1

Limitation of SDMA in Ultra-Dense Small Cell
Networks

arXiv:1801.00222v1 [] 31 Dec 2017

Junyu Liu, Min Sheng, Jiandong Li
State Key Laboratory of Integrated Service Networks, Xidian University, Xi’an, Shaanxi, 710071, China
Email: junyuliu@xidian.edu.cn, {msheng, jdli}@mail.xidian.edu.cn

Abstract—Benefitting from multi-user gain brought by multiantenna techniques, space division multiple access (SDMA) is
capable of significantly enhancing spatial throughput (ST) in
wireless networks. Nevertheless, we show in this letter that, even
when SDMA is applied, ST would diminish to be zero in ultradense networks (UDN), where small cell base stations (BSs) are
fully densified. More importantly, we compare the performance
of SDMA, single-user beamforming (SU-BF) (one user is served
in each cell) and full SDMA (the number of served users equals
the number of equipped antennas). Surprisingly, it is shown that
SU-BF achieves the highest ST and critical density, beyond which
ST starts to degrade, in UDN. The results in this work could shed
light on the fundamental limitation of SDMA in UDN.

I. I NTRODUCTION
While network densification is promising to improve network capacity, its limitation has been manifest in recent
research [1]–[3]. Depending on practical system settings, it
is shown that area spectral efficiency would decrease as base
station (BS) density grows in downlink cellular networks [2].
Worsestill, network capacity is shown to diminish to zero
when single-antenna BSs are over deployed [3]. To rejuvenate
the potential of network densification, space division multiple
access (SDMA), which is capable of simultaneously serving
multiple users within one cell over identical spectrum resources, is shown to be of great potential [4], [5]. Especially, it
is shown that single-user beamforming (SU-BF), a special case
of SDMA, could improve network capacity by tens of folds in
ultra-dense networks (UDN), compared to the single-antenna
regime [5]. However, only single user could be served within
one cell using SU-BF. Therefore, it is imperative to study
whether the multi-user gain of SDMA could be harvested to
further improve network capacity in UDN.
In this work, we investigate the performance of SDMA in
ultra-dense downlink small cell networks. As more than one
user is served within one cell using SDMA, spatial resources
could be fully exploited. However, it is unexpectedly shown
that network spatial throughput (ST) would monotonously
decrease with the number of served users within one cell when
BS density is sufficiently large, which radically contradicts
with the sparse deployment cases. Moreover, it is proved that
applying SU-BF could as well maximize the critical density,
beyond which network capacity starts to diminish. Therefore,
the results reveal the fundamental limits of SDMA and confirm
the superiority of SU-BF in UDN1 .
1 Different from [3], which investigates the limitation of network densification, we intend to unveil the limitation of SDMA in UDN in this work.

II. S YSTEM M ODEL
We consider a downlink small cell network, where the
locations of Na -antenna BSs (with constant transmit power
P ) and single-antenna users in a two-dimension plane are
modeled as two independent homogeneous Poisson Point
Processes (PPPs) ΠBS = {BSi } (i ∈ N) of density λ and
ΠU = {Uj } (j ∈ N) of density λU , respectively. If r is the
2-dimension distance of cellular link, the distance√
between the
antennas of BSs and the associated users is d = r2 + ∆h2 ,
where ∆h > 0 denotes the antenna height difference (AHD)
between BSs and users
We further assume that each user connects to the BS
providing the smallest pathloss. Accordingly, the probability
density function (PDF) of r is given by [6]
2

fr (x) =2πλxe−πλx . (x ≥ 0)

(1)

Denote NU (NU ≤ Na ) as the number of active users within
one cell. Meanwhile, a dense user deployment is considered,
i.e., λ ≫ λU , such that each full-buffer BS could select at
most Na users to serve using SDMA.
Channel gain consists of a pathloss component and a smallscale fading component. In particular, given transmission distance d, a generalized multi-slope pathloss model (MSPM) is
adopted to characterize differentiated signal power attenuation
rates within different regions, i.e.,


−1
(2)
;
d
= Kn d−αn , Rn ≤ d < Rn+1
lN {αn }N
n=0

where αn denotes the pathloss exponent, K0 = 1, Kn =
Q
αi −αi−1
n
(n ≥ 1), and 0 = R0 < R1 < · · · <
i=1 Ri
RN = ∞. Note that αi ≤ αj (i < j) and αN −1 > 2 for
practical concerns. Rayleigh fading is used to model smallscale fading for mathematical tractability. The rationality of
Rayleigh fading assumption in UDN has been verified via
experimental results in [7].
SDMA with zero-forcing precoding is applied, where perfect channel state information (CSI) from the BSs to the serving downlink users could be obtained to design the precoders.
According to [4], [8], [9], under Rayleigh fading channels, the
channel power gain caused by small-scaling fading of both
direct link and interfering link follows gamma distribution
when zero-forcing precoding is applied by the multi-antenna
technique. If denoting HU0 ,BS0 as the channel power gain
caused by small-scale fading from BS0 to U0 (typical pair)
and HU0 ,BSi (i 6= 0) as the channel power gain caused by
small scaling fading from BSi (interfering BS) to U0 , then

2

STN (λ) = NU λCPN (λ) log2 (1 + τ ) .

(3)

In (3), τ denotes the decoding threshold and CPN (λ) denotes
the coverage probability (CP) of the typical downlink users,
which is given by

0

10

SSPM
DSPM

−1

¡
¢
ST bits/ s · Hz · m2

HU0 ,BS0 ∼ Γ (Na − NU + 1, 1) and HU0 ,BSi ∼ Γ (NU , 1). It
is worth noting that SDMA degenerates into full SDMA when
NU = Na , while degenerates into SU-BF when NU = 1.
We use network ST for performance evaluation of SDMA
in UDN. In particular, ST is defined by

10

−2

10

21 folds

−3

10

−4

10

Na = 16, 8, 4, 1
1

CPN (λ) = P {SIRU0 > τ } .

IN

UDN

3

10

4

10
10
λ BS/km2

5

6

10

10

(a) Impact of Na .
2

10

SSPM
DSPM
¡
¢
ST bits/ s · Hz · m2

In (4), SIRU0 denotes the signal-to-interference ratio (SIR) at
U0 . Note that the subscript ‘N ’ denotes the number of slopes
in (2).
Notation: Denoting 2 F1 (·, ·, ·, ·) as Gaussian
hypergeomet

2
ric function, we denote ω (x, y, z) = 2 F1 x, − y , 1 − y2 , −z


N −1
in the remaining parts. Besides, lN {αn }n=0 ; d is replaced
by lN (d) for notation simplicity.
III. P ERFORMANCE E VALUATION OF SDMA

10

(4)

2

∆h = 0m

0

10

∆h = 2m
−2

10

∆h = 5m
−4

10

∆h = 10m

In this section, we evaluate the performance of SDMA in
terms of network ST, which depends on the distribution of
SIRU0 according to (3). In particular, SIRU0 is given by

−6

10

2

10

4

10
λ BS/km2

6

10

(b) Impact of AHD ∆h.

where IIC =

(5)
SIRU0 =P HU0 ,BS0 l (d0 ) /IIC ,
P
P HU0 ,BSi l (di ) is the inter-cell interfer-

BSi ∈Π̃BS

ence, di denotes the distance between the antennas of BSi and
U0 , and HU0 ,BSi denotes the power gain caused by smallscale-fading from BSi to U0 . Following SIRU0 in (5), we
present the results on CP and ST in the following corollary.

Corollary 1 (CP and ST in SDMA System). When SDMA is
applied under MSPM, ST in downlink small cell networks is
given by STN (λ) = NU λCPN (λ) log2 (1 + τ ), where
"N −N
#
a
X U (−s)k dk
CPN (λ) =E
LI (s) s= P l τ(d ) . (6)
0
N
k! dsk IC
k=0

In (6), LIIC (s) is the Laplace Transform of IIC evaluated at
s = P lNτ(d0 ) , which is given by
LIIC (s) =e

−2πλ

R∞
d0

x(1−(1+sP lN (x))−1 )dx

.

(7)

Proof. Following [5, Corollary 2], the results can be obtained
with easy manipulation. The detail is omitted due to space
limitation.
Corollary 1 provides a numerical approach to evaluate the
performance of SDMA in UDN. As well, the impact of key
parameters of SDMA, such as the number of antennas and
served users, on system performance could be captured. In
particular, we plot ST as a function of BS density in Fig. 1
under single-slope pathloss model (SSPM), i.e., N = 1 in (2),
and dual-slope pathloss model (DSPM), i.e., N = 2 in (2). It
can be seen from Fig. 1a that network ST could be significantly
improved by SDMA, compared to the single-antenna regime.

Figure 1. ST scaling laws. Set NU = 1 when Na = 1, while set NU = 2
when Na > 1. For system settings, set P = 23dBm and τ = 10dB. Set
α0 = 4 for SSPM, and α0 = 2.5, α1 = 4 and R1 = 10m for DSPM. In
(a), set ∆h = 2m. In (b), set Na =16. Lines and markers denote numerical
and simulation results, respectively.

Especially, the maximal ST could be improved by 21 folds
under DSPM when 16 antennas are equipped by each BS and
2 users are served within one cell. Meanwhile, we evaluate the
impact of AHD on the network ST in Fig. 1b. It is observed
that network ST would be significantly over-estimated without
considering the AHD, i.e., ST even linearly increases with BS
density. As the ∆h = 0m assumption is no longer impractical
when the distance from transmitters and the intended receivers
is small, this confirms the importance of modeling AHD when
evaluating the performance of UDN.
Taking ∆h > 0m into account, however, it is pessimistic
to observe that ST would asymptotically approach zero under
dense deployment even when SDMA is applied. Therefore, we
analytically study ST scaling law next.
Corollary 2 (CP and ST Scaling Laws in SDMA System).
In SDMA system, CP and ST scale with BS density λ as
CPN (λ) ∼ e−κλ and STN (λ) ∼ λe−κλ (κ is a constant),
respectively.
Proof. Given the number of antennas, the experience of single
user would be degraded if more users within one cell share the
antenna resources. Therefore, we have CPF
N (λ) ≤ CPN (λ) ≤
CPSN (λ), where CPF
(λ)
is
obtained
by
setting
NU = Na (full
N
SDMA) and CPSN (λ) is obtained by setting NU = 1 (SU-BF).

3

Hence, CPF (λ)

0

10

d0

(b)



−πλd20



(ω (Na , αN−1 , τ ) − 1)
= Er0 >RN −1 exp



2
2
− ∆h2
(c) exp −πλ ω (Na , αN−1 , τ ) RN−1 + ∆h
=
ω (Na , αN−1 , τ )
=CPL
N (λ) ,

(8)

Exact results
Approximate results

−1

¡
¢
ST bits/ s · Hz · m2



Z ∞ 
 
(a)
x 1 − (1 + sP lN (x))−Na dx
= Er0 exp −2πλ

 d0
Z ∞ 
 
x 1 − (1 + sP lN (x))−Na dx
>Er0 >RN −1 exp −2πλ

10

−2

10

−3

10

overlap

−4

10

Na = 16, 8, 4, 1

where s = P lNτ(d0 ) . In (8), (a) follows due to HU0 ,BS0 ∼
τ
Γ (1, 1) since Na = NU , (b) follows due to s =
−αN −1

2

4

10

P KN −1 d0

and lN (x) = KN −1 x−αN −1 when r0 > RN −1 , and (c)
follows due to the PDF of r0 given in (1). In consequence,
CPLN (λ) ∼ e−κλ holds. Besides, it follows from [5, Theorem
2] that CPS (λ) ∼ e−κλ . Therefore, CPN (λ) ∼ e−κλ .
Following the definition of ST, the proof is complete.

IV. SDMA, F ULL SDMA AND SU-BF: A S PECIAL C ASE
S TUDY
In this section, analytical comparison of SDMA, full SDMA
and SU-BF is made from the ST perspective. Especially, a
special case study is performed under SSPM given by
l1 (d) = d−α0 . d ≥ 0

0

10

SU − BF, NU = 1
SDMA, NU = 8
Full SDMA, NU = 16

Proposition 1. When SDMA is applied under SSPM, ST in
ultra-dense downlink small cell networks can be approximated
as ST†1 (λ) = λCP†1 (λ) log2 (1 + τ ), where
exp (−πλδ (α0 ))
,
ω (NU , α0 , τ S )


ω NU , x, τ S − 1 and τ S =

where δ (x) = ∆h2

¡
¢
ST bits/ s · Hz · m2

10

−4

10

DSPM
−6

10

SSPM
−8

10

1

10

2

3

10

10

4

10
λ BS/km2

5

10

6

10

7

10

(b) Comparison of SDMA, full SDMA and SU-BF.
Figure 2. ST scaling laws. In (a), set NU = 1 when Na = 1 and set
NU = Na /2 when Na > 1. In (b), set Na =16 and other system parameters
are identical to those in Fig. 1.

(9)

Nevertheless, the results on ST in Corollary 1 are much too
complicated even when SSPM is applied. To facilitate the comparison, we first present a simple but effective approximation
on ST in the Proposition 1.

CP†1 (λ) =

10

(a) Examine the approximation accuracy.

−2

Corollary 2 reveals the fundamental limitation of SDMA
in UDN. Meanwhile, it is intuitive to obtain that SU-BF
outperforms SDMA and full SDMA in terms of CP. Therefore,
it is interesting to analytically compare the performance of
these schemes in terms of ST as well, the detail of which is
presented in the following part.

6

10
λ BS/km2

exact and approximate results are small. They even overlap
when Na = 1. Moreover, it is observed that the critical
densities obtained via the approximate and exact results are
almost identical. Therefore, it is valid to use the approximate
results to derive critical density as follows.
Corollary 3. The critical density in SDMA downlink small
cell networks is given by
λ∗ =

(10)
τ
Na −NU +1 .

Proof.
 The key to the approximation is to use GU0 ,BS0 ∼
1
Exp Na −N
to replace HU0 ,BS0 ∼ Γ (Na − NU + 1, 1).
U +1
Accordingly, we have


 
R
−NU
−2πλ d∞ x 1−(1+s† P l1 (x))
dx
†
0
CP1 (λ) =Er0 e
h
i
S
2
=Er0 e−πλd0 [ω(NU ,α0 ,τ )−1] ,
(11)
p
S
where s† = P l1τ(d0 ) and d0 = r02 + ∆h2 . Aided by (11) and
the PDF of r0 given in (1), CP1 (λ) can be obtained.
We next verify the accuracy of the approximation in Proposition 1 using Fig. 2a. It can be seen that the gaps between the

where τ S =

1
,
π∆h2 (ω (NU , α0 , τ S ) − 1)

(12)

τ
Na −NU +1 .

Proof. The proof can be feasibly completed by solving
∂ST†1 (λ)
= 0.
∂λ
Corollary 3 identifies the relationship of key system parameters and critical density. For instance, it is apparent that λ∗
inversely increases with ∆h2 . For this reason, it is suggested
to lower the transmitter and receiver antenna heights and
reduce the AHD such that network densification could be
still effective in enhancing spectrum efficiency even when BS
density is large.
Next, we use the results in Corollary 3 to further evaluate
the performance of SDMA, full SDMA and SU-BF in UDN.
Before that, the results on ω (x, y, z) are given in Lemma 1.

4

Lemma 1. Given N > 1 (an integer), −1 < b < 0, c > 0
and z < 0, 2 F1 (N, b, c, z) is an increasing function of N .
Proof. The proof can be completed using mathematical induction. Assuming ∆1 = 2 F1 (N + 1, b, c, z) − 2 F1 (N, b, c, z) =
bz
c 2 F1 (N + 1, b + 1, c + 1, z) > 0, if the inequality ∆2 =
2 F1 (N + 2, b, c, z) − 2 F1 (N + 1, b, c, z) > 0 holds, the proof
is complete. By definition of hypergemetric function, we have
bz
2 F1 (N + 2, b + 1, c + 1, z)
c
b
=
[2 F1 (N + 1, b + 1, c, z) − 2 F1 (N + 1, b, c, z)] .
N +1
(13)

∆2 =

As N > 0 and b < 0, to prove ∆2 > 0, it is sufficient to show
(N, b + 1, c, z) − 2 F1 (N, b, c, z) < 0. Hence, we have

2 F1

∆3 =2 F1 (b + 1, N, c, z) − 2 F1 (b, N, c, z)
Nz
=
2 F1 (b + 1, N + 1, c + 1, z)
c
Nz
=
(14)
2 F1 (N + 1, b + 1, c + 1, z) .
c
According to the assumption that ∆1 > 0, we have
< 0
2 F1 (N + 1, b + 1, c + 1, z) > 0. Therefore, ∆3
holds.
Aided by Lemma 1, we provide the results on full SDMA
in the following theorem.
Theorem 1. When full SDMA is applied, the critical density
of downlink small cell networks is a decreasing function of
the number of antennas.
Proof. Applying full SDMA, NU = Na and the critical
density in Corollary 3 degenerates into
λ∗ =

1
.
π∆h2 (ω (Na , α0 , τ ) − 1)

(15)

Since ω (Na , α0 , τ ) is an increasing function of Na (Lemma
1), the proof is complete.
Theorem 1 evaluates the performance of full SDMA in
terms of critical density. More specifically, it is observed from
Fig. 2b that ST would begin to decrease at a smaller critical
density when more antennas are equipped. Following Theorem
1, we further compare SDMA, full SDMA and SU-BF in
Theorem 2.
Theorem 2. Given the number of antennas equipped on each
BS, the critical density achieved by SU-BF is greater than
those achieved by SDMA and full SDMA.
Proof. The results can be proved by showing λ∗ in Corollary 3
is a decreasing function of NU or equivalently ω NU , α0 , τ S
is an increasing function of NU . By making an extension
of

[3, Lemma 1], it can be shown that ω NU , α0 , τ S increases
1
with τ S = Na −N
, which monotonously increases with
U +1
NU . Therefore,
aided
by Lemma 1, it can be proved that

ω NU , α0 , τ S increases with NU .

Theorem 2 indicates that simultaneously serving multiple
users would enable ST to decrease at a smaller BS density.
In other words, compared to SU-BF, SDMA potentially degrades spectrum efficiency in UDN. This is inconsistent with

traditional understanding of SDMA in sparse deployment [4],
which shows higher spectrum efficiency could be achieved by
SDMA via serving more users. This can be verified via the
results in Fig. 2b. Specifically, although a higher ST could be
obtained by SDMA when NU is large in sparse scenario, it
would experience a notable decrease at a greater BS density,
compared to SU-BF. The intuition behind this is that the multiuser gain harvested by SDMA is ruined by the overwhelming
interference in UDN.
Besides, it is numerically shown in Fig. 2b that SU-BF
outperforms SDMA and full SDMA in terms of ST in UDN
as well. From this perspective, SU-BF is more favorable
to enhance network capacity in UDN. While the analytical
comparison is made based on SSPM, it could be easily tested
that the comparison results are still valid under MSPM. For
instance, the results under DSPM, i.e., N = 2 in (2), are
verified in Fig. 2b.
V. C ONCLUSION
In this letter, the performance of SDMA has been explored in ultra-dense downlink small cell networks. Specifically, SDMA, albeit incapable of improving network capacity scaling law, is proved to significantly enhancing spatial
throughput, compared to single-antenna regime. Moreover,
it is interestingly shown that network capacity and critical
density could be maximized when single user is served in each
small cell, which contradicts with the intuition. The primary
reason is that demerits of overwhelming inter-cell interference
dominate the benefits of multi-user gain of SDMA in UDN.
Therefore, the outcomes of this work could provide guideline
towards the application and optimization of SDMA in UDN
via reasonably selecting the number of users to serve.
R EFERENCES
[1] X. Zhang and J. G. Andrews, “Downlink cellular network analysis with
multi-slope path loss models,” IEEE Trans. Commun., vol. 63, no. 5, pp.
1881–1894, May. 2015.
[2] M. D. Renzo, W. Lu, and P. Guan, “The intensity matching approach: A
tractable stochastic geometry approximation to system-level analysis of
cellular networks,” IEEE Trans. Wireless Commun., vol. 15, no. 9, pp.
5963–5983, Sep. 2016.
[3] J. Liu, M. Sheng, L. Liu, and J. Li, “Effect of densification on cellular
network performance with bounded pathloss model,” IEEE Commun.
Lett., vol. 21, no. 2, pp. 346–349, Feb. 2017.
[4] H. S. Dhillon, M. Kountouris, and J. G. Andrews, “Downlink MIMO
hetnets: Modeling, ordering results and performance analysis,” IEEE
Trans. Wireless Commun., vol. 12, no. 10, pp. 5208–5222, Oct. 2013.
[5] J. Liu, M. Sheng, and J. Li, “MISO in ultra-dense networks: Balancing
the tradeoff between user and system performance,” submitted to IEEE
Trans. Wireless Commun., 2017. [Online]. Available: https://arxiv.org/
abs/1707.05957
[6] D. Stoyan, W. S. Kendall, J. Mecke, and L. Ruschendorf, Stochastic
geometry and its applications. Wiley Chichester, 1995, vol. 2.
[7] J. Liu, M. Sheng, L. Liu, and J. Li, “Network densification in 5G:
From the short-range communications perspective,” vol. abs/1606.04749,
2017. [Online]. Available: http://arxiv.org/abs/1606.04749
[8] N. Jindal, J. G. Andrews, and S. Weber, “Multi-antenna communication
in ad hoc networks: Achieving MIMO gains with SIMO transmission,”
IEEE Trans. Commun., vol. 59, no. 2, pp. 529–540, Feb. 2011.
[9] R. W. H. Jr, T. Wu, Y. H. Kwon, and A. C. K. Soong, “Multiuser MIMO
in distributed antenna systems with out-of-cell interference,” IEEE Trans.
Signal Process., vol. 59, no. 10, pp. 4885–4899, Oct 2011.

