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123
Jie Hu Kun Yang
School of Information and Communication School of Computer Science and Electronic
Engineering Engineering
University of Electronic Science and University of Essex
Technology of China Colchester
Chengdu, Sichuan UK
China
© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd., part of Springer Nature
2018
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Preface
v
vi Preface
vii
viii Contents
AC Alternative Current
AWGN Additive White Gaussian Noise
CDF Cumulative Distribution Function
CDMA Code Division Multiple Access
CICO-MC Continuous Input Continuous Output Memoryless Channel
DC Direct Current
DEIN Data and Energy Integrated communication Network
DIDO-MC Discrete Input Discrete Output Memoryless Channel
FDD Frequency Division Duplex
H-BS Hybrid Base Station
IoT Internet of Things
KKT Karush–Kuhn–Tucker
LPF Low-Pass Filter
MAC Medium Access Control
MIMO Multiple Input Multiple Output
MISO Multiple Input Single Output
mmW millimetre Wave
NASA National Aeronautics and Space Administration
NOMA Non-Orthogonal Multiple Access
NSFC National Natural Science Foundation of China
OFDMA Orthogonal Frequency Division Multiple Access
PAPR Peak to Average Power Ratio
PS Power Splitting
PSK Phase Shift Keying
QAM Quadrature Amplitude Modulation
QoS Quality of Service
RF Radio Frequency
SCMA Sparse Code Multiple Access
SER Symbol Error Ratio
SIMO Single-Input-Multiple-Output
ix
x Acronyms
SISO Signle-Input-Single-Output
SS Spatial Splitting
SVD Singular Value Decomposition
SWIPT Simultaneous Wireless Information and Power Transfer
TDD Time Division Duplex
TDMA Time Division Multiple Access
TS Time Switching
UE User Equipment
UESTC University of Electronic Science and Technology of China
WET Wireless Energy Transfer
WIT Wireless Information Transfer
WPCN Wireless Powered Communication Network
5G Fifth Generation
Chapter 1
Data and Energy Integrated
Communication Networks: An Overview
According to the prediction of the classic Moore’s Law, the density of transistors in an
integrated circuit doubles approximately every two years, which have been fuelling
the spectacular proliferation of electronic devices since the 1960s. Furthermore, con-
© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd., 1
part of Springer Nature 2018
J. Hu and K. Yang, Data and Energy Integrated Communication Networks,
SpringerBriefs in Computer Science, https://doi.org/10.1007/978-981-13-0116-2_1
2 1 Data and Energy Integrated Communication Networks: An Overview
sumer electronic devices are becoming shirt-pocket-sized and mobile. These devices
are normally powered by embedded batteries. However, as their functions become
ever more sophisticated, their thirst for abundant energy is not matched by the slow
progress of the batteries’ capacity. The situation in the communication industry is
even more daunting. Since the roll-out of the fifth-generation (5G) cellular system and
of the Internet of Things (IoT) is just around the corner, people’s appetite for super-
high data transmission rates, for high density of connectivity and for high mobilities
will indeed be satisfied to a large extent. A major portion of the future mobile data
traffic will be constituted by novel types of services, including high-definition stero-
scopic video streams, augmented/virtual reality, holographic tele-presence, cloud
desktops, as well as online games, etc. All these services require the user terminals
to be implemented with high computing capabilities for real-time signal process-
ing, which may quickly drain the embedded batteries. Furthermore, sensors will
be deployed in every corner of the future smart cities [1]. These sensors monitor
the environment and upload sensing results to central servers [2]. The life-span of
sensors and of sensing networks largely depends on the sensors’ battery capacity.
Regularly replacing the batteries may be an unrealistic or tedious task. Accordingly,
new sources of energy have to be explored to prolong the depletion period of con-
ventional batteries in order to relieve the energy concerns of various communication
devices.
Nowadays, resonant inductive coupling [3] and magnetic resonance coupling [4]
have emerged for remotely charging electronic devices in the near-field. Resonant
inductive coupling based wireless charging relies on the magnetic coupling that
delivers electrical energy between two coils tuned to resonate at the same frequency.
This technique has already been commercialised for some home electronic appliances
[5], such as mobile phones, electric toothbrushes and smart watches etc. However,
the coupling coils only support near-field wireless energy transfer (WET) over a
distance spanning from a few millimetres to a few centimetres [6], while achieving
a WET efficiency as high as 56.7%, when operating at a frequency of 508 kHz [7].
Furthermore, resonant inductive coupling requires strict alignment of the coupling
coils. Even a small misalignment may result in dramatic reduction of the WET
efficiency [8]. As a result, during the charging process, the electronic appliances
cannot be freely moved.
By contrast, magnetic resonance coupling [9] delivers electrical energy between
two resonators by exploiting evanescent-wave coupling. This technique has already
been adopted for charging the electric vehicles due to its high WET efficiency [10].
For example, magnetic resonance coupling is capable of achieving a WET efficiency
of 90% over a distance of 0.75 m [11]. Both its WET efficiency and its charging dis-
tance are much higher than that of the resonant inductive coupling. However, mag-
netic resonance coupling still belongs to the category of near-field wireless charging,
since its power transfer efficiency dramatically reduces to 30%, when the distance is
increased to 2.25 m [11]. Nonetheless, magnetic resonance coupling does not require
1.2 Near-Field Wireless Energy Transfer 3
strict alignment between the rechargeable device and the energy source. Hence, dur-
ing the charging process, the electronic appliances may be moved within the charging
area [12]. Furthermore, a multiple-input-multiple-output (MIMO) system, which has
already been widely adopted for improving the performance of the wireless commu-
nication, can also be introduced into the magnetic resonance coupling based WET
system for the sake of further enhancing the WET efficiency [13, 14].
Since RF signal based WET techniques require highly fexible beam directivity in
order to satisfy diverse charging requests, the best spectral band for steering energy
beams is in the range of 10 MHz to 100 GHz, which almost covers all the bands
allocated for wireless communication services. For example, TV/Radio broadcasting
services operate in the band spanning from 40 MHz to 220 MHz [24], the mobile
cellular communication system operates in the spectral band spanning from 800 MHz
to 3.7 GHz [25], while the WiFi communication system operates in the spectral
band spanning from 2.4 GHz to 6 GHz [26]. Furthermore, as a key technique in
the upcoming 5G era, millimetre wave (mmW) [27] may significantly increase the
achievable throughput of the air interface, which operates in the spectral band ranging
from 10 GHz to 100 GHz.
Although they both operate in the same RF band, WET and WIT still have the
following distinctive characteristics:
• They have different functional circuits. RF signals in the pass-band cannot be
directly invoked for both the information decoding and the energy harvesting.
For the information decoding, the RF signals in the pass-band have to be firstly
converted to the base-band, since all the signal processing has to be accomplished
in the base-band. By contrast, for the energy harvesting, the AC energy carried by
the RF signals has to be converted to the DC energy first, since only DC energy
can be stored in batteries or drive electronic loads. Specifically, during the AC-DC
conversion, the phase information carried by the RF signals is filtered.
• They require different absolute energy at receivers. The activation of the energy
harvesting circuits requires a relatively high energy carried by the received RF
signals, which is approximately on the order of −20 dBm. If the energy carried
by the received RF signal does not achieve the required activation threshold, none
of this energy can be harvested. By contrast, the successful information recovery
relies on the energy ratio between the received RF signal and the noise plus inter-
ference, not on the absolute energy carried by the received RF signal. As a result,
even a small amount of energy is capable of activating the information receiver,
which is approximately on the order of −80 dBm.
• They have different coverage. The RF signals are attenuated by hostile wireless
channels, such as the path loss, shadowing and multipath fading. Since the energy
harvesting requires a much higher absolute energy at the receivers than the infor-
mation decoding, the range of WET is accordingly much shorter than that of WIT.
Therefore, given the same set of transmitters and receivers, the resultant WET
network has a different topology with the WIT network.
• They treat noise and interference differently. The interference and noise ubiqui-
tously exist in any WIT system, which seriously impair the WIT performance.
Mitigating the performance degradation induced by the interference and noise is a
major challenge in the WIT system design. By contrast, WET systems may actu-
ally benefit from the interference and noise, since both of them are RF signals and
they both carry useful energy. The interference and noise can be jointly harvested
6 1 Data and Energy Integrated Communication Networks: An Overview
by the energy harvesting circuits, which may provide additional energy harvesting
gains for the energy requesters.
• They have different definitions in energy efficiency. The energy efficiency of WET
can be defined as the ratio of energy harvested by the receiver to the energy emitted
by the transmitter, which can be formulated as
1
ηW E T = · ρ (Pr + PI + PN ) (Watt/Watt), (1.1)
Pt
where ρ is the conversion rate from the received RF energy to the DC energy by
considering a linear RF-DC converter. By contrast, In the community of green
communications, the energy efficiency of WIT is defined as the ratio of spectral
efficiency to energy consumption, which is evaluated in the unit of bps/Hz/Watt or
bps/Hz/Joule. By exploiting the classic Shannon-Hartley theorem in an Additive-
White-Gaussian-Noise (AWGN) channel, the energy efficiency of WIT can be
expressed as
1 Pr
ηW I T = · log2 1 + (bps/Hz/Watt), (1.2)
Pt PI + PN
where Pt is the transmit power of the RF signal, Pr is the power received after
the signal being attenuated by the hostile wireless channel, PI is the aggregate
interference power and PN is the noise power at the receiver.
In Fig. 1.1, we exemplify the energy efficiency of WET and that of WIT, which
can be calculated by (1.2) and (1.1), respectively. Observe from Fig. 1.1a that in
our setting, the energy efficiency of WET reduces from 1.1% but converges to
1%, which is due to the channel attenuation incurred by the path loss between the
transmitter and receiver pair. Observe from Fig. 1.1b that the energy efficiency of
WIT gradually reduces from 35 bps/Hz/mW to 0 as the transmit power of the RF
signal increases. By contrast, WET and WIT operating in the same RF spectral
band may compete for the precious resources in the air interface and they may
thus impair each other’s performance to some extent. For example, WET requires
that the RF signals carry a high power to the receivers for the efficient energy
harvesting. However, the high-power RF signals of the WET system may impose
excessive interference on the WIT receivers, which may thus significantly degrade
the WIT performance attained. As a result, coordinating WET and WIT in the same
RF band imposes critical challenges on the RF circuit design, on the transceiver
design of the physical layer, on the resource scheduling/allocation schemes and
on the corresponding protocol design of the medium-access-control (MAC) layer.
Furthermore, integrated data and energy transfer in the RF band also requires
a joint networking concept for heterogeneous data and energy transceivers. All
these challenging issues require novel Data and Energy Integrated Communication
Networks (DEINs) [33].
1.5 Ubiquitous Architecture of the DEIN 7
(a) (b)
1.2 40
1.05 1.0 35
Energy efficiency of WET (mW/mW %)
0.6 20
0.45 15
0.3 10
0.15 5
0 0
10 15 20 25 30 35 40 45 50 10 15 20 25 30 35 40 45 50
Transmit power (dBm) Transmit power (dBm)
Fig. 1.1 Energy efficiency of WET (a) and that of WIT (b) against transmit power of RF signals.
The noise power is PN = −94 dBm, which is calculated by the power spectrum density of the
thermal noise −174 dBm/Hz and 100 MHz of the RF signals’ bandwidth. The aggregate interference
power at the receiver is set to be PI = −20 dBm, which appears in a heterogeneous cellular network
with the highest probability [28]. The distance between a transmitter and receiver pair is 10 m. The
path loss is calculated by the model invoked in [29–32], where the path loss exponent is 2. No fading
is assumed. The antenna gain in this example is set to be 40 dBi in order to counteract the path loss
DEINs are naturally heterogeneous in terms of all their technical aspects. We will
investigate the heterogeneity of the DEINs and synthesise a diverse range of WET
and WIT scenarios into its ubiquitous architecture, which is exemplified in Fig. 1.2.
First of all, there are various types of infrastructure elements in heterogeneous DEIN.
As portrayed in Fig. 1.2, we have generally three basic type of infrastructure in
DEINs, namely DEIN stations, WET stations and WIT stations/relays. DEIN stations
[34] are capable of operating both as information transmitter and as energy transmitter
for satisfying both of the user equipments’ (UEs’) data and energy requests. Thanks to
their powerful functionalities, DEIN stations are also capable of realising integrated
data and energy transfer for the sake of increasing the spectrum efficiency of the
congested RF band. Therefore, DEIN stations have to be connected to the core
communication network and they also have to be powered by stable energy sources,
such as large solar energy harvesters and the power grid. As illustrated in Fig. 1.2,
8 1 Data and Energy Integrated Communication Networks: An Overview
DEIN Devices
WIT-UE-2 for IoT
WIT
WIT
Narrow beam for point-to-
WIT WET WET point WET
WIT WET
DEIN- DEIN-UE-1 DEIN- (Directional Antenna)
WIT-Relay-1 Station-1 WIT WIT Station-2 WIT
WET
WIT-Relay-2 WIT
WET-Station-3 WET-UE-2
WIT Range WET
(Isotropic Antenna) DEIN-UE-2
WET-Station-1
WET-UE-1
DEIN-Station-1 may satisfy the integrated data and energy requests from the IoT
devices and those from DEIN-UE-1.
However, as we have discussed in Sect. 1.4, the reliable WET range is far shorter
than the reliable WIT range, as exemplified in Fig. 1.2. As a result, some blind
areas cannot be adequately covered by WET of DEIN stations. Furthermore, some
dedicated WET [35] stations are also deployed in order to supply energy to the
devices roaming in these blind areas. These WET stations are only connected to
energy sources, but they do not have to be connected to the core communication
network. As a result, they are dedicated for satisfying the UEs’ charging requests.
For instance, as shown in Fig. 1.2, three WET stations are deployed in order to supply
energy to the UEs beyond the WET range of the DEIN stations.
Apart from DEIN stations and WET stations, there are still many conventional
communication stations in heterogeneous DEINs, namely the classic femto-cellular
stations, pico-cellular stations and macro-cellular stations [36]. These communica-
tion stations have different levels of transmit power and coverage, which results
in obvious heterogeneity in DEINs. Sometimes, low-cost relay stations are also
deployed for forwarding the data packets to cell-edge UEs, as illustrated in Fig. 1.2.
However, small cellular stations and relay stations [37] are only capable of emitting
RF signals at a limited power. They are not suitable for carrying out sophisticated
WET tasks. Therefore, they are regarded as a dedicated communication infrastruc-
ture.
Apart from the heterogeneous infrastructure, our DEINs have to accommodate both
charging and communication requests from diverse types of UEs. We generally have
1.5 Ubiquitous Architecture of the DEIN 9
three types of UEs in DEINs, namely the WIT UEs, the WET UEs and the DEIN
UEs [38], as exemplified in Fig. 1.2. WIT UEs only require downlink and uplink
data transmission in DEINs. Since these UEs are always powered by stable energy
sources, they do not request any wireless charging from the DEIN stations. Laptops
and tablets are typical WIT UEs, which are either powered by high-capacity batteries
or are connected to the power grid. For example, as illustrated in the left part of
Fig. 1.2, WIT-UE-1 receives its requested data from DEIN-Station-1 with the aid of
two WIT relay stations, while WIT-UE-2 may consume its own energy for powering
its uplink information transmission.
By contrast, since WET UEs are not powered by stable energy sources, they
have to request additional energy supply either from the DEIN stations or from the
WET stations in order to support their basic functionalities, such as uplink informa-
tion transmissions and energy-consuming computations [39]. For instance, although
WET-UE-1 is beyond the WIT range of DEIN-Station-1, it may still establish reli-
able uplink transmissions with DEIN-Station-1 by exploiting the additional energy
received from WET-Station-1, as exemplified in Fig. 1.2. Similarly, the uplink trans-
mission of WET-UE-2 towards DEIN-Station-2 is powered by DEIN-Station-2 itself.
Miniature-sized IoT devices are typical WET UEs, since their functionalities are lim-
ited by the amount of energy stored in their batteries.
Furthermore, some UEs simultaneously request data and energy transmissions,
which are regarded as DEIN UEs [40]. For instance, in the right cell of Fig. 1.2, DEIN-
UE-1 simultaneously receives its requested data and energy from DEIN-Station-2,
while DEIN-UE-2 also simultaneously requests both downlink data transmission and
wireless charging. However, since DEIN-UE-2 is beyond the WET range of DEIN-
Station-1, it can only receive the requested data from DEIN-Station-2, but it can
receive energy from WET-Station-1. This energy may be exploited for supporting
DEIN-UE-2’s uplink data transmission to its associated DEIN-Station-2.
Sometimes, the functionalities of WIT relay stations are also limited by their
energy supply, especially when the WIT relay stations rely on energy gleaned from
batteries or harvested from renewable sources. As a result, they also need wireless
charging from DEIN stations or WET stations for powering their data packet for-
warding actions [41]. As a result, WIT relay stations can also be regarded as special
“DEIN UEs”. As portrayed in the left cell of Fig. 1.2, both data and energy are simul-
taneously transferred from DEIN-Station-1 to WIT-Relay-1. The energy harvested
by WIT-Relay-1 may be further exploited for forwarding the data packets to the
next hop. Since WIT-Relay-2 is beyond the WET range of DEIN-Station-2, it has to
request WET from the nearby WET-Station-2 and WET-Station-3. After receiving
the data packets from WIT-Relay-1 and gleaning sufficient energy from the WET
stations, the data packets are finally forwarded to their destination WIT-UE-1 by
WIT-Relay-2.
10 1 Data and Energy Integrated Communication Networks: An Overview
Our DEIN architecture has to accommodate both the WET and WIT in the same
RF spectral band. Although the WET and WIT both rely on the RF signal, they still
have distinctive features, as summarised in Sect. 1.4. Therefore, in order to satisfy
the UEs’ information and energy requests, the coexistence of WET and WIT in the
DEIN results in natural heterogeneity.
In order to guarantee the seamless WIT coverage, different techniques have to be
invoked. As exemplified in Fig. 1.2, when the omnidirectional antennas are adopted,
the boundary of a DEIN cell is determined by the WIT range of a DEIN sta-
tion. As a result, the UEs residing within the WIT range of a DEIN station may
receive their requested information via a single-hop cellular link. Furthermore, these
UEs are also capable of uploading information to their associated DEIN stations.
Observe from Fig. 1.2 that WIT-UE-2, DEIN-UE-1 and DEIN-UE-2 all receive their
requested information from the downlink WIT of their associated DEIN stations,
while WET-UE-2 and DEIN-UE-2 both upload their information to their associated
DEIN-Station-2. By contrast, in order to satisfy the information request of a UE
sitting beyond the WIT range of a DEIN station, multiple relay stations have to be
relied upon for forwarding the information from the DEIN station to the requester
or in a reverse direction via the multi-hop transmissions, such as the downlink trans-
mission from DEIN-Station-1 to WIT-UE-1 of Fig. 1.2. In addition, by exploiting
the extra energy supplied by the WET stations, a UE beyond the WIT range of a
DEIN station is also capable of uploading data to the DEIN station [39], such as the
uplink transmission of WET-UE-1 to DEIN-Station-1 in Fig. 1.2, which is powered
by WET-Station-1.
If we further look into the wireless charging actions in DEINs, various WET tech-
niques have to be invoked for satisfying diverse charging requirements. As illustrated
in Fig. 1.2, a DEIN station’s WET range is much shorter than its WIT range, when
the omnidirectional antenna is adopted. The reason is that for the successful WET,
the energy harvesting circuit of the receiver can only be activated by a high received
energy. As a result, the WET is more sensitive to the wireless channel attenuation,
which is dominated by the path loss. If a WET UE resides within the WET range of
a DEIN station, it may successfully harvest energy from the RF signal emitted by
this DEIN station. By contrast, when a WET UE is beyond the WET range, it has
to request energy from its nearby WET station. Directional antennas may enable a
DEIN station to focus its energy in the main-lobe, which substantially increases the
long-range WET efficiency in the direction of the main-lobe. However, the resultant
energy loss in the side-lobes may significantly reduce the WET efficiency in other
directions. If directional antennas are adopted by the DEIN stations, they may form
a narrow energy beam [42] for charging the WET UE beyond the WET range, which
is characterised by the omnidirectional antennas. As exemplified in the right DEIN
cell of Fig. 1.2, WET-UE-2 is still capable of receiving energy from the dedicated
narrow energy beam forming by DEIN-Station-2.
Furthermore, IoT devices will be pervasively deployed in the near future. Our
heterogeneous DEINs are also responsible for satisfying both of their communication
1.5 Ubiquitous Architecture of the DEIN 11
and energy demands. IoT devices normally are clustered in a specific area in order to
jointly carry out their tasks. As a result, for the sake of satisfying the charging requests
of the multiple IoT devices, DEIN stations may form wide-angle energy beams for
covering the cluster of requesters [43]. This technique may be regarded as energy
multicast. Moreover, this wide beam is also capable of transferring information and
energy together to the multiple requesters.
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Chapter 2
Fundamental of Integrated WET
and WIT
Abstract In order to realise integrated wireless energy transfer (WET) and wireless
information transfer (WIT), we have to revisit the information theory for finding
its performance limits, while redesigning the transceiver architecture in the physi-
cal layer for practical implementation. As a result, in this chapter, we impose the
energy delivery requirement on the channel output sequence, when maximising the
mutual information. The rate-energy tradeoff is studied from the information the-
oretical perspective for both the discrete-input-discrete-output channel and for the
continuous-input-continuous-output channel. Then we provide an overview on the
transceiver architecture in the physical layer by considering diverse signal splitters,
namely the spatial splitter, the power splitter and the time switcher. The resultant
integrated WET and WIT performance is then evaluated for different transceiver
architectures.
Keywords Continuous-Input-Continuous-Output-Channel
Discrete-Input-Discrete-Output Channel · Information Theory · Integrated WET
and WIT · Multiple-Input-Multiple-Output (MIMO) system · Mutual
Information · Power Splitting · Rate-Energy Tradeoff · RF based Wireless
Charging · Simultaneous Wireless Information and Power Transfer (SWIPT)
Spatial Splitting · Time Switching · Transceiver Architecture · Wireless Energy
Transfer (WET) · Wireless Information Transfer (WIT)
In this chapter, we will focus on the fundamental of the physical layer for implement-
ing the integrated wireless energy transfer (WET) and wireless information transfer
(WIT) of the point-to-point link. First of all, the information theoretical essence of the
integrated WET and WIT will be introduced. Key enabling modules of the generic
transceiver architecture for the integrated WET and WIT will also be included. Then,
we will cover the architectures of several popular receivers equipped with multi-
ple antennas for simultaneous information and energy reception, namely the ideal
receiver, the spatial splitting based receiver, the power splitting based receiver and
the time switching based receiver.
© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd., 15
part of Springer Nature 2018
J. Hu and K. Yang, Data and Energy Integrated Communication Networks,
SpringerBriefs in Computer Science, https://doi.org/10.1007/978-981-13-0116-2_2
16 2 Fundamental of Integrated WET and WIT
As previously discussed in Sect. 1.4, WET and WIT entail several conflicting speci-
fications, when they are coordinated in the same radio frequency (RF) spectral band.
As a result, theoretical investigations have to be carried out in order to reveal the
underlying relationship between the WET and WIT in data and energy integrated
communication networks (DEINs), which may provide researchers and engineers
with further valuable insights on improving the system-level performance of DEINs.
In this section, we will explore the information theoretical essence for DEIN and
reveal the natural contradiction between WET and WIT from an information theo-
retical perspective, which requires further efforts for jointly designing energy and
information transfer.
Note that the information theoretical exploration remain valid not only for the
integrated WET and WIT operating in the RF spectral band, but for all other integrated
data and power transfer scenarios, such as power line communication [1] and power
over Ethernet technique [2]. Classic information theoretical channel capacity analysis
has been dedicated to maximising the mutual information under the constraint of
specific input signals. By contrast, the pioneering work of Gastpar [3] has attempted
to maximise the mutual information under the constraint of specific output signals,
which aims for controlling the interference imposed by a communicating pair on
other peers. This piece of work may provide us with valuable hints for finding the
performance limits of integrated data and energy transfer.
As a result, the WIT performance limit can be formulated as the following optimi-
sation problem:
1
Subject to: · E[g(Yn )] ≥ β, (2.2a)
n
1
C(β) = sup · Cn (β), [bit/symbol] (2.3)
n n
which may be regarded as the rate-energy function. Observe from (2.3) that the rate-
energy function is a natural extension of the classic channel capacity concept. This
18 2 Fundamental of Integrated WET and WIT
1 1 1 1 1 1
(a) Noiseless Channel (b) Z Channel (c) Symmetric Channel
I Z (X ; Y ) = −(q +
q ω) log2 (q +
q ω) −
q
ω log2
q +
q ω log2 ω. (2.5)
When the source only sends the energy carrier symbol ‘1’, namely q = 0, the maxi-
mum energy carried by the output symbol at the information destination is βmax = ω
unit. In a nutshell, given a specific energy transfer requirement β, the maximum
achievable information rate of the Z channel can be formulated as [4]
1 ω
log 1 − ω 1−ω + ω 1−ω , 0 ≤ β ≤ (1 − ω)π ∗
C Z (β) = β
[bit/symbol],
H2 (β) − 1−ω
H2 (ω), (1 − ω)π ∗ < β ≤ 1 − ω,
(2.8)
As portrayed in Fig. 2.2c, in a symmetric channel, both the input binary symbols
‘0’ and ‘1’ can be erroneously delivered to the output end with a probability of ω,
while they can be correctly delivered with a probability of
ω = 1 − ω. Apart from
the energy loss incurred by the transition from the input symbol ‘1’ to the output
symbol ‘0’, the energy of the interference may change the input symbol ‘0’ to the
output symbol ‘1’, which results in the energy gain at the information destination.
The mutual information of the symmetric channel is expressed as
I S (X ; Y ) = − (q
ω + q ω) log2 (q
ω +
q ω) − (qω +
q
ω) log2 (qω +
q
ω)
+ω log2
ω + ω log2 ω. (2.10)
—Zult gij goed voor haar zorgen? vroeg Beaupré, terwijl hij Raffles
nogmaals de hand toestak. Bedenk, dat zij alles voor mij is, en dat ik
niets anders op de wereld heb om lief te hebben!
—Ik beloof u, dat ik zooveel als in mijn vermogen is, voor haar zal
waken!
Marthe Debussy boog zich over den gewonde en drukte hem een kus
op het voorhoofd. De twee geliefden wisselden fluisterend eenige innige
woorden, en daarop richtten de beide bezoekers zich naar de deur,
nadat Marthe beloofd had zoo spoedig, mogelijk te zullen terug komen,
en hem het adres had toegefluisterd, waar hij zou kunnen schrijven of
telefoneeren.
Als zij slechts eenige minuten langer waren gebleven, hadden zij
getuige kunnen zijn van een merkwaardig voorval.
—Het is zoo! Het gelaat van dien man kwam mij al aanstonds zoo
bekend voor! Het is een van de twee mannen, die hier gisternacht zijn
geweest, om Dubois te bezoeken!
Hij wendde het hoofd met een ruk naar de zijde van het bed, waar de
nieuwe patiënt lag, en trachtte zijn gelaatstrekken te zien.
Het was een van de beide bandieten, die hem gisteravond onder een
valsch voorgeven hadden bezocht, teneinde hem te bewegen, de
geheimen zijner eigen bende te verraden, in ruil voor het leven zijner
minnares, en te zweren, dat hij nimmer weder als mededinger van Dr.
Fox zou optreden!
Raffles had hem toch zooeven medegedeeld, dat hij die twee kerels, na
hen zoogenaamd te hebben gearresteerd, bewusteloos had gemaakt,
en hij zou toch wel zoo verstandig geweest zijn, dit niet op den
openbaren weg te doen?
Hoe was het mogelijk, dat die man nog altijd bewusteloos was, en dat
geen der doctoren nog naar hem was komen zien?
Hij zou spoediger antwoord op die vraag krijgen dan hij wel vermoedde,
en wel uit den mond van de hoofdverpleegster zelve, een praatgrage
dame, die volstrekt niet kon denken, dat Beaupré deel uitmaakte van
een dievenbende, en meende, dat hij inderdaad het slachtoffer [7]van
een laaghartige poging tot afzetterij was geweest—hetgeen in zekeren
zin ook de waarheid was!
Zij had het bed van den bewustelooze verlaten en trad nu snel op dat
van Beaupré toe.
Toen zij zag, dat hij klaar wakker en blijkbaar volkomen bij zijn
positieven was, begon zij:
—De politie!
Beaupré had dit antwoord verwacht, maar toch liep er een zenuwachtige
rilling over zijn bleek gelaat, welke hij niet geheel en al had kunnen
bedwingen.
—De politie? vroeg hij. Zeker in verband met het bezoek dier beide
kerels, die mij geld wilden afdreigen?
—Juist! Gij zult er van opzien, wat er met hen geschied is! Ik moet u
zeggen, dat ik er niets van begrijp—maar de detective, dien wij hier
verwachten zal wel licht in de duisternis brengen! Hij kan over een
kwartier hier zijn!
Wel hoopte hij, dat zijn gelaat, zooals het nu was, niet meer zou
overeenkomen met het signalement, dat ongeveer drie jaar geleden
door de Parijsche politie was gezonden aan alle groote politiebureaux
over de geheele wereld, vooral omdat hij zijn fraaien, blonden baard uit
dien tijd had afgeschoren, maar hij was daar verre van zeker van.
Hij wist zich echter te beheerschen en vroeg vrij kalm, alsof de zaak
hem eigenlijk niet al te veel belang inboezemde:
—Dat zal ik u zeggen! Het staat nu wel vast, dat de twee rechercheurs,
die hier gisteren de twee schavuiten in de vestibule van dit gebouw
hebben gearresteerd, in het geheel niet tot de politie behoorden, en ook
geen particuliere detectives waren!
—Maar met welk doel? riep Beaupré uit, die eens wilde zien wat de
politie wel, en wat zij niet wist.
—De vier mannen zijn hier voor de deur in een huurauto gestapt en wij
meenden natuurlijk niets anders of de gewaande detectives zouden
hunne arrestanten rechtstreeksch naar Scotland Yard brengen. Er
geschiedde echter iets geheel anders met de twee mannen die hier zijn
geweest! De beide bedriegers hadden hen op de een of andere wijze—
hoe, dat weten de geneesheeren nog niet—bewusteloos gemaakt, en
toen de auto stil stond, hebben zij hen naar een onbewoond huis aan de
Bishop Street gebracht, en daar opgesloten. Maar nu komt het mooie!
De chauffeur, die hen gereden had, was nieuwsgierig uitgevallen! Hij
had een buitengewoon hooge fooi gekregen, om zoo snel mogelijk te
rijden, en dat droeg er niet toe bij, hem te kalmeeren, dat begrijpt gij wel!
—Hij had duidelijk gezien, dat de twee mannen die hem besteld hadden
de beide anderen onder den arm [8]hadden moeten nemen, en dat die er
al heel gek uitzagen met hun wijd geopende oogen, die echter niets
schenen te zien, en hun automatische bewegingen! Hij kon niet
begrijpen, wat die mannen in dat huis gingen uitvoeren, en omdat hij,
zooals gezegd, heel nieuwsgierig was—zoo reed hij niet weg, maar
plaatste zijn auto om een hoek van een dwarsstraat en stelde zich
verdekt op in een donker portiek!
—Na een half uur kwamen er twee mannen uit het huis—en dat waren
de lieden, die geboeid waren binnengeleid! Maar van boeien was niets
te bekennen en zij liepen ook volkomen recht en natuurlijk! Zij schenen
groote haast te hebben en riepen een auto aan, die juist voorbij reed. En
de chauffeur was zoo overbluft, dat de wagen al uit het gezicht was,
voor hij er aan dacht hen met zijn eigen auto te volgen!
—Daar staat mijn verstand bij stil! riep Beaupré uit, ofschoon hij de
geheele zaak volkomen begreep. En waar is de andere?
—En.….. lukt het? vroeg Beaupré snel, terwijl hij de verpleegster met
zijn groote, zwarte oogen vorschend aankeek.
—Neen! Hij ligt daar nog even stil en schijnbaar levenloos, ofschoon het
lichaam warm is, als toen hij hier werd binnen gebracht.
—Kunt gij u in het geheel niet voorstellen wat dit alles te beteekenen
heeft en in welke verhouding die twee zoogenaamde detectives met de
mannen stonden, die u hier gisterenavond zijn komen bezoeken?
—Maar die twee bezoekers van gisteren—die kent gij toch wel?
Die man was James Sullivan, een der bekwaamste detectives van
Scotland Yard, die reeds eenige malen had deelgenomen aan de jacht
op den Grooten Onbekende, en tot zijn felste vijanden gerekend mocht
worden.
Toen trad hij snel op het bed toe en zeide tamelijk kortaf:
—Vergun mij een oogenblik, zuster.….. Dit is zeker de man, die hier
gisterenavond door messteken zwaar gewond werd binnen gebracht?
Sullivan trok haar een weinig terzijde en vroeg op zachten toon, zoodat
de zieke hem niet zou kunnen verstaan:
—Hebt gij dien man alles medegedeeld, wat u zooeven per telefoon is
gezegd?
—Ja, mijnheer! antwoordde de zuster aarzelend, en een weinig
schuldbewust, toen zij de ernstige, grijze oogen zoo strak op zich
gevestigd zag.
—Dat doet mij leed! hernam Sullivan en hij klemde de lippen opeen.
—Dat weet ik niet, maar hij speelt toch in dit alles een vrij dubbelzinnige
rol en het ware beter geweest, als gij hem onkundig hadt gehouden van
wat wij zoo pas ontdekt hebben! Nu, er is niets meer aan te doen—en ik
zou u nu wel gaarne verzoeken mij den man eenige vragen te laten
stellen. Hij schijnt sterk genoeg te zijn, om een kort verhoor te kunnen
ondergaan!
—Dat kan ik nu nog niet zeggen, miss! antwoordde Sullivan kortaf. Gij
kunt er trouwens bij tegenwoordig zijn, en goed opletten, of de man zich
zelf wellicht tegenspreekt!
De detective trad nu op het bed toe, zag den gewonde strak aan en
begon:
—Ik ben detective van Scotland Yard en aan mij is de taak opgedragen
onderzoek te doen, naar het geheimzinnig voorval, dat zich deels in dit
ziekenhuis, deels … ergens anders heeft afgespeeld en waarin gij
eveneens een rol hebt gespeeld, misschien ondanks uzelf! Hoe is uw
naam?
Hij kende den detective van aangezicht zeer goed en wist wie hij was—
een der beste speurneuzen van de politie der Engelsche hoofdstad.
Maar Sullivan herkende hem niet—dat was duidelijk—en dat was in
ieder geval een goede troef!
Dat was een vraag waarop de Franschman niet gerekend had! Want
inderdaad was hij pas drie jaren in Engeland, en daarvan had hij nog
eenige maanden in de Fransche hoofdstad doorgebracht, als chef eener
bende!
—Kunt gij het dan verklaren, hoe het komt, dat men u niets ontstolen
heeft? Uw beurs, horloge, uw zilveren sigarettenkoker zijn allen op uw
persoon gevonden!
Een ander zou door die vraag misschien in verwarring zijn gebracht,
maar niet aldus Beaupré!
Hij had zich nu hersteld en was vastbesloten zijn incognito tot het
uiterste te verdedigen.
—Dan hebben zij het zich eenvoudig verbeeld en waren het burgers die
naderden en die mij gevonden hebben! Gij moet mij de opmerking ten
goede houden, mijnheer, maar dit begint veel te gelijken op een verhoor!
Mag ik weten, wat gij eigenlijk denkt of vermoedt? [10]
Hij wantrouwde dezen man, dat was zeker, maar redenen, deugdelijke
redenen zou hij daarvoor niet kunnen opgeven.
—Ongetwijfeld! Als dit slechts in mijn vermogen is! Vraag vrij uit!
—Als gij mij dit toestaat dan zou ik u willen vragen: wie waren de twee
mannen, die u gisteravond kwamen bezoeken en wat wilden zij van u?
—Het zijn twee schurken, die iets uit mijn verleden weten, waarvan de
openbaarmaking mij groot nadeel zou kunnen berokkenen en daaruit
willen zij munt slaan! antwoordde Beaupré brutaal, ofschoon hij zijn hart
voelde kloppen bij het stellen van deze gevaarlijke vraag. Hun namen
wensch ik om begrijpelijke redenen niet te noemen.
—Maar dan hebben die mannen zich aan een strafbare zaak schuldig
gemaakt! riep Sullivan uit. En als gij een aanklacht in dient, kunnen wij
hen vervolgen wegens poging tot afpersing! Dat kunnen wij slechts dan
doen, als het slachtoffer zelf een klacht bij het parket indient!
—Uw stilzwijgen maakt onze taak niet gemakkelijker! zeide hij. Er heeft
hier een geheimzinnige gebeurtenis plaats gehad, waarin die twee
mannen een gewichtige rol vervullen. En het onderzoek naar de
identiteit van de beide gewaande detectives, die hen zijn komen
arresteeren—met een doel, dat ons volkomen onverklaarbaar is—zou
ons heel wat lichter worden gemaakt, als wij wisten, wie zij zijn!
—Wacht, tot zij uit hun bewusteloosheid ontwaakt zijn, kwam Beaupré
kortaf. Dan zullen zij wel spreken!
Hij had het stoutmoedig gezegd—maar bij zich zelf overwoog hij, dat het
voor hem wel eens zeer onaangename gevolgen zou kunnen hebben,
als de schurken inderdaad begonnen te spreken!
—Dat is ook juist een der meest verrassende zijden van deze gansche
geschiedenis! riep Sullivan uit. Geen der geneesheeren weet te zeggen,
welke eigenaardige verdooving de twee mannen heeft aangegrepen!
—Ik wil u thans niet langer lastig vallen, want gij zult wel rust behoeven.
Later echter hoop ik u nogmaals eenige vragen te mogen stellen.
Hij knikte Beaupré toe en stapte vervolgens op het bed toe waar de
bewustelooze ter neder lag.
Eenigen tijd keek hij onafgebroken naar het witte gelaat met de wijd
geopende oogen en toen schudde hij het hoofd en haalde de schouders
op.
—Ik begrijp er niets van! mompelde hij. Het lichaam is blijkbaar warm en
volstrekt niet stijf—het heeft niet weinig van schijndood!
[Inhoud]
HOOFDSTUK III.
Een raadselachtig geval.
Wat Beaupré betreft—hij voelde zijn hart in zijn keel kloppen, want als
deze geneesheeren er werkelijk eens in slaagden om den man weder
tot bewustzijn te brengen, dan liep zijn vrijheid groot gevaar!
Een der geneesheeren trad naast het hoofdeinde van het bed en trok
een der oogleden omlaag.
Even bleef het lid in dien zelfden toestand, maar toen schoof het uit zich
zelf langzaam weder naar boven.
De geneesheer lichtte een arm op en liet hem weder vallen, tastte den
pols, opende den mond, niet zonder moeite, en legde een thermometer
onder de tong van den bewustelooze.
Na eenigen tijd trok hij het instrument weder terug en raadpleegde het.
—Ik zou het bijzonder op prijs stellen, als ik u een vraag zou mogen
doen!
—Acht gij het mogelijk dat deze man zich uit zich zelf bewogen heeft?
—Omdat ik mij niet kan voorstellen hoe die beide mannen zonder
eenigen tegenstand te bieden, of tenminste hun verbazing te uiten, dat
donkere, onbewoonde huis in de Bishop Street binnen gingen! Zij
moesten immers verwacht hebben naar Scotland Yard of naar een Huis
van Bewaring te worden overgebracht?
—Als dat uw vaste overtuiging is—dan moet ik mij daar natuurlijk wel bij
neerleggen. Maar het maakt voor mij de zaak des te raadselachtiger.
Nu waren zijn twee doodsvijanden nog bewusteloos, maar wie weet hoe
lang dat zou duren?
Hij begon te kreunen, wentelde het hoofd van links naar rechts over zijn
kussen en het duurde niet lang of een der verpleegsters kwam
toeloopen, boog zich over hem heen, en vroeg:
—Vreeselijk, zuster! antwoordde Beaupré. Ik geloof dat het met mij ten
einde loopt.
—Maar uw toestand was van morgen redelijk, riep de zuster verschrikt
uit. Ik zal aanstonds de hoofdverpleegster roepen.
Deze werd gehaald en kwam haastig op het bed van den gewonde
toeloopen.
—Ik geloof dat het met mij mis loopt! Ik smeek u aanstonds mijn vriendin
te laten halen!
Zij legde den zieke den thermometer aan en bemerkte dat hij hooge
koorts had.
Een oogenblik stond zij in beraad, en toen nam zij een besluit en zeide:
—Geef mij het adres van uw vriendin—ik zal haar laten halen, maar gij
moogt volstrekt niet langer dan vijf minuten met haar spreken.
Hij noemde een afgelegen straat in een der Noordelijkste wijken van
Londen, een oogenblik later was er een telegram aan het adres van
Marthe Debussy gezonden.
Het zou niet veel helpen, als hij zich onder de dekens verborg, want „Big
Billy”, zoo was de naam van den bewustelooze, wist zeer goed wie er in
dat bed lag!
Het telegram had haar zeer ontsteld en zij meende niet anders of zij zou
haar minnaar stervende vinden.
Zij snelde op het bed van Beaupré toe, maar deze stelde haar
onmiddellijk met eenige gefluisterde woorden gerust, en hernam daarop
iets luider opdat de verpleegster hem zou kunnen verstaan:
—„Big Billy” is in deze zelfde zaal gebracht—hij ligt drie bedden van mij
af—kijk aanstonds eens voorzichtig!
Marthe Debussy kon met moeite een kreet van schrik weerhouden, want
ook zij had aanstonds het gevaar begrepen!
Zij kende Big Billy maar al te goed en zij wist dat hij geen medelijden
zou kennen, als hij zelf gearresteerd werd—hij zou trouwens overtuigd
zijn, zijn vriend Dr. Fox een grooten dienst te bewijzen als hij den
Franschen markies, diens mededinger, in het verderf stortte!