# Optimal Photodetector Size for High-Speed Free-Space Optics Receivers

Muhammad Salman Bashir,  _Senior Member, IEEE_ , Qasim Zeeshan Ahmed,
_Member, IEEE_ , and Mohamed-Slim Alouini,  _Fellow, IEEE_ Muhammad Salman
Bashir and Qasim Zeeshan Ahmed are with the School of Computing and
Engineering, University of Huddersfield, HD1 3DH, England, United Kingdom.
Mohamed-Slim Alouini is with Computer, Electrical and Mathematical Sciences
and Engineering (CEMSE) Division, King Abdullah University of Science and
Technology (KAUST), Thuwal, 23955-6900, Saudi Arabia.email:
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

The selection of an optimal photodetector area is closely linked to the
attainment of higher data rates in optical wireless communication receivers.
If the photodetector area is too large, the channel capacity degrades due to
lower modulation bandwidth of the detector. A smaller photodetector maximizes
the bandwidth, but minimizes the captured signal power and the subsequent
signal-to-noise ratio. Therein lies an opportunity in this trade-off to
maximize the channel rate by choosing the optimal photodetector area. In this
study, we have optimized the photodetector area in order to maximize the
channel capacity of a free-space optical link for a diverse set of
communication scenarios. We believe that the study in this paper in
general—and the closed-form solutions derived in this study in particular—will
be helpful to maximize achievable data rates of a wide gamut of optical
wireless communication systems: from long range deep space optical links to
short range indoor visible light communication systems.

###### Index Terms:

Channel capacity, free-space optics, modulation bandwidth, optical wireless
communications, photodetector area.

##  I Introduction

Due to the availability of large unregulated spectrum in the optical domain of
electromagnetic waves, free-space optics (FSO)—also commonly known as optical
wireless or laser communications—is an important candidate for supporting high
data-rates in the sixth generation (6G) and beyond wireless networks [1]. FSO
has already been a major player in non-terrestrial networks such as high data-
rate inter-satellite communications in low-Earth orbit, medium-Earth orbit,
geosynchronous-equatorial orbit, and high-Earth orbit. FSO has also found
extensive deployment in integrated space-to-ground and ground-to-space
networks [2]. The National Aeronautics and Space Administration (NASA)
successfully executed downlink data rates up to 622 Mbps for a Moon-to-Earth
link in the Aerospace Corporation’s Optical Communication and Sensor
Demonstration (OCSD) in 2017 [3]. Additionally, NASA’s TeraByte InfraRed
Delivery (TBIRD) system promises data rates of more than 200 Gbps for a
CubeSat-to-ground link using FSO communications [4, 3]. Due to narrow
beamwidths associated with optical signals, FSO is used exclusively for deep
space communications for link lengths that may stretch over a few hundred
thousand kilometers [2]. Recently, scientists at ETH Zürich were able to
achieve 1 Tbps speed with a single wavelength free-space laser link that
spanned $53$ kilometers from the Jungfraujoch to Bern in Switzerland [5].

###  I-A Trade-off Between Detector Bandwidth and SNR

Laser photodiodes are the basic detection elements in a modern optical
receiver that are used not only for detection of optical symbols but also
serve as position or angle-of-arrival sensors in a quad detector array
configuration [6, 7]. Photodiodes are classified as either P-N, P-I-N or
avalanche photodiodes [8]. As the name implies, a P-N photodiode is made up of
a P-N junction—a boundary between two types of semiconductor materials, the
P-type semiconductor and N-type semiconductor. A P-I-N diode consists of a
undoped intrinsic semiconductor region between the P-type and the N-type
semiconductor materials. P-I-N photodiodes have faster switching capability
compared to their P-N counterparts [9]. Avalanche photodiodes (APD) are either
based on Silicon, Germanium or InGaAs semiconducting materials. A
distinguishing feature of APD’s is that they are operated with a high reverse
voltage that allows them to detect very low levels of signal energy. This
feature endows them with high sensitivity and high signal-to-noise ratio
characteristics compared to P-N and P-I-N photodiodes [10, 11].

The existence of P-N junction in a photodiode leads to capacitive $C$, and
resistive $R$ effects that can limit the switching rate—or the response
speed—of a photodiode. This phenomenon is captured by $RC$ time constant of a
photodiode: a larger $RC$ time constant leads to a reduced response time or
modulation frequency (or cut-off frequency) of a photodiode and vice versa
[12]. In order to be able to transmit high-data rates in optical wireless
channels, a low $RC$ time constant or a high modulation frequency is highly
desirable. Since the capacitance of a diode is directly related to its active
area, photodiodes with a smaller area lead to a higher modulation frequency or
bandwidth. On the other hand, if the active area of the photodiode is too
small, the signal energy captured by the photodiode diminishes. This leads to
poor signal-to-noise ratio at the receiver photodiode and a poor signal
detection performance. Thus, a trade-off exists between diode bandwidth and
signal-to-noise ratio, and we ought to choose the optimal diode size or area
in order to maximize the communication performance of the optical receiver.

###  I-B Motivation of Current Study

The channel capacity is a useful metric that captures both the bandwidth as
well as the signal-to-noise ratio of a communication channel. This metric was
first employed in [13] to illustrate that the throughput (or capacity) of an
pulse amplitude modulation optical wireless channel is maximized at a
particular photodetector area. In order to demonstrate this concept, the
channel capacity of a coherent Gaussian optical wireless channel—denoted by
$\mathcal{C}$—is furnished by the well-known _Shannon-Hartley_ theorem [14]:

$\displaystyle\mathcal{C}$ $\displaystyle=W\mathcal{C}_{0},$
$\displaystyle=W\log_{2}\left(1+\frac{\bar{P}}{N_{0}W}\right),$ (1)

where $W$ is the bandwidth in Hz, $\mathcal{C}_{0}$ is channel capacity in
bits per channel use, $\bar{P}$ is the average received signal power in Watts
and $N_{0}$ is the two-sided power spectral density in Watts/Hz. Here, the
quantity $\frac{\bar{P}}{N_{0}W}$ is the signal-to-noise ratio. In (1), we
observe that the Gaussian channel capacity (measured in bits per second) is
the product of two terms—the bandwidth $W$ and the channel capacity in terms
of bits per channel use, $\mathcal{C}_{0}$. We also observe that the signal-
to-noise ratio in $\mathcal{C}_{0}$ is itself inversely proportional to the
bandwidth $W$. The bandwidth $W$ is inversely proportional to the area $A$ of
the photodiode, whereas signal-to-noise ratio is directly proportional to the
cube of photodiode active area: $A^{3}$. Thus, for a very small $A$, the
available modulation bandwidth will be quite large, whereas the signal-to-
noise ratio—and, therefore, the capacity $\mathcal{C}_{0}$—will diminish. On
the other hand, a large photodiode area $A$ leads to high capacity per channel
use $\mathcal{C}_{0}$ but negligible bandwidth $W$. This hints at the
existence of an optimal trade-off—in terms of best or optimal area $A$—between
bandwidth $W$ and $\mathcal{C}_{0}$ that leads to the maximization of the
product (of these two terms): the channel capacity in bits per second. This
phenomenon is illustrated in Fig. 1 where the bandwidth $W$ is monotonically
decreasing in $A$, the channel capacity $\mathcal{C}_{0}$ (bits per channel
use) is increasing monotonically with area $A$, and the product (channel
capacity in bits per seconds) is maximized at a certain (optimal) area $A$.

Figure 1: Channel capacity as a function of detector area.

Even though the study [13] hints at the existence of an optimal photodetector
area for maximization of channel capacity through a graphical illustration,
they did not provide any closed-form analytical solutions for the optimal
photodetector area. Moreover, the said study considers only one type of
channel: the Gaussian laser channel that is not affected by signal fading.
Since the optimization of photodetector area is an important problem in our
estimation, there is a strong motivation to look for optimal (or even
suboptimal) closed-form analytical solutions of the photodetector area for the
maximization of Gaussian channel capacity. Additionally, signal-dependent shot
noise appears in optical systems where the effect of background radiation can
be quite large, and this requires optimization of photodetector area for
signal dependent Gaussian noise channels. Finally, since detector arrays have
been shown to provide significant performance improvement over a single
detector for optical symbol detection [15], maximization of capacity of
channels involving detector array receivers—as a function of individual
detector size—is an important problem for future optical wireless
communication systems. All of the above mentioned considerations form the
motivation of the current study that deals with the optimization of
photodetector area for a diverse set of free-space optical communication
scenarios.

###  I-C Background Literature Review

In this paper, we examine the optimization of the photodetector area to
maximize the channel capacity of an optical wireless channel. For this
problem, we have classified the relevant literature review under four
categories: i) Relationship between modulation bandwidth of a photodiode and
its area, ii) signal dependent Gaussian noise models for optical wireless
channels, iii) pointing error/atmospheric turbulence-induced fading in optical
wireless channels and iv) optical wireless communications with a detector
array receiver.

There are a number of studies in current literature that discuss the
relationship between detector area and modulation bandwidth. For instance, the
paper [12] proposes a novel P-I-N photodiode based on InGaAsN lattice which is
matched to GaAs. The proposed photodiode consists of double transparent layers
for the depletion region and performs high speed photodetection compared to a
regular P-I-N photodiodes. The authors in [16] consider frequency response
optimization of P-I-N photodiodes based on GaN/InGaN semiconducting materials
that operate on $633$ nm wavelength. In the said study, the authors examine
frequency response optimization of the diode by using mixed depletion region
rather than a single absorbing depletion region. The study [17] demonstrates
design considerations and simulation aspects for multiple cladding layer GaAs-
based P-I-N waveguide photodiodes. The authors in this study demonstrated that
the theoretical cut-off frequency limit of $80$ GHz can be achieved by using a
thin absorption layer of $0.4~{}\mu$m. In [18], the authors consider design
and fabrication of high speed InGaN/GaN based photodiodes by reducing the
capacitive effects. The authors in this study demonstrated a cut-off frequency
of $300$ MHz using Noise measurement for $100\times 100~{}\mu$m2 photodiodes.
According to the authors, this result will enable a signficantly better higher
frequency response for photodiode dimensions up to $20~{}\mu$m.

We now consider some important references on the evaluation of channel
capacity as well as noise models for free-space optical links. The authors in
[19] derived several upper and lower bounds for a noncoherent intensity
modulated direct detection free-space optical link for constrained average and
peak power. Farid et al. [20] consider the design of nonuniform optical
intensity signaling that approaches capacity for constrained average and peak
signal amplitudes. In their work, they derive a simple expression for
capacity-approaching distribution via source entropy maximization. The study
[21] derives tight capacity bounds for indoor visible light communication
channels in the presence of signal dependent noise whereas the articles [22,
23] considers various capacity bounds for a free-space optical channel in the
presence of signal dependent Gaussian noise.

There are a number of studies on the modeling of pointing errors in free-space
optical communications. The authors in [24, 25] consider generalized models of
pointing error for capacity calculations, whereas [26, 27] consider pointing
error modeling along with turbulence induced fading in FSO links. The authors
in [28, 29] exclusively consider Rayleigh distribution-based pointing error
model for dual and multihop relaying FSO channels that involve hovering
unmanned aerial (UAV) based relays. The study [30] considers maximization of
outage capacity of an optical wireless link subject to pointing errors. Tsai
et al. [15] examine the effect of pointing error on the performance of a
detector array receiver in a free-space optical link.

We wrap up the literature review section with the mention of some references
on detector array receivers in free-space optical communications. The authors
consider detector arrays for joint beam tracking and symbol detection in [31,
32, 33, 34], whereas the studies [35, 36] consider FSO acquisition algorithms
based on detector array receiver. The article [37] consider time
synchronization of pulse position modulation signals based on an array of
detectors. The study [38] examines the effect of the number of array elements
on the performance of the maximal ratio combiner that is used to fuse the
outputs of the array detectors. The article [39] explores multiple-input-
single-output schemes for a detector array receiver in free-space optical
communications.

###  I-D Contributions of Our Study

With the exception of [13] and our understanding, there is no exiting research
or literature that analyzes the relationship between channel capacity and the
photodetector area of an optical wireless communication channel. Even though
the said paper highlights the fact that the channel capacity peaks at a
certain (optimal) photodetector area, a detailed analysis and derivations of
(even approximate) closed-form solutions are not provided. Additionally, the
focus of [13] is on just one type of channel: the signal independent thermal
noise channel that is modeled by a Gaussian distribution. In this study, we
carried out a comprehensive analysis on the optimization of photodetector area
for channel capacity maximization. Specifically, we i) derived closed-form
solution (in terms of Lambert W function) of optimal photodetector area for
the signal independent Gaussian noise channel as furnished by (23), ii)
provided analytical closed-form solutions of close-to-optimal area for
intensity fading channels as given by (42) and (44), iii) analyzed channels
based on a detector array receiver and provided closed-form optimal solutions
for equal gain combining and maximal ratio combining arrays as supplied by
(64) and (70), respectively, and iv) provided numerical solutions of optimal
detector area for shot noise limited regime that is modeled by signal
dependent Gaussian noise. A major conclusion of our study states that—for the
signal independent Gaussian noise case—the optimal area is proportional to the
factor $\left(\frac{\alpha}{\beta}\right)^{\frac{1}{m}}$ where the term
$\frac{\alpha}{\beta}$ is the inverse of the normalized signal-to-noise ratio
of the optical receiver. The constant $m=3$ for channels involving either the
single photodetector or the maximal ratio combining array, and $m=2$ for equal
gain combining array. This implies that as the signal-to-noise ratio improves,
the optimal detector area shrinks in order to exploit the extra bandwidth that
comes with a smaller area. Based on this discussion, we propose an adaptive
optical wireless communications receiver that consists of a number of
photodetectors with varying active areas. For such receivers, the “optimal”
photodetector based on the value of estimated signal intensity is chosen at
any given time in order to maximize the channel capacity. A conceptual example
of such an adaptive receiver is shown in Fig. 2 which consists of a signal
intensity estimation block, a selection block and three photodetectors of
different areas. The intensity estimation block operates on pilot symbols to
estimate signal intensity with the help of an array of detectors [34]. The
intensity estimate is then fed into a selection block that selects the best
photodetector to maximize channel capacity for incoming data symbols. This
adaptive receiver can effectively mitigate the effect of slow intensity fading
due to atmospheric turbulence or pointing error.

Figure 2: Block diagram of adaptive optical receiver.

###  I-E Paper Organization

This paper is organized as follows. Section II highlights the relationship
between detector area and bandwidth of an optical photodetector, and Section
III deals with the derivation and maximization channel capacity of a thermal
noise limited receiver. Section IV considers the photodetector area
optimization for signal intensity fading channels. In this section, two
scenarios are considered: one that deals with fading due to pointing error and
the other that examines signal fading due to strong atmospheric turbulence
that is modeled by negative exponential distribution. Section V deals with
channel capacity maximization of a detector array receiver for two combining
schemes: the _equal gain combining_ (EGC) and _maximal ratio combining_ (MRC).
Section VI deals with the general case where both the signal independent
thermal noise and signal dependent shot noise due to signal and background
radiation are considered. Section VII is dedicated to the interpretation and
commentary of experimental results, and the last section, Section VIII, sums
up the important results of this study.

##  II Relationship Between Detector Size and Bandwidth

Figure 3: Equivalent circuit representation of a photodiode.

###  II-A Detector Size and Bandwidth

Fig. 3 provides an equivalent circuit representation of a photodiode operated
in the photoconductive mode to detect light signal [40]. In this circuit, the
photodiode behaves as a photocontrolled current source in parallel with a
semiconductor diode. The circuit shows three current sources: the current due
to incident signal light denoted by $\mu_{s}$, current $\mu_{b}$ which is due
to (unwanted) background radiation or leakage current and the current
$I_{\textsf{th}}$ which is due to thermal noise in photodiode resistive
contacts. The sources $\mu_{b}$ and $I_{\textsf{th}}$ represent noise sources
in the circuit. The noise due to $\mu_{b}$ is the Poisson shot noise whose
variance depends on the value of $\mu_{b}$. The noise due to $I_{\textsf{th}}$
is modeled by a zero-mean Gaussian noise process with variance
$\sigma_{\textsf{th}}^{2}$.

The semiconductor diode in Fig. 3 is represented by its constituent elements:
the capacitance $C$ that represents the junction capacitance of the
photodiode, the photodiode shunt resistance $R_{\textsf{sh}}$ and the
resistance $R_{\textsf{s}}$ generated by photodiode contacts, wire bonds and
semiconductor material. The resistance $R_{\textsf{s}}$ is typically a few
tens of ohms. For silicon photodiodes, the resistance $R_{\textsf{sh}}$ is
usually in hundreds or thousands of megaohms, and indium gallium arsenide
based photodiodes have typically even higher shunt resistance values.
Therefore, in most practical scenarios, the shunt resistance $R_{\textsf{sh}}$
can be safely ignored [40].

The $3$-dB bandwidth (or cutoff frequency) of a photodiode—denoted by
$f_{c}$—is given by [12, 16]

$\displaystyle
f_{c}=\frac{f_{t}f_{\textsf{RC}}}{\sqrt{f_{t}^{2}+f_{\textsf{RC}}^{2}}},$ (2)

where $f_{t}$ is the _transit-time_ cutoff frequency and $f_{\textsf{RC}}$ is
the _capacitive_ cut-off frequency. The transit-time limited bandwidth is
determined by the electron and hole transit times through the depletion
region. The transit-time limited bandwidth is expressed by the equation [17]

$\displaystyle f_{t}\approx\frac{0.45\bar{\nu}}{d},$ (3)

where $\bar{\nu}$ is the mean carrier drift velocity and $d$ is the thickness
of depletion region. Since the holes’ mobility is quite limited compared to
free electrons, it is typically the holes that determine the transit-time
limited bandwidth [41].

The capacitive cut-off frequency is given by [18]

$\displaystyle f_{\textsf{RC}}=\frac{1}{2\pi RC},$ (4)

where $R$ is the sum of photodiode series resistance and load resistance,
i.e., $R=R_{\textsf{s}}+R_{\textsf{L}}$, and $C$ is the photodiode junction
capacitance [40]. The junction capacitance is given by

$\displaystyle C=\dfrac{\epsilon_{0}\epsilon_{r}A}{d},$ (5)

where $A$ is the active area of photodiode, $d$ is the depletion region
thickness, $\epsilon_{0}$ is the permittivity of free space and $\epsilon_{r}$
is the relative permittivity of photodiode depletion region [12]. By combining
(5) and (4), we have that

$\displaystyle
f_{\textsf{RC}}=\frac{d}{2\pi\epsilon_{0}\epsilon_{r}RA}=\frac{\alpha}{A},$
(6)

where

$\displaystyle\alpha\coloneqq\frac{d}{2\pi\epsilon_{0}\epsilon_{r}R}.$ (7)

It has been demonstrated in [12] that for detector areas greater than
$100~{}\mu m^{2}$, the capacitive effect dominates the transit-time cutoff. In
this case, the overall cut-off frequency $f_{c}$ is majorly determined by
$f_{\textsf{RC}}$. Thus, when $f_{\textsf{RC}}\ll f_{t}$ for $A>100\,\,\mu
m^{2}$, we have that

$\displaystyle
f_{c}=\frac{f_{t}f_{\textsf{RC}}}{\sqrt{f_{t}^{2}+f_{\textsf{RC}}^{2}}}\approx\frac{f_{t}f_{\textsf{RC}}}{\sqrt{f_{t}^{2}}}=f_{\textsf{RC}}.$
(8)

##  III Channel Capacity Maximization of Thermal Noise Limited Receiver

###  III-A Signal Model

The signal power captured by the photodetector is given by

$\displaystyle\mu_{s}$
$\displaystyle\coloneqq\iint_{\mathcal{D}}\Lambda_{s}(x,y)\,dy\,dx,$ (9)

where $\Lambda_{s}$ is the signal intensity measured in Watts/m2 and
$\mathcal{D}$ is the region of the circular detector. The intensity in focal
plane is modeled by a two-dimensional circularly symmetric Gaussian
distribution [6]:

$\displaystyle\Lambda_{s}$
$\displaystyle\coloneqq\frac{I_{0}}{2\pi\rho^{2}}\exp\left(-\frac{(x-x_{0})^{2}+(y-y_{0})^{2}}{2\rho^{2}}\right).$
(10)

Here, $(x_{0},y_{0})$ is the center of the beam, $\rho$ is the beam or spot
radius and $I_{0}$ is the total power in the beam measured in Watts. Fig. 4
shows the Gaussian beam impinging on a circular photodetector.

Figure 4:  Gaussian beam footprint projected onto a circular detector in the
focal plane.

We assume that the detector is placed at the origin and that the center of the
beam and the center of the detector coincide111This condition corresponds to
the condition of absence of pointing error in the channel and maximum power
capture by the detector.. For $A\ll\pi\rho^{2}$, we have that

$\displaystyle\mu_{s}\approx\mu_{0}A,$ (11)

where

$\displaystyle\mu_{0}\coloneqq\frac{I_{0}}{2\pi\rho^{2}}$ (12)

is the peak intensity projected onto the focal plane.

###  III-B Channel Capacity

If we assume that the only major source of noise in the circuit is the
_thermal noise_ with variance $\sigma_{\textsf{th}}^{2}$, we have that
$\sigma_{\textsf{th}}^{2}=N_{0}W$ where $W$ is the photodiode bandwidth and
$N_{0}$ is one-sided noise spectral density measured in W/Hz. Here, we assume
that the photodiode resistance is negligible compared to circuit resistance,
and therefore $\sigma_{\textsf{th}}^{2}$ is generated mainly by the circuit
resistive elements.

The Gaussian channel capacity of a coherent optical channel is reproduced
below as

$\displaystyle\mathcal{C}=\mathcal{W}\log_{2}\left(1+\frac{\mu_{s}^{2}}{\sigma_{\textsf{th}}^{2}}\right),$
(13)

where the bandwidth

$\displaystyle\mathcal{W}\coloneqq f_{\textsf{RC}}=\frac{\alpha}{A}.$ (14)

By substituting the expression for $\mu_{s}$ and $\sigma_{\textsf{th}}$ in
(13), we have that

$\displaystyle\mathcal{C}$
$\displaystyle=\mathcal{W}\log_{2}\left(1+\frac{\mu_{0}^{2}A^{2}}{N_{0}\mathcal{W}}\right).$
(15)

After substituting the expression of $\mathcal{W}$ in (15), the final
expression for channel capacity is

$\displaystyle\mathcal{C}=\frac{\alpha}{A}\log_{2}\left(1+\frac{\beta_{0}}{\alpha}A^{3}\right),$
(16)

where

$\displaystyle\beta_{0}\coloneqq\frac{\mu_{0}^{2}}{N_{0}}.$ (17)

The factor $\dfrac{\beta_{0}}{\alpha}A^{3}$ is the signal-to-noise ratio (SNR)
of the optical channel, and the term $\dfrac{\beta_{0}}{\alpha}$ is the
normalized SNR value.

###  III-C Optimal Photodetector Area

In order to find the optimal photodetector area that maximizes the channel
capacity furnished by (16), we rewrite the capacity in terms of natural log
function as

$\displaystyle\mathcal{C}=\frac{\alpha}{A\ln(2)}\ln\left(1+\frac{\beta_{0}}{\alpha}A^{3}\right).$
(18)

Taking the partial derivative of (18) with respect to area $A$ and setting it
equal to zero, we obtain

$\displaystyle\frac{\partial\mathcal{C}}{\partial A}=0$
$\displaystyle\implies\left(1+\frac{\beta_{0}}{\alpha}A^{3}\right)\ln\left(1+\frac{\beta_{0}}{\alpha}A^{3}\right)-3\frac{\beta_{0}}{\alpha}A^{3}=0$
(19)

Subtracting three from both sides of (19), we obtain

$\displaystyle\left(1+\frac{\beta_{0}}{\alpha}A^{3}\right)\ln\left(1+\frac{\beta_{0}}{\alpha}A^{3}\right)-3\frac{\beta_{0}}{\alpha}A^{3}-3=-3$
$\displaystyle\implies\left(1+\frac{\beta_{0}}{\alpha}A^{3}\right)\ln\left(1+\frac{\beta_{0}}{\alpha}A^{3}\right)-3\left(1+\frac{\beta_{0}}{\alpha}A^{3}\right)=-3$
$\displaystyle\implies\left(1+\frac{\beta_{0}}{\alpha}A^{3}\right)\left(\ln\left(1+\frac{\beta_{0}}{\alpha}A^{3}\right)-3\right)=-3.$
(20)

Dividing both sides of (20) with $\exp(3)$ and noting that $x=\exp(\ln(x))$,
we have that

$\displaystyle\left(\ln\left(1+\frac{\beta_{0}}{\alpha}A^{3}\right)-3\right)\exp\left(\ln\left(1+\frac{\beta_{0}}{\alpha}A^{3}\right)-3\right)=-3\exp(-3).$
(21)

Now, using the fact that the solution of the equation of the form

$\displaystyle x\exp(x)=y\implies x=W(y),$ (22)

where $W(\cdot)$ is the Lambert W function, we have that (21) can be solved as

$\displaystyle\ln\left(1+\frac{\beta_{0}}{\alpha}A^{3}\right)-3=W\left(-3\exp(-3)\right)$
$\displaystyle\implies
1+\frac{\beta_{0}}{\alpha}A^{3}=\exp\left(W\left(-3\exp(-3)\right)+3\right)$
$\displaystyle\implies
A^{*}=\left(\exp\left(W\left(-3\exp(-3)\right)+3\right)-1\right)^{\frac{1}{3}}\left(\frac{\alpha}{\beta_{0}}\right)^{\frac{1}{3}}.$
(23)

Let us define the factor $\gamma_{0}$ as

$\displaystyle\gamma_{0}\coloneqq\left(\exp\left(W\left(-3\exp(-3)\right)+3\right)-1\right)^{\frac{1}{3}}.$
(24)

To show that the solution (23) is indeed a maximizer, we evaluate the second
derivative of capacity $\mathcal{C}$ at $A=A^{*}$. We have that

$\displaystyle\frac{\partial^{2}\mathcal{C}}{\partial
A^{2}}=\frac{1}{\ln(2)}\left(\frac{-9\beta_{0}^{2}A^{3}/\alpha}{\left(1+\frac{\beta_{0}}{\alpha}A^{3}\right)^{2}}+\frac{2\alpha}{A^{3}}\ln\left(1+\frac{\beta_{0}}{\alpha}A^{3}\right)\right),$
(25)

and

$\displaystyle\left.\frac{\partial^{2}\mathcal{C}}{\partial
A^{2}}\right|_{A=A^{*}}=\frac{\beta_{0}}{\ln(2)}\left(\frac{-9\gamma_{0}}{(1+\gamma_{0})^{2}}+\frac{2}{\gamma_{0}}\ln(1+\gamma_{0})\right)\approx-\frac{2.0947\beta_{0}}{\ln(2)}.$
(26)

Since the second derivative is negative at $A^{*}$, we have that $A^{*}$
represents global maximizer of channel capacity. Therefore, from (23), we note
that the optimal area varies as the cube root of the factor
$\frac{\alpha}{\beta_{0}}$ which is the inverse of normalized SNR value.

##  IV Ergodic Capacity Maximization Under Intensity Fading in Free-Space
Optical Channels

###  IV-A Channel Capacity Under Intensity Fading

The erogodic capacity of a free-space optical channel under fading is
described by

$\displaystyle\mathcal{C}=\int_{0}^{\infty}\mathcal{W}\log_{2}\left(1+\frac{\mu_{s}^{2}h^{2}}{N_{0}\mathcal{W}}\right)f_{\mathbbm{h}}(h)\,dh,$
(27)

where $f_{\mathbbm{h}}(h)$ is the distribution of random channel coefficient
$\mathbbm{h}$, and $\mu_{s}$ is the maximum signal power captured by the
optical receiver under no fading. The signal does not experience fading when
the channel coefficient $\mathbbm{h}=1$ with probability one, i.e.,
$\mathbbm{P}(\\{\mathbbm{h}=1\\})=1.$

###  IV-B Close-to-Optimal Detector Area Under Pointing error

Assuming that the pointing error (due to angle-of-arrival fluctuations) is
distributed as a Rayleigh random variable [28, 29, 15], the received signal
power—denoted by $P_{s}$—captured by the detector is furnished by the
relationship:

$\displaystyle P_{s}=\mu_{s}\mathbbm{h}_{\mathcal{R}},$ (28)

where

$\displaystyle\mathbbm{h}_{\mathcal{R}}\coloneqq\exp\left(-\frac{\mathcal{R}^{2}}{2\rho^{2}}\right),$
(29)

where $\mathcal{R}$ is a Rayleigh random variable that models the pointing or
misalignment error in the focal plane. The probability density function (pdf)
of $\mathcal{R}$ is defined as

$\displaystyle
f_{\mathcal{R}}(r)\coloneqq\dfrac{r}{\sigma_{p}^{2}}\exp\left(-\dfrac{r^{2}}{2\sigma_{p}^{2}}\right)\cdot\mathbf{1}_{[0,\infty)}(r),$
(30)

where $\sigma_{p}$ is the _scale parameter_ of the pdf $f_{\mathcal{R}}(r)$.
By noting that $h_{\mathcal{R}}$ is the realization of
$\mathbbm{h}_{\mathcal{R}}$, the ergodic capacity—under pointing error—is
given by

$\displaystyle\mathcal{C}$
$\displaystyle=\int_{0}^{\infty}\mathcal{W}\log_{2}\left(1+\frac{\mu_{s}^{2}h_{\mathcal{R}}^{2}}{N_{0}\mathcal{W}}\right)f_{\mathcal{R}}(r)\,dr,$
(31)
$\displaystyle=\int_{0}^{\infty}\mathcal{W}\log_{2}\left(1+\frac{\mu_{s}^{2}\exp\left(-\frac{r^{2}}{\rho^{2}}\right)}{N_{0}\mathcal{W}}\right)\dfrac{r}{\sigma_{p}^{2}}\exp\left(-\dfrac{r^{2}}{2\sigma_{p}^{2}}\right)\,dr$
(32)
$\displaystyle=\int_{0}^{\infty}\frac{\alpha}{A}\log_{2}\left(1+\frac{\mu_{0}^{2}\exp\left(-\frac{r^{2}}{\rho^{2}}\right)}{N_{0}\alpha}A^{3}\right)\dfrac{r}{\sigma_{p}^{2}}\exp\left(-\dfrac{r^{2}}{2\sigma_{p}^{2}}\right)\,dr$
(33)
$\displaystyle=\int_{0}^{\infty}\frac{\alpha}{A}\log_{2}\left(1+\frac{\beta(r)}{\alpha}A^{3}\right)\dfrac{r}{\sigma_{p}^{2}}\exp\left(-\dfrac{r^{2}}{2\sigma_{p}^{2}}\right)\,dr,$
(34)

where

$\displaystyle\beta(r)\coloneqq\frac{\mu_{0}^{2}\exp\left(-\frac{r^{2}}{\rho^{2}}\right)}{N_{0}},r>0.$
(35)

Let us denote the optimal photodetector area $A^{*}$ for a realization $r$ of
random variable $\mathcal{R}$ as $A^{*}(r)$ where

$\displaystyle
A^{*}(r)=\gamma_{0}\left(\frac{\alpha}{\beta(r)}\right)^{\frac{1}{3}}=\gamma_{0}\left(\frac{\alpha
N_{0}}{\mu_{0}^{2}\exp\left(-\frac{r^{2}}{\rho^{2}}\right)}\right)^{\frac{1}{3}}.$
(36)

The (suboptimal) average area in this case is

$\displaystyle
A^{\star}=\int_{0}^{\infty}A^{*}(r)\dfrac{r}{\sigma_{p}^{2}}\exp\left(-\dfrac{r^{2}}{2\sigma_{p}^{2}}\right)\,dr$
(37) $\displaystyle=\int_{0}^{\infty}\gamma_{0}\left(\frac{\alpha
N_{0}}{\mu_{0}^{2}\exp\left(-\frac{r^{2}}{\rho^{2}}\right)}\right)^{\frac{1}{3}}\dfrac{r}{\sigma_{p}^{2}}\exp\left(-\dfrac{r^{2}}{2\sigma_{p}^{2}}\right)\,dr$
$\displaystyle=\gamma_{0}\left(\frac{\alpha
N_{0}}{\mu_{0}^{2}}\right)^{\frac{1}{3}}\int_{0}^{\infty}\frac{r}{\sigma_{p}^{2}}\exp\left(-r^{2}\left(\frac{1}{2\sigma_{p}^{2}}-\frac{1}{3\rho^{2}}\right)\right)\,dr,$
(38)

where the integral in (38) converges if the factor

$\displaystyle\frac{1}{2\sigma_{p}^{2}}-\frac{1}{3\rho^{2}}>0\implies\rho>\sqrt{\frac{2}{3}}\sigma_{p}.$
(39)

This condition implies that the spot radius has to be at least
$\sqrt{\frac{2}{3}}$ times the pointing error scale factor $\sigma_{p}$. This
condition is almost always satisfied since the beam footprint is made large
enough to accommodate any deviations due to misalignment or pointing error. In
most scenarios, a suboptimal value of beam radius $\rho$ (in relation to
$\sigma_{p}$) is given by $\rho=\sqrt{2}\sigma_{p}$ [29].

When the condition (39) holds, we have that

$\displaystyle A^{\star}=\gamma_{0}\left(\frac{\alpha
N_{0}}{\mu_{0}^{2}}\right)^{\frac{1}{3}}\int_{0}^{\infty}\frac{r}{\sigma_{p}^{2}}\exp\left(-r^{2}\left(\frac{3\rho^{2}-2\sigma_{p}^{2}}{6\rho^{2}\sigma_{p}^{2}}\right)\right)\,dr$
$\displaystyle=\gamma_{0}\left(\frac{\alpha
N_{0}}{\mu_{0}^{2}}\right)^{\frac{1}{3}}\int_{0}^{\infty}\frac{r}{\sigma_{p}^{2}}\exp\left(-\left(\frac{r^{2}}{2\left(\frac{3\rho^{2}\sigma_{p}^{2}}{3\rho^{2}-2\sigma_{p}^{2}}\right)}\right)\right)\,dr.$
(40)

Dividing and multiplying (40) by the factor
$\dfrac{3\rho^{2}}{3\rho^{2}-2\sigma_{p}^{2}}$, we obtain the suboptimal
photodetector area as

$\displaystyle A^{\star}$ $\displaystyle=\gamma_{0}\left(\frac{\alpha
N_{0}}{\mu_{0}^{2}}\right)^{\frac{1}{3}}\left(\frac{3\rho^{2}}{3\rho^{2}-2\sigma_{p}^{2}}\right)\int_{0}^{\infty}\frac{r}{\frac{3\rho^{2}\sigma_{p}^{2}}{3\rho^{2}-2\sigma_{p}^{2}}}\exp\left(-\left(\frac{r^{2}}{2\left(\frac{3\rho^{2}\sigma_{p}^{2}}{3\rho^{2}-2\sigma_{p}^{2}}\right)}\right)\right)\,dr$
(41) $\displaystyle=\gamma_{0}\left(\frac{\alpha
N_{0}}{\mu_{0}^{2}}\right)^{\frac{1}{3}}\left(\frac{3\rho^{2}}{3\rho^{2}-2\sigma_{p}^{2}}\right),$
(42)

since the integral in (41) evaluates to unity.

###  IV-C Suboptimal Detector Area Under Strong Atmospheric Turbulence

Assuming the case of strong atmospheric turbulence that is modeled by the
negative exponential distribution, we note that the suboptimal average area in
this case is

$\displaystyle A^{\star}$
$\displaystyle=\int_{0}^{\infty}A^{*}(h)\frac{1}{\eta}\exp\left(-\frac{1}{\eta}h\right)\,dh,$
$\displaystyle=\int_{0}^{\infty}\gamma_{0}\left(\frac{\alpha}{\beta(h)}\right)^{\frac{1}{3}}\frac{1}{\eta}\exp\left(-\frac{1}{\eta}h\right)\,dh,$
(43)

where $\eta>0$ is the mean of negative exponential distribution and
$\beta(h)=\dfrac{(\mu_{0}h)^{2}}{N_{0}}$. This implies that

$\displaystyle A^{\star}$
$\displaystyle=\frac{\gamma_{0}\alpha^{\frac{1}{3}}N_{0}^{\frac{1}{3}}}{\eta\mu_{0}^{\frac{2}{3}}}\int_{0}^{\infty}h^{-\frac{2}{3}}\exp\left(-\frac{1}{\eta}h\right)\,dh$
$\displaystyle=\frac{\gamma_{0}\alpha^{\frac{1}{3}}N_{0}^{\frac{1}{3}}}{\eta\mu_{0}^{\frac{2}{3}}}\eta^{\frac{1}{3}}\int_{0}^{\infty}t^{-\frac{2}{3}}\exp(-t)\,dt\,dt$
$\displaystyle=\frac{\gamma_{0}\alpha^{\frac{1}{3}}N_{0}^{\frac{1}{3}}\Gamma(\frac{1}{3})}{\mu_{0}^{\frac{2}{3}}\eta^{\frac{2}{3}}},$
(44)

where the gamma function $\Gamma(\cdot)$ is defined as
$\Gamma(z)\coloneqq\int_{0}^{\infty}t^{z-1}\exp(-t)\,dt.$

##  V Channel Capacity Maximization of Detector Array Receivers

Figure 5: Gaussian beam projected onto a $4\times 4$ detector array.

In this section, we consider the channel capacity of a detector array receiver
in free-space optical communications. Detector arrays are commonly used for
tracking the angle-of-arrival of laser beams by virtue of estimation of spot
position on a detector array in focal plane [7, 31]. However, detector arrays
can also be used in a joint beam tracking/symbol detection configuration which
simplifies receiver hardware by getting rid of a separate beam tracking
assembly dedicated exclusively to beam tracking[15].

As shown in Fig. 5, the detector array considered in this study is of a square
configuration where the individual elements or detectors are also of a square
shape and are arranged in an $n\times n$ arrangement for $n$ a positive
integer. Therefore, the total number of individual detectors in the array is
$M=n^{2}$. We further assume that the area of the array is $\mathcal{A}$ m2.
Without any loss of generality, we assume that the array is centered at the
origin $(0,0)$.

For thermal noise limited regime, the output $Y_{m}$ of the $m$th element of
the detector array is given by

$\displaystyle Y_{m}=\Lambda_{m}+X_{m},\quad m=1,2,\dotsc,M,$ (45)

In (45), the signal component $\Lambda_{m}$ is defined as

$\displaystyle\Lambda_{m}\coloneqq\iint_{A_{m}}\Lambda_{s}(x,y)\,dy\,dx,$ (46)

where $\Lambda_{s}$ is furnished by (10). The noise component $X_{m}$ is
distributed as a Gaussian random varaible:
$\mathcal{N}(0,\sigma_{\textsf{th}}^{2}=N_{0}W)$. We note that $X_{i}$ and
$X_{j}$ are independent and identically distributed random variables for
integers $i$ and $j$ such that $i\neq j$ and $1\leq i,j\leq M$.

In order to form the sufficient statistic with an array of detectors, we look
at two signal combining schemes: the _equal gain combiner_ (EGC) and the
_maximal ratio combiner_ (MRC). The EGC combines the output of each detector
of the array with uniform weights, whereas the MRC weights the output of each
detector with the optimal intensity value before combining [15, 33]. It is
important to note here that the intensity in each detector is not the same as
the intensity in every other detector since the intensity of the laser spot in
the focal plane varies according to Gaussian distribution as given by (10). In
the presence of random fluctuation of angle-of-arrival, the spot location on
the array will change randomly. In such a scenario, diversity schemes such as
the MRC are quite effective in countering the effect of misalignment provided
that the spot location on the array can be tracked effectively in real time
[7, 15]. Here, we also remind the reader that the MRC algorithm, discussed in
this paper, does not take into account the phase of the optical signal since
we are dealing with intensity modulated/direct detection (IM/DD) scheme in
this study.

In the upcoming sections, we look at the ECG and MRC combining schemes for a
detector array receiver. We will also consider the optimization of individual
detector area of the array that maximizes the capacity that is achieved with
these combining schemes.

###  V-A Equal Gain Combining

An equal gain combiner simply adds output from each element of the array to
form the sufficient statistic $Y$ as follows:

$\displaystyle Y$ $\displaystyle=\sum_{m=1}^{M}Y_{m}=S+X,$ (47)

where $S=\sum_{m=1}^{M}\Lambda_{m}$ is the signal component and
$X=\sum_{m=1}^{M}X_{m}$ is the noise component. Since the noise in each
detector of the array is assumed to be independent, the variance of $X$ is
$MN_{0}W$. The channel capacity of this scheme is

$\displaystyle\mathcal{C}$
$\displaystyle=\mathcal{W}\log_{2}\left(1+\frac{\left(\sum_{m=1}^{M}\Lambda_{m}\right)^{2}}{MN_{0}\mathcal{W}}\right)$
(48)
$\displaystyle=\mathcal{W}\log_{2}\left(1+\frac{\mu_{s}^{2}}{MN_{0}\mathcal{W}}\right)$
(49)

where $\mu_{s}\coloneqq\sum_{m=1}^{M}\Lambda_{m}$ is the total energy captured
by the detector array. The bandwidth $\mathcal{W}$ in (49) is a function of
number of detectors $M$ in the array provided the array area $\mathcal{A}$ is
fixed. It is given by

$\displaystyle\mathcal{W}=\frac{d}{2\pi\epsilon_{0}\epsilon_{r}RA}=\frac{dM}{2\pi\epsilon_{0}\epsilon_{r}R\mathcal{A}},$
(50)

where $A$ is the area of a single detector in the array.

###  V-B Maximal Ratio Combining

The output $Y_{m}$ of the $m$th element of the array is weighted by the signal
component $\Lambda_{m}$ before fusion takes place for the formation of the
sufficient statistic. The resulting sufficient statistic, $Y$, is

$\displaystyle Y=\sum_{m=1}^{M}Y_{m}\Lambda_{m}=S+X,$ (51)

where the signal component $S=\sum_{m=1}^{M}\Lambda_{m}^{2}$ and the noise
component $X=\sum_{m=1}^{M}\Lambda_{m}X_{m}$. We note that $X$ is a zero-mean
Gaussian random variable with variance
$\sigma_{X}^{2}=N_{0}W\sum_{m=1}^{M}\Lambda_{m}^{2}$. The channel capacity of
this scheme is

$\displaystyle\mathcal{C}$
$\displaystyle=\mathcal{W}\log_{2}\left(1+\frac{S^{2}}{\sigma_{X}^{2}}\right)=\mathcal{W}\log_{2}\left(1+\frac{\left(\sum_{m=1}^{M}\Lambda_{m}^{2}\right)^{2}}{N_{0}\mathcal{W}\sum_{m=1}^{M}\Lambda_{m}^{2}}\right)$
$\displaystyle=\mathcal{W}\log_{2}\left(1+\frac{\sum_{m=1}^{M}\Lambda_{m}^{2}}{N_{0}\mathcal{W}}\right),$
(52)

where $\mathcal{W}$ is given by (50).

###  V-C Optimization Problem

For maximization of channel capacity with an array of detectors, we optimize
the area $A$ of the single detector element while we keep the total area of
the array constant. Thus, we consider the optimization problem of the
following type:

$\displaystyle\underset{A}{\text{maximize}}$ $\displaystyle\mathcal{C}$ (53)
subject to $\displaystyle\,\mathcal{A}=\mathcal{A}_{0}.$

In this optimization problem, $\mathcal{C}$ is furnished by (49) for the equal
gain combiner and (52) for maximal ratio combining. The factor
$\mathcal{A}_{0}$ is a positive constant.

####  V-C1 Optimization of Photodetector Area for EGC Array Receiver

In order to obtain a closed-form solution to (53) for the EGC array, we
rewrite (49) as

$\displaystyle\mathcal{C}$
$\displaystyle=\mathcal{W}\log_{2}\left(1+\frac{\mu_{s}^{2}}{MN_{0}\mathcal{W}}\right)$
(54)
$\displaystyle=\frac{\alpha}{A}\log_{2}\left(1+\frac{\mu_{s}^{2}}{\mathcal{A}N_{0}\alpha}A^{2}\right),$
(55)

where we used the fact that the number of detectors $M=\frac{\mathcal{A}}{A}$,
and that the bandwidth of the array is the same as the bandwidth of each
individual element or detector of the array: $\mathcal{W}=\frac{\alpha}{A}$.
Let us denote the factor

$\displaystyle\beta_{1}\coloneqq\frac{\mu_{s}^{2}}{N_{0}}=\frac{\left(\sum_{m=1}^{M}\Lambda_{m}\right)^{2}}{N_{0}}.$
(56)

Then,

$\displaystyle\mathcal{C}$
$\displaystyle=\frac{\alpha}{A}\log_{2}\left(1+\frac{\beta_{1}}{\mathcal{A}\alpha}A^{2}\right)$
(57)
$\displaystyle=\frac{\alpha}{A\ln(2)}\ln\left(1+\frac{\beta_{1}}{\mathcal{A}\alpha}A^{2}\right).$
(58)

The quantity $\frac{\beta_{1}}{\mathcal{A}\alpha}$ is the normalized SNR of
the EGC array. Here, we note that when the individual detector area $A$ is
fixed, increasing array area $\mathcal{A}$ implies increasing the number of
detectors $M$ in the array. Thus, for fixed $A$, increasing $\mathcal{A}$
implies increasing the total noise in the array since noise power increases
linearly with the number of elements $M$ of the array.

In order to find the maximum of $\mathcal{C}$ in (58) as a function of area
$A$, we take the derivative of $\mathcal{C}$ as a function of $A$ and set it
equal to zero to obtain

$\displaystyle\left(1+\frac{\beta_{1}}{\mathcal{A}\alpha}A^{2}\right)\ln\left(1+\frac{\beta_{1}}{\mathcal{A}\alpha}A^{2}\right)-2\frac{\beta_{1}}{\mathcal{A}\alpha}A^{2}=0.$
(59)

Subtracting two from both sides of (59), we obtain

$\displaystyle\left(1+\frac{\beta_{1}}{\mathcal{A}\alpha}A^{2}\right)\left(\ln\left(1+\frac{\beta_{1}}{\mathcal{A}\alpha}A^{2}\right)-2\right)=-2.$
(60)

Representing $\left(1+\frac{\beta_{1}}{\mathcal{A}\alpha}A^{2}\right)$ by
$\exp\left(\ln\left(1+\frac{\beta_{1}}{\mathcal{A}\alpha}A^{2}\right)\right)$
and dividing both sides of (60) by $\exp(2)$, we have that

$\displaystyle\exp\left(\ln\left(1+\frac{\beta_{1}}{\mathcal{A}\alpha}A^{2}\right)-2\right)\left(\ln\left(1+\frac{\beta_{1}}{\mathcal{A}\alpha}A^{2}\right)-2\right)=-2\exp(-2)$
(61)
$\displaystyle\implies\ln\left(1+\frac{\beta_{1}}{\mathcal{A}\alpha}A^{2}\right)-2=W\left(-2\exp(-2)\right)$
(62) $\displaystyle\implies
1+\frac{\beta_{1}}{\mathcal{A}\alpha}A^{2}=\exp\left(W\left(-2\exp(-2)\right)+2\right)$
(63) $\displaystyle\implies
A^{*}=\left(\exp\left(W\left(-2\exp(-2)\right)+2\right)-1\right)^{\frac{1}{2}}\left(\frac{\alpha\mathcal{A}}{\beta_{1}}\right)^{\frac{1}{2}}.$
(64)

Let us define
$\gamma_{1}\coloneqq\left(\exp\left(W\left(-2\exp(-2)\right)+2\right)-1\right)^{\frac{1}{2}}$.
In order to show that $A^{*}$ is indeed a global maximizer, we evaluate the
second partial of channel capacity at $A=A^{*}$:

$\displaystyle\left.\frac{\partial^{2}\mathcal{C}}{\partial
A^{2}}\right|_{A=A^{*}}$
$\displaystyle=\frac{\beta_{1}^{\frac{3}{2}}}{\alpha^{\frac{1}{2}}\mathcal{A}^{\frac{3}{2}}\ln(2)}\left(-\frac{4\gamma_{1}}{\left(1+\gamma_{1}^{2}\right)^{2}}-\frac{2}{\gamma_{1}\left(1+\gamma_{1}^{2}\right)}+\frac{2\ln\left(1+\gamma_{1}^{2}\right)}{\gamma_{1}^{3}}\right)$
(65)
$\displaystyle\approx-\frac{0.1218\beta_{1}^{\frac{3}{2}}}{\alpha^{\frac{1}{2}}\mathcal{A}^{\frac{3}{2}}\ln(2)}$
(66)

which is strictly negative at $A=A^{*}$. Hence, $A^{*}$ in (64) is indeed a
global maximizer of the channel capacity achieved with an EGC array.

From (64), we observe that the optimal area is proportional to the square root
of the inverse of normalized SNR which is in contrast to the cube root
relationship for a single-detector optical receiver (see (23)).

####  V-C2 Optimization of Photodetector Area for MRC Array Receiver

In order to maximize capacity of an MRC array—given by (52)—as a function of
photodetector area, we first define the average signal intensity
$\Lambda_{s}^{(m)}$ (measured in Watts/unit area) on the $m$th detector of the
array:

$\displaystyle\Lambda_{s}^{(m)}\coloneqq\frac{\Lambda_{m}}{A}.$ (67)

We now rewrite (52) as a function of $\Lambda_{s}^{(m)}$ as

$\displaystyle\mathcal{C}$
$\displaystyle=\mathcal{W}\log_{2}\left(1+\frac{A^{2}\sum_{m=1}^{M}\left(\Lambda_{s}^{(m)}\right)^{2}}{N_{0}\mathcal{W}}\right)$
(68)
$\displaystyle=\frac{\alpha}{A}\log_{2}\left(1+\frac{\beta_{2}}{\alpha}A^{3}\right)$
(69)

where
$\beta_{2}\coloneqq\dfrac{\sum_{m=1}^{M}\left(\Lambda_{s}^{(m)}\right)^{2}}{N_{0}}$.
We note that the channel capacity of an MRC array has the same form as the
channel capacity of a single detector (given by (16)). Therefore, the optimum
detector area for the MRC array is

$\displaystyle
A^{*}=\left(\exp\left(W\left(-3\exp(-3)\right)+3\right)-1\right)^{\frac{1}{3}}\left(\frac{\alpha}{\beta_{2}}\right)^{\frac{1}{3}}.$
(70)

Here, the optimal photodetector area varies as cube root of normalized SNR as
opposed to square root relationship for EGC array.

##  VI Channel Capacity Under Signal Dependent Shot Noise

In this section, we consider the channel capacity maximization for the general
case where the channel suffers both from shot noise and thermal noise. Let the
number of received signal photons be equal to $N_{s}$. Then,
$\mathbbm{E}[N_{s}]=\mu_{s}$ where $\mathbbm{E}[\cdot]$ is the ensemble
average operator. The average background radiation intensity (giving rise to
dark current) is denoted by $\lambda_{b}$ Watts/m2. The total noise power due
to background radiation is $\mu_{b}\coloneqq\lambda_{b}A$. Let the average
number of noise photons be $N_{b}$. In this case,
$\mathbbm{E}[N_{b}]=\mu_{b}$.

Here, for the sake of simplicity, let us consider the popular on-off keying
(OOK) signaling. For the “on” state, let the total number of photons (signals
plus noise ) be denoted by $N_{t}=N_{s}+N_{b}$. In low-to-medium photon rate
regime, the random variable $N_{t}$ is modeled by a Poisson distribution with
parameter $(\mu_{s}+\mu_{b})$. In high photon rate regime, $N_{t}$ converges
(in distribution) to a Gaussian random variable:
$N_{t}\sim\mathcal{N}(\mu=\mu_{s}+\mu_{b},\sigma_{\textsf{sh}}^{2}=\mu_{s}+\mu_{b})$
when the signal pulse is present. When no signal is present,
$N_{t}\sim\mathcal{N}(\mu=\mu_{b},\sigma_{\textsf{sh}}^{2}=\mu_{b})$. Here,
$\sigma_{\textsf{sh}}^{2}$ represents the (signal dependent) power in _shot
noise_.

In the high photon rate regime, the signal-to-noise ratio (denoted by the
symbol SNR) is given by

SNR
$\displaystyle=\frac{\mu_{s}^{2}}{\sigma_{\textsf{sh}}^{2}+\sigma_{\textsf{th}}^{2}}=\frac{\mu_{s}^{2}}{\mu_{s}+\mu_{b}+\sigma_{\textsf{th}}^{2}}$
$\displaystyle\overset{A\ll\pi\rho^{2}}{\approx}\frac{\mu_{0}^{2}A^{2}}{\mu_{0}A+\lambda_{b}A+\sigma_{\textsf{th}}^{2}}=\frac{\mu_{0}^{2}A}{\mu_{0}+\lambda_{b}+\frac{\sigma_{\textsf{th}}^{2}}{A}}.$
(71)

Thus, the $\textsf{SNR}\to 0$ as $A\to 0$.

For the binary OOK channel, the output $Y$ is related to random input $X$ as

$\displaystyle Y=X+N_{\textsf{sh}}+N_{\textsf{th}},$ (72)

where the random variable $N_{\textsf{sh}}$ corresponds to signal dependent
shot noise with distribution $\mathcal{N}(0,\sigma_{\textsf{sh}}^{2})$, and
$N_{\textsf{th}}$ represents thermal noise with distribution
$\mathcal{N}(0,\sigma_{\textsf{th}}^{2})$. We assume that
$N_{\textsf{sh}}\perp N_{\textsf{th}}$. For this channel, we obtain the
capacity as the maximization of the difference of entropy $H(X)$ and
conditional entropy $H(X|Y)$:

$\displaystyle\mathcal{C}_{0}\coloneqq\max_{p(x)}\left(H(X)-H(X|Y)\right)$
(73)

where $\mathcal{C}_{0}$ is capacity in bits per channel use and $p(x)$ is the
distribution of input $X$. The capacity $\mathcal{C}$ in bits per second is
given by

$\displaystyle\mathcal{C}=W\mathcal{C}_{0}.$ (74)

In (73), the input $X$ is a discrete random variable and the output $Y$ is
sampled from a continuous distribution. The quantity
$H(X)\coloneqq-\sum_{x\in\mathcal{X}}p(x)\log_{2}p(x),$ is the entropy of the
input $X$ where the set $\mathcal{X}\coloneqq\\{0,1\\}$. The conditional
entropy is

$\displaystyle H(X|Y)$
$\displaystyle\coloneqq-\sum_{x\in\mathcal{X}}\int_{y}p(x)p(y|x)\log_{2}\frac{p(y|x)p(x)}{p(y)}\,dy,$
(75)

where $p(y)\coloneqq\sum\limits_{x\in\mathcal{X}}p(y|x=i)p\left(x=i\right)$.
Moreover,

$\displaystyle p(y|x=1)$
$\displaystyle=\frac{1}{\sqrt{2\pi(\mu_{0}A+\lambda_{b}A+\sigma_{\textsf{th}}^{2})}}\exp\left(-\frac{(y-\mu_{0}A-\lambda_{b}A)^{2}}{2(\mu_{0}A+\lambda_{b}A+\sigma_{\textsf{th}}^{2})}\right),$
$\displaystyle p(y|x=0)$
$\displaystyle=\frac{1}{\sqrt{2\pi(\lambda_{b}A+\sigma_{\textsf{th}}^{2})}}\exp\left(-\frac{(y-\lambda_{b}A)^{2}}{2(\lambda_{b}A+\sigma_{\textsf{th}}^{2})}\right).$
(76)

The closed-form solution of optimal detector area that maximizes channel
capacity in (74) is not easy to achieve. Therefore, for signal dependent shot
noise scenario, we resort to finding the peak of channel capacity numerically
(graphically) in Section VII.

##  VII Experimental Results and Discussion

In this section, we explain and interpret the experimental results obtained
for this study. The default parameter values are highlighted in Table. I.

TABLE I: Default parameter values for simulations

Parameter | Symbol | Default value
---|---|---
Diode junction thickness | $d$ | 0.1 $\mu$m
Permittivity (free space) | $\epsilon_{0}$ |  $8.854\times 10^{-12}$ F/m
Permittivity (relative) | $\epsilon_{r}$ | 12.95
Noise PSD | $N_{0}$ |  $4.11\times 10^{-21}$ W/Hz
Photodiode resistance | $R$ | 10 ohms
Peak received intensity | $\mu_{0}$ | 10 mW
Background radiation intensity | $\lambda_{b}$ |  $\frac{\mu_{0}}{20}$ to $\frac{\mu_{0}}{3}$
Focal spot radius | $\rho$ | 2 mm
Number of detectors in the array | $M$ | 16
Photodetector area | $A$ |  $10^{-9}$ m2
Photodetector array area | $\mathcal{A}$ | 4 mm2
Pointing error scale factor | $\sigma_{p}$ | 0.01–8 mm
Exponential fading distribution mean | $\eta$ | 0.1–0.8

Fig. 6 depicts the channel capacity as a function of photodetector area for
the thermal noise limited regime where the noise statistics is assumed to
follow the Gaussian distribution. This figure plots the channel capacity as a
function of photodiode area $A$ for different values of received signal power
$\mu_{0}$. We observe that the channel capacity is maximized at a certain
(optimal) value of photodiode area. We also note from this figure that as the
received signal power $\mu_{0}$ is increased, the optimal area shifts to the
left towards zero. This is explained by the inverse dependence of the optimal
photodetector area on the signal-to-noise ratio as highlighted by (23).

Figure 6: Channel capacity for thermal noise dominated regime.

Fig. 7 depicts channel capacity for different values of pointing error
(modeled by Rayleigh distribution) scale factor $\sigma_{p}$. Here, we observe
that the optimal photodiode area depends on the value of $\sigma_{p}$. A
larger value of $\sigma_{p}$ minimizes the average signal-to-noise ratio, and
this fact implies that the optimal area should shift to the right to
compensate for the reduced signal-to-noise ratio when the scale factor
$\sigma_{p}$ is large. A related figure, Fig. 8 depicts channel capacity for
signal fading due to strong turbulence that is modeled by the negative
exponential fading. These two figures are also compared with the suboptimal
solutions presented by (42) for the channel suffering from pointing error
exclusively, and (44) for strong turbulence condition. The suboptimal
solutions are shown by circular markers of the same color as the capacity
curves. We note that the suboptimal solutions are quite close to the actual
optimal solution (the area value right below the peak capacity) especially for
the pointing error case. We also note that the approximation with suboptimal
solution gets better with increasing SNR at the receiver.

Figure 7: Channel capacity as a function of pointing error standard deviation
$\sigma_{p}$. Figure 8: Channel capacity maximization with respect to
photodetector area under atmospheric turbulence conditions.

For optical receivers based on detector arrays, Fig. 9 and Fig. 10 correspond
to the channel capacity maximization problems for EGC and MRC arrays. Here, we
have assumed a square array with square individual detectors as shown in
Fig.5. Fig. 9 shows the channel capacity plots for the EGC array receiver as a
function of individual detector area $A$ when the array area $\mathcal{A}$ is
fixed at 4 square millimeters. In Fig. 10, the channel capacity is plotted as
a function of total number of detectors $M$ in the array for the EGC and MRC
arrays for a fixed array area $\mathcal{A}$. Here, for a particular value of
$M$, the area of each detector is
$A=\frac{\mathcal{A}}{M}=\frac{{4\text{mm}^{2}}}{M}$. We note that there is a
certain number of elements $M^{*}$ (or $A^{*}$) at which the channel capacity
is maximized. Moreover, since the MRC minimizes the effect of noise—or
alternatively, maximizes the signal-to-noise ratio—the optimal area $A^{*}$
shifts to the left (relative to optimal area for EGC) in order to maximize the
bandwidth and the resulting channel capacity. Since $\mathcal{A}$ is constant,
this implies that the optimal $M^{*}$ should shift to the right because
$M^{*}=\frac{\mathcal{A}}{A^{*}}$. We also note that the channel capacity
degrades if the number of detectors or elements in the array is too large.
This is due to the fact that each detector contributes thermal noise because
of the resistance generated by detector contacts and circuit wires, and the
variance of this noise is independent of the area of each detector. If the
number of detectors is two large, the noise contribution grows linearly with
the number of detectors and the resulting signal-to-noise and capacity
degrades significantly.

Figure 9: Channel capacity as a function of element or detector area $A$ for
EGC array. Figure 10: Channel capacity as a function of number of detectors
$M$ in the array for MRC and EGC combining techniques.

Fig. 11 shows the channel capacity for the general noise (Section VI) case
where the total noise is the sum of both the signal dependent and signal
independent Gaussian noise components. Here, the capacity curves are plotted
for different values of background radiation power $\lambda_{b}$, and for this
set of curves, we assume that the shot noise generated by background radiation
dominates thermal noise. Here, we observe that the channel capacity decreases
monotonically with $\lambda_{b}$. We also note that the optimal area decreases
(shifts to the left) with increasing value of $\lambda_{b}$. This is due to
the fact that the total background noise power—$\lambda_{b}A$—is a function of
area $A$. In order to minimize the effect of large $\lambda_{b}$, the area has
to shrink in order to maximize channel capacity. This is in contrast to the
thermal noise limited regime case in Fig. 6 where the noise power was
independent of detector area $A$, and the optimal area shifted to the right
with increasing noise power.

Figure 11: Channel capacity for background radiation dominated shot noise
regime.

##  VIII Conclusion

In this paper, we carried out a detailed analysis of the relationship between
the photodetector area and the capacity of a free-space optical channel. For
the signal independent Gaussian noise optical channel, we discovered that the
analytical solutions for optimal area are functions of the term
$\left(\frac{\alpha}{\beta}\right)^{\frac{1}{m}}$ where $\frac{\beta}{\alpha}$
is the normalized SNR. This result indicates that the optimal photodetector
area is inversely proportional to signal-to-noise ratio at the receiver. We
also carried out channel capacity maximization for intensity fading channels
and derived suboptimal solutions that provide quite a good approximation to
optimal solutions, especially at high SNR. For the signal dependent shot noise
channel that is dominated by noise from background radiation, we discovered
that the optimal area increases with increasing SNR at the receiver, a fact
that is opposite to the channel capacity behavior for signal independent
Gaussian noise. The results of this study can be used to devise a novel
optical wireless receiver that comprises a number of photodetectors of
different areas (see Fig. 2). For such a receiver, a photodetector of
appropriate area can be chosen—based on the estimated value of signal
intensity—in order to mitigate the effect of slow intensity fading either due
to atmospheric turbulence or pointing error.

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# Information Extraction through AI techniques: The KIDs use case at CONSOB

Domenico Lembo1 Alessandra Limosani2 Francesca Medda2
Alessandra Monaco1 Federico Maria Scafoglieri1 1Sapienza Università di Roma
2Commissione Nazionale per le Società e la Borsa 1{lembo, monaco,
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

In this paper we report on the initial activities carried out within a
collaboration between Consob and Sapienza University. We focus on Information
Extraction from documents describing financial instruments. We discuss how we
automate this task, via both rule-based and machine learning-based methods,
and provide our first results.

##  1 Introduction

In Italy, _Commissione Nazionale per le Società e la Borsa (Consob)_ is the
supervisory and regulatory authority of the financial market. Among the
several functions, Consob has the role of monitoring and supervising any
financial instruments that are issued, with the ultimate aim of detecting and
enforcing against illicit conduct. In this project, Consob, rather than
outsourcing the digital activity to a specific operator, has taken the
decision to develop the project through a collaboration between expertise from
Academia and its own expertise.

The objective of such an approach is twofold: first to develop a bottom-up
approach driven by the actual needs of supervisory activity; and second, to
explore and experiment in innovative ways the improvement of supervisory
effectiveness. The results to date show not only the digital improvement but
also a direct involvement and participation of the Consob teams and thus
effective knowledge spill-overs from one Division to another have taken place.

The specific project we will discuss in the following, besides Consob,
involves Sapienza University of Rome, and namely a team from the department of
Computer, Control, and Management Engineering, with expertise in Artificial
Intelligence and Information Management. To support Consob in its supervision
activities, and in particular in the task of verifying the correctness and
completeness of the information about financial instruments, within this
collaboration a solution for the automatic extraction of structured
information from free text documents is being developed. Through such
solution, relevant data contained in the documents are not to be manually
identified and extracted, but can be collected through an automated process
which make them available in a machine processable format. To this aim,
essentially two approaches to automatic Information Extraction (IE) have been
adopted: rule-based and machine learning (ML)-based. In the rest of the paper
we describe them, explain why both are useful to achieve our goals, and report
on the first results we obtained in the project, which have been partially
reported, limited to the rule-based approach, in [?].

Figure 1: Portion of a KID by CreditSuisse.

##  2 The use case

In the EU, the creators of financial products (a.k.a. financial manufacturers)
are obliged by law111PRIIPs Regulation n. 1286/2014 to make information
related to so-called PRIIPs (Packaged Retail Investment and Insurance-based
Investments Products) publicly available. The NCAs (National Competent
Authorities) have supervisory duties on such products, so that they can be
safely placed on the respective national markets. The legislation requires
information about PRIIPs to be communicated to NCAs through documents called
KIDs (Key Information Documents). In the practice, this means that features to
be checked are cast into text reports, typically formatted as pdf files (cf.
Figure 1), and extracting structured data from them (to bootstrap control
activities), is actually in charge to the authority (In Italy, CONSOB). Due to
the massive amount of documents to be analyzed (e.g., $\sim$700.000 KIDs
received by CONSOB in 2019, more than 1 million in 2020), this process cannot
be carried out manually, but still it is only partially automated to date.

##  3 The approach

The KIDs present key information about financial products through free text or
tables. We decided to extract relevant data contained in the free text part of
the documents through a rule-based mechanism using language that is an
evolution of regular expressions [?; ?]. This choice lies in the fact that the
KIDs follow a quite rigid template imposed by the European authority, a
characteristic that makes domain experts able to express quite precise
patterns for the identification of wanted data, and AI experts to encode them
into extraction rules.

However, rule-based mechanisms do not lend themselves well to the extraction
of data contained in tables, which are particularly difficult to identify in
pdf documents, depending also on the way in which the table is originally
formatted. For this reason the chosen extraction approach is based on learning
systems that treat the pdf as if it were an image and aim to identify the set
of pixels that are tables, reconstructing the cells from which the information
sought will be identified. In the following we will discuss in more detail the
two approaches highlighting their complete pipelines and showing the results
obtained.

###  3.1 Rule-based IE

The rules we defined deal with the extraction of 16 information fields from
the free text contained in a KID. Among them we mention the product name, the
product ID (a.k.a. ISIN), the manufacturer, the investment risk, and so on.

The rule-based extraction, is organized in three macro modules described in
the following, each of them dealing with a specific task.

Data Preparation: The Data Preparation Module transforms the PDF into plain
text and clean it from errors. This step is very critical, and if the PDF-to-
text transformation produces bad quality results, many errors do actually
occur in the subsequent phases. For this reason, we performed it using
PDFBox222https://pdfbox.apache.org by Apache, a state-of-the-art and highly
customizable library for working with PDF documents. PDFBox presents a large
number of options to manage such transformation, including those for dealing
with different types of characters, font normalization, white space
management, reading modes, etc. We set them in order to minimize the number of
errors incoming from the transformation.

Annotation: The annotation task is carried out through CoreNLP, an open-source
JAVA toolkit for natural language processing, whose first release dates back
to 2006, and which is still maintained and evolving. This toolkit provides
core NLP services accessible through APIs that can be combined to generate a
pipeline for text annotation. Among these services, we mention those related
to tokenization, lemmatization, POS tagging, all used in our pipeline.

Another core component of CoreNLP is TokensRegex [?], which extends
traditional regular expressions on strings by working on tokens instead of
characters and defining pattern matching via a stage-based application. These
extensions of regular expressions allow for writing _extraction rules_ , i.e.,
rule-based extractors matching on additional token-level features, such as
part-of-speech annotations, named entity tags, and custom annotations.

We introduce the syntax of a rule written in TokensRegex through the following
example and refer the reader to [?] for a complete treatment. Consider the
following TokensRegex extraction rule useful for extracting International
Securities Identification Numbers (ISINs) from (the free text portion of)
KIDs:

$StartISIN = (/ISIN/ /:/ | /Codice/ /del/ /Prodotto|prodotto/ /:/ |...)$EndISIN = (/*/ ...)$code = "/([A-Za-z][A-Za-z][0-9]{10})/"{ ruleType: "tokens", pattern: ( ($StartISIN) (?$CodeISIN [{word:$code} & {SECTION:"SECTION_PRODUCT"}]+?) ($EndISIN) ), action: ( Annotate($CodeISIN, ISIN, "ISIN"))}

The ISIN is an alphanumeric sequence of 12 characters identifying a PRIIP.
This sequence begins with two characters, identifying the country code of the
product, followed by ten numbers. In our TokensRegex rule, the ISIN format is
compiled into the regular expression assigned to $code. Then, $StartISIN and
$EndISIN represent the set of sequences of tokens (separated from each other
through the symbol |) preceding or following the ISIN code, respectively. The
JSON-like structure at the end of the rule is its main part. In a nutshell,
ruleType specifies how the pattern should be used. In our case, the rule works
on tokens specified by "tokens" value. pattern contains the pattern to be
matched over the text, built over the tokens using groups like the ones in
POSIX (but also with names such as ?$CodeISIN). In our example, our pattern
specifies how to find the token such that there is a matching with the regex
$code among all the tokens in the product section (i.e. annotated with
SECTION_PRODUCT), and such that it is preceded by the tokens in $StartISN and
followed by the ones in $EndISIN. Finally action describes what should happen
when the pattern is matched, precisely, what annotation to apply. In our
example, we annotate the token identified by the group $CodeISIN with the
annotation ISIN.

The result of the application of this rule over the text in Figure 1 is the
annotation of CH0524993752 with ISIN.

Exporter: The purpose of the Exporter module is to prepare the output in the
desired form. Actually, this module is able to transform the text annotations
into two possible formats (_i_) CSV (Excel) file or (_ii_) database. In
particular, option (_ii_) allows to arrange data in various formats, and paves
the way for the construction of knowledge graphs on top of the extracted data,
e.g., via an OBDM approach [?].

###  3.2 Machine Learning-based IE

Figure 2: Pipeline for table extraction

We used ML techniques to properly deal with tabular data, and in particular to
extract data from the following tables:

  * •

Performance scenarios: It shows the potential refund and the average yield per
year in all the possible scenarios: stress, unfavorable, moderate, favorable,
each at fixed time periods (initial, intermediate, recommended detention).

  * •

Costs evolution: It includes data about total costs and reduction in yield
(RIY) in percentage at the same time periods considered for the performance
scenarios.

  * •

Costs composition: It describes the RIY for different costs categories and
subcategories: input (una tantum costs), output (una tantum costs), wallet
transactions (recurrent costs), others (recurrent costs), performance fees
(accessory charges), overperformance fees (accessory charges).

We modeled the problem as a Computer Vision task, converting each page of the
KID into of images. In such a setting, the Extraction involves a Table
Detection task, aiming at detecting bordered or borderless/semi-bordered
tables, a Table Structure Recognition task, able to identify the cells
belonging to the detected table, and a Text Extraction task, to get the
textual information contained inside each cell.
We propose an automatic, learning-based approach, exploiting the power of Deep
Learning and Artificial Neural Networks (ANN) for image processing. In
particular, for subtasks 1-2 we use CascadeTabNet [?], a recent implementation
of a ANN model that benefits from convolutional layers to learn image features
at different level of abstractions, detecting table areas and related cells.
The model, that has proved to achieve the state-of-the-art for the considered
task, combines two existing CNNs for object detection and image segmentation
with a 2-stage transfer learning and image augmentation techniques. In
details, HRNet is used as a feature extractor, while Cascade R-CNN predicts 3
classes of objects: (i) bordered table, (ii) borderless table and (iii) cell.
In order to adapt the object detection capability of the network to the
desired domain (tables inside images), the combined network, initially pre-
trained on ImageNet dataset, is retrained on images containing tabular data;
the transfer learning is applied firstly on a large, general dataset, allowing
the network to learn table masks, and then on a smaller dataset. Depending on
the specific dataset used for the transfer learning, different models are
developed; we use the Modern Table Structure Recognition model, trained on
ICDAR-19.
For subtask 3 instead, we use Tesseract, a popular Optical Character
Recognition (OCR) engine, originally developed at HP and later by Google. The
original Tesseract engine (’legacy engine’) was based on multiple steps such
as page segmentation, outline and blob detection, textlines breaking, 2-pass
word recognition with a trained classifier. From Tesseract version 4, an
improved engine is available, fully based on LSTM neural networks, and many
new languages are supported.
A complete view of the pipeline of the proposed approach is shown in Figure 2
and its module are:

  * •

A Page Identification Module uses a set of strings for which we should find a
match in one of the pages. For each table type (i.e. performance scenarios,
costs evolution, costs composition) we select a different set of strings that
identify the precise section containing the table we aim to extract.

  * •

The Table Detection and Table Structure Recognition Module uses the pre-
trained CascadeTabNet model to identify the masks (i.e. bounding boxes) of
borderless/bordered tables and cells, with a certain level of confidence. The
masks are returned if and only if their level of confidence is higher than the
input threshold, that we experimentally tuned to 0.6. Cells are then assigned
to the correspondent table depending on their positions.

  * •

The Table Identification and Row Extraction Module applies the OCR to each
cell bounding box, configuring Tesseract with the LSTM engine, the english and
italian languages, and assuming a single uniform block of text for page
segmentation. Again, a string match identifies the table of interest, since
many tables may be present in a single page. If the table is founded, the
algorithm returns a set of rows. To assign each cell to its row we use the top
coordinate of the cell mask but, since very often the bounding boxes of cells
belonging to the same row are not precisely aligned (i.e. one top value is
slightly different than the other), we defined an alignment factor such that
two cells belong to the same row if their top value differs from at most the
alignment factor.

  * •

Rows are arranged into fields in the Row Processing and Cleaning Module,
assigning the numerical cells to their field type. A final Cleaning step,
fully regex-based, deals with missing punctuations, extra space removal, OCR
error correction and, mostly important, standardizes the numerical formats:
many countries, indeed, use a comma to separate the integer from the decimal
part, while others use a point. Currency units of measure are also removed
from the costs, extracting just the numerical information.

The effectiveness of the results often depends on the table template: there
are few tables that the algorithm fails to recognize at all, or for which some
cells are not detected (mostly the table headers). Table/cell detection is
indeed a challenging task, due to different reasons. Firstly, the
heterogeneity of table templates adopted by the PRIIP authors makes the task
harder for the neural network; templates are slightly different from the ones
in ICDAR-19, because they are modern, often semi-bordered or borderless, semi-
coloured, with multi-line cells. The same table type may present different
number of rows or columns, just because the PRIIP type requires it, or because
of a different arrangement of the information in the table (for instance, a
single cell contains both a cost and a RIY). Moreover, we noticed issues in
identifying multi-line cells: few times true multi-line cells are detected as
two distinct cells or, on the contrary, distinct cells that appear one under
the other, probably very close, are detected as a single multi-line cell. We
addressed this last case in the Rows Cleaning Module: using our knowledge of
the table structures, knowing which field types appear one under the other, we
split the wrongly multi-line cells and assign the data to the proper field.
Regarding cell masks, we noticed also that, in few cases, the detected
bounding box was quite inaccurate, cropping the text and consequently causing
troubles in OCR. In order to avoid it, we consider an enlargement factor such
that we run the OCR on a slightly larger area. The correctness of the results
indeed, depends both on the CascadeTabNet and OCR performances. Our first
results showed few inaccuracies in recognizing some characters; as an example,
many ’7’ were recognized as ’/’, or some digits were recognized as alphabetic
characters. Changing the OCR engine type from LSTM to legacy gave no
improvement at all. Running experiments on the documents that caused the
trouble, we discovered that they mostly share the same table type (performance
scenarios), the same author and, therefore, the same template, that consisted
of a coloured and saturated background with a white text. Since the latest
versions of Tesseract has some troubles in detecting white text on a darker
background, we addressed this issue by improving Image Pre-Processing. In
particular, for each cell, we checked if it has white text on a coloured
background and if so we inverted the image. An important improvement has been
obtained moving in this direction, removing more than half of the OCR errors
in the troubling documents; the remaining errors are all of the same type (’7’
as ’/’). Grayscaling and adding more contrast to the image allowed to further
reduce the errors.

##  4 Results

Table 1 provides some information regarding the two datasets on which we
tested our approach.

Dataset | Total KIDs | Size | Manufacturers
---|---|---|---
DATASET-1 | 1240 |  250MB | 36
DATASET-2 | 7736 |  3GB | 52
Table 1: Dataset Info

Dataset-1 was created specifically as a test scenario. It is representative of
document heterogeneity and includes KIDs from the years 2018-2020. Dataset-2
was selected randomly from a sample of KIDs from the first half of 2021.

Table 2 reports an average of precision and recall over all information
extracted through the rule-based approach.

Dataset | Precision | Recall | F-Measure
---|---|---|---
DATASET-1 |  98% |  96% |  97%
DATASET-2 |  98% |  94% |  96%
Table 2: Results rule-based approach

Table 3 shows the number of tables extracted through our ML-based approach for
each table type.

| Dataset
---|---
DATASET-1 | DATASET-2
Performance Scenario | Extracted | 1218 | 6435
Missing | 22 | 1301
Cost Evolution | Extracted | 752 | 3925
Missing | 488 | 3811
Cost Composition | Extracted | 1153 | 6886
Missing | 87 | 850
Table 3: Results of the Extractor for the different table types.

We obtained the worst recall for the Costs Evolution table; this is probably
due to the fact that the extraction of this table mostly relies on just two
header cells in the Table Identification step, therefore, if the model fails
for any reason in detecting just those two cell masks, the table is not
extracted even though the numerical cell masks are often correctly detected.
This issue is less critical for the other table types, since more cells can be
used for Table Identification.

##  5 Conclusion

We are currently working to complete IE from KIDs, by attacking some of their
portions that have not been so far involved in the extraction process. The
next project steps concern with post-extraction data analysis, aimed to
identify anomalous documents, correct errors and missing mandatory data,
discover interesting correlations among data.

## References

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