1

Ergodic Spectral Efficiency in MIMO
Cellular Networks

arXiv:1607.04352v4 [] 3 Aug 2017

Geordie George, Ratheesh K. Mungara, Angel Lozano, and Martin Haenggi

Abstract
This paper shows how the application of stochastic geometry to the analysis of wireless networks
is greatly facilitated by (i) a clear separation of time scales, (ii) the abstraction of small-scale effects
via ergodicity, and (iii) an interference model that reflects the receiver’s lack of knowledge of how
each individual interference term is faded. These procedures render the analysis both more manageable
and more precise, as well as more amenable to the incorporation of subsequent features. In particular,
the paper presents analytical characterizations of the ergodic spectral efficiency of cellular networks
with single-user multiple-input multiple-output (MIMO) and sectorization. These characterizations, in
the form of easy-to-evaluate expressions, encompass the coverage, the distribution of spectral efficiency
over the network locations, and the average thereof.

Index Terms
Stochastic geometry, cellular networks, ergodic spectral efficiency, MIMO, sectorization, Poisson
point process, shadowing, interference, SINR

I. I NTRODUCTION
Stochastic geometry is quickly becoming an indispensable instrument in wireless network analysis. By mapping the empirical distribution of transmitter and receiver locations to appropriate
point processes, it becomes possible to apply a powerful and expanding toolkit of mathematical
G. George, R. K. Mungara and A. Lozano are with the Department of Information and Communication Technologies,
Universitat Pompeu Fabra (UPF), 08018 Barcelona, Spain. E-mail: {geordie.george, ratheesh.mungara, angel.lozano}@upf.edu.
Martin Haenggi is with the University of Notre Dame, Notre Dame, IN 46556, USA. E-mail: mhaenggi@nd.edu.
This work was supported by Project TEC2015-66228-P (MINECO/FEDER, UE), by the European Research Council under
the H2020 Framework Programme/ERC grant agreement 694974, and by the U.S. NSF through award CCF 1525904. This paper
is accepted for presentation in part at the 2017 IEEE Int’l Conference on Communications (ICC).

2

results. This offers a complement, and increasingly even an outright alternative, to the MonteCarlo simulations that have long been the workhorse of wireless network design.
Although a stochastic modelling of transmitter and receiver locations may seem mostly amenable
to ad hoc networks, which are devoid of fixed infrastructure, a seminal paper by Andrews et al. [1]
proved the truly remarkable effectiveness of stochastic modelling also in cellular networks—even
with simple Poisson point processes (PPPs). Indeed, while it may appear that more sophisticated
spatial distributions could better capture the relative regularity of actual base station (BS) placements, because of shadowing it is the case that PPPs lead to remarkably precise characterizations
of signal strengths and interference, and thus of all ensuing performance measures. In fact, as
shown in [2], [3] and expounded later in this paper, PPP-based characterizations represent the
limit to which actual behaviors converge as the shadowing strengthens.
Altogether, the irruption of stochastic geometry is a transcendent development in wireless
research, and it is reasonable to expect its importance to grow even further as networks become
denser and more heterogeneous [4], [5]. Important contributions to the advancement of the
discipline in the context of wireless networks include [6]–[27].
The present paper deals with the ergodic spectral efficiency of Poisson cellular networks, a
quantity already tackled in works dating back to [1]. Our analysis, however, relies on a different
modeling approach for the interference. This takes us on a different route, one that proves greatly
advantageous because it yields solutions that are both simpler and more precise than previous
ones, and, most importantly, because it opens the door to accommodating key ingredients—such
as MIMO and cell sectorization—that seemed previously elusive. The modeling approach that
unlocks these new analytical possibilities is not arbitrary, but rather based on sound arguments:
1) A clean separation between small- and large-scale channel features, in recognition that the
phenomena that give rise to these features are distinct.
2) An unwavering embrace of ergodic performance metrics with respect to the small-scale
features, in recognition that such small-scale ergodicity reflects well the operating conditions of modern wireless systems [28], [29].
3) The admission that each receiver can track the fading of its intended signal, but not the
fading of each individual interference term.
With small- and large-scale features decoupled, ergodicity enables abstracting out the former
so as to focus the stochastic geometry analysis where it makes a difference (on the large-scale
aspects), reaping the most out of its potent machinery. As mentioned, this allows advancing

3

the analysis on all fronts: tractability, accuracy, and generality. In particular, and to the extent
of our knowledge, the spectral efficiency expressions obtained in this paper are the first such
characterizations that incorporate MIMO spatial multiplexing. Likewise, sectorization is also
readily included.
Besides providing new expressions for quantities of interest, the present paper seeks to promote
the importance of keeping the two foregoing arguments present when applying stochastic geometry to wireless networks. For conceptual clarity, these arguments are herein elaborated on the
basis of a cellular network where each user is served by a single BS, and where only the downlink
is considered. However, the arguments apply equally to networks featuring BS cooperation, and
to the uplink, only with certain expressions suitably replaced by generalizations or counterparts.
II. N ETWORK M ODELLING
A. Separation of Scales
Rooted in extensive propagation measurements, the separation between large-scale propagation
phenomena (distance-dependent path loss and shadowing) and small-scale multipath fading
has been instrumental in the study of wireless networks since their onset, greatly facilitating
characterizations that would otherwise be unwieldy [30]. The premise of this separation is that,
over suitably small distances (tens to hundreds of wavelengths), the large-scale phenomena
remain essentially unchanged and only small-scale variations transpire. This allows delineating
local neighborhoods around transmitters and receivers wherein the small-scale channel behavior
conforms to a stationary random process whose distribution is dictated by the large-scale features.
Then, through the user velocity, this space-domain random process maps to a time-domain
process. Moreover, under the very mild condition that the Doppler spectrum be free of delta
functions, this time-domain process is ergodic.
For frequencies and velocities in the widest possible range of interest, the dwell time in a local
neighborhood is far longer than the extension of signal codewords. Thus, large-scale features
can be regarded as constant over an individual codeword. Alternatively, the small-scale fading
may or may not remain constant over a codeword, depending on the coherence of such fading
and on how codewords are arranged in time and frequency, and this dichotomy gives rise to two
classic information-theoretic idealizations of the channel over the horizon of a codeword:
•

Nonergodic. Fading random, but fixed over the codeword.

4

•

Ergodic. Fading random and exhibiting sufficiently many values over the codeword to
essentially reveal its distribution.

These two idealizations, in turn, map respectively to outage and ergodic definitions for the
spectral efficiency. Although both are useful, the ergodic definition is the most representative in
modern systems where codewords can be interspersed over very wide bandwidths, across hybridARQ repetitions, and possibly over multiple antennas, and they can be subject to scheduling and
link adaptation. As argued in [28], [29], the balance of these mechanisms is indeed best abstracted
by ergodic spectral efficiencies involving expectations over the small-scale fading, with the largescale features held fixed. It is at this point that stochastic geometry should enter the analysis,
when the small-scale effects have been abstracted out and we can zoom out to cleanly examine
the large-scale ones.
B. Large-scale Modeling
We consider the downlink of a cellular network, initially with omnidirectional antennas (to
be generalized to sectorized antennas in Section VIII), where the signals are subject to path loss
with exponent η > 2 and shadowing.
Suppose that the BS positions are agnostic to the radio propagation. It has been recently shown
[2], [3] that, regardless of what those BS positions are (under only a very mild homogeneity
condition), an increasing shadowing standard deviation σdB renders the network progressively
PPP-like from the vantage of any given user, i.e., it makes the powers that a user receives
from any population of BSs look as if they originated from PPP-distributed BSs. This important
observation, upheld even if the shadowing is correlated [31], strongly justifies the modelling
assumption of PPP-located BSs. Ironically then, shadowing, a nuisance in the study of regular
geometries, simplifies the stochastic modelling of networks by making them all look alike
propagation-wise regardless of their underlying geometry. And, although the convergence to
a PPP behavior is asymptotic in σdB , values of interest suffice for networks to look essentially
Poissonian. In particular, it is shown in [2] that, for a regular lattice of BSs spawning hexagonal
cells and η = 4, σdB = 12 dB suffices to render the received powers indistinguishable (with
99% confidence in a Kolmogorov-Smirnov test) from those in a Poisson network, 90% of the
time. In Section IX we provide further evidence supporting the suitability of a PPP model for
the analysis of lattice networks with relevant values of σdB .

5

The foregoing convergence is a powerful argument in favor of a PPP model for the spatial
distribution of BSs, say a process Φb ⊂ R2 , without the need for explicit modeling of the
shadowing as it is already implicitly captured by the Poisson nature of the network. The density
of Φb , say λb , depends on the type and strength of the shadowing in addition to the actual
positions of the BSs [32]–[35].
Turning now to the spatial distribution of users, a good starting point is to model it as an
independent PPP Φu with density λu . (This model could be refined to incorporate user clustering
tendencies as well as dependences between the positions of users and BSs [36]–[39].)
Without loss of generality, the analysis can be conducted from the perspective of a user at the
origin, which becomes the typical user under expectation over Φb . Denoting by rk the distance
between such user and the kth BS, whose location—recall—is distributed according to Φb , we
index the BSs in increasing order of rk , i.e., such that rk < rk+1 for k ∈ N0 . Since, in terms
of Φb , the only large-scale propagation mechanism at play is path loss, the user at the origin
receives the strongest power from the BS at r0 , which we deem the serving BS.
C. Small-scale Modeling
Let the communication be SISO for now, i.e., with BSs and users having a single antenna,
and further let each receiver be privy to the fading of only its intended signal. Denoting by P
the power measured at 1 m from a BS transmitter, at symbol n the user at the origin observes
q
(1)
y[n] = P r0−η H0 [n] s0 [n] + z[n],
where the leading term is the intended signal from the serving BS while
∞ q
X
P rk−η Hk [n] sk [n] + v[n]
z[n] =

(2)

k=1

is the aggregate interference from all other BSs, plus thermal noise v. In turn, sk is the signal
transmitted by the kth BS and Hk is the associated fading coefficient.
The fading coefficients {Hk }∞
k=0 are independent and of unit power, but they are otherwise
arbitrarily distributed, whereas v ∼ NC (0, N0 ). The signal is sk ∼ NC (0, 1), a choice that is
justified later.
Conditioned on {rk }∞
k=0 , which are fixed as far as the small-scale modeling is concerned, the
instantaneous SINR at symbol n is
SINR[n] =

P r−η |H0 [n]|2
.
P∞ 0−η
P k=1 rk |Hk [n]|2 + N0

(3)

6

III. I NTERFERENCE M ODELING
With H0 [1], . . . , H0 [N ] known at the receiver, the mutual information (in bits/symbol) over
codewords spanning N symbols is

1 
I s0 [1], . . . , s0 [N ]; y[1], . . . , y[N ] H0 [1], . . . , H0 [N ], {rk }
N
which, with IID codeword symbols, becomes
N

1 X
I s0 [n]; y[n] H0 [n], {rk }∞
k=0 .
N n=1

(4)

As argued earlier, codewords are nowadays long enough—thousands of symbols—and arranged
in such a way—interspersed in time, frequency, and increasingly across antennas—so as to
experience sufficiently many fading swings for an effective averaging of the mutual information
to take place over the small-scale fading. From the stationarity and ergodicity of the small-scale
fading over the codeword, the averaging in (4) becomes an expectation over H0 and confers the
significance of the ergodic spectral efficiency (in bits/s/Hz)
h
i
Cexact = EH0 I s0 ; y H0 , {rk }∞
k=0

  q
−η
∞
= EH0 I s0 ; P r0 H0 s0 + z H0 , {rk }k=0 .

(5)
(6)

This quantity, a baseline in the sequel, does not admit explicit expressions. Rather, the evaluation
of Cexact requires computationally very intensive Monte-Carlo simulations (cf. App. A) and a
64-core high-performance computing cluster is employed to generate the corresponding results
throughout the paper; for all such results, 99% confidence intervals are given.
Let us examine the local distribution, for some given {rk }∞
k=1 , of the interference-plus-noise
z as defined in (2). The first thing to note is that, without further conditioning on {Hk }∞
k=1 , i.e.,
without the receiver knowing the fading coefficients from all interfering BSs, the distribution
of z over the local neighborhood is generally not Gaussian. Conditioned only on {rk }∞
k=1 , the
distribution of z is actually highly involved; in Rayleigh fading, for instance, it involves products
of Gaussians. While the non-Gaussianity of z is irrelevant to the SINR, since only the variance
of z matters in that respect, it is relevant to information-theoretic derivations and chiefly that of
the spectral efficiency, which does depend on the distribution of z.
It is nevertheless customary to analyze Cexact in the form it would have if {Hk }∞
k=1 were
actually known by the receiver at the origin and z were consequently Gaussian, namely the form



P r0−η |H0 |2
Cub = E{Hk }∞
log2 1 + P∞ −η
(7)
k=0
P k=1 rk |Hk |2 + N0

7

where the tacit—and seldom made explicit—redefinition of z as Gaussian is unmistakable from
√
I(s0 ; γs0 + z) = log2 (1 + γ), which holds only when s0 and z are Gaussian. As it turns out,
a Gaussian modeling of z is not unreasonable because, if a decoder is intended for Gaussian
interference-plus-noise (either by design or because the distribution thereof is unknown), then the
spectral efficiency is precisely as if the interference-plus-noise were indeed Gaussian, regardless
of its actual distribution [40]. And, once z is taken to be Gaussian, the capacity-achieving signal
distribution is also Gaussian, validating our choice for s0 . At the same time, the granting of
{Hk }∞
k=1 as additional side information to the receiver renders (7) an upper bound to Cexact ,
hence the denomination Cub .
While much more tractable than Cexact , the form of Cub has the issue of depending not only
on H0 , but further on {Hk }∞
k=1 . This still clutters its analysis considerably, as will be seen later.
Alternatively, what we propound in this paper is to model z as Gaussian, but forgoing the
small-scale variations in its power, i.e., to use
z ∼ NC

0, P

∞
X

!
rk−η

+ N0

.

(8)

k=1

The closeness between this distribution for z and its original brethren in (2) has been tightly
bounded [41], [42].
The model proposed in (8) has the virtue of rendering z Gaussian without the strain of gifting
the receiver with {Hk }∞
k=1 and, as in other contexts where it has been tested [7], [43], [44], the
result of this restrain turns out to be gratifyingly good for cellular network analysis. With z as
in (8), the instantaneous SINR then becomes
SINR =

P r−η |H0 |2
,
P∞0 −η
P k=1 rk + N0

and the corresponding ergodic spectral efficiency is



P r0−η |H0 |2
C = EH0 log2 1 + P∞ −η
,
P k=1 rk + N0

(9)

(10)

which is the quantity we shall work with.
Since, with Gaussian codewords and a given variance for z, the mutual information is minimized when z is Gaussian [45], we have that C ≤ Cexact . The similarity between C and
Cexact , with the former characterized analytically and the latter obtained through Monte-Carlo
simulation, is illustrated throughout the paper.

8

Contrasting the instantaneous SINR expressions in (3) and (9), the analytical virtues of our
model for z become evident once we rewrite the latter as SINR = ρ |H0 |2 , where
ρ=

P r−η
P∞ 0−η
P k=1 rk + N0

(11)

is the local-average1 SINR at the origin, fixed over any entire codeword and cleanly separated
from the fluctuant term |H0 |2 ; this reflects the decoupling between the large- and small-scale
dependences. In interference-limited conditions (P/N0 → ∞), the local-average SINR specialP
−η
izes to ρ = r0−η / ∞
k=1 rk . For given BS and user locations, ρ becomes determined and the
conditional distribution of the instantaneous SINR is then given directly by that of |H0 |2 , i.e.,
by the CDF F|H0 |2 (·), while the ergodic spectral efficiency of a user with local-average SINR ρ
is
h
i
C(ρ) = EH0 log2 1 + ρ |H0 |2
Z ∞
=
log2 (1 + ρ ξ) dF|H0 |2 (ξ),

(12)
(13)

0

which, through ρ, sets the stage for further computations involving the geometry of the network. As anticipated, it is here where stochastic geometry can be applied with all its potency,
undistracted by lingering small-scale terms.
Example 1. In Rayleigh fading, F|H0 |2 (ξ) = 1 − e−ξ , from which
 
1
1/ρ
log2 e,
C(ρ) = e E1
ρ
R∞
where En (x) = 1 t−n e−x t dt is an exponential integral.

(14)

For fading distributions other than Rayleigh, or with MIMO or other features, corresponding
forms can be obtained for C(ρ), always with the key property of these being a function of ρ
and not of the instantaneous fading coefficients.
1

The term “local-average” indicates averaging over the small-scale fading only. It follows that the “local-average SINR” is

its average value over a region small enough for the path loss (and the shadowing, if applicable) not to change noticeably.

9

10

as in (14)

Spectral efficiency (bits/s/Hz)

8

as in (15)

6

4

2

0
150

200

250

300
(m)

350

400

450

√
Fig. 1: Spectral efficiency vs. r0 for λb = 2 BSs/km2 , η = 3.8 and rk = Γ(k + 1.5)/( πλb Γ(k + 1)) for k = 1, . . . , 100.
The 99% confidence intervals around Cexact range from ±0.029 at r0 = 150 m to ±0.009 at r0 = 450 m, for both SISO and
MIMO.

Example 2. Let transmitter and receiver be equipped, respectively, with Nt and Nr antennas. If
H0 is replaced by an Nr × Nt channel matrix H 0 having IID Rayleigh-faded entries, then [46]
(


2j
m−1
i X
XX
2i − 2j
2j + 2n − 2m
MIMO
Nt /ρ
CNr ×Nt (ρ) = e
i−j
2j − `
i=0 j=0 `=0
 )
n−m+`
(−1)` (2 j)! (n − m + `)! X
Nt
· 2i−`
Eq+1
log2 e
(15)
2
j! `! (n − m + j)! q=0
ρ
with m = min(Nt , Nr ) and n = max(Nt , Nr ).
Example 3. Let us consider the application of (14) to an interference-limited network with 100
interfering BSs. To typify the network, we set rk to the expected value of the distance to the
kth nearest point in a PPP:
rk = √

Γ(k + 1.5)
πλb Γ(k + 1)

10

for k = 1, . . . , 100 [47]. We further set η = 3.8 and λb = 2 BSs/km2 and neglect the noise.
Shown in Fig. 1 is C(ρ) in (14) compared against Cexact . The same comparison is provided for
MIMO with Nt = Nr = 2. In both cases the differences are minute, supporting the interference
modeling approach propounded in this paper.
IV. D ISTRIBUTION OF THE L OCAL -AVERAGE SINR
Thus far, the link distances {rk } have been conditioned upon. Once they are released, ρ
becomes itself a random variable whose distribution is induced by the process Φb . The corresponding CDF, Fρ (·), is to be a central ingredient in our analysis of the ergodic spectral efficiency.
This important function, extensively utilized by system designers, has traditionally been obtained
by means of simulation over lattice networks with shadowing and random translations [48].
For hexagonal cells in particular, and without shadow fading, an infinite series solution is also
available [49].
Here we set out to characterize Fρ (·) for Poisson networks, so as to implicitly incorporate
shadowing, and specifically for interference-limited Poisson networks. Interestingly, with PPPpopulated BSs, the powers received by any given user are statistically invariant—save for a
scaling of the BS density if noise were not negligible—to the distribution of the channel gains
[32]–[34] and thus Fρ (·) is equivalent to what FSIR (·) would look like if users connected to the
BS with the strongest instantaneous link. This distribution was established in [14], [50]–[52],
very compactly for arguments above 1 and in a still manageable form for arguments between 1/2
and 1, but in an accelerating cumbersome fashion (involving progressively higher-dimensional
integrations) as the argument of the CDF dips below 1/2 [51, Sec. V.A], [52, Cor. 19]. As
alternatives, [19], [50], [53] derived F1/ρ (·) in the Laplace domain, which would then require
numerical inversion, while [54] showed that the lower tail (θ → 0) of Fρ (θ) satisfies
log Fρ (θ) =

s?
+ o(1)
θ

(16)

with s? < 0 being the solution to
s?δ Γ̄(−δ, s? ) = 0

(17)

where Γ̄ is the lower incomplete gamma function and, for compactness, we have introduced the
shorthand notation δ = 2/η. Since it only depends—through δ—on the path loss exponent η,
the parameter s? can be precomputed for all relevant values thereof by solving (17) using any
standard software package. Values of s? for some typical η are listed in Table I.

11

TABLE I: Parameter s? and the corresponding δ for typical values of the path loss exponent η.
η

δ

s?

η

δ

s?

3.5

0.571

-0.672

3.9

0.513

-0.819

3.6

0.556

-0.71

4

0.5

-0.854

3.7

0.540

-0.747

4.1

0.488

-0.888

3.8

0.526

-0.783

4.2

0.476

-0.922

The approach we take is to apply the exact form down to θ = 1/2 and then (16) for θ < 1/2.
This combination gives

s? /θ

 Fρ (θ) ' e



 F (θ) ≈ A
ρ

0≤θ<
s?
log Aδ

δ



Fρ (θ) = 1 − θ−δ sinc δ + Bδ




Fρ (θ) = 1 − θ−δ sinc δ

θ
1−θ



s?
log Aδ

≤ θ < 1/2

(18)

1/2 ≤ θ < 1
θ ≥ 1,

where ”'” indicates an approximation with asymptotic (θ → 0) equality while
Aδ = 1 − 2δ sinc δ + Bδ (1)

(19)


δ sinc2 (δ) Γ2 (δ + 1) 2 F1 1, δ + 1; 2 δ + 2; −1/x
Bδ (x) =
x1+2 δ Γ(2 δ + 2)

(20)

and

with 2 F1 the Gauss hypergeometric function. When the path loss exponent is η = 4, we have
that δ = 1/2 and the above specialize to


Fρ (θ) ' e−0.854/θ




 F (θ) ≈ 0.154
ρ



Fρ (θ) = 1 −




Fρ (θ) = 1 −

0 ≤ θ < 0.457
0.457 ≤ θ < 1/2

√
4 θ−θ−1
πθ

1/2 ≤ θ < 1

2
√
π θ

θ ≥ 1.

(21)

An even simpler, slightly less accurate expression for Fρ (·) is obtained using the exact form
only down to θ = 1 while stretching the lower tail expansion in (16) up to θ = 1. This gives


s?
s? /θ

0 ≤ θ < log(1−sinc

δ)
 Fρ (θ) ' e
Fρ (θ) ≈ 1 − sinc δ




 Fρ (θ) = 1 − θ−δ sinc δ

s?
log(1−sinc δ)

θ ≥ 1.

≤θ<1

(22)

12

V. D ISTRIBUTION OF THE S PECTRAL E FFICIENCY
The randomness that ρ acquires once the BS positions are randomized is then inherited by C,
and the corresponding CDF provides a complete description of the ergodic spectral efficiency
offered by the network over all locations [8]. By mapping the applicable function C(ρ) onto
the expressions for Fρ (·) put forth in the previous section, FC (·) is readily characterized. In
interference-limited conditions, such FC (·) depends only on the path loss exponent, η.
A. SISO
In Rayleigh-faded SISO channels, C(ρ) is given by (14). By resorting to the invertible
approximation [55]


0.82
e E1 (ν) log2 e ≈ 1.4 log 1 +
ν
ν

it becomes possible to write ρ ≈

(23)

eC/1.4 −1
0.82

and subsequently, by means of (18),
 γ

e 1.4 − 1
FC (γ) ≈ Fρ
0.82

which can be expressed

0.82 s?
exp(γ/1.4)−1

e






Aδ



 γ/1.4 
e
−1
sinc δ
FC (γ) ≈
δ + Bδ
1 −  γ/1.4
1.82−eγ/1.4

e
−1

0.82




sinc


1 −  γ/1.4 δ δ


e
−1

(24)



s?
0 ≤ γ < 1.4 log 1 + 0.82
 logAδ

?
s
≤ γ < 0.48
1.4 log 1 + 0.82
logAδ
0.48 ≤ γ < 0.84

(25)

γ ≥ 0.84,

0.82

whose accuracy is validated in the following example.
Example 4. Consider an interference-limited network with η = 3.8. Shown in Fig. 2 are (25),
as well as the numerical mapping of C(ρ)—without the bypass of its invertible approximation—
onto (18), both solutions contrasted against Cexact . For the computation of Cexact via Monte-Carlo,
here and in all examples involving simulations hereafter, PPP-populated BSs (1000 on average)
are dropped in a disk centered at the receiver.

13

1

8
0.8
0.05
0.04

0.6
6

0.03
0.02

0.4
4

0.01
0
0

0.2

0.4

0.6

0.2
2
(25)
Mapping of
0
0

2

4

6

onto
onto (18)
(18)
8

10

(bits/ s/ Hz)
Fig. 2: CDF of ergodic spectral efficiency for η = 3.8. In the inset, a zoom-in of the lower tail.

For η = 4, (25) specializes to

0.7

e− exp(γ/1.4)−1





0.154
q
FC (γ) ≈
0.82
4

1 − π eγ/1.4
+

−1


q


0.82
1 − 2
π
eγ/1.4 −1

0 ≤ γ < 0.44
0.44 ≤ γ < 0.48
eγ/1.4 −0.18
π (eγ/1.4 −1)

0.48 ≤ γ < 0.84

(26)

γ ≥ 0.84.

In contrast to these pleasing results, without the model for z propounded in this paper the
distribution of the spectral efficiency over the network locations is far more inaccessible. Indeed,
the corresponding Cub can be rewritten (cf. App. B) as
Z ∞ −x rη N0 /P Y
∞
e 0
1
Cub = log2 e
dx,
1 + x k=1 1 + x (r0 /rk )η
0

(27)

which is no longer a function of singly ρ, whose distribution was established in the previous
section; rather, (27) is a more involved function of {rk }∞
k=0 and offers no obvious way of

14

disentangling these dependences. Faced with this obstacle, some authors choose to instead
characterize the distribution of

log2

P r0−η |H0 |2
1 + P∞ −η
P k=1 rk |Hk |2 + N0



∞
over {Hk }∞
k=0 as well as {rk }k=0 [56] [57, Sec. VII.A]. However, the mixing of small- and large-

scale variations within this quantity clutters potential observations. Moreover, the generalization
to more involved settings, say MIMO, appears arduous or outright hopeless. Indeed, existing
stochastic geometry analyses of spectral efficiency featuring MIMO are restricted to beamforming
or space-division multiple access, rather than spatial multiplexing [22]–[26], [56].
B. MIMO
Our approach, in contrast, only requires mapping the appropriate C(ρ) onto Fρ (·). Whenever
C(ρ) does not lend itself to inversion, even approximately, it is straightforward to perform this
mapping numerically.
Example 5. Reconsider Example 4, but now with Nt = Nr = 2 such that, from (15),
  
 
2
2
MIMO
2/ρ
C2×2 (ρ) = 2 e
E1
+ E3
log2 e
ρ
ρ
 


 
2
2
2
2/ρ
= 2 e E1
1+ 2 + 1−
log2 e.
ρ
ρ
ρ

(28)
(29)

MIMO
(ρ) onto (18), as well as the corresponding Cexact .
Fig. 2 depicts the numerical mapping of C2×2

C. Lognormal Fit
Inspecting the distribution of C(ρ) we observe that, interestingly, it closely resembles a
lognormal function, i.e., that log C(ρ) admits a rather precise Gaussian fit. This opens the door
to an alternative to FC (·) as obtained by mapping C(ρ) onto Fρ (·), namely the alternative
log C(ρ) ∼ N (µ, σ 2 ) with
Z

∞

µ=

log C(θ) dFρ (θ)

(30)

0
2

Z

σ =

∞



2
log C(θ) dFρ (θ) − µ2 .

(31)

0
MIMO
Example 6. Let η = 4. For C2×2
(ρ) as given in (29) and Fρ (·) as given in (21), the numerical

integrations in (30)–(31) yield µ = 0.92 and σ 2 = 0.8. Fig. 3 presents the empirical PDF of
MIMO
log C2×2
(ρ), generated via Monte-Carlo, and a Gaussian PDF with µ = 0.92 and σ 2 = 0.8.

15

Fig. 3: Empirical PDF of the logarithm of the 2 × 2 MIMO ergodic spectral efficiency for η = 4 and its normal fit.

Thanks to this lognormal behavior, which holds for both SISO and MIMO, a Gaussian
distribution with mean µ and variance σ 2 can provide a quick idea of the disparity of the
user experiences throughout the network. This complements the coverage analysis presented in
the next section, which is more precise but valid only for the lower tail. The lognormal fit, rather,
may be used to determine the fraction of users whose spectral efficiency lies within a certain
interval of the average, or it may simplify further calculations that require averaging with respect
to the distribution of C(ρ).
VI. C OVERAGE
While the entire CDF of ergodic spectral efficiency is relevant from a network-design perspective, the lower tail is especially important as it determines the coverage, i.e., the share of network
locations in which a minimum level of service can be provided. The shape of the lower tail,
in particular, reveals the improvement in coverage with a diminishing service requirement or,
equivalently, the sacrifice in coverage that is necessary to guarantee a better minimum service.

16

The behavior of this tail (which, for η = 3.8, is detailed in the inset of Fig. 2) warrants attention,
and this section is devoted precisely to its analysis and to gleaning coverage insights from it.
A. SISO
For SISO, the CDF lower tail is seen from (24) to behave as
0.82 s?

FC (γ) ≈ e exp(γ/1.4)−1 .

(32)

Using eγ/1.4 − 1 = 0.72 γ + O(γ 2 ), the tail behavior can be re-expressed as
FC (γ) ≈ e1.15 s

? /γ

(33)

where, recall, s? is negative. By inverting (33), the spectral efficiency γ achievable on a share
1 − ξ of the network is seen to satisfy (for ξ ≤ Aδ )
γ≈

1.15 s?
.
loge ξ

(34)

For η = 4, we can recall s? = −0.854 and the above direct and inverse expressions then
specialize, respectively, to
FC (γ) ≈ e−1/γ

(35)

1
.
loge 1/ξ

(36)

and (for ξ ≤ 0.154)
γ≈

Example 7. For SISO and η = 4, (35) is depicted in Fig. 4 alongside previously obtained
expressions for FC (·) as well as the exact tail computed via Monte-Carlo.
Example 8. For SISO and η = 4, the spectral efficiency γ achievable in 99% of the network
satisfies
γ≈

1
= 0.22
loge 100

whereas the exact value, obtained numerically, equals 0.24 bits/s/Hz.

(37)

17

B. MIMO
With Nt transmit and Nr receive antennas, the ergodic capacity expands as [58]
C(ρ) = Nr ρ log2 e + O(ρ2 )

(38)

indicating a linear scaling with Nr and no dependence on Nt . This low-SNR behavior is very
robust, holding irrespectively of the fading distribution and in the face of antenna correlation,
signifying that what matters in this regime is only the receiver’s ability to capture power.
Applying the linear scaling of the low-SNR capacity with Nr to (33) and (34) we obtain
FC (γ) ≈ e1.15 s

?N

r /γ

(39)

and
γ≈

1.15 s? Nr
.
loge ξ

(40)

which can be readily specialized to η = 4 as well.
Example 9. For Nt = Nr = 2 and η = 4, (39) is depicted in Fig. 4 alongside previously obtained
expressions for FC (·) as well as the exact tail computed via Monte-Carlo.
Since coverage is not gained or lost in reference to small-scale fades, which are highly localized
in space, time and frequency, care must be exercised not to infer coverage from the distribution
of instantaneous SINR or of instantaneous spectral efficiency, both of which are dominated
by small-scale fading. With Rayleigh fading and η = 4 in particular, the distribution of the
instantaneous SIR throughout the network equals [1]
FSIR (θ) = 1 −

1+

√

1
√ 
θ arctan θ

(41)

whose lower tail behaves as FSIR (θ) = θ + O(θ2 ). This linear decay is drastically different from
the exponential ones derived in this section on the basis of FC (·). At 99% coverage, FSIR (·)
would map to a value of 0.014 bits/s/Hz, far from the exact value of 0.24 and from the value
of 0.22 we obtained in Example 8. It is therefore important that coverage be gleaned from
the ergodic spectral efficiency, which is impervious to small-scale fluctuations, rather than from
quantities subject to those fluctuations.

18

0.05
Tail behaviors in (35) and (39)
onto (21)

Mapping of

0.04

0.03

0.02

0.01
(35)

0
0

0.1

(39)

0.2

0.3
0.4
(bits/ s/ Hz)

0.5

0.6

0.7

Fig. 4: Lower tail of the ergodic spectral efficiency CDF for η = 4.

VII. S PATIAL AVERAGE OF THE S PECTRAL E FFICIENCY
Sometimes, it is of interest to condense FC (·) down to a single quantity, and in that case the
average is the logical choice. Under spatial ergodicity, which holds for the PPP and many other
point processes [9], this quantity equals the average of all per-user spectral efficiencies in any
realization of the network.
Here again, the approach propounded in this paper proves advantageous. For any setting for
which C(ρ) is available, our expressions for Fρ (·) enable computing
Z ∞
C̄ =
C(θ) dFρ (θ),

(42)

0

which, in interference-limited conditions, again depends only on the path loss exponent. Remarkably, the above integration can be solved for SISO and for (at least some) MIMO settings, and
the expressions obtained involve only readily computable special functions. Although devoid of
insight, these expressions allow circumventing large-scale Monte-Carlo simulations.

19

A. SISO
In Rayleigh-faded SISO channels, with (14) and (22) plugged into (42), the integration yields
(cf. App. C)


 

 
?
?
1+s
0,
1
−
δ
s
1
sin(πδ) log2 e 2,2 
−s log2 e
 (43)
E1 −
− e Dδ E1
+
G2,3 1
C̄ ≈
?
1+s
Dδ
Dδ
π
0, 0, −δ
?

where

z
Gm,n
p,q

a1 , ..., an , an+1 , ..., ap
b1 , ..., bm , bm+1 , ..., bq




(44)

is the Meijer-G function while Dδ = s? / log(1 − sinc δ). An even more precise, albeit also more
involved expression for C̄ can be obtained using (18) in lieu of (22).
Example 10. Fig. 5 compares (43) against C̄exact , the Monte-Carlo average of Cexact , for 0.48 ≤
δ ≤ 0.57 corresponding to 3.5 ≤ η ≤ 4.2. For η = 4 in particular,


log2 e 2,2  0, 1/2
 = 1.99,
C̄ ≈ 0.187 +
G2,3 1
π
0, 0, −1/2

(45)

while C̄exact is 2.01
It is worthwhile to contrast (43) with its counterpart obtained without the model for z
propounded in this paper, namely the average of Cub given by [1]
Z ∞
log2 e
R∞
C̄ub =
dt
t
δ
1 + (e − 1) (et −1)−δ 1+x11/δ dx
0
or, by means of Pfaff’s transformation [59], equivalently by
Z ∞
log2 e

 dγ.
C̄ub =
γ
0
2 F1 1, 1; 1 − δ; 1+γ

(46)

(47)

Example 11. Included in Fig. 5, alongside its SISO counterparts C̄ and C̄exact , is also C̄ub .
Besides being further from Cexact than our solution C̄, and requiring either a double integration
or a single integral over a hypergeometric function, neither of the expressions for C̄ub offers a
viable path to MIMO generalization. With our approach, in contrast, the analysis of the spatial
average becomes feasible also with MIMO.

20

Average spectral efficiency (bits/s/Hz)

5

4

3

2

1

as in (43)
as in (48)
as in (50)

0
3.5

3.6

3.7

3.8

3.9

4

4.1

4.2

Fig. 5: Spatially averaged ergodic spectral efficiency as a function of η for SISO and for 2 × 2 MIMO. The 99% confidence
interval around the Monte-Carlo results ranges from ±0.008 at η = 3.5 to ±0.01 at η = 4.2 for SISO and from ±0.015 at
η = 3.5 to ±0.02 at η = 4.2 for MIMO. The SISO upper bound C̄ub is computed via (47).

B. MIMO
MIMO
(ρ) as given in (29) over Fρ (·) as given in (22)
For 2 × 2 MIMO, the integration of C2×2

returns (cf. App. C)
2+s? 
 
? Dδ
2s
e
2(2 + s? )2 − 4(2 + s∗ )Dδ + 8 + s? (4 + s? ) Dδ2 E1 (2/Dδ )
MIMO
C̄2×2 ≈
(48)
(2 + s? )3 Dδ2 log 2



?
s? Dδ 2 8 + s? (4 + s? ) Dδ E1 (−s? /Dδ ) − es /Dδ (2 + s? ) (6 + s? )Dδ − 2(2 + s? )
+
(2 + s? )3 Dδ2 log 2






0, −1 − δ
sin(πδ) log2 e  2,2  0, 1 − δ 
2,2 
 + 1 − δ .
+
2 G2,3 1
+ 4 G2,3
1
π
(1 + δ) δ
0, 0, −δ
0, 0, −2 − δ
An alternative expression not involving the Meijer-G function can be obtained using
E3 (x) ≈ e−3x/2 /2

21

and
2 e2/ρ E1 (2/ρ) log2 e ≈ 2.8 log(1 + 0.41 ρ)
MIMO
(ρ) in (28) into
to simplify C2×2
MIMO
(ρ) ≈ 2.8 log(1 + 0.41 ρ) + e−1/ρ log2 e.
C2×2

(49)

With this form in place of (29) in the integration in (42) we obtain, as detailed again in App. C,

s? log2 e
MIMO
+ 1 F1 δ, 1 + δ, −1 sinc(δ) log2 e
C̄2×2 ≈
s? −1
(s? − 1) e Dδ
h
i sinc(δ)

+ 2.8 2 F1 1, δ, 1 + δ, −2.44 + δ log(1.41)
δ


?
E1 (−s (0.41 + 1/Dδ ))
?
s? /Dδ
log(1 + 0.41 Dδ ) .
(50)
− E1 (−s /Dδ ) + e
+ 2.8
e0.41 s?
Example 12. Fig. 5 compares (48) and (50) against C̄exact . For η = 4 in particular, (50) returns


i
√
log e
5.6 h√
MIMO
C̄2×2
≈ 0.26 + √2 erf(1) +
0.41 π − 2 arctan( 0.41) + log 1.41
(51)
π
π
= 3.84
while the Monte-Carlo average of its Cexact counterpart is 3.87.
Combining Examples 10 and 12, two-antenna single-user MIMO is seen from our analysis
to provide a 93% increase in the spectral efficiency of an entire interference-limited network, a
determination that would classically have entailed very extensive simulations.
(ρ) over Fρ (·) in (22) yields the singleFor higher-order MIMO, the integration of CNMIMO
r ×Nt
integral solution
C̄

MIMO
Nr ×Nt

2j
m−1
i X
XX

≈ log2 e

i=0 j=0 `=0

·

n−m+`
X Z ∞
q=0

0

(


2 j + 2 n − 2 m (−1)` (2 j)! (n − m + `)!
22i−` j! `! (n − m + j)!
2j − `

 )
? −γ N
?
s
t
s
Γ̄(δ, γ Nt )
dγ .
e Dδ + δ sinc(δ)
?
s − γ Nt
(γ Nt )δ

2i − 2j
i−j

1
(1 + γ)q+1



(52)

Although, for some antenna configurations beyond Nt = Nr = 2, it may be possible to express
this integral via special functions, it is beside the point once the expressions become overly
intricate. Rather, it seems preferable to directly integrate numerically for each specific path loss
exponent of interest.
Example 13. Shown in Table II is C̄NMIMO
computed via (52) for different values of Nt and Nr
r ×Nt
with the path loss exponent η = 4.

22

MIMO
TABLE II: Spatially averaged ergodic spectral efficiency C̄N
with η = 4.
r ×Nt

Nr

1

2

3

4

1

1.99

2.76

3.25

3.62

2

2.13

3.84

4.79

5.48

3

2.18

4.11

5.71

6.75

4

2.21

4.24

6.05

7.59

Nt

(a) No sectorization (S = 1).

(b) Tri-sectorized cells (S = 3).

Fig. 6: Tesselation of a Poisson network, with and without sectorization.

VIII. S ECTORIZATION
Let us now incorporate cell sectorization to the model. Each BS, now allowed to comprise S
sector antennas uniformly staggered in azimuth, communicates with one user per channel and
per sector (cf. Fig. 6). From the vantage of the user at the origin, the downlink signals from the
sectors of any given BS undergo the same path loss and shadowing but different antenna gains.
Given an arbitrary azimuth pattern for the sector antennas, [19] characterized the local-average
SINR distribution in the Laplace domain. Differently, in this paper, we provide direct expressions
that rely only on the antenna front-to-back ratio Q ≥ 1 and on the number of sectors, S. Given
an in-sector gain G =

QS
Q+S−1

and an out-of-sector gain g =

pattern as a function of the azimuth φ is (cf. Fig. 7)

S
,
Q+S−1

our model for the antenna

23

Fig. 7: Antenna pattern for S = 3 with in-sector gain G and out-of-sector gain g.


 G
gS (φ) =
 g

φ0 − π/S ≤ φ < φ0 + π/S

(53)

elsewhere,

where φ0 indicates the orientation of the antenna. This way, it is ensured that the total radiated
power is preserved, i.e., that
Z2π

gS (φ)
dφ = 1.
2π

(54)

0

Setting S = 1 we recover an unsectorized network where g1 (φ) = 1. In turn, for Q → ∞, the
S sectors become ideal as G → S and g → 0. Under the foregoing model, the intended signal
from the serving sector has gain G while the (S − 1) interfering transmissions from other sectors
of the same BS have gain g. The S transmissions from every other BS add to the interference.
The small-scale fading in the link from each sector is independent and of unit-power, with
the receiver knowing only the fading experienced by the intended signal.
A. Local-Average SINR
With P now denoting the per-sector transmit power, the total power that each interfering BS

launches towards the user at the origin is P G + (S − 1) g = P S. Since the useful signal
launched by the intended BS is P G, the local-average SINR at the origin is
P G r0−η
ρS =
P
−η
P (S − 1) g r0−η + P S ∞
k=1 rk + N0

(55)

irrespective of how the sectors are oriented at each BS. (This orientation-invariance is not an
artifact of our model; rather, it has been shown that the distribution of ρS is insensitive to the
sector orientations regardless of the antenna patterns [19].)

24

1

=3
=1

0.8

0.6

0.4

0.2
Simulation
(60)
0
10

5

0

5
(dB)

10

15

20

Fig. 8: Fρ (·) without sectorization and Fρ3 (·) with Q = 20 dB, both for η = 4.

In interference-limited conditions, the local-average SINR becomes
ρS =

Q r0−η
P
−η ,
(S − 1) r0−η + (Q + S − 1) ∞
k=1 rk

(56)

which, with ideal sectorization, i.e., for Q → ∞, converges to its unsectorized self, ρ =
P
−η
r0−η / ∞
k=1 rk . It follows that, under ideal sectorization, all the results derived for unsectorized
networks continue to apply, only on a per-sector rather than a per-BS basis. Conversely, under
nonideal sectors, the CDF of ρS can be obtained as


Q r0−η
FρS (θ) = P
P
−η < θ
(S − 1) r0−η + (Q + S − 1) ∞
k=1 rk


r0−η
Q+S−1
= P P∞ −η <
Q/θ − S + 1
k=1 rk

h
i
Q
 Fρ Q+S−1
0 ≤ θ < S−1
Q/θ−S+1
=
 1
θ≥ Q ,
S−1

which is capped at

Q
S−1

because of interference among same-BS sectors.

(57)

(58)

25

Plugging (18) into (58) gives

Q/θ−S+1
s? Q/ log Aδ
s? Q+S−1

F
(θ)
'
e
0 ≤ θ < Q + (1 +

ρ
S

s? / log Aδ )(S−1)


? Q/ log A

s
δ

FρS (θ) ≈ Aδ
≤ θ < 2 Q + 3Q(S−1)

Q + (1 + s? / log Aδ )(S−1)

h
i


Q+S−1
Q
sinc δ
FρS (θ) = 1 − Q+S−1
≤ θ < Q + 2Q(S−1)
δ + Bδ Q (1/θ−1)−2 (S−1)
2 Q + 3 (S−1)
(
)
Q/θ−S+1



Q
Q
sinc δ

FρS (θ) = 1 − Q+S−1
≤ θ < S−1

δ

Q
+
2
(S−1)

( Q/θ−S+1 )



Q
 F (θ) = 1
,
θ ≥ S−1
ρS
(59)
which, for η = 4, specializes to

Q/θ−S+1


FρS (θ) ' e−0.854 Q+S−1





 FρS (θ) ≈ 0.154


q
4
FρS (θ) = 1 − π Q/θ−S+1
+

q Q+S−1




FρS (θ) = 1 − π2 Q/θ−S+1

Q+S−1



 Fρ (θ) = 1
S

0≤θ<
Q 1+1/θ
π Q+S−1

0.457 Q
Q+1.457 (S−1)

0.457 Q
≤ θ < 2 Q + 3Q(S−1)
Q+1.457 (S−1)
Q
≤ θ < Q + 2Q(S−1)
2 Q + 3 (S−1)
Q
Q + 2 (S−1)

θ≥

≤θ<

(60)

Q
S−1

Q
.
S−1

Alternatively, plugging (22) into (58), a simpler and slightly less accurate expression is obtained

Q/θ−S+1
s? Q/ log(1−sinc δ)
s? Q+S−1

F
(θ)
'
e
0
≤
θ
<

ρ
S
Q + [1 + s? / log(1−sinc δ)](S−1)



s? Q/ log(1−sinc δ)

 FρS (θ) ≈ 1 − sinc δ
≤ θ < Q + 2Q(S−1)
Q + [1 + s? / log(1−sinc δ)](S−1)
(61)
Q
Q
sinc δ

FρS (θ) = 1 − Q+S−1
≤ θ < S−1
δ

Q
+
2
(S−1)

( Q/θ−S+1 )



 F (θ) = 1
θ≥ Q .
ρS

S−1

Example 14. Let η = 4. Shown in Fig. 8 are Fρ (·) without sectorization and Fρ3 (·) with Q =
20 dB, which is a rather typical front-to-back ratio.
Although, as Fig. 8 visualizes, nonideal sectorization puts a hard ceiling on the SIR, this only
affects the distribution modestly. In exchange, the available bandwidth gets to be reused S times
per cell and thus sectorization is decidedly advantageous.
B. Ergodic Spectral Efficiency
As in Section V, the CDF of the ergodic spectral efficiency FC (·) is determined by mapping
the applicable function C(ρS ) onto FρS (·). For Rayleigh-faded SISO channels in particular, we
can invoke (24) to explicitly express this mapping as

 γ
e 1.4 − 1
.
FC (γ) ≈ FρS
0.82

(62)

26

1

0.8

0.6

0.4

0.2
(62)
Mapping of
0
0

2

4

6
(bits/ s/ Hz)

onto (59)
8

10

Fig. 9: CDF of ergodic spectral efficiency for η = 3.8, S = 3 and Q = 20 dB.

Example 15. Reconsider Examples 4 and 5, but now with S = 3 sectors having front-to-back
ratio Q = 20 dB. Shown in Fig. 9 are (62), the numerical mapping of C(ρ) onto (59) for SISO
and MIMO, and the simulated Cexact .
For any setting for which C(ρS ) is available, we can also compute the spatially averaged
ergodic spectral efficiency as
Z

Q
S−1

C(θ) dFρS (θ).

C̄ =

(63)

0

Example 16. Let η = 3.8, S = 3 and Q = 20 dB. For SISO, C̄ = 1.53 b/s/Hz per sector, which
adds up to 4.59 b/s/Hz per BS. For S = 1, in contrast, a read-out of Fig. 5 gives C̄ = 1.84
b/s/Hz per BS. With the sector nonideality accounted for, the overall spectral efficiency of a
SISO network gets multiplied by 2.5 when cells are split in three sectors.
For 2 × 2 MIMO, in turn, C̄ = 2.95 b/s/Hz per sector adding up to 8.85 b/s/Hz per BS. The
sectorization multiplier is 2.49, almost unchanged from its SISO value.

27

1

0.8

(26)

0.6

= {0, 10, 14}

0.4

(26), SIR scaled by 2.188
Simulation,
hexagonal grid

0.2

0
0

1

2

3
(bits/ s/ Hz)

4

5

6

Fig. 10: CDF of SISO ergodic spectral efficiency with no sectorization (S = 1): PPP-based analytical result vs Monte-Carlo over
a lattice network of hexagonal cells with σdB = 0 dB, 10 dB and 14 dB. The dashed curve corresponds to using Fρ (θ/2.188)
in lieu of Fρ (θ) and proceeding as if the network conformed to a PPP. In all cases, η = 4.

IX. A PPLICATION TO L ATTICE N ETWORKS
Before wrapping up, we close the loop and verify the initial premise whereby a PPP model was
invoked for the BS locations, confirming that such model is representative because of shadowing.
To that end, we compare our PPP-based analytical results with Monte-Carlo simulations for
settings at the extreme end of the point process spectrum, namely regular lattices.
Example 17. Shown in Fig. 10 is FC (·) as given in (26) for SISO, in comparison with the results
for a lattice of 977 hexagonal cells with η = 4 and various values of σdB . The convergence is
conspicuous and, most importantly, the agreement is excellent for typical outdoor values of σdB ,
in the range of 10–12 dB.

Example 18. The comparison of Fig. 10 is repeated, for S = 3 sectors and Q = 20 dB, in Fig.

28

1

0.8

Mapping of

onto (60)

0.6

= {0, 10, 14}

0.4
Simulation,
hexagonal grid

0.2

0
0

1

2

3
(bits/ s/ Hz)

4

5

6

Fig. 11: CDF of SISO ergodic spectral efficiency with S = 3 sectors and Q = 20 dB: PPP-based analytical result vs Monte-Carlo
over a lattice network of hexagonal cells with σdB = 0 dB, 10 dB and 14 dB. In all cases, η = 4.

11, with similar observations in terms of the convergence.
Example 19. Shown in Fig. 12 is C̄ as given in (43), for SISO, in comparison with the results
for a lattice of 977 hexagonal cells with various values of σdB and η.
Further reinforcing the relevance of PPP-based results to lattice networks, it has been argued in
[20], [60], [61] that the SINR distribution of a shadowless lattice network is essentially a shifted
version of its PPP counterpart, i.e., a shifted version of the SINR distribution for asymptotically
strong shadowing. Moreover, this shift does not depend on the path loss exponent, but only on
the type of lattice.
Example 20. For a triangular lattice (hexagonal cells), the shift to apply is 10 log10 2.188 = 3.4
dB [20], [60], [61]. Shown in Fig. 10 is FC (·) recomputed with Fρ (θ/2.188) in lieu of Fρ (θ),
and the agreement with FC (·) for a network of hexagonal cells and no shadowing is highly

29

Simulation,
hexagonal grid

= {0, 10, 14}

as in (43)

Fig. 12: Spatially averaged SISO ergodic spectral efficiency as function of η with no sectorization (S = 1): PPP-based analytical
result vs Monte-Carlo on a lattice network of hexagonal cells with σdB = 0 dB, 10 dB and 14 dB. The error bars around the
simulation results indicate the 99% confidence interval.

satisfactory.
Altogether then, PPP results enable characterizing the distributions of SINR and spectral
efficiency both with (asymptotically strong) shadowing and without, bracketing what a given
network can exhibit over all possible shadowing strengths.
X. S UMMARY
By decoupling small- and large-scale channel features and abstracting the former via local
ergodicity, the stochastic geometry analysis of wireless networks can focus crisply on the largescale properties. Jointly with a Gaussian model for the aggregate interference that recognizes
that the fading of each term therein is unknown, this enables circumventing analytical roadblocks
and deriving expressions that are more open to generalization, readily accommodating aspects

30

such as MIMO or sectorization. The obtained spectral efficiencies lower-bound the exact values
with a degree of accuracy that justifies writing C . Cexact .
Thanks to the PPP-likeness brought about by shadowing, the obtained expressions apply to any
stationary and ergodic network model exhibiting reasonable shadowing strengths. Furthermore,
with a proper shift, these expressions apply to shadowless networks, which are relevant insofar as
they can model planned deployments where the BS locations do depend on the radio propagation.
Thus, PPP analysis can serve to bracket the entire performance range in a given environment.
The approach propounded in this paper is extensible to settings where noise is nonnegligible,
and the accuracy of the results could then only improve further since our interference model is
an exact match for Gaussian noise devoid of fading.
Besides the addition of noise, numerous other extensions are invited, for instance multiuser
MIMO or BS cooperation [62]. When dealing with MIMO, care must be exercised whenever
Nr > Nt and, especially, whenever Nr  Nt ; then, if the fading of dominant interferer(s) can be
tracked, spatial color can be exploited [63], [64]. This can be accounted for by circumscribing
our unfaded interference model to the rest of the interference, separately incorporating the terms
that correspond to interferers whose fading is known.
ACKNOWLEDGMENT
Motivating discussions with Prof. Jeffrey G. Andrews are gratefully acknowledged. The efficient editorial handling by Dr. Bruno Clerckx and the excellent feedback provided by the
reviewers are also gratefully acknowledged.
A PPENDIX A
C OMPUTATION OF Cexact
For given {rk }k∈N0 and H0 , the exact mutual information under the non-Gaussian z in (2) is

 q
−η
I s0 ; P r0 H0 s0 + z
q

q

−η
−η
=h
(64)
P r0 H0 s0 + z − h
P r0 H0 s0 + z s0
q

−η
=h
P r0 H0 s0 + z − h(z)
(65)
where we have introduced the differential entropy h(x) = −E [log fx0 (x0 )] with x0 = [<(x) =(x)]T .
The expectations are evaluated via Monte-Carlo over the random variables {sk }k∈N0 and {Hk }k∈N ,

31

and averaged over multiple realizations of H0 to obtain Cexact in (5). The large-scale distribution
of Cexact , and its average C̄exact , are obtained via multiple realizations of {rk }k∈N0 .
With MIMO, the foregoing computation involves channel matrices and signal vectors, with the
differential entropy of a vector x ∈ Cn being h(x) = −E [log fx00 (x00 )] with x00 = [<(xT ) =(xT )]T .
A PPENDIX B
D ERIVATION OF (27)
With SINR given by (3),


E log2 1 + SINR |{rk }∞
k=0
Z ∞



=
P log2 1 + SINR |{rk }∞
k=0 > ν dν
Z0 ∞
log2 e c
=
F
(x) dx
1 + x SINR|{rk }
0

(66)
(67)

c
where (67) follows from the variable change y = log2 (1 + x) and the CCDF FSINR|{r
(·) can
k}

be computed as
P r0−η |H0 |2
> x {rk }∞
=P
P
k=0
−η
2+N
r
P ∞
|H
|
k
0
k=1 k
"
#
η P∞
−η
2
= E e−x r0 ( k=1 rk |Hk | +N0 /P ) {r }∞

c
FSINR|{r
(x)
k}





k k=0

−x r0η

=e

N0 /P

"∞
#
Y
η −η
2
E
e−x r0 rk |Hk | {rk }∞
k=0

(68)

(69)

k=1
−x r0η N0 /P

=e

∞
Y

1
1 + x (r0 /rk )η
k=1

(70)

where (68) follows from the exponential distribution of |H0 |2 and the expectation is over
∞
{Hk }∞
k=1 . In turn, (70) follows from the fact that {Hk }k=1 are IID.

A PPENDIX C
D ERIVATIONS OF (43), (48)

AND

(50)

Plugging the PDF obtained by differentiating (22) into (42),
s?
Z
Z ∞
?
log(1−sinc δ)
−es /θ s?
δ sinc δ
C̄ ≈
C(θ)
dθ +
C(θ) δ+1 dθ.
2
θ
θ
0
1

(71)

32

From C(·) as given in (14) and (29), the above integrals yield (43) and (48), respectively.
R ∞ −t
These integrations are facilitated by invoking E1 (x) = −Ei (−x), where Ei (x) = −x −et dt, in
conjunction with the identities given in [65] with appropriate variable changes.
Similarly, from C(·) as given in (49), integration by parts in (71) using the identities [66,
2.325.6], [66, 2.325.7], [66, 2.728.1] and [66, 3.194.2] with appropriate variable changes gives
the expression claimed in (50).
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