Novel Expressions for the Outage Probability and Diversity Gains in Fluid Antenna System

Abstract

The flexibility and reconfigurability at the radio frequency (RF) front-end offered by the fluid antenna system (FAS) make this technology promising for providing remarkable diversity gains in networks with small and constrained devices. Toward this direction, this letter compares the outage probability (OP) performance of non-diversity and diversity FAS receivers undergoing spatially correlated Nakagami-m fading channels. Although the system properties of FAS incur in complex analysis, we derive a simple yet accurate closed-form approximation by relying on a novel asymptotic matching method for the OP of a maximum-gain combining-FAS (MGC-FAS). The approximation is performed in two stages, the approximation of the cumulative density function (CDF) of each MGC-FAS branch, and then the approximation of the end-to-end CDF of the MGC-FAS scheme. With these results, closed-form expressions for the OP and the asymptotic OP are derived. Finally, numerical results validate our approximation of the MGC-FAS scheme and demonstrate its accuracy under different diversity FAS scenarios.

Full text

1
Novel Expressions for the Outage Probability and Diversity
Gains in Fluid Antenna System
Jos´
e David Vega-S´
anchez, Member, IEEE, Arianna Estefan´
ıa L´
opez-Ram´
ırez, Luis Urquiza-Aguiar, Member,
IEEE, and Diana Pamela Moya Osorio, Senior Member, IEEE
Abstract—The flexibility and reconfigurability at the radio
frequency (RF) front-end offered by the fluid antenna system
(FAS) make this technology promising for providing remarkable
diversity gains in networks with small and constrained devices.
Toward this direction, this letter compares the outage probability
(OP) performance of non-diversity and diversity FAS receivers
undergoing spatially correlated Nakagami-mfading channels.
Although the system properties of FAS incur in complex analysis,
we derive a simple yet accurate closed-form approximation by
relying on a novel asymptotic matching method for the OP of a
maximum-gain combining-FAS (MGC-FAS). The approximation
is performed in two stages, the approximation of the cumulative
density function (CDF) of each MGC-FAS branch, and then the
approximation of the end-to-end CDF of the MGC-FAS scheme.
With these results, closed-form expressions for the OP and the
asymptotic OP are derived. Finally, numerical results validate
our approximation of the MGC-FAS scheme and demonstrate
its accuracy under different diversity FAS scenarios.
Index Terms—Asymptotic matching, maximum-gain
combining-FAS (MGC-FAS), nakagami-mfading, spatial
correlation, outage probability.
I. INT RODU CT IO N
Multiple-input multiple-output (MIMO) has been a funda-
mental part of the evolution of 5G to realize advancements in
data rates and spectral efficiency. With MIMO, diversity gain
is guaranteed as long as the antennas are separated by at least
half wavelength. However, this may be challenging in very
small devices of some Internet of Things (IoT) applications.
Recently, a technology that uses liquid metals (e.g., gallium-
indium eutectic) to design a software-controllable fluidic struc-
ture that, in its most basic implementation with only one radio
frequency (RF) chain, allows a fluid radiator to switch among
different positions in a small linear space, which has been
referred to as a fluid antenna system (FAS) [1].
Manuscript received MONTH xx, YEAR; revised XXX. The review of
this paper was coordinated by XXXX. The work of Luis Urquiza-Aguiar was
supported by the Escuela Polit´
ecnica Nacional. The work of D. P. M. Osorio
was partially supported by Academy of Finland, project FAITH under Grant
334280, and by ELLIIT funding endowed by the Swedish government.
(Corresponding author: Jos´
e David Vega-S´
anchez)
J. D. Vega-S´
anchez is with the Faculty of Engineering and Applied
Sciences (FICA), Telecommunications Engineering, Universidad
de Las Am´
ericas (UDLA), Quito 170124, Ecuador (E-mail:
jose.vega.sanchez@udla.edu.ec).
A. E. L´
opez-Ram´
ırez, and L. Urquiza-Aguiar are with the Departamento de
Electr´
onica, Telecomunicaciones y Redes de Informaci´
on, Escuela Polit´
ecnica
Nacional (EPN), Ladr´
on de Guevara E11-253, Quito, 170525, Ecuador. (e-
mail: cecilia.paredes@epn.edu.ec; luis.urquiza@epn.edu.ec).
D. P. Moya Osorio is with the Communication Systems Division, De-
partment of Electrical Engineering, Link¨
oping University, Link ¨
oping 581 83,
Sweden (e-mail: diana.moya.osorio@liu.se). She was also with the Centre for
Wireless Communications, University of Oulu, Oulu 90014, Finland (e-mail:
diana.moyaosorio@oulu.fi).
The performance of FAS has been investigated in a number
of works. For instance, in [1], Wong et al. introduced the
concept of a single-antenna FAS over correlated Rayleigh
channels inspired by the advancement in mechanically flexible
antennas. Afterward, in [2], Mukherjee et al. proposed a
framework for the evaluation of the second-order statistic of
the FAS by considering time-varying channels. In [3], Wong
et al. revealed how the ergodic capacity scales with the system
parameters of the FAS. In [4], Tlebaldiyeva et al. derived a
single-integral form of the outage probability (OP) of a single-
antenna FAS over correlated Nakagami-mfading channels.
A concept of fluid antenna multiple access (FAMA) was
proposed in [5], which takes advantage of the deep fades
suffered by the interference to attain a good channel condition
without complex signal processing. In [6], Skouroumounis et
al. presented a framework based on stochastic geometry for
evaluation the performance of large-scale FAS-aided cellular
networks. In [7], New et al. investigated the limit of FAS
performance and the diversity gain. In [8], Tlebaldiyeva et
al. compared non-diversity and diversity FAS receivers over
α-µfading channels.Specifically, the diversity FAS scheme
considers enabling multiple ports of a fluid antenna and using
a combining technique with multi-port signals to enhance FAS
performance further. Therein, a maximum-gain combining-
FAS (MGC-FAS) scheme was investigated via Monte Carlo
simulations due to the mathematical complexity for the un-
derlying MGC-FAS.
Motivated by the potential of the FAS schemes to further
enhance the capacity of future networks, with a great potential
for IoT scenarios, we approximate the OP and asymptotic OP
for the MGC-FAS scheme in a closed-form fashion, which
is useful for evaluations of this scheme. For this purpose,
we first approximate the cumulative density function (CDF)
of each MGC-FAS branch, and then, the CDF of the MGC-
FAS over correlated Nakagami-mfading is derived. In both
stages, the fitting parameters are estimated by employing the
asymptotic matching method, proposed in [9] that render a
simple yet accurate approximation. To the best of the author’s
current knowledge, no prior work has provided a closed-
form expression for the OP of the MGC-FAS scheme in the
literature.
II. SY ST EM AND CHANNEL MOD EL S
Consider a point-to-point FAS where the transmitter is
equipped with a traditional antenna and the receiver with a
fluid one with/without a diversity scheme, as described below.

2
Transmitter
ports
ports
1
1
L
L
W×λ
a)M= 1 order FAS,Non −diversity
b)M−order MGC −FAS
W
M×λ
gFAS = max (|g1|,··· ,|gL|)
gFAS
1= max g(L/M)(j−1)+1,···,g(L/M)j
gFAS
M= max g(L/M)(j−1)+1,···,g(L/M)j
gMGC
FAS =PM
j=1 gFAS
j
Fig. 1. System model for FAS-enabled communication with a)MGC-FAS
scheme and b)non-diversity FAS configuration.
A. Non-Diversity FAS Receiver
Here, the FAS-receiver is built of a fluid antenna that can
move freely along L-ports equally distributed along a linear
dimension of length W λ, with Wbeing the antenna size and λ
the wavelength of the carrier frequency, as illustrated in Fig.
1a. We assume that the FAS can always switch to the best
port for reaching the maximum received signal-to-noise ratio
(SNR). Thus, the channel gain for the FAS is given by
gFAS = max (|g1|,··· ,|gL|),(1)
where gi=|hi|2for i∈ {1,··· , L}denotes the channel gain
of each port in the FAS, with hibeing modeled as correlated
Nakagami-mfading because the antennas are located very
close to each other in the linear space. The received SNR
for non-diversity FAS receiver can be formulated as
γ=P|gFAS|
N0
=γ|gFAS|,(2)
where γ=P
N0is the average transmit SNR, with Pbeing
the transmit power and N0the additive white Gaussian noise
(AWGN) power.
B. Diversity FAS Receiver
The entire FAS with W×λis split into Msub-FAS branches
with a size of W×λ/M, so the MGC-FAS comprises L/M
ports per FAS branch, as shown in Fig. 1b. In the MGC-FAS
scheme, the end-to-end channel is the sum of the strongest
channels of each FAS tube as
gMGC
FAS =
M
X
j=1
gFAS
j,(3)
where gFAS
j= max g(L/M)(j−1)+1,··· ,g(L/M)j, and
gk=|hk|2for k∈ {(L/M)(j−1) + 1,··· ,(L/M)j}, with
hksubject to correlated Nakagami-mfading. The received
SNR for MGC-FAS receiver can be expressed as
γMGC =PgMGC
FAS 
N0
=γgMGC
FAS .(4)
In the following sections an approximate statistical model
for gMGC
FAS will be obtained, then the OP distribution can be
obtained.
III. PER FO RM AN CE ANALYSI S
The OP is considered to assess the performance of the FAS,
which is defined as the probability that the received SNR is
less than a threshold rate γth.
A. Exact OP Distributions
Diversity case: From [4, Eq. (10)] and applying the re-
lationship FgFAS
j(xj) = Pout(xjγ)for j∈ {1,··· , M }, the
CDF of the end-to-end channel gain in each sub-FAS branch
of (3) is given by
FgFAS
j(xj) = 2mm
Γ(m)Ω2m
1Z√xj
0
r2m
1exp −mr2
1
Ω2
1
×
L/M
Y
k=2 1−Qmr2mµ2
kr2
1
Ω2
1(1−µ2
k),r2mxj
Ω2
k(1−µ2
k)dr1,
(5)
where Γ(·)denotes the Gamma function, Qm(·,·)is the m-
order Marcum Q-function, mis the fading parameter, and Ω2
k
indicates the average channel power. Motivated by [10], we
assume that the spatial correlation coefficient, denoted by, µi,
is given by
µ2=
2
L(L−1)
L−1
X
i=1
(L−i)J02πiW
L−1,for µ=µi∀i,
(6)
where all the ports don’t have a reference port or any port is a
reference to any other port. Also, J0(·)is the zero-order Bessel
function of the first kind. Then, the probability density function
(PDF) of (5), i.e., fgFAS
j(xj)can be obtained by computing
the respective derivative. To obtain the CDF of (3), we use
Brennan’s approach1, which argued that when all the sub-
FAS branches involved in the CDF sum of the MGC-FAS
scheme are non-negative random variables (RVs) (like our case
of fading envelopes), the traditional convolution to computed
such a CDF can be reformulated within its limits of integration
employing a geometric approach as an M-fold integral in
terms of the joint PDF of the correlated branches. Specifically,
the distribution of the sum of MRVs is the integral of the joint
correlated density function over the M-dimensional volume
bounded by the hyper-plane x1+x2+···+xM=xand the
coordinate hyper-planes. Based on this, the end-to-end CDF
of the MGC-FAS in (3) is expressed as
FgMGC
FAS (x) = Zx
0Zx−xM
0
... Zx−PM
i=3 xi
0Zx−PM
i=2 xi
0
×fgFAS
j,...,gFAS
M(x1, ..., xM)dx1dx2...dxM−1dxM,
(7)
where fgFAS
j,...,gFAS
M(x1, ..., xM)is the joint PDF of the corre-
lated branches. Finally, the OP for the MGC-FAS is computed
as PMGC
out (γth) = FgMGC
FAS γth
γ.
Non-diversity case: From (5), the OP for non-diversity FAS
over correlated Nakagami-mRVs is given as Pout(γth) =
FgFAS
jγth
γby setting M= 1.
B. Proposed OP Approximation
It is noteworthy that the multi-fold integral in (7) is quite
intricate, thus the derivation of a closed-form solution appears
to be unfeasible. To overcome such limitation, an accurate
approximation is proposed for the PMGC
out (γth), which is
1Interested readers can refer to [11, Secs. IV-V] for detailed information
on Brenann’s approach.

3
obtained via the asymptotic matching method introduced in
[9], as stated in the following proposition.
Proposition 1. The OP expression of FAS undergoing corre-
lated Nakagami-mRVs can be approximated by
PMGC
out ≈Υ(αMGC,γth
βMGCγ)
Γ(αMGC),(8)
where is Υ(·,·), is the lower incomplete gamma function [12,
Eq. (6.5.2)] and
αMGC =L
M
M
X
j=1
mj, βMGC =

1
QM
j=1 1
(βFAS
j)αFAS
j



1/αMGC
,
(9a)
αFAS
j=L
Mmj, βFAS
j= 1
Γ(αFAS
j)a0,j αFAS
j!1/αFAS
j
,
(9b)
a0,j =mmj−1
j
Γ(mj)Ω2mj
1,j (mj!) L
M−1
L
M
Y
k=2 
mj
Ω2
k,j 1−µ2
k,j 

mj
.
(9c)
Proof. See Appendix A.
Remark 1. Notice that the OP expression is a novel and
simple approximation that does not need to solve any involved
integrals regarding the joint distribution of correlated fading
channels of the MGC-FAS scheme.
In order to attain more insights into the influence of sys-
tem parameters for the MGC-FAS performance, an asymp-
totic closed-form expression for the OP is derived. To this
end, the asymptotic OP is developed in the form OP∞≃
Gcγ−Gd[13], where Gcand Gdis the array gain and the
diversity order, respectively. The asymptotic OP is stated in
the following Proposition.
Proposition 2. The asymptotic OP expression for the proposed
MGC-FAS over correlated Nakagami-mRVs is given by
PMGC
out (γth)≃(γth
βMGCγ)αMGC
αMGCΓ(αMGC ),(10)
Proof. By using Υ (a, x)≃xa/a as x→0into (8), (10) is
obtained straightforwardly.
Remark 2. From (10) and (9a), it is clear that the diversity
order reduces to Gd=Lm when all the sub-FAS tubes
experience the same fading, i.e., m=mi,∀i. Moreover, Gd
is directly influenced by the number of ports and the fading
severity.
IV. NUM ERICA L RE SU LTS A ND D IS CUSSI ON S
In this section, the impact of the system model parameters
(e.g., the number of Msub-FAS branches, the size of the
antenna W, and the severity of fading) on the OP performance
is investigated, as well as the accuracy of the proposed
approximations through illustrative examples. Unless stated
otherwise, Ωk= 1,∀kis considered for all plots, and the
spatial correlation model is computed with the help of (6).
-10 -5 0 5 10 15 20 25
10-8
10-6
10-4
10-2
100
Outage Probability
MGC-FAS, M = 2
FAS (non - diversity)
Asymptotic MGC-FAS, M = 2
Fig. 2. OP vs. γ, for different numbers of ports by assuming W= 2,m= 1,
and γth = 5 dB. Markers denote the proposed approximation given in (8),
whereas the solid and dotted lines represent the analytical and the asymptotic
solutions computed via (7) and (10), respectively.
For the sake of comparison, i)the traditional maximal ratio
combining (MRC) technique with uncorrelated antennas and
ii)the non-diversity FAS receiver, are included as a baseline
in the OP analysis.
In Fig 2, we show the OP versus the γfor W= 2,
m= 1,γth = 5 dB and by varying the number of ports
L={10,200}in the non-diversity FAS receiver. Hence, when
setting M= 2-order MGC-FAS, it means that there are two
sub-FAS tubes with L
M={5,100}ports each. In this figure,
the accuracy of the proposed approach in (8) to approximate
the exact solution computed via (7) is evaluated. Note that
all approximate curves are tight to the analytical solutions
for the entire average SNR range. It is worth noting that
as M(i.e., the FAS tubes) increases, the exact formulation
in (7) becomes computationally hard to compute, prone to
convergence, or even unworkable. Hence, our simple and
accurate approximation, with negligible computational cost,
proves useful for the performance analysis for diversity FAS
schemes. Considering the asymptotic curves, note that the OP
decline is steeper (i.e., good OP performance) as the number of
ports in the sub-FAS tubes or the severity fading mincreases.
Contrariwise, the OP is affected when the number of sub-FAS
ports or the mparameter decreases, so the OP slope is is less
pronounced. These results are in coherence with the insights
examined in Remark 2. On the other hand, it is observed
that the asymptotic OP of the MGC-FAS quickly matches the
diversity order of the exact solution for L
M= 5. Conversely,
for the scenario with L
M= 5, the asymptotic OP fits the correct
asymptotic behavior for relatively lower operational OP values.
Henceforth, the approximation curves are represented in
all plots with solid lines for visibility purposes. In Fig. 3,
the OP is depicted as a function of the number of ports of
the FAS scheme. For instance, the fixed value L= 100 on
the x-axis for a non-diversity FAS scheme corresponds to
L
M= 50 in the 2-order MGC-FAS technique. The remaining
parameters are set to: γ= 1 dB, m= 1, and γth = 2 dB.
In these results, the influence of changing the antenna size W

4
101102
10-8
10-6
10-4
10-2
100
Outage Probability
Fig. 3. OP vs. number of ports by varying Wfor γ= 1 dB, m= 1, and
γth = 2 dB. Markers denote Monte Carlo simulations, whereas the solid
lines denote the proposed approximations given in (8).
101102
10-8
10-6
10-4
10-2
100
Outage Probability
Fig. 4. OP vs. number of ports with M={2,4}-order MGC-FAS for
W= 3,m= 1,γ= 1 dB, and γth = 2 dB. Markers denote Monte Carlo
simulations, whereas the solid lines denote the proposed approximations given
in (8).
on the OP behavior for MGC-FAS receivers is investigated.
Note that large Wvalues (i.e., more space in the MGC-FAS)
favor the OP performance compared to the non-diversity FAS
scheme. In particular, obtaining a higher performance gain
from MGC-FAS over non-diversity FAS highly depends on
the size coefficient Wfor a fixed number of ports. Moreover,
non-diversity/diversity FAS schemes beat the MRC method.
For instance, the OP for the 9-antenna MRC is exceeded when
the FAS is deployed by assuming W= 5,L= 91, and
L
M= 24 ports for non-diversity FAS and MGC-FAS schemes,
respectively.
In Fig. 4, the OP is displayed as a function of the number
of ports, as explained in Fig. 3. Herein, the achievable OP is
examined by comparing M={2,4}-order MGC-FAS with
the non-diversity FAS receiver for W= 3,m= 1,γ= 1,
and γth = 2 dB. All plots show that increasing the sub-FAS
tubes (i.e., M) greatly benefits the performance of the OP
compared to the non-diversity FAS receiver. In fact, this gain
1 2 3 4 5 6 7 8 9 10
10-8
10-6
10-4
10-2
100
Outage Probability
Fig. 5. OP vs. the number of sub-FAS branches Mby varying the antenna
size Wfor L= 200,γ= 0 dB, and m= 1. Markers denote Monte Carlo
simulations, whereas the solid lines denote the proposed approximations given
in (8).
gap between M-order MGC-FAS and the FAS could be further
boosted by increasing the size of the antenna W, as explained
in Fig. 3. Furthermore, 9-antenna MRC is surpassed when the
FAS is assumed with L
M= 20,L
M= 34 and L= 132 ports
for 4-order MGC-FAS, 2-order MGC-FAS, and non-diversity
FAS schemes, respectively. This fact confirms the importance
of using diversity in the FAS receiver.
Fig. 5 illustrates the OP versus the number of sub-FAS
branches2, i.e., Mby varying the antenna size Wfor L= 200,
γ= 0 dB, and m= 1. Here, our primary goal is to provide
a general thought on determining the more suitable number
of branches in the FAS diversity scheme to further improve
OP’s performance. Overall, it can be noticed that increasing
the number of sub-FAS branches Mleads to a remarkable
improvement in the OP’s performance. However, this OP’s
behavior is up to a break-point where increasing Minstead
of helping harms the OP. This is because large numbers of
sub-FAS branches (M↑)distributed into a small antenna size
(W↓)lead to highly correlated ports, which causes the gain of
the combining selection technique to be negligible. In addition,
it can be observed that the appearance of such a break-point
occurs at very low OP values when dealing with large W
values (e.g., W= 6,7). Finally, MRC configurations are
beaten more quickly using higher-order MGC-FAS schemes
as long as the break-point is not exceeded.
V. CONCLUSIONS
In this letter, we examined the OP performance of a point-
to-point FAS by assuming non-diversity and diversity FAS
receivers undergoing correlated Nakagami-mfading channels.
Specifically, a novel asymptotic matching method is employed
to approximate the CDF of the MGC-FAS receiver in an
analytically tractable way without incurring multi/single-fold
integrals. With this by-product, a simple closed-form expres-
sion of the OP for the MGC-FAS scheme was derived. Fur-
2It is worth mentioning that for both the Monte Carlo simulations and the
analytical approximations, ⌊L/M⌋is assumed, where ⌊·⌋ is the floor function.

5
thermore, useful insights were provided concerning how the
antenna size Winfluences the OP performance of the MGC-
FAS. Specifically, the MGC-FAS scheme provided remarkable
gains in terms of the OP over the non-diversity FAS when the
Wvalues are large enough.
APP EN DI X A
PROO F OF PROP OS IT ION 1
In the first stage, a suitable approximation for the CDF
of each sub-FAS branch given in (5) is obtained by using
a Gamma distribution. Toward that [14, Eq. (3)] is replaced
into (5), which is re-expressed as
FgFAS
j(xj)≈2mjmj
Γ(mj)Ω2mj
1,j Z√xj
0
r2mj−1
1exp −mjr2
1
Ω2
1,j 
|{z }
I1
×
L
M
Y
k=2


 mjxj
Ω2
k,j (1−µ2
k,j )!mj
exp −
mjµ2
k,j r2
1
Ω2
1,j (1−µ2
k,j )!
mj!

dr1
|{z }
I1
.
(11)
With the aid of [15, Eq. (3.381.1)], I1can be evaluated in
closed-fashion in terms of the incomplete Gamma function,
i.e., Υ(·,·). Then, by applying, Υ (a, x)≃xa/a as x→0,
the asymptotic behavior of the CDF of each sub-FAS branch
in the form FgFAS
j(xj)≃a0xb0
j, can be formulated as
FgFAS
j(xj)≃mmj−1
jmj!1−
L
M
Γ(mj)(Ω1,j )2mj
L
M
Y
k=2 mj
Ω2
k,j (1−µ2
k,j )mj
|{z }
a0
x
b0
z}|{
mjL
M
j.
(12)
To find the shape parameters of the Gamma distribution to
approximate (5), the asymptotic Gamma CDF3of each sub-
FAS branch is used to obtain the following expression
e
FgFAS
j(xj)≃1
βFAS
j
αFAS
jαFAS
jΓ(αFAS
j)
|{z }
ea0
x
e
b0
z}|{
αFAS
j
j.(13)
Then, by applying the asymptotic matching [16], i.e., a0=ea0
and b0=e
b0, the shape parameters αFAS
jand βFAS
jof the
Gamma distribution to approximate (5) can be expressed
as (9b). In the second stage, we approximate the CDF of
the MGC-FAS in (7) by using again the Gamma distribution
via the asymptotic matching technique. For this purpose, the
approximate PDF and CDF of the MGC-FAS can be expressed
as
e
fgMGC
FAS (x) = xe
d0
z}| {
αMGC −1
Γ (αMGC)βMGC αMGC
|{z }
ec0
exp −x
βMGC (14a)
e
FgMGC
FAS (x) =Υ(αMGC ,x
βMGC )
Γ(αMGC),(14b)
3To asymptotically approximate FgFAS
j(xj)≈
Υ αFAS
j,xj
βFAS
j!
Γ(αFAS
j), the
relationship Υ (a, x)≃xa/a as x→0, is employed.
where ec0and e
d0are the linear and the angular coefficients that
capture the asymptotic behavior of the approximate distribu-
tion, i.e., e
fgMGC
FAS (x). Now, we are interested in the asymptotic
behavior of the PDF of the sum fgMGC
FAS (x)given in (3).
Hence, by appropriately substituting the shape parameters of
the summands, i.e., αFAS
jand βFAS
jinto [16, Eq. (4)] and
after some manipulations, the linear and angular coefficients
that govern the asymptote of the sum, fgMGC
FAS (x)≃c0xd0, are
given by
c0=QM
j=1 1
Γ(αFAS
j)βFAS
j
αFAS
j
ΓPM
j=1 αFAS
j, d0=−1 +
M
X
j=1
αFAS
j.
(15)
Next, by matching c0=ec0and d0=e
d0, the fitting parameters
of (14) are found straightforwardly. Finally, (8) is obtained
with the help of (14b) by setting PMGC
out (γth)≈e
FgMGC
FAS γth
γ.
This completes the proof.
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