Tracy–Widom distribution,”Journal of Multivariate Analysis, vol. 129, pp. 69–81, 2014.


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