Performance Analysis of High Throughput Satellite Systems with Optical Feeder Links

Abstract

In this paper, we present the performance analysis of a multibeam high throughput satellite (HTS) system with optical feeder link. We consider the forward link, or downlink, with a fixed gain transparent satellite payload, where the optical feeder link, i.e. the link between gateway and satellite, is impaired by turbulence fading and the RF user link, i.e. the link between satellite and user terminals, is degraded by rain fading. In multibeam HTS systems, frequency re-use in the user link creates interbeam interference which needs to be mitigated to avoid performance degradation. For this purpose, we consider our recently proposed novel zero-forcing (ZF) precoder that does not require channel state information at the gateway and compare its performance with the conventional ZF precoder. We derive statistics of the end-to-end signal-to-noise ratio to approximate performance measures including outage probability, average bit error rate and ergodic capacity. These approximations provide valuable insights into the system analysis. Monte Carlo simulations are used to validate the accuracy of our analysis.

Full text

Performance Analysis of High Throughput Satellite
Systems with Optical Feeder Links
Ishtiaq Ahmad, Khoa D. Nguyen and Nick Letzepis†
Institute for Telecommunications Research, University of South Australia, Adelaide SA 5095
†Defence Science and Technology Group (DSTG), West Avenue, Edinburgh SA 5111
Email: ishtiaq.ahmad@mymail.unisa.edu.au, khoa.nguyen@unisa.edu.au, †nick.letzepis@ieee.org
Abstract—In this paper, we present the performance analysis of
a multibeam high throughput satellite (HTS) system with optical
feeder link. We consider the forward link, or downlink, with a
fixed gain transparent satellite payload, where the optical feeder
link, i.e. the link between gateway and satellite, is impaired by
turbulence fading and the RF user link, i.e. the link between
satellite and user terminals, is degraded by rain fading. In
multibeam HTS systems, frequency re-use in the user link
creates interbeam interference which needs to be mitigated to
avoid performance degradation. For this purpose, we consider
our recently proposed novel zero-forcing (ZF) precoder that
does not require channel state information at the gateway and
compare its performance with the conventional ZF precoder.
We derive statistics of the end-to-end signal-to-noise ratio to
approximate performance measures including outage probability,
average bit error rate and ergodic capacity. These approximations
provide valuable insights into the system analysis. Monte Carlo
simulations are used to validate the accuracy of our analysis.
I. INTRODUCTION
Satellite communication (SatCom) systems are evolving
from traditional TV broadcasting systems to broadband Sat-
Com systems to meet the demands of future remote high-
bandwidth multimedia applications and to integrate with the
terrestrial 5G technology [1]. Towards this end, multibeam
SatCom systems proved to be a most potent solution1[3–
8]. In multibeam SatCom systems, a large number of beams
can be deployed for transmission in the user link, i.e. the
link between the satellite and the user terminals (UTs). To
achieve high spectral efficiency, an aggressive or full frequency
re-use is desirable in the user link, i.e. all beams use the
same frequency band, and hence creating a multiuser multiple-
input single-output (MISO) link. Due to the side lobes in the
satellite’s antenna radiation pattern, this aggressive frequency
re-use suffers from interbeam interference (IBI). The IBI can
be alleviated using precoding techniques in the forward link
(or downlink) and multiuser detection techniques in the reverse
link (or uplink).
In multibeam SatCom systems, the bandwidth requirement
for the feeder link, i.e. the link between the gateway (GW)
and the satellite, increases proportionally with the number of
beams. Considering full frequency re-use factor in the user
link, the bandwidth of the feeder link is 𝑁times the bandwidth
of the user link [5], where 𝑁is the number of beams in the
1Multibeam broadband SatCom systems are coming into practice whose
recent example is Australia’s Sky Muster systems [2].
user link. It is estimated that next generation SatCom systems
will require capacity of one Terabits/s (1000 Gbps) by 2020
[9], and the feeder link, currently operating in radio-frequency
(RF) Ka band, will find it extremely difficult to fulfil this
requirement.
In the literature, various solutions to address feeder link
bandwidth issues are proposed [5, 9–11]. In [9, 10], the
authors proposed to move the feeder link from Ka band
to the Q/V band (40/50 GHz) and W band (70/80 GHz)
where larger bandwidth is available but these bands are more
susceptible to atmospheric attenuations. Another solution is to
use multiple GWs in the feeder link so that the available Ka
band bandwidth can be reused among multiple gateways [5].
This solution requires large overheads, as data and channel
state information (CSI) must be exchanged among multiple
GWs, and suffers from feeder link interference [12]. Free
space optical feeder links represent another solution and has
various potential advantages with respect to the RF links such
as, 100 to 1000 times higher available bandwidth, unlicensed
spectrum allocation, immunity to interference, and highly
secure connection [11].
Various studies related to SatCom systems with optical
feeder links have been conducted in the literature2[13–16].
In [14], the authors presented various transmission and coding
schemes for geostationary earth orbit (GEO) SatCom systems
with optical feeder links. Diversity techniques for optical
feeder links SatCom systems are presented in [15]. Exper-
imental demonstrations to validate the feasibility of optical
feeder link GEO SatCom systems have also been conducted
in [16]. All these studies concluded that optical feeder links
provide promising results and can be the potential candidate
for the feeder link of next generation HTS SatCom systems.
On the other hand, the user link (satellite to UTs) operates
in the RF Ka band and, as discussed above, is a multiuser
MISO link. For the mitigation of IBI in this multiuser MISO
link we use multiuser precoding at the GW. Conventional
approaches of designing precoder for the mitigation of IBI in
the user link require complete CSI at the GW [5–7]. Acquiring
complete CSI at the GW is generally an expensive process
both in terms of latency and signalling overheads, and suffers
from quantisation noise and outdated CSI [8]. Recently, we
proposed a zero-forcing (ZF) precoding technique for fixed
2See [13] for a detailed study on the history of optical feeder links in
SatCom systems.

satellite services (FSS) multibeam SatCom systems that makes
use of users’ physical location and antenna beam radiation
pattern available at the GW and does not require any or less
CSI [17]. We use this ZF precoder here for the mitigation of
IBI.
This work analytically investigates the performance of
multibeam HTS systems with optical feeder link. In partic-
ular, we derive statistics including the cumulative distribution
function (CDF), the probability density function (PDF) and
the moments of the end-to-end signal-to-noise ratio (SNR) of
such systems. We then apply this statistical characterisation of
the SNR to derive approximations of performance measures
including outage probability, average bit error rate (BER) of
𝑀-ary quadrature amplitude modulation (𝑀-QAM) and 𝑀-
ary phase shift keying (𝑀-PSK), and ergodic capacity.
Throughout the paper, we use 𝔼[⋅],tr[⋅],∣∣⋅∣∣,(⋅)T, and
(⋅)†for expectation, trace of a matrix, Euclidian norm, trans-
pose and Hermitian operator, respectively. ln (.)is the natural
logarithm, ∣𝑎∣denotes the absolute value of a complex number
𝑎and diag(a)represents a diagonal matrix with diagonal
elements a=[𝑎1, ..., 𝑎𝑛].I𝑁represents an 𝑁×𝑁identity
matrix and ℕdenotes the set of indices {1,2, ..., 𝑁 }.ℝand
ℂdenote the field of real and complex numbers, respectively.
II. SYSTEM AND CHANNEL MODEL
We consider the forward link of a HTS system employing
multibeam technology, where a single GW simultaneously
serves multiple fixed single-antenna UTs via multiple beams
as shown in Figure 1. The feeder link, which is a single-input
single-output (SISO) link, operates in the optical band while
the user link, which is multiuser MISO link, operates in the
RF Ka band. Similar systems have been considered in the
literature, see e.g. [15, 16].
A. Optical Feeder Link Channel model
Since the user link is a multiuser MISO link, the data
intended for different beams (UTs) are first precoded at the
GW. In order to maintain low-complexity signal processing
on-board the satellite, we consider an analog transparent dense
wavelength division multiplexing (DWDM) system where the
precoded RF streams are modulated onto the optical carriers
using intensity modulation [16]. The individual optical carriers
are then wavelength-division multiplexed into a single fiber
and sent through the telescope towards the satellite. At the
satellite, the incoming optical signal is collected by a tele-
scope and then demultiplexed to obtain the individual DWDM
carriers. The DWDM optical carriers are then converted to
electrical signals representing the corresponding RF streams
using direct detection. The electrical RF signal y1∈ℂ𝑁×1at
the satellite can be modelled as
y1=𝑔x+n1,(1)
where x∈ℂ𝑁×1is the precoded transmit signal vector with
a total power constraint of 𝔼x†x≤𝑃𝑔,n1∈ℂ𝑁×1is
an additive noise vector consisting of circularly symmetric
complex Gaussian entries with zero-mean and variance 𝜎2
1,
i.e. 𝒞𝒩 0,𝜎
2
1, and 𝑔is the optical channel fading gain also
Gateway
Multibeam Satellite
Optical Feeder Link
Multibeam Coverage
User Terminal
024−2
−4×105
×105
0
4
−4
Ka band RF User Link
Optical
Turbulent
Channel
Fig. 1: Geometrical representation of the multibeam SatCom system
with optical feeder link.
known as scintillation fading.3This scintillation fading is typ-
ically modelled by lognormal (LN) or gamma-gamma fading
distributions with Scintillation Index (SI)𝜎2
SI =𝔼[𝑔2]/𝔼[𝑔]2−
1[18]. The SI provides a simple measure of fading severity.
In this work we assume the scintillation fading 𝑔to follow
a lognormal distribution, which is commonly used to model
weak atmospheric turbulence in SatCom optical feeder links
[14].
B. Processing at the Satellite
Since we consider transparent satellite payload, the RF pre-
coded signal y1is amplified with a constant gain matrix F=
𝛼I𝑁, i.e. y𝑠=Fy1, and transmitted towards UTs via multiple
beams. The amplification parameter 𝛼=𝑃𝑠/𝑁
𝑃𝑔𝔼[∣𝑔∣2]+𝜎2
1
is
selected to ensure that the total transmit power constraint at
the satellite 𝔼∣∣Fy1∣∣2≤𝑃𝑠is met [19].
C. RF User Link Channel model
The satellite payload, equipped with an array fed reflector
antenna consisting of 𝑁feeds, generates 𝑁fixed adjacent
beams (single feed per beam) on earth. We assume full fre-
quency re-use so that all beams operate at the same frequency.
The signal received at all the UTs can be stacked into a vector
y2∈ℂ𝑁×1written as
y2=Hy𝑠+n2=𝛼𝑔Hx +𝛼Hn1+n2,(2)
where n2∈ℂ𝑁×1is the noise vector with elements drawn
from 𝒞𝒩 (0,1)4and the matrix H∈ℂ𝑁×𝑁represents the
channel gains between the 𝑁feeds and the 𝑁users. Hcan
be decomposed as [3–7]:
H=DB,(3)
3In this work, for simplicity, we assume that the point error impairments
caused by beam wandering are perfectly mitigated.
4We assume unit variance noise in the user link since, as shown in Section
II-C1, the multibeam gain matrix Bwill be normalised to the receiver noise.

where D∈ℂ𝑁×𝑁is a random fading matrix containing the
rain attenuation coefficients and the phase rotations due to
different propagation paths, and B∈ℝ𝑁×𝑁models the
beam radiation pattern and path losses. These matrices are
interpreted as follows:
1) Multibeam Gain matrix B:The multibeam gain matrix
B∈ℝ𝑁×𝑁incorporates the satellite antenna radiation pattern
and the path loss. The (𝑖, 𝑗)-th entry of Bis given by [3–7],
[B]𝑖,𝑗 =𝑣𝐺𝑖
𝑅𝐺𝑗
𝑇𝑎𝑖,𝑗
4𝜋𝑓𝑟𝑖√𝜅𝐵𝑇𝐵
𝑊
,(4)
where 𝑟𝑖is the distance between the satellite and the 𝑖-th
UT (slant-range), 𝑓is the carrier frequency, 𝑣is the speed
of light, 𝜅𝐵is the Boltzman constant, 𝑇is the receiver noise
temperature, 𝐵𝑊is the carrier bandwidth, 𝐺𝑖
𝑅is the receiver
antenna gain at the 𝑖-th UT, 𝐺𝑗
𝑇is the satellite transmit antenna
gain for the 𝑗-th beam (feed) and 𝑎𝑖,𝑗 is the normalized beam
radiation gain between 𝑗-th on board feed and the 𝑖-th UT.
Hence, the multibeam gain depends on the antenna radiation
pattern and on the UT location (or position). For simplicity,
we assume all UTs have a common receive antenna gain
𝐺𝑖
𝑅=𝐺𝑅, all satellite feeds have a common transmitter
antenna gain 𝐺𝑗
𝑇=𝐺𝑇, and negligible earth curvature so
that 𝑟𝑖=𝑟. With this, (4) simplifies to
[B]𝑖,𝑗 =𝜉√𝑎𝑖,𝑗 ,(5)
where 𝜉=𝑐√𝐺𝑅𝐺𝑇
4𝜋𝑓𝑟√𝜅𝐵𝑇𝐵
𝑊. Let us define an angle 𝜃𝑖,𝑗 between
the 𝑖-th UT and 𝑗-th beam boresight with respect to the
satellite and let 𝜃𝑗,3dB denotes the 3-dB angle for the 𝑗-th
beam. Then for a typical tapered-aperture antenna on-board
the satellite, the beam gain from the 𝑗-th feed to the 𝑖-th UT
is approximated by [4, 7, 17]:
[B]𝑖,𝑗 =𝜉𝐽1(𝑢𝑖,𝑗 )
2𝑢𝑖,𝑗
+36𝐽3(𝑢𝑖,𝑗 )
𝑢3
𝑖,𝑗 ,(6)
where 𝑢𝑖,𝑗 =2.07123 sin(𝜃𝑖,𝑗)
sin(𝜃𝑗,3dB), and 𝐽1(.)and 𝐽3(.)are the
first-kind Bessel function of order 1 and 3 respectively. Notice
that for fixed UTs’ location on earth, the beam gain for all
satellite feed-UT pairs is fixed and hence the matrix Bis
deterministic [3–7, 17].
2) Fading matrix D:A distinctive characteristic of the
multibeam SatCom system is that for any given user the same
fading gain is observed across all feeds on-board the satellite
antenna. It is due to the fact that the distance between the
feeds is small compared to the distance between a user and the
satellite [3–7, 17]. Therefore, the fading matrix D∈ℂ𝑁×𝑁is
a diagonal matrix with diagonal entry 𝑑𝑖,𝑖 ∈ℕrepresenting
the complex fading gain for the 𝑖-th UT.
In Ka and higher frequency bands, rain attenuation causes
the largest degradation in system performance [4–9, 17]. The
rain attenuation in dB, 𝑙dB
𝑖, is typically modelled with a
lognormal distribution as specified in ITU-R Recommendation
P.1853 [20]. Consequently, the rain fading channel gain ampli-
tude in natural units is ∣𝑑𝑖∣=10
−𝑙𝑑𝐵
𝑖
20 and the corresponding
fading channel complex coefficient is 𝑑𝑖=∣𝑑𝑖∣exp (𝑗𝜙𝑖),
where 𝜙𝑖represents a random phase and is uniformly dis-
tributed between 0and 2𝜋.5The rain fading channel gain
amplitude ∣𝑑𝑖∣,𝑖 ∈ℕfollows a double-lognormal or log-
lognormal distribution whose PDF and CDF are given by [17,
Eq. 17] and [17, Eq. 18], respectively.
III. ZERO-FORCING PRECODING
A precoded transmit signal x∈ℂ𝑁×1can be written as
x=Ts,(7)
where T∈ℂ𝑁×𝑁is a precoding matrix and s∈ℂ𝑁×1
represents the UTs data symbols at the GW. The UTs data
symbols are assumed to be independent with unit energy such
that 𝔼ss†=I𝐾and the transmit power constraint becomes
𝔼∣∣x∣∣2=𝔼∣∣Ts∣∣2=tr TT†≤𝑃𝑔.(8)
Using (3) and (7), the received signal vector (2) becomes
y2=𝛼𝑔DBTs +𝛼DBn1+n2.(9)
The ZF precoder aims to completely eliminate the IBI among
the UTs. Conventional approach of designing ZF precoder for
the user link requires complete channel knowledge at the GW
and is given by [22, 23]
Tconv
ZF =𝑐conv
ZF B†D†DBB†D†−1,(10)
where 𝑐conv
ZF is chosen to satisfy the total transmit power
constraint. Note that Tconv
ZF (10) requires the channel matrices
Dand B. Conventionally Dand Bare estimated at the UTs
using pilot sequences [8], and then fed back to the GW. This
acquisition process is expensive both in terms of time and
signalling overheads. Recently, we proposed to design ZF
precoder based solely on deterministic multibeam matrix B,
which can be computed at the GW if the antenna radiation
pattern and the UT locations are known [17]. The design of
ZF precoder based on this deterministic matrix Bis given by
TZF =√𝑐ZFB†BB†−1,(11)
where 𝑐ZF is chosen to satisfy (8), i.e.
𝑐ZF =𝑃𝑔
tr (BB†)−1.(12)
Utilising precoder (11) in (9), the received signal and the
received SNR at the 𝑖-th UT can be written, respectively, as
𝑦ZF,𝑖 =√𝑐ZF 𝛼𝑔𝑑𝑖𝑠𝑖+𝛼𝑑𝑖bT
𝑖n1+𝑛2,𝑖,(13)
𝛾ZF,𝑖 =1
∣∣bT
𝑖∣∣2
𝛾1𝛾2,𝑖
𝛾2,𝑖 +𝐶,(14)
where 𝛾1=𝑃𝑔
𝜎2
1tr[(BB†)−1]∣𝑔∣2=𝛾1∣𝑔∣2and 𝛾2,𝑖 =
𝑃𝑠∣∣bT
𝑖∣∣2
𝑁∣𝑑𝑖∣2=𝛾2,𝑖∣𝑑𝑖∣2represents the 𝑖-th UT SNR in the
feeder link and the user link, respectively, bT
𝑖denotes the
𝑖-th row of matrix Band 𝐶=𝑃𝑠
𝑁𝛼2𝜎2
1
is a constant for
fixed gain 𝛼. Notice from (13) that, to detect 𝑠𝑖the 𝑖-th UT
requires √𝑐ZF𝛼𝑔 𝑑𝑖which is a complex coefficient and can be
estimated using a common pilot sequence for all UTs.
5𝜙𝑖represents the signal phase due to the channel fading and the propaga-
tion paths. Phase shifts due to on-board payload imperfections are omitted here
by assuming ultra-stable payload oscillators. This assumption is commonplace
in related literature [3, 4, 6, 21].

IV. END-TO-END SNR STATI ST I CS
Analysis of performance measures such as outage probabil-
ity, error rate and ergodic capacity require knowledge of the
statistics of the end-to-end SNR 𝛾ZF,𝑖,𝑖 ∈ℕ.
As discussed in Section II-A, the scintillation fading in the
optical feeder link follows lognormal distribution. Therefore
the PDF of the SNR 𝛾1, utilising simple random variable
transformation, is obtained as
𝑓𝛾1(𝛾)= 1
𝛾𝜎opt√2𝜋exp −(ln(𝛾)−𝜇opt)2
2𝜎2
opt ,(15)
where 𝜇opt =2𝜇opt +ln(𝛾1)and 𝜎opt =2𝜎opt.𝜇opt and
𝜎2
opt are the mean and the variance of the corresponding
normal distribution respectively, and are related to the SI via
𝜇opt =−ln 1+𝜎2
SIand 𝜎2
opt =ln
1+𝜎2
SI[18].
Similarly, due to double-lognormal fading in the RF user
link as discussed in Section II-C2, the SNR 𝛾2,𝑖 follows the
following PDF
𝑓𝛾2,𝑖 (𝛾)=−1
√2𝜋𝜎RF,𝑖 𝛾ln 𝛾
ˆ𝛾2,𝑖 
×exp 



−ln −ln 𝛾
ˆ𝛾2,𝑖 −ln(2) −𝜂RF,𝑖 2
2𝜎2
RF,𝑖 



,(16)
where 0<𝛾
2<𝛾2,𝑖 and 𝜂RF,𝑖 =𝜇RF,𝑖+ln (ln (10))−ln (20).
Here 𝜇RF,𝑖 is the lognormal location parameter and 𝜎RF,𝑖 is
the scale parameter that models the severity of the rain fading
attenuation [24].
Theorem 1. The CDF, PDF and moments of the received SNR
at the 𝑖-th UT 𝛾ZF,𝑖 (14) are
𝐹𝛾ZF,𝑖 (𝛾)=+∞
−∞
exp −𝑥2
√𝜋Φln(𝛾)−𝑎𝑖(𝑥)
𝜎opt 𝑑𝑥,
(17)
𝑓𝛾ZF,𝑖 (𝛾)=+∞
−∞
exp −𝑥2
√2𝜋𝛾𝜎opt
exp −(ln(𝛾)−𝑎𝑖(𝑥))2
2𝜎2
opt 𝑑𝑥,
(18)
𝔼𝛾𝑛
ZF,𝑖=+∞
−∞
exp −𝑥2
√𝜋exp 𝑛𝑎𝑖(𝑥)+ 𝑛2𝜎opt
2𝑑𝑥,
(19)
where,
𝑎𝑖(𝑥)
Δ
=𝜇opt −ln ∣∣bT
𝑖∣∣2−ln 1+𝑁𝑃𝑔(𝜎2
SI +1)+𝑁𝜎2
1
𝑃𝑠∣∣bT
𝑖∣∣2𝜎2
1
×exp exp √2𝑥𝜎RF,𝑖 +ln(2)+𝜂RF,𝑖,(20)
and Φ(.)is the CDF of the standard normal distribution.
Proof: The complete proof of this Theorem is omitted
due to space limitations. The main steps involve a change of
random variables, integration by parts and [25].
0 0.5 1 1.5 2 2.5 3 3.5
γ
0
0.2
0.4
0.6
0.8
1
fγZF,i
(γ)
Analytical approximation
Monte Carlo simulation
Fig. 2: Comparison between PDFs obtained via analytical approxi-
mation (22) and Monte Carlo simulation for rain fading attenuation
of 3dB (𝜇RF =0.6,𝜎
RF =1),𝜎2
SI =0.1and 𝑃𝑔=𝑃𝑠=15dB.
Remark: To the best of the authors’ knowledge, the
integrals in (17)-(19) have no solutions in exact closed-form,
however an accurate approximations to these integrals are
provided in the following.
Using Gauss Hermite quadrature integration [26,
Eq. (25.4.46)] , the CDF, PDF and moments of the
received SNR at the 𝑖-th UT 𝛾ZF,𝑖 (14) can be efficiently
approximated by
𝐹𝛾ZF,𝑖 (𝛾)≈1
√𝜋
𝐾

𝑘=1
𝑤𝑘Φln(𝛾)−𝑎𝑖(𝑥𝑘)
𝜎opt ,(21)
𝑓𝛾ZF,𝑖 (𝛾)≈
𝐾

𝑘=1
𝑤𝑘
√2𝜋𝛾𝜎opt
exp −(ln(𝛾)−𝑎𝑖(𝑥𝑘))2
2𝜎2
opt ,
(22)
𝔼𝛾𝑛
ZF,𝑖≈
𝐾

𝑘=1
𝑤𝑘
√𝜋exp 𝑛𝑎𝑖(𝑥𝑘)+ 𝑛2𝜎opt
2,(23)
where 𝑤𝑘and 𝑥𝑘,𝑘 =1,2, ..., 𝐾 are the weights and the
abscissas of the 𝑛-th order Hermite polynomial that are
tabulated in [26, Table 25.10].
Fig. 2 compares the approximate PDF obtained with (22)
for 𝐾=10with the Monte Carlo simulation. Clearly, there
is a good agreement between our derived approximation and
simulation and hence proving the accuracy of our approxima-
tion.
V. A PPLICATIONS TO PERFORMANCE ANALYSIS
This section utilises the statistical characterisation of the
received SNR 𝛾ZF,𝑖 (14), derived in Section IV, to analyse
the following performance measures.
A. Outage Probability
The outage probability of the 𝑖-th beam when communi-
cating at rate ℛ(in nats/s/Hz) is defined as the probability
that the instantaneous mutual information of the 𝑖-th beam,

i.e. ℐ𝑖=ln(1+𝛾ZF,𝑖), falls below ℛ[25]. Mathematically
speaking,
𝑃out,𝑖(ℛ)=Pr[ℐ𝑖<ℛ]=𝐹𝛾ZF,𝑖 (exp {ℛ} − 1) .(24)
Using (17) and (21) in (24) gives the exact and approximate
expressions of the outage probability of the 𝑖-th beam, respec-
tively.
B. Average BER
The average BER of the 𝑖-th beam can be obtained as [25]
𝑃BER,𝑖 =∞
0
𝑃BER,𝑖 (𝛾)𝑓𝛾ZF,𝑖 (𝛾)𝑑𝛾, (25)
where 𝑃BER,𝑖 (𝛾)is the instantaneous BER, which depends on
the modulation scheme. For 𝑀-QAM and 𝑀-PSK modulation
schemes, the instantaneous BER expressions can be written
using [27, Eq. (32)] and [27, Eq. (27)], respectively as
𝑃MQAM
BER,𝑖 (𝛾)≃Θ𝑀
√𝑀
2

𝑙=1
𝑄(𝒜𝑙√𝛾)(26)
𝑃MPSK
BER,𝑖 (𝛾)≃Φ𝑀
max(𝑀/4,1)

𝑙=1
𝑄ℬ𝑙2𝛾(27)
where Θ𝑀=(4/log2(𝑀)) 1−1/√𝑀,𝒜𝑙=
(2𝑙−1) 3/(𝑀−1),Φ𝑀=2/max (log2(𝑀),2),
ℬ𝑙=sin
(2𝑙−1)𝜋
𝑀and 𝑄(.)is the Gaussian Q-function.
An approximation of the BER can be obtained using the
exact PDF of SNR by substituting (18) and (26) or (27)
into (25). Similarly by using (22) and (26) or (27) into (25)
and evaluating their respective integrals give the following
approximations of the average BER for 𝑀-QAM and 𝑀-PSK
modulation schemes,
𝑃MQAM
BER,𝑖 ≈Θ𝑀
𝜋
√𝑀
2

𝑙=1
𝐾

𝑘1=1
𝐾

𝑘2=1
𝑤𝑘1𝑤𝑘2
𝑄𝒜𝑙exp √2𝑥𝑘2𝜎opt +𝑎𝑖(𝑥𝑘1)(28)
𝑃MPSK
BER,𝑖 ≈Φ𝑀
𝜋
max(𝑀/4,1)

𝑙=1
𝐾

𝑘1=1
𝐾

𝑘2=1
𝑤𝑘1𝑤𝑘2
𝑄ℬ𝑙2exp √2𝑥𝑘2𝜎opt +𝑎𝑖(𝑥𝑘1)(29)
where 𝑤𝑘and 𝑥𝑘,𝑘 =1,2, ..., 𝐾 are the weights and the
abscissas of the 𝑛-th order Hermite polynomial [26, Table
25.10].
C. Ergodic Capacity
The ergodic channel capacity of the 𝑖-th beam, in
(nats/s/Hz), is defined as [25]
𝐶𝑖
Δ
=𝔼[ln (1 + 𝛾ZF,𝑖 )] = ∞
0
ln (1 + 𝛾)𝑓𝛾ZF,𝑖 (𝛾)𝑑𝛾. (30)
Substituting (18) into (30) gives an exact ergodic capacity of
the 𝑖-th beam. The integrals in (30) appears to have no closed-
form solution. 𝐶𝑖can be efficiently approximated as
𝐶𝑖≈
𝐾

𝑘1=1
𝐾

𝑘2=1
𝑤𝑘1𝑤𝑘2
𝜋ln 1+exp2𝜎2
opt𝑥𝑘2
+𝑎𝑖(𝑥𝑘1),(31)
where 𝑤𝑘and 𝑥𝑘,𝑘 =1,2, ..., 𝐾 are the weights and the
abscissas of the 𝑛-th order Hermite polynomial [26, Table
25.10]. Next, a lower bound on the ergodic capacity of the
𝑖-th beam is obtained as
𝐶𝑖≥𝐶𝑖,LB
Δ
=1
𝜋∞
−∞
exp −𝑥2𝑎𝑖(𝑥)𝑑𝑥. (32)
Note that (32) is obtained by using the bound ln (1 + 𝛾)≥
ln (𝛾)in (30) and evaluating the integral. The bound is shown,
in Section VI, to be asymptotically tight at high SNR. The
integral in (32) appears to have no closed-form solution, how-
ever it can be efficiently approximated using Gauss Hermite
quadrature integration as
𝐶𝑖,LB
>
≈
𝐾

𝑘=1
𝑤𝑘𝑎𝑖(𝑥𝑘)
√𝜋.(33)
Similarly the ergodic capacity of the 𝑖-th beam can be upper
bounded as 𝐶𝑖≤𝐶𝑖,UB
Δ
=ln(1+𝔼[𝛾ZF,𝑖]), where we used
the well-known Jensen’s inequality for the bound.
Remarks:
∙Notice that the above derived expressions of outage
probability, average BER and ergodic capacity are all
functions of 𝑎𝑖(𝑥)(20) which provides useful insights
into the system analysis. For example increasing the SI,
which increases the optical link turbulence, decreases
𝑎𝑖(𝑥)which consequently increases outage probability
(24) and average BER (28), (29), and decreases the er-
godic capacity (31). Similar observations can be observed
for the rain fading in the RF user link.
∙Increasing the number of spot beams while keeping the
average transmit power at the satellite constant decreases
𝑎𝑖(𝑥)which in turn increases outage probability and
average BER and decreases the ergodic capacity.
∙Furthermore, asymptotic approximations of the outage
probability, average BER and ergodic capacity of the 𝑖-th
beam can also be obtained for the cases when (i) both
𝑃𝑔and 𝑃𝑠are large, i.e. 𝑃𝑔,𝑃
𝑠→∞(ii) 𝑃𝑔is large, i.e.
𝑃𝑔→∞and 𝑃𝑠is kept fixed and (iii) 𝑃𝑠is large, i.e.
𝑃𝑠→∞and 𝑃𝑔is kept fixed.
VI. SIMULATION RESULTS
In this section, simulation results are presented to corrobo-
rate the analytical derivations and approximations. A standard
Monte Carlo methodology is used for generating simulation
results. We consider a cluster of 𝑁=7beams with fixed UTs
locations as depicted in Fig. 1, and generated the multibeam
gain matrix Busing (6). The system parameters used are listed
in Table I [7, 17, 22]. For generating numerical results, we

TABLE I: System Parameters
Parameter Value
Orbit GEO (𝑟𝑖= 35786 Km)
Frequency band Ka-band (𝑓= 20 GHz)
Number of beams 𝑁7
Beam radius 250 Km
Boltzmann’s constant 𝜅𝐵1.38 ×10−23 J/m
Noise bandwidth 𝐵𝑊50 MHz
Satellite antenna max. gain 𝐺𝑖
𝑇52 dBi
UT antenna max. gain 𝐺𝑖
𝑅38.16 dBi
Clear sky receiver temp. 𝑇207𝑜K
UTs location distribution Fixed
3dB angle 𝜃3dB =0.4𝑜
Avg. rain attenuation 3dB(𝜇RF =0.6,𝜎RF =1)
5.5 dB (𝜇RF =1.2,𝜎RF =1.01)
0 102030405060
P [dB]
10-3
10-2
10-1
100
Outage probability, P out
Rain attenuation = 5.5 dB (Analytical)
Rain attenuation = 3 dB (Analytical)
Rain attenuation = 5.5 dB (Conventional ZF precoder)
Rain attenuation = 3 dB (Conventional ZF precoder)
Monte Carlo simulation
Fig. 3: Outage probability of the central beam with 𝜎2
SI =1and
ℛ=2nats/s/Hz.
0 1020304050
P [dB]
10-2
10-1
100
Average Bit Error Rate
16-PSK (Analytical)
16-QAM (Analytical)
16-PSK (Conventional ZF precoder)
16-QAM (Conventional ZF precoder)
Monte Carlo simulation
Fig. 4: Average BER of the central beam with 𝜎2
SI =1and average
rain attenuation of 5.5 dB.
consider the central beam as it will receive the maximum IBI.
We also consider two rain fading scenarios in the RF user link,
one featuring (𝜇RF =0.6,𝜎RF =1) models an average rain
attenuation of 3 dB and the other (𝜇RF =1.2,𝜎RF =1.01)
models an average rain attenuation of 5.5 dB representing
0 10 20 30 40 50 60
0
1
2
3
4
5
6
7
8
9
P [dB]
Ergodic Capacity
SI = 0.5 (Analytical)
SI = 1 (Analytical)
Lower bound (Analytical)
SI = 0.5 (Conventional ZF precoder)
SI = 1 (Conventional ZF precoder)
Monte Carlo simulation
Fig. 5: Ergodic capacity of the central beam for different values of
SI under rain fading with average rain attenuation of 5.5 dB.
heavy rain. All results are plotted across 𝑃=𝑃𝑔=𝑃𝑠where
𝜎2
1=1. Moreover, we also included results obtained with the
conventional ZF precoder (10) for comparison. These results
are plotted using Monte Carlo simulation.
The outage probability of the central beam is plotted in Fig.
3. Analytical results are generated by using (21) in (24) with
𝐾=10and are perfectly matching with the Monte Carlo
simulation, thus proving the accuracy of our derived approx-
imations. Notice from this figure that both the scintillation
fading in the optical link and the rain fading in the RF user
link, as expected, increase the outage probability. Interestingly,
it can be observed that with our proposed ZF precoder the
central beam achieves lower outage probability than with the
conventional precoder. Specifically at 𝑃out =0.1, compared
to the conventional ZF precoder, the central beam with a
proposed precoder achieves a gain of approximately 2.5 dB
and 8.2 dB with an average rain fading attenuation of 3 dB
and 5.5 dB, respectively.
Fig. 4 demonstrates the average BER of the central beam
for 16-PSK and 16-QAM modulations schemes. The analytical
approximations (28) and (29) are plotted for 𝐾=10and are
perfectly matching with the Monte Carlo simulations. Notice
that for both modulation schemes the system with the pro-
posed precoder outperforms the system with the conventional
precoder. For example, at an average BER of 10−116-PSK
and 16-QAM provide a gain of approximately 6.7 dB and 6.5
dB respectively.
Finally, the ergodic capacity of the central beam is illus-
trated in Fig. 5. The analytical approximation is computed
for 𝐾=10and is once again clearly matching with the
Monte Carlo simulation. We also included in this figure a
lower bound on the ergodic capacity (33). Notice that this
bound asymptotically becomes exact at high SNR. Similar
to the outage probability and average BER, the central beam
with the proposed precoder achieves a higher ergodic capacity
than with the conventional ZF precoder and provides a gain of
approximately 5.1 dB at 5 nats/s/Hz for 𝜎2
SI =1and average
rain attenuation of 5.5 dB. It is due to the fact that conventional

ZF precoders rely on channel inversion power allocation which
results in equal SNR at all UTs. This approach suffers a
large capacity penalty in severe fading [28]. In the proposed
ZF precoder, we allocate uniform power across all beams
irrespective of the channel fading gains which outperforms
channel inversion power allocation.
VII. CONCLUSION
In this work, for the first time, we analytically investigated
the performance of multibeam HTS systems where the feeder
link operates in the optical band and the user link operates
in the RF Ka band. Taking into account turbulence fading in
the optical feeder link, rain fading in the RF user link and
fixed gain transparent satellite payload, we derived analytical
approximations for performance measures including outage
probability, average BER and ergodic capacity. These ap-
proximations provide useful insights into the system analysis.
Simulation results are presented to valid the accuracy of our
derived approximations. Moreover, we have also observed
that our proposed ZF precoder outperforms conventional ZF
precoder in terms of all considered performance measures.
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