JAIST Repository
https://dspace.jaist.ac.jp/

Title

Interim Report of Detailed Optimisation Algorithm
and Performance Comparisons

Author(s)Matsumoto, Tad; Qian, Shen; Hou, Jiancao; Yi, Na;
Matth´e, Maximilian; Wolf, Albrecht; Tervo,
Valtteri; Zhou, Xiaobo

Citation

ICT‑619555 RESCUE D2.2.1 Version 1.0: 1‑49

Issue Date

2015‑04

Type

Research Paper

Text version

publisher

URL

http://hdl.handle.net/10119/13798

RightsThis material is posted here by permission of the
EU FP7 RESCUE Project. http://www.ict‑rescue.eu/
RESCUE is founded by the European Commission
under the 7th Framework Programme,Theme 3‑"ICT"call FP7‑ICT‑2013‑11,Work Programme Topic
1.1"Future Networks"

Description

Japan Advanced Institute of Science and Technology

ICT-619555 RESCUE
D2.2.1 Version 1.0
Interim Report of Detailed Optimisation Algorithm and
Performance ComparisonsContractual Date of Delivery to the CEC:
Actual Date of Delivery to the CEC:
Editor:Authors:
Participants:
Work package:
Security:
Nature:Version:Total number of pages:

04/2015 (M18)
04/2015
Jiancao Hou
Tad Matsumoto, Shen Qian, Jiancao Hou, Na Yi, Maximilian Matthé, Albrecht Wolf, Valtteri Tervo, Xiaobo Zhou
JAIST, UNIS, TUD, UOULU
WP2 – Code and algorithm design and optimization
PU
R
1.0
49

Abstract: One of the main tasks of RESCUE project is to exploit the optimal power allocation of the transmission nodes in the network to improve the system performance. In this deliverable, the intermediate results on the physical layer power optimisation algorithms with respect to different toy scenarios described inD1.2.1 are provided. Start from the toy scenario one, the outage probability based power allocation for thebasic lossy-forward relaying system is investigated. Moreover, the system model is extended to multi-antennacase, where the relay-oriented source power allocation for the erroneous decode-and-forward relaying system is provided. For the toy scenario two, apart from the outage probability based power allocation, the joint
relay nodes power allocation to equalize the signal to noise ratios limits for each relay decoder is also considered. Furthermore, the optimal power allocation algorithm for orthogonal multiple access relay channel
with intra-link errors is proposed. Simulation results are provided to illustrate the improved performance of
all the proposed algorithms by comparing with the conventional schemes.

Keyword list: power allocation, relay channel, erroneous decode-and-forward, outage probability, multiple
antennas, chief executive officer, multiple accessDisclaimer:

RESCUE

D2.2.1 Version 1.0

Executive Summary
The integrated concept “links-on-the-fly” of the RESCUE project allows relays decode-and-forward (DF) data
symbols with specified level of loss/distortion, which aims to create efficient and robust information transfer
through multi-path networks. In this case, the optimal power allocation on each link/node can significantly enhance the efficiency and robustness of the network and keep the rate distortion below a certain specified limit. In
order to present the merit of power allocation strategy in the RESCUE project, in this interim report, some detailed
power optimization algorithms are evaluated using different toy scenarios provided in D1.2.1.
For the toy scenario one (TS1), the theoretical outage bound of the lossy-forwarding system is derived over block
Rayleigh fading channels and the existing results are just obtained by numerical computations for multi-fold integrals with equal power allocation at each transmitter. By assuming high SNR values, the closed-form expressionof the outage probability can be mathematically derived. Thus, the optimal power allocation in order to minimize
the outage probability subject to the total power constraint can be formulated as convex optimization problem andsolved by using a convex optimization tool. Moreover, without the global channel knowledge to be assumed ateach node, the conventional multi-antenna based erroneous DF relaying network normally allows the source optimize its transmit power on its direct channel links to its served users. Meanwhile, the relay node comes to help
if its received signal to interference plus noise ratios of certain data streams can pass the predetermined threshold.
However, such strategy may loss the system performance if the source node has the location and power budget
information of the relay node. This motivates the relay oriented source power allocation algorithm to be proposed
to improve the system performance.
The toy scenario two (TS2) assume that multiple relays are deployed to help the information transmission fromthe source node to the destination node. Without the direct link between the source node and the destination
node (i.e., S-D link) and all the source node to relay node links (i.e., S-R links) are lossy, the system is known
as the chief executive officer (CEO) problem. As the way of analysing TS1, the outage probability based power
allocation algorithm for CEO problem is also investigated. Moreover, according to the theoretical analysis, therelation between bit flipping probabilities and mutual information values can be built, where any relay decoderwith low bit flipping probability benefits to a greater extent from the high mutual information. Here, the bit
flipping probabilities can be controlled by the power allocation on the relay nodes. Hence, with the proposed joint
relay nodes power allocation algorithm, the signal to noise ratios limits for each relay decoder can be equalized,
such that the ramp region of the joint decoder can be removed or reduced.
As mentioned in D1.2.1, the toy scenario three (TS3) is an extension of TS1, and it can be deemed as combining
multiple independent TS1 together. In this case, based on the different channel conditions of S-R links, the power
allocation algorithms proposed for TS1 can be used to solve TS3 problem via proportionally splitting the totaltransmit power for each relay node. Moreover, TS3 can be composed by the TS2 plus the S-D link, therefore
additional information can be involved in the decoding process to further improve the system performance.
Consider the toy scenario four (TS4), where the orthogonal multiple access relay channel allows that the intra-link
errors exist. It has been found that even the relay node cannot correctly decode the estimates, at the destinationnode, the forwarded erroneous estimates are still helpful for the reconstruction of the information sequence sentfrom the source nodes. Moreover, by forwarding the specific level of incorrectly decoded estimates, the systemoutage performance can be improved especially when one of the source nodes is quite far away from both the relay
node and the destination node. Under such circumstances, the optimal power allocation among the source nodes
and the relay node subject to lossy rate constraints is exploited. The system performance is improved by comparing
with traditional DF in terms of total power consumption especially in the cases where the relay node is close to thedestination node.
To sum up, the above described power allocation algorithms for different toy scenarios are only the intermediate research results of the RESCUE project, where the detailed algorithm description and comparison have been
included in this deliverable. More comprehensive analytical results and computer simulation of all the proposed
power optimization algorithms will be carried out in the final report.

Page 2 (49)RESCUE

D2.2.1 Version 1.0

Authors
PartnerName

E-mail / Phone

Japan Advanced Instituteof Science and Technology(JAIST)

Tad Matsumoto
Shen Qian

matumoto@jaist.ac.jp
shen.qian@jaist.ac.jp

University of Surrey(UNIS)

Jiancao Hou
Na Yi

j.hou@surrey.ac.uk/+44 1483 682229
n.yi@surrey.ac.uk

Technical University Dresden(TUD)

Maximilian MatthéAlbrecht Wolf

maximilian.matthe@tu-dresden.dealbrecht.wolf@tu-dresden.de

University of Oulu
(UOULU)

Valtteri Tervo
Xiaobo Zhou

valtteri.tervo@ee.oulu.fi
xiaobo.zhou@ee.oulu.fi

Page 3 (49)RESCUE

D2.2.1 Version 1.0

Table of Contents
Executive Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2Authors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5

List of Acronyms and Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7

2. Power Allocation for Basic Erroneous Decode-and-Forward Relaying Network 9
2.1

2.2

Outage Based Power Allocation for a Lossy-Forwarding Relaying System . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.1 Motivation and Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.2 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.3 Outage Probability Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.4 Optimal Power Allocation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.4.1 Proof of Convexity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.4.2 Optimal Power Allocation: Total Power Fixed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.4.3 Optimal Power Allocation: Outage Probability Requirement Fixed . . . . . . . . . . . . . . . .
2.1.5 Impact of LOS component on optimal power allocation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Relay-Oriented Source Power Allocation for MIMO Selective-DF Relaying System . . . . . . . . . . . . . . . . .
2.2.1 Motivation and Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.2 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.3 Precoding/Decoding Matrices Design and Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.4 The Proposed Relay-Oriented Source Power Allocation Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.5 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9
9
9
10
12
13
14
15
16
18
20
20
20
21
22
24
26

3. Power Allocation for Chief Executive Officer (CEO) Problem . . . . . . . . . . . . . . . . . . . . . . . . 27
3.1

3.2

Outage Based Power Allocation for CEO Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.1 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.2 Optimal Power Allocation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.2.1 Proof of Convexity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.2.2 Simulation Settings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.2.3 Total Power Fixed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.2.4 Outage probability requirement fixed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
SNR Limit Based Power Allocation Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.1 Motivation and Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.2 State-of-the-Art . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.3 Implemented Power Allocation Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.4 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

27
27
28
28
29
29
30
32
32
33
35
36
39

4. Power Allocation for Orthogonal Multiple Access Relay Channel Allowing
Intra-link Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
4.1
4.2
4.3
4.4

Motivation and Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Conditions for Reliable Transmission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Power Minimization with Rate Constraints. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

40
40
41
42

Page 4 (49)RESCUE

4.5
4.6

D2.2.1 Version 1.0

4.4.1 Successive Convex Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

5. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
6. References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

Page 5 (49)RESCUE

D2.2.1 Version 1.0

List of Acronyms and Abbreviations
Term
AWGN
BCJR
BER
BSC
CCC
CEO
CPU
CSI
DF
EP
FDD
i.i.d.
JD
KPI
LOS
LosFoR
MARC
MIMO
OFDMA
OPA
PDF
R-D
RF
RESCUE
SCA
SCS
S-DF
SDR
SINR
SNR
S-D
S-R
TDD
TS
WSN
WSRM
ZF

Description
Additive White Gaussian Noise
Bahl Cocke Jelinek Raviv
Bit Error Rate
Binary Symmetric Channel
Constellation Constrained Capacity
Chief Executive Officer
Central Processing Unit
Channel State Information
Decode-and-Forward
Equal Power
Frequency-Division Duplex
Independent and Identically Distributed
Joint Decoder
Key Performance Indicator
Line of Sight
Lossy-Forwarding Relay
Multiple Access Relay Channel
Multiple Input Multiple Output
Orthogonal Frequency Division Multiple Access
Optimal Power Allocation
Probability Density Function
Relay-Destination
Rayleigh Fading
Links-on-the-fly Technology for Robust, Efficient, and Smart Communication in Unpredictable Environments
Successive Convex Approximation
Source-Channel Separation
Selective Decode-and-Forward
Software Defined Radio
Signal-to-Interference-plus-Noise Ratio
Signal-to-Noise Ratio
Source-Destination
Source-Relay
Time-Division Duplex
Toy Scenario
Wireless Sensor Network
Weighted Sum Rate Maximization
Zero-Forcing

Page 6 (49)RESCUE1.

D2.2.1 Version 1.0

Introduction

In conventional communication systems, accurate/strict link-budget allocation, strong error-correction coding andretransmission protocols are needed to keep the communications with high transmission rate and the outage probability in an acceptable range. However, such kind of system design sacrifices the energy/spectral efficiencies dueto the sophisticated decoding and channel estimation techniques plus a large amount of signalling overhead. Moreover, it performs poorly or completely fail the communications if the transmission environment is unpredictable,
such as in the presence of earthquakes or tsunamis. These motivated the RESCUE project to develop robust,
efficient, and smart communication systems by using the concept “links-on-the-fly”.
The basic idea of “links-on-the-fly” is to allow the specified level of erroneously decoded data streams at therelay node to be forwarded to the destination node. Thus, the required power to be allocated to each link can besignificantly reduced in the network as a whole. Moreover, such decoding errors can be estimated at the destination
node through the proposed decoding techniques which were introduced in D2.1.1. Here, apart from the decoding
techniques, another major physical layer research task is the transmit power optimization under various constraints.
In detail, the optimization issues can be mainly categorized into two parts: One is the total transmit power of allnodes minimization subject to the fixed distortion or rate constraints; The other is the distortion minimization or
the rate maximization subject to the total power constraint. As we can see, the two objectives are dual to eachother, and the tools from joint convex and combinatorial optimization theory can be used to solve the problems.
In this case, proposing the complexity-reduced power optimization algorithms plays a major role of enhancing theenergy efficiency of the system.
As discussed in D1.2.1, the theoretical analysis and performance evaluation of the RESCUE network model
presents a lot of challenges due to its complicated and dynamic network topology. Instead, we decompose thegeneral network into four basic toy scenarios in order to facilitate our analysis. Such strategy is also applied to thepower optimization algorithms design in this deliverable, and this motivated us to categorize the work based on
different toy scenarios (TSs):
TS1 Firstly, a sufficiently accurate closed-form expression of the outage probability is derived by taking the geometrical gain into account, and the optimal power allocation scheme can be formulated to the convex problem
for minimizing the outage probability while keeping the total transmit power constraint or minimizing thetransmit power while keeping the outage probability fixed. Then, we extend the system model to the multiple input multiple output (MIMO) case to exploit the multiplexing gain. Without the channel knowledgeof relay-destination (R-D) link at the source node, the conventional relaying strategy allocates the source’s
power based on the source-destination (S-D) link, and the system performance is not enhanced if the source
node has the location power budget information of the relay node. This motivated the relay-oriented source
power allocation algorithm to be proposed to improve the system performance.
TS2 According to the “Slepian-Wolf Admissible Rate Region based Outage Probability Derivation for CEO”
of deliverable D1.2.1, the approximated closed-form expression of the outage probability for high signal
to noise ratio (SNR) of the system model is provided, and the optimal power allocation scheme can also
be formulated as a convex problem for minimizing the outage probability while keeping the total transmit
power constraint or minimizing the transmit power while keeping the outage probability fixed. Then, weinvestigate a power allocation algorithm to equalize the SNR limits for each relay decoder, such that theramp region of the joint decoder at the destination node can be removed or reduced. Here, the SNR limit isdefined as the bit error rate (BER) curve reaches its error floor for each relay decoder.
TS3 As we explained in D1.2.1, TS3 is an extension of TS1, where multiple relays coexist in the system tohelp the information transmission from the source node to the destination node. In this case, following the“links-on-the-fly” concept, different relay nodes can base on their different source-relay (S-R) links statistic
properties to do the power allocation. In other words, TS3 can be divided into several independent andparallel TS1s, and by given the total transmit power budget, we can propositionally allocate the power to thedifferent transmission nodes. According to the above discussion, the power allocation strategies for TS3 canbe refer to the power allocation strategies for TS1.
TS4 Consider the orthogonal multiple access relay channel (MARC), where two source nodes are communicating with a common destination node with the assist of a relay node. Comparing with the conventional

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decode-and-forward (DF) relaying strategies, it has been shown in the literature that even the estimates areincorrectly decoded at the relay node, the forwarded erroneous estimates are still helpful for the destination
node to reconstruct the information sequences sent from the source nodes. This motivates us to investigate
the power minimization problem subject to lossy rate constraints for MARC by comparing with the lossless
DF. Such a problem has been deemed as a non-convex problem, and the successive convex approximation(SCA) algorithm can be implemented to solve it.
In addition, in order to evaluate the performance of the proposed power allocation algorithms which are mentioned
above, apart from the physical-layer computer simulations, implementing the algorithms to the small-scale software defined radio (SDR) platform (e.g., 4 USRP2 boards) is also very important, where the interaction with WP3
is required. In detail, we have agreed to produce some interfaces of the algorithms, and then the blackbox based
simulation can be built up to link the proposed power algorithms with the SDR devices. In this case, one of the
key steps is to provide the channel requirements for each proposed algorithm. For example, for outage probability minimization problem, the statistical characterization of the channels is only needed to optimally allocate
the power, where the statistical channel information can be obtained through the long-term observation for either
time-division duplex (TDD) or frequency-division duplex (FDD) models. Or, considering to solve the chief executive officer (CEO) problem, the bit flipping probabilities is only needed for the joint relay nodes power allocation,
where such kind of information can be linked with some quantized information, e.g., the SNRs at the relay node,and stored at the predetermined look-up table.
It is worthwhile to note that, in this deliverable, we also produce some power allocation algorithms which need theinstantaneous local or global channel knowledge at each node. Such algorithms requires a central processing unit
(CPU) to collect the information, perform the processing and report the allocated powers back to each transmission
nodes. Although the algorithms with instantaneous channel are not quite practical to be implemented, they canbe good baselines for us to further investigate the power allocation algorithms of the erroneous DF based relaying
network.

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D2.2.1 Version 1.0

2.

Power Allocation for Basic Erroneous Decode-and-Forward
Relaying Network

2.1

Outage Based Power Allocation for a Lossy-Forwarding Relaying System

2.1.1

Motivation and Objective

In [AM12], a novel lossy-forwarding relay (LosFoR) system allowing erroneous information to be transmitted by
the relay is proposed. The information sequence, obtained as the result of relay decoding, is interleaved, re-encoded
and transmitted to the destination, even it includes errors occurring in the S-R transmission (lossy-forwarding). At
the destination, the S-R link error probability is estimated, and used as a correlation knowledge between the information sent from source and relay. The theoretical outage bound of the LosFoR system is derived over block
Rayleigh fading channels [ZCH+14], which assumes that the bit error probability after decoding at the relay is
a random variable which changes according to the S-R variation. By imposing Shannon’s lossy source-channel
separation (SCS) theorem, the relationship between the S-R channel error probability and its corresponding instantaneous SNR is established. However, only equal power allocation is adopted for each transmitter (at both S andR) in [ZCH+14], and the results are just obtained by numerical computations for multi-fold integrals. Moreover,another important performance metric − geometrical gain, which is very important for protocol design − has notbeen related to power allocation analysis of the LosFoR scheme.
The major purpose of this section is to analyse the optimum power allocation of the LosFoR system in morepractical scenarios, where all channels experience fading variation. By assuming high SNR values for such asystem scenario, closed-form expression of the outage probability is mathematically derived. It is shown thatthe power allocation for the proposed LosFoR system can be formulated as a convex optimization problem. The
optimum power allocation scheme for minimizing the outage probability is then presented, where geometrical
gain is taken into account. We found that proposed optimal power allocation scheme improve outage performancecompared with the equal power allocation, when R is located close to D. Also, this section investigates the impactof line-of-sight (LOS) component on optimum power allocation of the LosFoR system.
2.1.2

System ModelIn this section, a very simple three-node relay transmission model is considered, as shown in Fig. 2.1. During
the first time slot, the original information sequence b1 is broadcasted from the S to both R and D. As in the DF
scheme, the original information sequence is decoded and re-constructed at R, before being forwarded as b2 . In
contrast to the conventional DF system, our proposed technique always interleaves the re-constructed sequence at
R, and then re-encodes and maps it for the second time slot transmission. It should be noted that the re-constructed
bit sequence may contain some errors, due to the imperfectness transmission of S-R channel, but it is still highly
correlated with the original one sent from S.
After receiving signals from both S and R, joint decoding process takes place at D. It is already known that by
exploiting the correlation knowledge between the bit sequences sent from S and R, the minimum rate required forlossless transmission is no longer equal to the entropy of each source, but could be even lower, as stated by the(!!!


b1









!

!




!

 
 !!


(

 







b2

Figure 2.1: Single relay transmission system.

Page 9 (49)RESCUE

D2.2.1 Version 1.0

Slepian-Wolf theorem. In other words, if we keep transmitting the redundancy (the correlated portion), the wholesystem performance will benefit from utilizing the correlation knowledge.
All the three transmission channels are assumed to suffer from independent block fading. Hence, it is reasonable
to use the binary symmetric channel (BSC) model [GZ05], with the crossover probability pe as a parameter within
one block. Hence, b2 = b1 ⊕ e, where ⊕ denotes modulo-2 addition and e is a binary error vector. Pr(e = 1) = pe
is fixed within each block, and the pe value changes block-by-block. pe = 0 indicates the perfect decoding at R,
while pe 6= 0 indicates there exist S-R link errors which could not be eliminated by the decoding process at R.
In this section, all the links are assumed to be suffering from block Rayleigh fading whose probability density
function (PDF) with instantaneous SNR γi is given byp(γi ) =γi
1exp(− ), (i = 1, 2, 3),
Γi
Γi

(2.1)

where Γi represent received average SNRs of the S-D, R-D and S-R channels, respectively.
With di (i=1,2,3) representing the length of S-D, R-D and S-R channels, respectively. With the geometrical gain of
the S-D channel, G1 , being normalized to one, the geometric gains of R-D and S-R channels, G2 and G3 , can be
defined as [LNH10]
 α
d1
,
G2 =
d2
 α
d1
G3 =
,
(2.2)
d3
where the path-loss factor α is set to 3.52 [YG11]. For sake of simplicity, the variation due to shadowing is ignoredin this section.
2.1.3

Outage Probability Derivation

Based on Shannon’s lossy SCS theorem, for S-R channel, we have R3 (D)Rc3 6 C(γ3 ), where R3 (D) is the ratefunction of distortion D with D = pe , Rc3 denotes the spectrum efficiency including the channel coding schemeand the modulation multiplicity and C(γ3 ) is the capacity with instantaneous SNR γ3 of S-R channel. For hamming
distortion measure, R3 (D) = 1 − H(D).
With the relationship between pe and γ3 , the rate constrains supported by the Slepian-Wolf theorem are given by[ZCH+14]
R1 > H(b1 | b2 ),R2 > H(b2 | b1 ),
−1 C(γ3 ),R1 + R2 > H(b1 , b2 ) = 1 + H(pe ) = 1 + H R3
Rc3
(2.3)
where R1 and R2 are source rate for b1 and b2 , respectively. H(·|·) and H(·, ·) denote the conditional entropy andthe joint entropy between the arguments, respectively.
Actually, the Slepian-Wolf theorem, which intends to recover two correlated sources, is not perfectly suitable
for the system setup in this section since the destination only aims to reconstruct the information sent from S.
However, the difference between exact outage derived from theorem for source coding with side information andthe approximation outage derived from Slepian-Wolf theorem is negligible [ZCH+14]. For simplicity reason on
calculation, in this section, the outage probability is calculated based on Slepian-Wolf theorem.
Eq. (2.3) can be understood from the viewpoint of the Slepian-Wolf rate region presented in Fig. 2.2 and Fig. 2.3.
According to the value of R3 (D), the rate constrains can be divided into two cases for discussing:Page 10 (49)RESCUE

D2.2.1 Version 1.0

R2
H(b1 , b2)

H(b2)

H(b2|b1)

1
H(b1|b2)

H(b1)

H(b1 , b2)

R1

Figure 2.2: Lossy transmission of the S-R channel, 0 < pe 6 0.5.

R2
H(b1 , b2)

H(b2)

H(b2|b1)

2
3

1

H(b1|b2)

H(b1)

H(b1 , b2)

R1

Figure 2.3: Slepian-Wolf rate regions of the proposed relay system.
1. R3 (D) ≥ 1 (pe = 0), implying that perfect decoding is conducted at R. In this case, the outage event happens
when R1 and R2 are within Area 1 shown in Fig. 2.2. The probability which R1 and R2 fall into Area 1 isdenoted as P1 ;2. 0 6 R3 (D) < 1 (0 < pe 6 0.5), indicating that decoding at the relay is imperfect. S-R channel SNR cannot
support error-free transmission. Then, the outage event occurs when R1 and R2 are inside Areas 2 and 3
shown in Fig. 2.3, which the probabilities are denoted by P2 and P3 , respectively. Hence, the total outage
probability of the LosFoR system can be defined by

Pout = P1 + P2 + P3 .

(2.4)

It is found that the outage probability can be calculated by using a triple integral with respect to the joint pdf of theinstantaneous SNRs p(γ1 , γ2 , γ3 ) [ZCH+14], given the range defined in (2.3). Since we assume that the variationof each link’s instantaneous SNR is statistically independent,p(γ1 , γ2 , γ3 ) = p(γ1 )p(γ2 )p(γ3 ).

(2.5)

With the assumption that Gaussian codebook is used, the relationship between the instantaneous channel SNR γi

Page 11 (49)RESCUE

D2.2.1 Version 1.0

and its corresponding source rate Ri is given by


2γi
En
log 1 + n , (i = 1, 2, 3)Ri ≤2Rci 2E(2.6)

where E n denotes the signalling dimensionality. In this section, we assume E n = 2. Rci denotes the systemefficiency of the transmission chain representing the coding rate and multiplicity of the modulation for i-th (i = 1,2, 3) channel.
In practical systems, the channel capacity is upper bounded by the multiplicity of modulation. However, thedifference between the constellation constrained capacity (CCC) and Gaussian capacity is very small in low SNR
region. While in high SNR region, distortion level D is very close to 0, resulting in pe = 0. The difference between
CCC and Gaussian capacity does not have any significant impact on D for the entire SNR region. Therefore, it isreasonable to replace CCC by Gaussian capacity for outage derivation.
By converting the rate constrains into the instantaneous SNR constrains [CAM13b], the theoretical calculation of
the outage probability P1 , P2 and P3 can then be mathematically expressed as,
P1 =Pr [0 < R1 < H(b1 ), 0 < R2 < H(b1 , b2 ), R3 ≥ 1]=

Z 21−log2 (1+γ1 ) −1 Z ∞Z 1p(γ1 )p(γ2 )p(γ3 )dγ1 dγ2 dγ3!# Z γ =0
 "
1
γ1
1
1
21−log2 (1+γ1 )) − 1
exp −= exp −
· 1 − exp −
dγ1 ,
Γ1
Γ3
Γ1
Γ2
1
γ1 =0 γ2 =0

γ3 =1

(2.7)

P2 =Pr [0 < R1 < H(b1 | b2 ), R2 > 0, 0 ≤ R3 < 1]=Z 2H(pe ) −1 Z ∞ Z 1

p(γ1 )p(γ2 )p(γ3 )dγ1 dγ2 dγ3
!#

 "
Z
γ3
21−log2 (1+γ3 ) − 1
1 1
exp −
· 1 − exp −dγ3 ,=Γ3 γ3 =0
Γ3
Γ1
γ1 =0

γ2 =0 γ3 =0(2.8)

P3 =Pr [H(b1 | b2 ) < R1 < H(b1 ), 0 < R2 < H(b1 , b2 ) − R1 , 0 ≤ R3 < 1]=

Z 21+H(pe )−log2 (1+γ1 ) −1

Z 1

γ1 =2H(pe ) −1 γ2 =0

Z 1

γ3 =0

1 1
=
Γ1 Γ3

p(γ1 )p(γ2 )p(γ3 )dγ1 dγ2 dγ3 ,Z 1
γ1 =21−log2 (1+γ3 )−1

!#



 "
γ1
γ3
22−log2 (1+γ3 )−log2 (1+γ1 ) − 1
exp −
exp −
· 1 − exp −dγ1 dγ3 .
Γ1
Γ3
Γ2
γ3 =0

Z 1

(2.9)
The values of P1 , P2 and P3 can be calculated by numerical method [Sha08], with sufficiently accurate numericalcalculation error control.
2.1.4

Optimal Power Allocation

The goal of this section is to minimize the outage probability while the total transmit power ET is fixed, and tominimize ET while keeping the outage probability constant. The noise variance σn 2 of each channel is normalized
to unity. The transmit powers, which are now equivalent to their corresponding average SNRs, allocated to S and
R are denoted as kET and (1 − k)ET , respectively, where k ( 0 < k < 1) is power allocation ratio.

Page 12 (49)RESCUE

D2.2.1 Version 1.0−2

10

Original, ET = 24 dB
Approximated, E = 24 dB
T

Original, ET = 20 dB−3outage probability

10

Approximated, E = 20 dB
T

−4

10

−5

10

−6

10

0

0.2

0.4
0.6
Distance ratio dr

0.8

1

Figure 2.4: Accuracy of the approximated closed-form outage expression.
While keeping the ratio k constant and bringing the average SNRs Γ1 , Γ2 and Γ3 to infinity, by invoking theproperty of exponential function e−x ≈ 1 − x for small x, at the high SNR regime, the outage probability (2.4) canbe approximated to a closed-form expression, as,
Pout ≈

0.5+

ET2 k2 G21

0.5
ET2 k2 G23+

1.3863ET2 k2 G1 G3+

0.3863ET2 k(1 − k)G1 G2.

(2.10)

To identify the accuracy of the approximation, outage probability curves obtained by using the approximated expression (2.10), and by numerically calculating (2.7)-(2.9), are presented in Figure 2.4. The relay node is assumed
to be allocated in the line between the source and the destination, and the distance ratio is defined by dr = d2 /d1 .
Good matching is illustrated between the curves of approximation and numerical calculation, which indicates sufficient accuracy of the approximation. Then, the optimal power allocation problem is formulated by a standard
optimization problem, which can be proven to be convex.

2.1.4.1

Proof of Convexity

Here we give the proof of the convexity of the approximated outage probability expression, Eq. (2.10), with regard
to variables ET and k. If each of the four terms in Eq. (2.10) can be proven to be convex, (2.10) is also convex
because it is a sum of the convex terms. The Hessian matrix of the first term E 20.5
can be calculated as
k 2 G2T
H

0.5
ET2 k2 G21


=

" 3

1
ET2 G21 k2

k2
2
ET k

1

#
2
ET k
3
ET2

.

(2.11)The eigenvalues of (2.11)


s
2
1
3
3
3
20
3
 +
±
+
− 2 2
λ1,2 =
k2 ET22ET2 G21 k2 k2 ET2
ET k(2.12)are clearly non-negative. Therefore, the Hessian matrix of E 20.5
is positive semi-definite and hence its convexity
k 2 G2T

has been proven.

1

can be calculated as
The Hessian matrix of the second term E 20.5
k 2 G2
T

3

" 3

0.5
1
2
H 2 2 2 = 2 2 2 k2
ET k G3
ET G3 k ET k


2
ET k
3
ET2

#
.

(2.13)

Page 13 (49)RESCUE

D2.2.1 Version 1.0

The eigenvalues of (2.13)


s
2
3
1
3
3
3
20
 +
λ1,2 =+±− 2 2
k2 ET22ET2 G21 k2 k2 ET2ET k(2.14)are clearly non-negative. Therefore, the Hessian matrix of E 20.5
is also positive semi-definite and hence its
k 2 G2T

convexity has been proven.

3

can be calculated as
Similarly, the Hessian matrix of the third term E 21.3863
k2 G GT

3 1" 8.3178

1.3863
1
k2
H 2 2
= 2
5.5452
ET k G3 G1
ET G3 G1 k2 ET k


5.5452ET k
8.3178ET2#.

(2.15)The eigenvalues of (2.15)

λ1,2 =

1  8.3178 8.3178
+
±
2
k2
ET2

s

8.3178 8.3178
+
k2
ET2



2
−

153.7468 
ET2 k2(2.16)are clearly non-negative. Therefore, the Hessian matrix of E 21.3863
is positive semi-definite and hence its convexk2 G G
T

ity has been proven.

3 1

0.3863
The Hessian matrix of the fourth term E 2 k(1−k)G
can be calculated as
G
1 2

T


 2 2
1
0.7726
ET k(2k2 − 3k + 1)
ET (3k − 3k + 1)
= 3
.
·
H 2
ET k(1 − k)G1 G2
k (1 − k)3 ET4 G1 G2 −ET k(2k2 − 3k + 1) 3k2 (k2 − 2k + 1)


(2.17)Let u(k) = 3k3 (k2 − 2k + 1) + ET2 (3k2 − 2k + 1) and v(k) = 3k2 ET2 (3k3 − 3k + 1)(k2 − 2k + 1), the eigenvalues of(2.17) can be calculated aspu(k) ± u(k)2 − 4v(k)λ1,2 =.
(2.18)
2
For 0 < k < 1, the value of the quadratic polynomials k2 − 2k + 1 and 3k2 − 2k + 1 in u(k), and the quadratic
polynomials 3k3 − 3k + 1 and k2 − 2k + 1 in v(k) are larger than 0. This indicates the non-negativity of u(k) and
0.3863v(k). Hence, the Hessian matrix of E 2 k(1−k)Gis proven to be positive semi-definite and hence its convexity has
G
T

1 2

been proven. Therefore, the outage probability expression (2.10) is a convex function.

2.1.4.2

Optimal Power Allocation: Total Power Fixed

In this subsection, we minimize the outage probability while the total power ET is fixed. The convex problem canbe formulated asminimize Pout (k, ET )k
(2.19)
subject to k − 1 < 0−k < 0.
The solution to the optimization problem can be obtained by using a convex optimization tool, and optimal valuesof k are listed in Table 2.1. Obviously, the larger the dr value, the more transmit power should be allocated to S.
Theoretical results for the outage performance with and without optimal power allocation are shown in Fig. 2.5,
where the total transmit power ET is set at 16 dB. First of all, it can be obviously seen that with equal power
allocation, the outage curve is symmetric to the mid-point between S and D, which yields the lowest outage

Page 14 (49)RESCUE

D2.2.1 Version 1.0

Table 2.1: Optimal power ratio kdr

optimal k (ET = 16 dB)

0.1

0.500

0.2

0.502

0.3

0.520

0.4

0.574

0.5

0.666

0.6

0.774

0.7

0.871

0.8

0.943

0.9

0.985

−2

10

outage probability

with optimal allocation
equal Tx power allocation

−3

10

−4

10

0

0.2

0.4
0.6
Distance ratios dr

0.8

1

Figure 2.5: Outage probabilities with and without optimal power allocation, total transmit power fixed.
probability. This finding is different from the conventional DF case, of which R should be located at a pointcloser to S in order to achieve the best performance [ZCH+14]. The optimal power allocation provides the lowest
outage probability at the point where the contributions of both S-R and R-D channels to outage are balanced. In
other words, the proposed technique enables the transmitter to find proper relays for cooperation in a broader area.
It should be noted that, by selecting the optimal power allocation ratio k, the system can much reduce the outage
probability compared with the case of the equal power allocation when the relay is allocated near to the destination,
while they achieve almost the same outage when relay is near to the source.

2.1.4.3

Optimal Power Allocation: Outage Probability Requirement Fixed

In this sub-section, we investigate how much total transmit power can be saved through optimal power allocation,
given a fixed outage probability requirement. The problem can be formulated asminimizek,ETET

subject to Pout (k, ET ) − Tout ≤ 0
k−1 < 0−k < 0−ET < 0,(2.20)

where Tout is required outage probability. The convexity of (2.20) is proven in sub-section 2.1.4.1.
Tables 2.2 and 2.3 show the required total transmit power with equal and optimal power allocations for outage requirements, where dr is 0.6 and 0.7, respectively. The outage probability requirements are set as Pout =Page 15 (49)RESCUE

D2.2.1 Version 1.0010

ET = 12.45 dB
ET = 11.35 dB−1

10

E = 17.45 dB
T

ET = 16.40 dB
E = 22.50 dB

Outage probability

−2

10

T

ET = 21.42 dB
−3

10

−4

10

−5

10

−6

10

0

0.1

0.2

0.3

0.4
0.5
0.6
Transmit power ratio k

0.7

0.8

0.9

1

Figure 2.6: Theoretical outage probabilities versus the optimal power ratio k with different total transmit
powers, dr = 0.6.
010

E = 13.25 dB
T

ET = 11.47 dB−1

10

ET = 18.30 dB
ET = 16.42 dB−2

ET = 23.31 dB

outage probability

10

ET = 21.44 dB
−3

10

−4

10

−5

10

−6

10

0

0.1

0.2

0.3

0.4
0.5
0.6
Transmit power ratio k

0.7

0.8

0.9

1

Figure 2.7: Theoretical outage probabilities versus the optimal power ratio k with different total transmit
powers, dr = 0.7.
10−5 , 10−4 , 10−3 , respectively. It is clearly seen that by selecting the optimal k values, roughly 1∼2 dB gain canbe achieved by optimal power allocation compared with that with equal power allocation. It is also found that,given the relay location, the transmit power gains with the optimal power allocation over the equal power allocation are almost the same, regardless of the outage requirements. As shown in Fig. 2.6 and Fig. 2.7, the optimal k
values corresponding to outage requirements are exactly consistent to the the data obtained as the solution to the
optimization problem.
2.1.5

Impact of LOS component on optimal power allocation

In this sub-section, we focus on the impact of the LOS components on the outage probability of the LosFoR systemdefined in Fig. 2.1. The S-D link is assumed to suffer from block Rayleigh fading while the R-D and S-R links
from block Rician fading. The pdf of the instantaneous SNR γi (i = 2, 3) following Rician distribution is given
by
 s





−Ki
(1
+
K
)
e
(1
+
K
)γ
K
(1
+
K
)
γ
i
i i
i
i i
pRici (γi ) =
exp −
I0 2
,
(2.21)
Γi
Γi
Γiwhere Γi (i = 2, 3) represent received average SNRs of the R-D and S-R channels, respectively. I0 (·) is the zero-thorder modified Bessel function of the first kind, and Ki denotes the ratio of the LOS component power-to-non-LOS
components average power for the corresponding links. Ki represents the severity of fading. With Ki = ∞, thePage 16 (49)RESCUE

D2.2.1 Version 1.0

Table 2.2: Optimal power ratio k with fixed Pout (dr =0.6)
required ET
required ET
required Pout (equal power)
Gain(optimal power)
0.001

12.45 dB

11.35 dB (k=0.774)

1.1 dB

0.000117.45 dB

16.40 dB (k=0.774)

1.05 dB

0.0000120.50 dB

21.42 dB (k=0.774)

1.08 dB

Table 2.3: Optimal power ratio k with fixed Pout (dr =0.7)
required ET
required ET
required Pout (equal power)(optimal power)
Gain
0.001

13.25 dB

11.47 dB (k=0.871)

1.88 dB

0.000118.30 dB

16.42 dB (k=0.872)

1.88 dB

0.00001

23.31 dB

21.44 dB (k=0.872)

1.87 dBchannel is equivalent to a static additive white Gaussian noise (AWGN) channel, and with Ki = 0 the channel
reduces to Rayleigh fading channel.
Based on the same definition of outage probability in previous section, the outage probabilities of the LosFoR
system with the fading variation of the S-R and R-D links following the Rician distribution can then be expressed
P1Rici =Z 21−log2 (1+γ1 ) −1 Z ∞Z 1p(γ1 )pRici (γ2 )pRici (γ3 )dγ1 dγ2 dγ3





s
s


Z γ1 =01−log2 (1+γ1 ) − 1
p
p
1
γ
2
2(1
+
K
)
1
3 
 dγ1 ,exp −· 1 − Q1  2K2 , 2(1 + K2 )=Q1  2K3 ,
Γ3
Γ1
Γ1
Γ2
1
γ1 =0 γ2 =0

γ3 =1(2.22)

P2Rici =Z 2H(pe ) −1 Z ∞ Z 1

p(γ1 )pRici (γ2 )pRici (γ3 )dγ1 dγ2 dγ3
 s
 "
!#



Z 1
−K
1−log2 (1+γ3 ) − 1
3(1 + K3 ) e
K
(1
+
K
)
γ
2
(1 + K3 )γ3333 · 1 − exp −
I0 2
dγ3 ,
=
exp −
Γ3
Γ3
Γ3
Γ1
γ3 =0
γ1 =0

γ2 =0 γ3 =0(2.23)

P3Rici =Z 1

Z 21+H(pe )−log2 (1+γ1 ) −1 Z 1

γ1 =2H(pe ) −1 γ2 =0

γ3 =0p(γ1 )pRici (γ2 )pRici (γ3 )dγ1 dγ2 dγ3 ,

 s






−K
3
1
γ1
(1 + K3 )γ3(1 + K3 ) e
K3 (1 + K3 ) γ3 
=
exp −
exp −
I0 2
Γ1 γ1 =21−log2 (1+γ3 )−1 γ3 =0
Γ1
Γ3
Γ3
Γ3




s
p
22−log2 (1+γ3 )−log2 (1+γ1 ) − 1 · 1 − Q1  2K2 , 2(1 + K2 )dγ1 dγ3 ,(2.24)
Γ2
Z 1

Z 1

where Q1 (·, ·) is the Marcum Q-Function.
We follow the same strategy as in previous section, where all links are assumed Rayleigh fading channels, tominimize the outage probability. The total transmit power ET is fixed and the noise variance σn 2 of each channel

Page 17 (49)RESCUE

D2.2.1 Version 1.0
0

10

K3=0
K3=5

−1

10

K3=10

ET/mn2=16 dB−2outage Probablity

10

−3

10

−4

10

−5

10

−6

10
0 %

10 %

20 %

30 %

40 %

50 %

60 %

70 %

80 %

90 %

100%

Transimit Power ratio k

Figure 2.8: Theoretical outage probabilities versus the optimal power ratio k with different Rician factor K3
in S-R channel.
is normalized to unity. The transmit power, which is equivalent to their corresponding average SNR, allocated to
S and R are denoted as kET and (1 − k)ET , respectively, where k ( 0 < k < 1) is power allocation ratio.
Since the presence of Marcum Q-Function and zero-th order modified Bessel function in outage expressions,derivation of the explicit closed-form expression of (2.22), (2.23) and (2.24) may be excessively complex. The
recursive adaptive Simpson quadrature algorithm [GG00] can be used to numerically calculate the integrals of
P1Rici , P2Rici and P3Rici .
Fig. 2.8 illustrates the outage probability curves with different Rician factor K3 values for S-R channel while Rician
factor K2 for R-D equals to zero, and ET /σn 2 = 16 dB. The x-axis represents the ratio k of transmit power allocatedto S. The 100% point indicates that transmit power is totally allocated to S, and 50% point represents the transmit
power is allocated to both sources equally. It is found from Figure 2.8 that the optimal power ratio k that achieves
the minimum outage probability decreases as the value of K3 increases. This is reasonable because when the fading
variation of S-R channel reduce, we should allocate more power to the R which decodes b2 with higher successful
ratio.
Figure 2.9 shows the power allocation ratio k versus the outage probability with the Rician factor K2 values for
R-D channel as a parameter for ET /σn 2 = 16 dB, while Rician factor K3 = 0. It should be noted that the optimal
k value, that achieves the smallest outage probability is increased as the K2 value is increased. This indicates thatthe larger the ratio of LOS component in R-D channel, the less the outage probability. However, more transmit
power needs to be allocated to S even though R-D channel is under milder fading condition. This is because thedeep fading in S-R channel (K3 = 0) results in high decoding error in R. b2 is transmitted successfully over R-D
having strong LOS component even though b2 6= b1 . This indicates that allocating more power to R cannot help
to improve the outage performance. On the contrary, it is more effective to allocate more power to S than to R insuch scenario.
Figure 2.10 shows the impact of the Rician factor K2 and K3 on the outage probability simultaneously. It is foundthat by increasing the values of K2 and K3 simultaneously, significant outage gain can be achieved. Similarly to thecase when only K3 becomes larger, optimal power ratio k that achieves the minimum outage probability becomes
small as the value of K2 and K3 become larger simultaneously. This is because the decrease of fading variationin S-R improve the decoding success ratio at R. Furthermore, b2 benefit from the increase of LOS power ratio in
R-D channel. Therefore the lowest outage probability can be achieved when the contribution of the S-R and R-Dchannels are balanced.
2.1.6

SummaryThis section has investigated the optimal power allocation for a LosFoR system based on outage probability analysis. In the proposed system model, besides the S-D and R-D channels, the S-R channel is also assumed to suffer

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0

10

K2=0
K2=5
K =10

−1

10

2

ET/mn2=16 dB−2outage Probablity

10

−3

10

−4

10

−5

10

−6

10
0 %

10 %

20 %

30 %

40 %

50 %

60 %

70 %

80 %

90 %

100%

Transimit Power ratio k

Figure 2.9: Theoretical outage probabilities versus the optimal power ratio k with different Rician factor K2
in R-D channel.
010

K =K =0
2

−1

10

3

K2=K3=5

ET/mn2=16 dB

K =K =102

3−2

outage Probablity

10

−3

10

−4

10

−5

10

−6

10
0 %

10 %

20 %

30 %

40 %

50 %

60 %

70 %

80 %

90 %

100%

Transimit Power ratio k

Figure 2.10: Theoretical outage probabilities versus the optimal power ratio k with different Rician factor
K3 (= K2 ) in S-R and R-D channel.
from block Rayleigh fading and is modelled by the rate distortion function in lossy scenario. An approximatedclosed-form expression of outage probability for the LosFoR system has been derived, where the geometrical gain
is taken into account. As shown in the results, the proposed approximation can ensure sufficient accuracy of theoutage probability calculation. Based on the expression, the optimal power allocation scheme has been formulated
as a convex problem for minimizing the outage probability while keeping the total transmit power constant, and forminimizing the transmit power while keeping the outage probability fixed. Compared with equal power allocation,by adjusting the power allocated to S and R, lower outage probability can be achieved. Moreover, it has beenshown that when R is near to D, the proposed optimization scheme can achieve much lower outage probability
than that with the equal power allocation. This section also has investigated the impact of the LOS component on
optimal power allocation for the LosFoR system.

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2.2

Relay-Oriented Source Power Allocation for MIMO Selective-DF Relaying System

2.2.1

Motivation and Objective

In this section, we exploit a relay-oriented source power allocation algorithm for MIMO selective-DF (S-DF)
relaying network in order to enhance the system spectrum efficiency. Consider a two-hop MIMO relaying network,where the system consists of a multi-antenna based source node, a multi-antenna based relay node, and K singleantenna based destination nodes. Without the global channel state information (CSI) to be assumed at each node,joint power allocation and beamforming design in order to maximize the system performance seems extremely
difficult. In this case, the conventional MIMO S-DF relaying strategy normally allows the source node optimize
its precoding matrix based on its direct channel links to its served destination nodes. Meanwhile, if the SNR or
the signal-to-interference-plus-noise ratio (SINR) of certain received data streams at relay node can pass the predetermined threshold, the relay node will forward the data streams to the destination nodes in order to help thesource node improve the system performances [LTW04; NBK04; BK13]. However, such strategy may loss thesystem optimality if the source node has the location and power budget information of the relay node.
Motivated by the above discussion, with only the local CSI to be assumed at each node, a relay-oriented source
power allocation algorithm is proposed in order to improve the system performance. In detail, instead of optimizing
the transmission power to increase the performance between its direct links to the destination nodes, the source
node firstly allocates its transmission power aiming to allow as much as the data streams’ SNR/SINR at the relay
node pass the predetermined threshold. Then, if there is still some power left, the source node will allocate theresidual power for its direct links to the destination nodes in order to further increase the system performance.
Subsequently, the relay node will formulate the precoding matrix for its selected data streams and forward themessages to the corresponding destination nodes. Simulation results show that the proposed power allocation
algorithm can exhibit better sum-rate and bit error rate performances by comparing with the conventional MIMO
S-DF relaying scheme, especially when the relay node is placed in the middle of source node and the destination
nodes.
2.2.2

System Model

R

.. ..

.. ..

An example of a two-hop MIMO S-DF relaying network is given by Fig.2.11. As shown in Fig. 2.11, the source

1

.. ..

S

2

K

Figure 2.11: System model of MIMO S-DF relaying network.
node (S) deployed with Mb transmit antennas serves K single-antenna based destination nodes (Di , i = 1, . . . , K)
with the help of Mr antennas based relay node (R). The relay node is placed in a position between the source nodeand the destination nodes, and its operation is performed in a half duplex mode, where the entire communication
process can be divided into two phases. In the first phase, the source node broadcasts multiple data streams to the
K destination nodes and the relay node; In the second phase, the relay node decodes certain received signal andforwards the re-encoded symbols to the corresponding destination nodes. Here, the threshold based S-DF strategy
is implemented at the relay node, and both phases span the same time duration.
Consider that the system works in narrowband. Each the node can only obtain the CSI regarding to its own links

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D2.2.1 Version 1.0(i.e., the local CSI), where the CSI is obtained through either the channel reciprocity for the TDD mode or thelimited feedback for the FDD mode. Denote that W with the size of Mb × K (Mb ≥ K) is the precoding matrix at
the source node. In the first phase, the discrete-time equivalent form of received signal at the kth destination node
can be expressed asyk,b = hTk,b Wsb + v1 ,(2.25)
where sb ∈ RK×1 is the vector of transmitted data streams subject to the transmission power constraint Pb , i.e.,
2 ≤ P , here, σ 2 is the transmission power allocated to the kth data stream at the source node;E{ksb k2 } , ∑Kk=1 σk,b
b
k,b
M
×1
hk,b ∈ C b is the channel vector from the source node to the kth destination node; v1 is the independent andidentically distributed (i.i.d.) AWGN with zero mean and variance σ02 . Meanwhile, the discrete-time equivalent
form of received signal after the decoding process at the relay node can be expressed aseyrb= UH yrb= UH Hrb Wsb + UH vr ,(2.26)

where Hrb ∈ CMr ×Mb is the channel matrix from the source node to the relay node; U with the size of Mr × K(Mr ≥ K) is the decoding matrix at the relay node; vr is the i.i.d. AWGN vector with zero mean and covariancematrix σ02 I.
Denote that T with the size of Mr × K (Mr ≥ K) is the precoding matrix at the relay node. Hence, in the second
phase, the discrete-time equivalent form of received signal at the kth destination node then can be expressed asyk,r = hTk,r Tsr + v2 ,(2.27)

where sr ∈ RK×1 is the vector of transmitted data streams subject to the transmission power constraint Pr , i.e.,
2 ≤ P , here, σ 2 is the transmit power allocated to the kth data stream at the relay node;E{ksr k2 } = ∑Kl=1 σk,r
r
k,r
M
×1
r
hk,r ∈ C
is the channel vector from the relay node to the kth destination node; v2 is the i.i.d. AWGN with zeromean and variance σ02 . It is worthwhile to note that, following the S-DF strategy, the transmit power of unselected
data streams at the relay node need to be set up to zeros, and the corresponding destination node in the second
phase needs to stop receiving signals.
2.2.3

Precoding/Decoding Matrices Design and Problem Formulation

The precoding and decoding matrices as mentioned above are used to mitigate the multiuser/inter-stream interference and achieve the system sum-rate performance. Such matrices can be designed based on the dirty paper
coding [Cos83] and/or maximum-likelihood principle [Ver98] at the cost of exponentially complexity, which is notefficient for the hardware implementation [ZMG+09]. In order to reduce the precoding and decoding complexity, the linear based precoding and decoding design are favoured, e.g., zero-forcing (ZF). In this section, we aremainly focusing on ZF based precoding and decoding matrices design for the multiuser/inter-stream interference
mitigation.
Start from the precoding matrix W formulation at the source node. Our aim is to use it to completely eliminate themultiuser interference among all K destination nodes in the first phase. With the channel information hk,b , ∀k, atthe source node, such precoding matrix can be formulated as
H
W = HHb Hb Hb
{z
|

−1

β,

(2.28)

}

,f
W

i
h
where Hb , hT1,b ; . . . ; hTK,b is the channel matrix from the source node to all K destination nodes; β =diag {β1 , . . . , βK } is a diagonal matrix for achieving the power constraint on the precoding matrix W, here, wehaves1 , k = 1, . . . , K,
βk =(2.29)
e kw
eH
trace wke k is the kth column of fwhere w
W.

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According to (2.26), the received signal at the relay node needs to go though the decoding matrix U to eliminate
the inter-stream interference at the relay node. Denote that A , Hrb W, hence, the decoding matrix U can beformulated as−1.
(2.30)U = A AH A
Then, according to the S-DF strategy, the SNR of the kth received data stream after the decoding process at therelay node can be expressed as
2σk,b,(2.31)
SNRk =
σ02 kuk k2
where uk is the kth column of the decoding matrix U at the relay node. By setting up the pre-determined threshold,we aim at comparing each data stream’s SNR (i.e., see (2.31)) with the pre-determined threshold, and only the data
streams whose SNRs can pass the pre-determined threshold will be re-encoded and forwarded to the corresponding
destination nodes. Here, the precoding matrix T at the relay node can be formulated by following the way in (2.31)
and (2.32), and as we mentioned, the transmit power of all the unselected data streams in this case will be set up tozero.
Following the above described precoding/decoding matrices design, all the multiuser/inter-stream interference canbe completely removed. Then, the efficient power allocation algorithm needs to be designed to guarantee thesystem performance. In other words, the main objective of the work is to!
maximize
2 ,σ 2σk,bk,rsubject to

1 K
2 ∑k=1 log

1+

2 |hT w |2 +σ 2 |hT t |2
σk,bk,r k,r k
k,b k

σ02

2 ≤ P , K σ2 ≤ P ,∑Kk=1 σk,br
b ∑k=1 k,r(2.32)

where wk and tk with unit norm are the kth column of W and T, respectively.
2.2.4

The Proposed Relay-Oriented Source Power Allocation Algorithm

Assume that each node can only has the CSI related to its own links. In this case, the conventional MIMO SDF relaying strategy allows the source node optimizing its transmission power based on the direct channel links
between itself and its served destination nodes. Here, the relay node might help if its received signal strength fromthe source node can pass the pre-determined threshold, where the decoding errors can be controlled. In general,such strategy may lose the system optimality if the source node has the location information about the relay node.
For example, if the relay node is in the middle of the source node and the destination nodes, why not allocate thesource’s transmission power to let more data streams’ SNRs at the relay node pass the pre-determined threshold,which allow the relay node to provide its maximum help.
Motivated by this, the proposed relay-oriented source power allocation strategy is to maximize the number of data
streams whose SNRs at the relay node can pass the pre-determined threshold, and utilize the residual power for thesource’s direct link optimization. In detail, start from sorting the inverse of the noise power of all the data streams
received at the relay node after the decoding process. Without loss of generality, we give an example assumingthat
1
1≥ ... ≥.
(2.33)
σ02 ku1 k2
σ02 kuK k2
As we know, each node can only have its related channel knowledge. Hence, after the sorting process, the source
node can allocate its transmission power Pb starting from the data stream, which has the largest value of the inverseof the noise power (i.e., the 1st term in (2.33)). Assume that SNRT is the pre-determined threshold that is used to
select the data stream(s) at the relay node. Hence, we suppose to allocate the transmission power to the 1st data
stream in order to satisfy2
σe1,b= SNRT,(2.34)
σ02 ku1 k2
2 denotes the transmission power allocated to the 1st data stream in order to pass SNRT at the relay node.
e1,bwhere σ2 < P , the residual power (P − σ2 ) will be used to allocate to the next data stream which has
e1,b
e1,b
In this case, if σbb
2 > P , which means that P cannot let
e1,b
the second largest value of the inverse of the noise power in (2.33); If σbbPage 22 (49)RESCUE

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even one data stream pass SNRT, thus the system will not let the relay help. To sum up, there are some cases could
happen after the relay-oriented source power allocation process:Case 1: There are some/all data streams’ SNRs at the relay node passing SNRT, but all the transmission power at
the source node has been used;
Case 2: There are some/all data streams’ SNRs at the relay node passing SNRT, and there is still some residual
power left;
Case 3: There is no data stream’s SNR at the relay node passing SNRT.
For Case 1, all the transmission power has been used to let some/all data streams’ SNRs at the relay node passing
SNRT, and there is no additional power left to be allocated to further enhance the channel links between the source
node and the destination nodes. For Case 3, the relay node cannot help, hence, the conventional water-filling power
allocation algorithm (e.g., see [Gol05]) for the channel links between the source node and all the destination nodes
can be implemented in order to maximize the system performance.
For Case 2, in one scenario, the SNRs of some data streams at the relay node can pass SNRT, but the residual power
cannot let an additional data stream’s SNR pass SNRT; in the other scenario, the SNRs of all the data streams at therelay node can pass SNRT, and there is still some residual power left. For both scenarios, the residual power canbe used to further enhance the performance of the channel links between the source node and its served destination
nodes. Specifically, assume that there are L (≤ K) number of the data streams’ SNRs at the relay node that canpass SNRT, and following the assumption in (2.33), the residual power can be formulated by
L

Pres,b2el,b= Pb − ∑ σ
l=1
L= Pb − ∑ σ02 kul k2 · SNRT,(2.35)

l=1

where Pres,b can be used to further enhance the performance of the channel links between the source node and the
2 transmission power to let L data streams’
el,bdestination nodes. In detail, after the source node allocating ∑Ll=1 σ
SNRs pass SNRT, the corresponding allocated power for the data streams will be fixed, and accompanied with the(K − L) data streams which have not been allocated by any power, we will use the residual power Pres,b to
2 |hT w |2el,bεl |hT wl |2 +σεl |hTl wl |2
l l
K
+
log
1
+
maximize ∑Ll=1 log 1 + l
∑
l=L+1
σ2
σ2
εl

subject to00∑Kl=1 εl = Pres,b ,(2.36)

where εl is the additional power spend on the l th data stream, and the total power spend on the l th data stream at
2 =σ
2 + ε for l ∈ {1, . . . , L} or σ 2 = ε for l ∈ {(L + 1), . . . , K}. Here, the problem (2.36)
el,bthe source node is σl,bl
ll,bcan be solved based on the Lagrangian duality theory in [BV04; Gol05].
To sum up, the proposed relay-oriented source power allocation algorithm can be described as:Page 23 (49)RESCUE

D2.2.1 Version 1.0

I.
II.

III.
VI.
V.

Relay-Oriented Source Power Allocation Algorithm
Sort 1/(σ02 kuk k2 ), ∀k, in a descent order, and set i = 1;
2 > P , Case 3 happens, the water-filling power allocation algorithm in [Gol05] is
ei,b
If σ
b
implemented for the channel links between the source node and the destination nodes,
return;2 = P , Case 1 happens, return;
ei,b
If σ
b
2
2 <P ,σ
2
2
ei,b
If σ
b ei,b = σ0 kui k · SNRT;
While i ≤ K:
1. Increasing i by 1;i−1 e 2
2 > P − i−1 σ
2
2. If σei,bb ∑l=1 el,b , Case 2 happens, let Pres,b = Pb − ∑l=1 σl,b ,solve (2.36), break;
2 = P − i−1 σ
2
3. If σei,bb ∑l=1 el,b , Case 1 happens, break;i−1
2 <P −2 ,σ
2 = σ 2 ku k2 · SNRT, goto Step V-1;
ei,b
el,b
ei,b
4. If σib ∑l=1 σ
0
End While

The worst-case computationalcomplexity of proposed power allocation algorithm can be analyzed as: For Step
I, the sorting process takes O K 2 complexity, where K is the total number of the data streams; For StepII, if
3
the condition is satisfied, the conventional water-filling algorithm is implemented,
 3 and the algorithm takes O K
complexity as shown in [Gol05]; For Step V-2, solving (2.36) also takes O K complexity; Apart from these, 3allthe rest steps are just some linear additions and scalar multiplications, where
theircomplexity
are
less
than
O
K .
 3
Hence, the computational complexity of the proposed algorithm is O K , which is the cubic complexity.
2.2.5

Simulation Results

Computer simulations were used to evaluate the proposed relay-oriented source power allocation algorithm in termsof sum-rate and bit error rate. All the channels were generated as independent Rayleigh fading, which remained
static over the transmission of each block. Assumed that the source node was equipped with Mb = 4 transmit
antennas serving K = 4 destination nodes with the help of Mr = 4 antenna based relay node. The SNR was definedas the transmission power normalized by noise power ratio, where both the source node and the relay node havethe same power constraint (i.e., Pb = Pr ). There were two baselines for comparison: For the first one (Baseline I),
the source node optimized its transmission power for the direct link to its served destination nodes, and the relay
node would help if its received data streams’ SNRs can pass the pre-determined threshold (i.e., SNRT=1 dB); For
the second one (Baseline II), equal power allocation was implemented to all activated data streams at both source
and relay nodes.
(k)Given the distance between the source node and the destination nodes, e.g., dsd = d, ∀k, the sum-rate performancesof different schemes are presented based on two relay locations. For Location A, the distance between the source(k)node and the relay node (e.g. dsr ), and the distance between the relay node and the destination nodes (e.g., drd , ∀k)(k)

are the same as d, where dsr = drd = d, ∀k; For Location B, the relay node is placed in the middle of the source(k)node and the destination nodes, where dsr = drd = d/2, ∀k. Following the work in [YG11], the SNRs of the three(k)types of channels, e.g., SNRsd , SNRsr and SNRrd , ∀k, for the two relay locations can be approximated by: (A)
(k)

(k)

SNRsr = SNRrd = SNRsd , ∀k; (B) SNRsr = SNRrd = SNRsd + 10.6 dB, ∀k.

Fig. 2.12 shows the sum-rate performances of different schemes with the two relay locations. As shown in the
figure, for Location A, the sum-rate performance of proposed algorithm is near the same as the one of Baseline I
when the SNRsd values are low. This is because, in this case, few data streams’ SNRs at the relay node can pass
SNRT, and all the transmission power at the source will be allocated to the channel links between the source nodeand the destination nodes. In contrast, at high SNRsd range, the proposed algorithm for Location A outperforms
Baseline I & II, and, as expected, the performances of Baseline I & II are converged to each other; For Location
B, the proposed algorithm outperforms both baselines around 2 bit/s/Hz throughout the whole given SNRsd range.
The reason is because that, in this case, the SNRs of many data streams at the relay node can pass SNRT, where
the proposed algorithm is able to be implemented to let the relay node provide the maximum help.
Fig. 2.13 shows the bit-error-rate performances of different schemes with the two relay locations. Referring

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25

Baseline I (A)Baseline II (A)The proposed algm. (A)Baseline I (B)Baseline II (B)The proposed algm. (B)Sum−rate (bit/s/Hz)

20

Location B
15

10

5
Location A

0

0

2

4

6

8

10

12

14

16

18

20

The source transmission power normalized by noise (dB)Figure 2.12: Sum-rate vs. the source’s transmission power normalized by noise. Comparing the different
schemes for two relay locations.
010

Baseline I (A)Baseline II (A)The proposed algm. (A)Baseline I (B)Baseline II (B)The proposed algm. (B)−110Bit error rate

Location A−2

10

Location B

−3

10

0

2

4

6

8

10

12

14

16

18

20

The source transmission power normalized by noise (dB)Figure 2.13: Bit-error-rate vs. the source’s transmission power normalized by noise. Comparing the different schemes for two relay locations.
to D2.1.1, here, we also include turbo-like channel coding and quadrature phase-shift keying modulation in thesystem. Specifically, the 1/2 rate serial concatenated convolutional code is implemented at the source node, where
the first encoder is the non-recursive non-systematic convolutional code with a generator polynomial G = ([3, 2])8 ,after the random interleaver with the length of 1000, the second encoder is the doped-accumulator with a doping
rate Ps = 1. At the relay node, after passing U, the selected data streams are firstly decoded according to single
iteration based turbo decoders with Bahl-Cocke-Jelinek-Raviv (BCJR) algorithm [BCJ+74], and then the same
encoders as used at the source node are implemented to re-encode and forward the selected data streams to theusers. Each destination node, after the soft demodulation, linearly combines the received bits’ log likelihood ratiosfrom the source node and the relay node, and then decodes the data streams according to up to 50 iteration basedturbo decoders with BCJR algorithm.
As shown in Fig 2.13, for Location A, similar as the sum-rate performance, the proposed algorithm has the same
bit error rate performance as Baseline I when SNRsd is low. However, when SNRsd is high, the bit error rate
performance of the proposed algorithm is a bit worse than Baseline I & II. This is because that, in order to simplify
the decoding process, only one iteration is implemented for turbo decoding at the relay node. In this case, even the
proposed algorithm tries to allow more data streams to pass the pre-determined threshold SNRT, there still exist
certain amount of decoding errors and will be forwarded to the destination nodes. Such kind of error propagation
can hardly happen when the relay node is placed in Location B. That is why the proposed algorithm in Location B,

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as expected, has much better bit-error-rate performance by comparing with the two baselines throughout the whole
given SNRsd range.
2.2.6

Summary

In this section, a relay-oriented source power allocation algorithm for MIMO S-DF relaying has been developed
to maximize the system performance under the total power constraints. The proposed algorithm featured the cubic
computational complexity. Simulation results have shown that, without the global channel knowledge at the source
node, the proposed algorithm outperforms the conventional MIMO S-DF relaying schemes in terms of sum-rateand bit-error-rate, especially when the relay node is placed in the middle of the source node and the destination
nodes.

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3.

Power Allocation for Chief Executive Officer (CEO) Problem

3.1

Outage Based Power Allocation for CEO Problem

In this section, we minimize the outage probability for TS2 also referred to as CEO problem [BZH96]. It is shownthat the outage probability can be approximated as a convex function and thus, convex optimization can be applied
to find the optimal solution.
3.1.1

System Model

Fig. 3.1 represents the system model, where an i.i.d. binary Bernoulli information sequence b0 is originated by
the source (S) with Pr[b0 = 0] = Pr[b0 = 1] = 0.5. The information sequence is corrupted by independent binary
Bernoulli distributed errors ei via BSCs. The information sequence bi = b0 ⊕ ei , i ∈ {1, 2, 3} is observed by relay(R) i and can be associated with the bit flipping probability pi = Pr[ei = 1] and pi ∈ (0, 0.5). ”⊕” denotes thebinary exclusive OR operation. The correlated information sequences bi , ∀i are transmitted via Rayleigh fading
channels to the destination (D). The PDF of the Rayleigh fading channel is given byp(γi ) =γi
1exp(− ), ∀i,
Γi
Γi

(3.1)

Ei −η
·d ,
N0 i

(3.2)

with instantaneous SNR γi and average SNRΓi =where Ei is the transmit power of relay i, N0 is the variance of AWGN, di the distance between relay i and destination and η the path loss exponent. Best performed recovery of b̂0 at the destination can by achieved if all
information sequences b̂1 , b̂2 and b̂3 are recovered error free [WMF15].

b1
R
p1
b0

b2
p2

S
p3

R
b3

RF 1
RF 2

D

b̂1 , b̂2 , b̂3

RF 3

R
Figure 3.1: System model for CEO Problem.
If at least one relay information sequence cannot be recovered error-free, the information recovery of the source isconsidered to be unsuccessful, also referred to as outage. In chapter ”Slepian-Wolf Admissible Rate Region Based
Outage Probability Derivation for CEO Problem” of deliverable D1.2.1, the outage probability for high-SNR of
the system model is derived and can be approximated as
Pout ≈

C1 − 1 C2 − 1 C3 − 1 C1C2 −C1,2 C2C3 −C2,3 C1C3 −C1,3
+
+
+
+
+
Γ1
Γ2
Γ3
Γ1 Γ2
Γ2 Γ3
Γ1 Γ3

(3.3)

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where the constants are defined as
C1 ,2Rc H(b1 |b2 ,b3 ) ,(3.4)

Rc H(b2 |b1 ,b3 )

,

(3.5)

C3 ,2Rc H(b3 |b1 ,b2 ) ,

(3.6)



C2,3 ,2Rc H(b2 ,b3 |b1 ) − 2Rc H(b2 ,b3 |b1 ) ln 2Rc H(b1 |b2 ) − ln 2Rc H(b2 |b1 ,b3 ) ,
C1,2 ,2Rc H(b1 ,b2 |b3 ) − 2Rc H(b1 ,b2 |b3 ) ln 2Rc H(b1 |b3 ) − ln 2Rc H(b1 |b2 ,b3 ) ,
C1,3 ,2Rc H(b1 ,b3 |b2 ) − 2Rc H(b1 ,b3 |b2 ) ln 2Rc H(b2 |b3 ) − ln 2Rc H(b3 |b1 ,b2 ) ,

(3.7)

C2 ,2

(3.8)
(3.9)

where H represent the entropy function and Rc the spectrum efficiency of the Rayleigh fading channel, considering
the channel coding scheme and the modulation multiplicity [CAM13a].
3.1.2

Optimal Power Allocation

By redistribution of power among the relays, a reduction of outage probability can be achieved. The total powerof all relays represented by ET is allocated by αi ∈ [0, 1] and ∑3i=1 αi = 1 to the corresponding relay i, ∀i. Consequently, the average SNR can be replaced by
Γi = αi ET .

(3.10)

with noise variance N0 being normalized to unity for all channels and distances di , ∀i are assumed to be equal.
Finally, Eq. (3.3) can be expressed in dependence on αi and ET as
Pout ≈

C1 − 1 C2 − 1 C3 − 1 C2 ·C3 −C2,3 C1 ·C2 −C1,2 C1 ·C3 −C1,3
+
+
+ 2
+ 2
+ 2
.
ET · α1 ET · α2 ET · α3ET · α2 · α3ET · α1 · α2ET · α1 · α3(3.11)

If Eq. (3.11) can be proven to be convex in αi , ∀i and ET , the outage probability can be minimized by convex
optimization.

3.1.2.1

Proof of Convexity

The approximation of the outage probability can be rewritten as
6aPout ≈ ∑ cn · α1a1n · α2a2n · α3 3n · ETa4n(3.12)

n=1

with
n

cn

a1n

a2n

a3n

a4n

1
2
3
4
5
6

C1 − 1C2 − 1C3 − 1C2 ·C3 −C2,3
C1 ·C2 −C1,2
C1 ·C3 −C1,3

−1
0
0
0
−1
−1

0
−1
0
−1
−1
0

0
0
−1
−1
0
−1

−1
−1
−1 .
−2
−2
−2

If and only if cn > 0 and ain ∈ R, then Eq. (3.12) is a posynomial function [BV10, Chap. 4.5]. For (i, j, l) ∈{(1, 2, 3), (2, 1, 3), (3, 1, 2)} and with Eq. (3.4) - Eq. (3.6) the constants are
ci = Ci − 1 = 2Rc H(bi |b j ,bl ) − 1.

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Where H(bi | b j , bl ) > 0, since the information sequences bi , b j and bl are correlated but not identical due to(pi , p j , pl ) ∈ (0, 0.5)3 . Consequently,
ci > 0.
For (k, i, j, l) ∈ {(4, 2, 3, 1), (5, 1, 2, 3), (6, 1, 3, 2)} and with Eq. (3.7) - Eq. (3.9) the constants are
ck = Ci ·C j −Ci, j= 2Rc (H(bi |bl ,b j )+H(b j |bl ,bi )) + 2Rc H(bi ,b j |bl ) log 2Rc (H(bl |bi )−H(bi |bl ,b j ) − 1= 2Rc (H(bi ,b j |bl )−H(bl |b j )+H(b j |bl ,bi )) + 2Rc H(bi ,b j |bl ) log 2Rc (H(bl |bi )−H(bi |bl ,b j ) − 1= 2Rc (H(bi ,b j |bl ) 2Rc (H(b j |bl ,bi ))−H(bl |b j )) − log 2Rc (H(bi |bl ,b j )−H(bl |bi ) − 1

since bi ,b j and bl are correlated it holds H(b j | bl , bi ) < H(bl | b j ). It is known that x − log(x) > 1 if 0 < x < 1,
as a resultck > 0.
Having proven Eq. (3.12) to be a posynomial function, the minimization of the outage probability can be performed
by geometric programming. It is shown that geometric program can be transformed to a convex optimization
problems [BV10, Chap. 4.5].

3.1.2.2

Simulation Settings

In the simulations two scenarios with different bit flipping probabilities are investigated• Scenario 1 : p1 = 0.35 p2 = 0.01 p3 = 0.01,• Scenario 2 : p1 = 0.02 p2 = 0.10 p3 = 0.01.
The convex optimization problem is solved with CVX [GB15] in geometric programming mode and the solution
is verified with Monte-Carlo simulation. We assume binary phase shift keying as modulation and a channel code
rate of 1/2, thus Rc = RM1·Rcc = log 12·1/2 = 2. Furthermore the distance between relay and destination is di = 0.5, ∀i
2and the path loss exponent η = 4.

3.1.2.3

Total Power Fixed

In this subsection we aim to minimize the outage probability while the total power is fixed. The convex optimization problem can be formulated asminimize Pout (α1 , α2 , α3 )subject to 0 ≤ αi ≤ 1, ∀i(3.13)∑i αi = 1.

Tab. 3.1 shows the scenarios where ET ∈ {20 dB, 30 dB, 40 dB}. The outage probability with optimal power
allocation (OPA) Pout,OPA shows a reduction by 29.5% (scenario 1) resp. 9.6% (scenario 2) with respect to theoutage probability with equal power (EP) distribution Pout,EP . Fig. 3.2 (a) shows the outage probability for ET ∈[10 dB, 40 dB]. A reduction of the outage probability with OPA can be observed in the whole range of ET . The
convex optimization result is verified by the Monte-Carlo simulation. Furthermore the OPA performance is close
the Monte-Carlo simulation in low- and medium-SNR region. Fig. 3.2 (b) shows OPA for the ET range. The
outage probability is minimized, if more power is allocated to the relays with higher bit flipping probabilities.
pi < p j −→ αi < α j(3.14)

With increasing ET , the allocation converges eventually.

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Table 3.1: Total power fixed
Scenario

ET in dB

1

20
30
40

2

20
30
40

Pout,OPA

Pout,EP

Gain in %4.0938 · 10−3
4.0731 · 10−4
4.0718 · 10−55.7863 · 10−3
5.7801 · 10−4
5.7795 · 10−5

29.25
29.53
29.55

2.5231 · 10−3
2.5218 · 10−4
2.5216 · 10−5

10−12.7904 · 10−3
2.7897 · 10−4
2.7896 · 10−5

9.58
9.60
9.61

0.7

α1
α2
α3

EP
OPA
Simulation EP

0.6
Allocation

Simulation OPA

10−2
Pout

Scenario 1

10−3

Scenario 2

2 · 10−3

0.5
0.4
0.3

10−4

0.2

10

5 · 10−4

−5

10

25

15
26

20

25
27

30

35

40

10

15

20

25

30

35

40

ET in dBET in dB(b)(a)

Figure 3.2: Outage probability (a) and allocation (b) with fixed total power.
3.1.2.4

Outage probability requirement fixed

In this subsection the minimum value of ET is desired which copes with an outage probability requirement Cout .
The minimization of ET is achieved by OPA. We can formulate the convex optimization problem as
minimize ET
subject to Pout (ET , α1 , α2 , α3 ) −Cout ≤ 0(3.15)

0 ≤ αi ≤ 1, ∀i∑i αi = 1.

Tab. 3.2 shows the two scenarios where outage probability requirement is Cout ∈ {10−3 , 10−4 , 10−5 }. The representative value for total power with OPA ET,OPA shows a reduction by 1.5 dB (scenario 1) resp. 0.45dB (scenario 2)
with respect to the representative value for total power with EP distribution ET,EP . Fig. 3.3 (a) shows ET for outage
probability requirement Cout ∈ [10−5 , 10−1 ] range. A gain of ET in the hole range of outage probability requirement
can be observed. The convex optimization result is verified by the Monte-Carlo simulation. Furthermore the OPA
performance is close to the Monte-Carlo simulation in low- and medium-SNR region as well. Fig. 3.3 (b) shows
the OPA for outage probability requirement range. With decreasing Cout , the allocation converges eventually.

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Table 3.2: Outage probability requirement fixed
Scenario

CoutET,OPA in dB

ET,EP in dB

Gain in dB

1

10−3

10−4
10−5

26.12
36.10
46.10

27.62
37.62
49.62

1.50
1.52
1.52

10−3
10−4
10−5

24.05
34.02
43.93

24.46
34.46
44.46

0.41
0.44
0.52

2

50

0.7

EP
OPA

0.6

Simulation OPA

40

30

Scenario 1

32

20

Allocation

ET in dB

Simulation EP

α1
α2
α3

0.5
0.4

1.54

0.3

Scenario 2

30

0.55

0.2
28

10 −5
10

3 · 10−4

10−4
7 · 10−4

10−3

10−2

10−5

10−4

10−3

Cout

Cout

(a)

(b)

10−2

Figure 3.3: Outage probability (a) and allocation (b) with outage probability requirement fixed.

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3.2

SNR Limit Based Power Allocation Algorithm

3.2.1

Motivation and Objective

We propose an algorithm for distribution of transmit power among all relays observing the same source in a wirelesssensor network. Since each relay can only access erroneous observations of the source, an estimation of theunderlying correct source information is performed at the destination, to which each relay reports its observations.
Hence, the system follows TS 2, which can be adequately modeled by the CEO problem [BZH96]. An abstractsystem model is shown in Fig. 3.4. The binary source S is erroneously observed by the K relays, which is modeledby the random variables Ek which follow a Bernoulli distribution with Pr(Ek = 1) = pk . At the relays, eachobserved data stream is encoded and transmitted over an AWGN channel to the destination, where the underlying
binary sequence is estimated by jointly decoding [WMF15] all observed streams.
The task of the power allocation algorithm is to define the transmit power of each relay, so that certain requirements
such as a minimum BER at the joint decoder is achieved. Since the quality of observations of each relay is describedby pk , it is intuitive to provide a power allocation scheme as a function of the BER of the different relays.

E1

N1
U1

S

Encoder1

X1

EK

Joint
Decoder

NK
UK

BSC

Y1

EncoderK
Relays

XK

Ŝ

YK

AWGN

Figure 3.4: System model: Estimating a binary source observed by multiple relays.
Clearly, a good power allocation can reduce the required overall transmit power in a network, so that batteries
last longer and the overall power consumption is reduced. Also, with smart resource management it is possible toreduce the outage probability of the entire network.
Four key performance indicators (KPIs) were identified that can be used to describe the performance of the joint
decoder at the destination. These are• The error floor (pEF ) at the destination. Due to the structure of the system model, S cannot be restoredcompletely, but an error floor remains [WMF15]. This error floor should be educed to as low as possible.
• The SNR limit (γlim ) at the destination where the BER curve reaches its error floor. Clearly, a lower γlim
reduces the overall required transmit power of the system.
• The amount of active relay nodes (kact ). With a reduced number of relays both the transmit power anddecoding quality can be improved. Furthermore, if not all relays are used for transmission, the protocol andsignalling overhead in the network can be significantly reduced.
• The total transmit power (eT ) is the fourth KPI. Clearly, the total transmit power directly influences the total
power consumption in the network and should hence be kept as low as poosible.
When looking at the definitions of the KPIs it is clear that one cannot optimize at the same time. For example,
with a reduced number of relay nodes, the error floor will certainly increase. Instead, the objective is to optimize
one KPI or, if possible, two KPIs and satisfy thresholds for other KPIs. We have formulated three optimization
problems as follows:

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1.
minimize

γlim
pEF = pallsubject toeT = ET
kact = KThis objective minimizes the SNR limit by redistributing the total transmit power to all available relays. ET
represents the total transmit power budget and pall the lowest possible error floor including all relays.
2.
minimizef (eT , γlim )subject topEF = pallkact = K

In this objective, a joint metric of total transmit power and SNR limit is optimized, given that all available
relays are used and hence the error floor is kept at its minimum.
3.
minimizef (kact , γlim )subject topEF = palleT = ETThis metric optimizes the number of used relays. Within the proposed algorithm, the number of switchedoff relays is given, and the algorithm chooses which relays to switch off. This can be extended to run thealgorithm for different numbers of switched-off relays and compare the results.
3.2.2

State-of-the-Art

For the Slepian-Wolf problem in data gathering wireless sensor network (WSN), a power allocation algorithm is
described in [CAM13a]. The proposed algorithm is implemented for block Rayleigh fading channels. For the CEO
problem with AWGN channels, no power allocation algorithm is yet available, which is focused in this section.
When looking at the typical BER curve of the proposed joint decoder in Fig. 3.5, a relatively wide ramp region
appears. The simulation settings for this particular figure were set to p = [0.17 0.19 0.03 0.19 0.13 0.02 0.06]
but the principle behaviour is general. The ramp region, as illustrated in Fig. 3.5, can be comprehended byexamining the performance of the relay decoders. The relays with low bit flipping probabilities such as relay 3
and 6 achieve arbitrarily low error rates at approximately Eb/N0 = −5dB and outperform the relays 1, 2 and 4 wichsuffer from large bit flipping probabilities by approximately 3dB. This phenomenon is caused by the different
mutual information between one relay k and the remaining relays.
I(Uk ;Uk,κ ) = H(Uk ) − H(Uk |Uk,κ )(3.16)

where set Uk,κ = {Uκ | κ ∈ K \k} and set K = {1, 2, ..., K}. Uk is binary Bernoulli random variable such that Eq.
(3.16) can be reformulated toI(Uk ;Uk,κ ) = 1 − H(Uk |Uk,κ )(3.17)

A causal relation between bit flipping probabilities and mutual informations holds. If
pl < pm < ... < pn

(3.18)

with l 6= m 6= n and l, m, n ∈ K , the conditional entropies are relate to each other as
H(Ul |Ul,κ ) < H(Um |Um,κ ) < ... < H(Un |Un,κ ).

(3.19)

Without loss of generality, we prove the above statement for p1 < p2 < p3 :Page 33 (49)RESCUE

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Performance
10

0
Joint Decoder
Relay Decoder 1
Relay Decoder 2
Relay Decoder 3

10−1

Relay Decoder 4
Relay Decoder 5

BER

Relay Decoder 6
Relay Decoder 7

10−2

10−3
−8

−6

−4

−2

0

2

Eb/N0 in dB

Figure 3.5: Performance of joint decoder and relay decoders.
Proof.
H(U1 |U1,{2,3} ) = H(U1 ,U2 ,U3 ) − H(U2 |U3 ) − H(U3 )= H(U1 ,U2 ,U3 ) − H(p2 + p3 − 2 · p2 · p3 ) − 1> H(U1 ,U2 ,U3 ) − H(p1 + p3 − 2 · p1 · p3 ) − 1 = H(U2 |U2,{1,3} )> H(U1 ,U2 ,U3 ) − H(p1 + p2 − 2 · p1 · p2 ) − 1 = H(U3 |U3,{1,2} )As a consequence, the mutual information can be ordered according to Eq. (3.17) asI(Ul ;Ul,κ ) > I(Um ;Um,κ ) > ... > I(Un ;Un,κ ).

(3.20)

This implies, that any relay decoder with low bit flipping probability benefits to a greater extent from the information exchange executed via the global iteration (GI). It therefore achieves an arbitrarily low error rate, despite thefact that the AWGN channel is suffering from significant noise. A relay decoder with a high bit flipping probability
depends more on reliable information from the relay transmitted over the AWGN channel since the informationexchange from the GI is not as beneficial. Finally, at γlim = −2 dB all relays can be decoded correctly and hence
the decoder reaches its error floor.
Hence, in order to lower γlim , it is necessary that the relay decoders for the high bit flipping probabilities are
decoded at a lower SNR. Naturally, this can be achieved by distributing more power to these relays. On the other
hand, relays with low bit flipping probability can transmit with a lower transmit power, since they benefit morefrom the other transmitting relays. In the end, the SNR limits for each relay decoder should fall together in orderto reduce the overall SNR limit as much as possible. This is illustrated in Fig. 3.6, where the SNR limits per relay
decoder are depicted as vertical dashed lines.
Fig. 3.7 illustrates the power distribution used in Fig. 3.5, i.e. equal power allocation, which is depicted by equally
sized blocks for each relay also refereed to as the reference case. In order to describe different power allocations,
we introduce the variable power allocation variable αk which describes the ratio of the overall transmit power thatis transmitted by the kth relay. Hence, the transmit energy of the kth relay is given by
Ek = αk ET /K(3.21)

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Performance
10

0
Joint Decoder

BER

10−1

10−2

γk (p1 , ..., p7 , E1 , ..., E7 )

10−3
−8

−6

−4

−2

0

2

4

Eb/N0 in dB

Figure 3.6: Performance of joint decoder and SNR limits of relay decoders.
ET

E1

0

E2

1

E3

2

E4

3

E5

4

E6

5

E7

6

7

Figure 3.7: Equal transmit power distribution among relays.
where ET is the total available transmit energy and∑ αk = K.

(3.22)k

For a given set of bit flipping probabilities, the SNR limits for the relay decoders are a function of {αk }. With
the knowledge of this function, the power allocation can be optimized. However, the analytical derivation of thisexpression remains an open problem up to now and the following algorithm is based on a numerical evaluation.
3.2.3

Implemented Power Allocation Algorithm

Algorithm 1 optimizes the transmit power for each relay such that the SNR limits of each relay decoder alignat one common SNR. Initially, it equally distributes the available transmit power to all relays and runs the joint
decoder (JD) at the destination. Then, for each relay decoder it is evaluated at which SNR γk,T it reaches a given
BER TBER . This information is then translated into an adaptation of the allocated transmit power for each relay bycomparing γk,T to the mean of all SNR limits. However, the maximum change in the transmit power for each relay
and iteration is given by the parameter β . β adapts to the improvement of the relay decoders and finally reaches asmall value such that the convergence of the algorithm is ensured. Finally, the parameter ε describes the required
accuracy of the algorithm, i.e. how close the SNR limits for the relay decoders should be to each other and hence
provides a trade-off between accuracy and complexity.

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D2.2.1 Version 1.0

Algorithm 1: Power allocation
Input: ε: tolerated range; β : redistribution maximum; M: maximum iteration;
K: number of relays; ET : total transmit power
Output: αk : power distribution
Initialization: αk (1) = 1/K , ∀k;for m = 1 to M do
Run JD with Ek = αk (m) · ET , ∀k in SNR range;
Output BERk for relay decoder;Calculate γk,T = γk (BERk = TBER );Calculate γdif = mink (Tγk ) − maxk (Tγk );if γdif ≤ ε then
αkfinal = αk (m), ∀k;Exit for;else
γ −E(γ )αk (m + 1) = αk (m) − β · k,TE(γ )·,T , ∀k;·,T

end
if m=M then
αkfinal = αk (M), ∀k;
end
end

3.2.4

Simulation Results

In this section, the performance of the power allocation algorithm and the resulting decoding performance at thedestination are evaluated. Simulations are carried out for three and seven relays with bit flipping probabilities given
in Tab. 3.3. However, for the three-relay case it only makes sense to optimize for objective one. The resulting
optimal power allocations are given as the third column in this table.
Table 3.3: Simulation Settings

3 Relays
7 Relays

Bit flipping probabilities p

Objectives

power distribution α0.11 0.17 0.04
0.17 0.19 0.03 0.19 0.13 0.02 0.06

1.
1.
2.
3.

0.987 1.119 0.894
1.234 1.308 0.623 1.317 1.125 0.575 0.815
0.896 0.942 0.537 0.926 0.812 0.497 0.637
2.306 0.872 1.880 0.773 1.167

 0.03  0.13 0.02 0.06
0.17 
0.19
0.19
1. Objective
minimizesubject toγlim
pEF = palleT = ET
kact = K

Fig. 3.8 shows the power redistribution among seven relays (cf. Tab 3.3). The allocation assigns more power torelays with high bit flipping probabilities. Relays with low bit flippling probabilities are allocated less power.
For the three-relay case, the behaviour is similar, as shown in Tab. 3.3. Fig. 3.9 shows the performance of the joint
decoder when using three relays. Obviously, the SNR limits for the relay decoders overlap and hence the ramp
region in the BER curve is shorter. However, the ramp region starts at higher SNR compared to the equal power
distribution case and in overall, the performance of BER curve is not strongly influenced by the different power
allocation.
For seven relays a strong alignment of relay decoder SNR limits is shown in Fig. 3.10 (b). More power is assigned
to the relays with higher bit flipping probabilities which improves the performance of corresponding decoders andPage 36 (49)RESCUE

D2.2.1 Version 1.0
ET

E1

0

1

E2

E3

E4

E5

E6

2

3

4

5

6

E7

7

Figure 3.8: Redistributed transmit power among relays by means to fulfil first objective.
100

100
α equaly distributed
α allocatedα equaly distributed
α allocated

BER

BER

10−1

10−1

10−2

10−3

10−2

−5

−4

−3

Eb/N0 in dB

−2

10−4

−1

−5

−4

−3

Eb/N0 in dB

(a)

−2

−1

(b)

Figure 3.9: Three relays - performance of joint decoder (a) and relay decoders (b) for first objective.
100

100
α equaly distributed
α allocatedα equaly distributed
α allocated
10−1
BER

BER

10−1
10−2

10−2
10−3

10−3
−7

−6

−5

−4

Eb/N0 in dB

(a)

−3

−2

10−4
−7

−6

−5

−4

Eb/N0 in dB

−3

−2

(b)

Figure 3.10: Seven relays - performance of joint decoder (a) and relay decoders (b) for first objective.
consequently a significant improvement of γlim can be observed. As a result the BER curve shows a smaller ramp
region in Fig. 3.10 (a) compared to the reference case. However relay decoders with low bit flipping probabilities
experience a significant drop in performance and thus, a shift of the waterfall region towards a higher SNR value
appears.
2. Objective
minimizesubject tof (eT , γlim )pEF = pallkact = K

Fig. 3.11 shows the power redistribution among seven relays subject to a reduction of total transmit power by 25%(cf. Tab 3.3) compared to the reference case.

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ET

E1

E2

0

1

E3

E4

E5 E6 E7

2

3

4

5

6

7

Figure 3.11: Redistributed transmit power among relays by means to fulfil second objective.
100

100
α equaly distributed
α allocatedα equaly distributed
α allocated
10−1
BER

BER

10−1
10−2

10−2
10−3

10−3
−7

−6

−5

−4

Eb/N0 in dB

−3

−2

10−4
−7

−1

−6

−5

(a)

−4

Eb/N0 in dB

−3

−2

−1

(b)

Figure 3.12: Seven relays - performance of joint decoder (a) and relay decoders (b) for second objective.
Fig. 3.12 (b) shows a strong alignment of relay decoder SNR limits for power redistribution among the sevenrelays. Despite the reduction of total transmit power the SNR limit can remain the same compared to the referencecase. As a result the BER curve hits the error floor at the same SNR value as the reference case, depicted in Fig.
3.12 (a). Moreover the BER curve shows a smaller ramp region due to the strong alignment of relay decoder SNR
limits. However, a significant loss in the waterfall region appears due to the reduced transmit power of all relays.
3. Objective
minimizesubject tof (kact , γlim )pEF = palleT = ET

Fig. 3.13 shows the power redistribution among all relays subject to a reduction of active relays by two (cf. Tab
3.3) compared to the reference case. Since two relays are no longer required to use power resources, more transmit
power is allocated to the remaining relays.
ET

E1

0

1

E3

2

3

E5

E6

4

5

E7

6

7

Figure 3.13: Redistributed transmit power among relays by means to fulfill third objective.
A tremendous reduction of γlim is achieved by the redistribution of the power and reducing the active relays.
Furthermore the relay decoder SNR limits can be aligned. Consequently the ramp region is reduced as shown in
Fig. 3.14 (a) compared to the reference case. The BER curve hits the possible error floor for five relays at a lower
SNR limit compared to the reference case. However, the error floor of five relays is higher compared to the error
floor of seven relays [WMF15].

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100100
α equaly distributed
α allocatedα equaly distributed
α allocated
10−1
BER

BER

10−1
10−2

10−2
10−3

10−3

−7

−6

−5

−4

Eb/N0 in dB

(a)

−3

−2

10−4

−7

−6

−5

−4

Eb/N0 in dB

−3

−2

(b)

Figure 3.14: Seven relays - performance of joint decoder (a) and relay decoders (b) for third objective.
3.2.5

Summary

A power allocation algorithm has been described that aims to equalize the SNR limits for each relay decoder, suchthat the ramp region of the joint decoder can be removed or reduced. The proposed algorithm shows significant
improvements in the SNR limit for the relay decoders, and the BER curve of the joint decoder qualitatively follows
as expected. However, no significant improvements for the BER of the joint decoder could be achieved with theproposed approach.

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4.

Power Allocation for Orthogonal Multiple Access Relay Channel
Allowing Intra-link Errors

4.1

Motivation and Objective

Decode-and-forward (DF) [NHH04] is one of the classical relaying strategies applied in cooperative wireless networks. The signals transmitted from source nodes are able to be re-generated in the DF strategies such that thenoise propagation can be avoided. Quite recently, several resource allocation problems for DF-based orthogonal
MARC systems have been investigated. For example, in [EZL+10] and [SAK12], the sum rates of the DF-based
orthogonal MARC systems are maximized at the destination by optimally allocating the power for the different
nodes and carriers with the total and individual power constraints, respectively. In [AB14], weighted sum rate maximization (WSRM) is studied for a two-user DF-based MARC system with using orthogonal frequency divisionmultiple access (OFDMA).
However, it has been found in [LZA+14] that even the estimates are incorrectly decoded at the relay, the forwarded
erroneous estimates are still helpful for the reconstruction of the information sequences sent from the source nodes
at the destination. Moreover, it is shown in [LZM14], that the outage performance is improved by forwarding theincorrectly decoded estimates, especially when one of the source nodes is quite far away from both the relay anddestination.
The main contributions of this section are summarized as follows: A power minimization problem subject to lossy
rate constraints is formulated for MARC. Successive convex approximation (SCA) [MW78] algorithm is used forsolving the non-convex problem. Furthermore, convergence of the algorithm is justified.

4.2

System Model

Block diagram of an orthogonal MARC is shown in Fig. 4.1, where two source nodes S1 and S2 are communicatingwith a common destination D with the assist of a relay node R. It is assumed that all the nodes are equipped withsingle antenna, and R works in a half-duplex mode. The K-bit length i.i.d. binary information sequence generated
from S1 and S2 are denoted as u1 = {u1 (i)}Ki=1 and u2 = {u2 (i)}Ki=1 . There are three time slots in one transmission
1
cycle. In the first two time slots, S1 and S2 encode u1 and u2 , and broadcast the coded sequence x1 = {x1 (i)}Ni=1
N2
and x2 = {x2 (i)}i=1 to both R and D, respectively.
In the third time slot, R first tries to reconstruct u1 and u2 that transmitted from S1 and S2 , of which the estimates
are denoted as ũ1 and ũ2 , respectively. Different from the conventional DF protocol, R always combines the two
estimates as ur = ũ1 ⊕ ũ2 in the system considered even though ũ1 and ũ2 may contain errors. Here ⊕ indicatesr
modulus-2 addition. ur is then interleaved, re-encoded into coded sequence xr = {xr (i)}Ni=1, and forwarded toD. Finally, after receiving signals from S1 , S2 and R, joint decoding is performed at D to retrieve the original
information sequences u1 and u2 .
1st time slot

3rd time slot

2nd time slot

Figure 4.1: The block diagram of orthogonal MARC.

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Encoder
Decoder
Encoder

Encoder
Figure 4.2: Abstract model for lossless source coding of u1 and u2 with the help of ur .
The code rate of S1 , S2 and R are defined as R1 = K/N1 , R2 = K/N2 and Rr = K/Nr , respectively. Let Li, j , i ∈{S1 , S2 , R}, j ∈ {R, D}, i 6= j, denote the link between node i and j. The received signal of Li, j can be expressedasp√yi, j = Gi, j · Pi · xi + ni, j ,(4.1)
where Gi, j is the geometric-gain of Li, j , Pi is the average transmit power at node i, and ni, j is the vector of independent zero-mean complex AWGN with variance σi,2 j = N0 /2 per dimension. Let di, j denote the distance of Li, j
and GS1 ,D being normalized to unity. Then, Gi, j can be defined asGi, j ,

dS1 ,D
di, j

l
,

(4.2)

where l is the pathloss exponent.

4.3

Conditions for Reliable Transmission

In this section, we investigate the conditions that guarantee reliable transmission from S1 and S2 to D. First we
look at the transmission from S1 to R. As described above, errors are allowed at R after decoding. According to
Shannon’s lossy source-channel separation theorem [Sha59; VFC+14], u1 can be transmitted from S1 to R with adistortion level D1 if the following inequality is satisfied:R(D1 ) · R1 ≤ C(

GS1 ,R P1
),
N0

(4.3)

where R(D1 ) and C(x) = log2 (1 + x) are the rate-distortion function and the Gaussian capacity function, respectively. With hamming distortion measure, D1 = K1 ∑Ki=1 d(u1 (i), ũ1 (i)), where
1, if u1 (i) 6= ũ1 (i),d(u1 (i), ũ1 (i)) =(4.4)
0, if u1 (i) = ũ1 (i),and R(D1 ) = 1 − H2 (D1 ), where H2 (x) = −x log2 x − (1 − x) log2 (1 − x) is the binary entropy function. In thesimilar way, the condition for transmitting u2 from S2 to R with a distortion level D2 isR(D2 ) · R2 ≤ C(

GS2 ,R P2
).
N0

(4.5)

Next we consider the transmission of u1 , u2 and ur to D from S1 , S2 and R, respectively. Note that ur is not to be
recovered at D, in other words, ur serve as a helper in recovering u1 and u2 at D, as shown in Fig. 4.2. This problem
falls exactly into the category of lossless source coding with one helper. Assume u1 , u2 and ur are first describedwith rates Rs1 , Rs2 and Rsr , respectively. According to the theorem in [GK11, Theorem 10.4], the admissible rate
region is specified as
Rs1 ≥ H(u1 |u2 , ûr ),
Rs2 ≥ H(u2 |u1 , ûr ),(4.6)
s
R + Rs2 ≥ H(u1 , u2 |ûr ),


 1
s
Rr ≥ I(ur ; ûr ),Page 41 (49)RESCUE

D2.2.1 Version 1.0

where ûr is the estimate of ur at the final output, as shown in Fig. 4.2. Let D3 denote the hamming distortion
between ûr and ur . By using the similar approach as in [LZM14], (4.6) can be further expressed as
Rs1 ≥ H2 (D1 ∗ D2 ∗ D3 ),
Rs2 ≥ H2 (D1 ∗ D2 ∗ D3 ),(4.7)
s
R + Rs2 ≥ 1 + H2 (D1 ∗ D2 ∗ D3 ),


 1
s
Rr ≥ 1 − H2 (D3 ),where x ∗ y = x(1 − y) + (1 − x)y is the binary convolution.
On the other hand, according to Shannon’s separation theorem, if the total information transmission rates over thethree independent channels satisfy Rs R1 ≤ C( GS1 ,D P1 ),
N0 1
Rs2 R2 Rs Rr r

GS2 ,D P2
N0 ),GR,D Pr
C( N0 ),

≤ C(

≤

(4.8)

the message error probability can be made arbitrarily small.
In order to achieve reliable transmission of u1 and u2 , the conditions shown in (4.3), (4.5), (4.7) and (4.8) should
be satisfied simultaneously. These conditions are reformulated and summarized as
GS ,R P1

1 − H2 (D1 ) ≤ R11 C( N1 0 ),
GS ,R P2

1 − H2 (D2 ) ≤ R12 C( N2 0 ),




G Pr

1 − H2 (D3 ) ≤ R1r C( R,D
N0 ),
(4.9)
GS1 ,D P11
H2 (D1 ∗ D2 ∗ D3 ) ≤ R1 C( N0 ),GS ,D P2
H2 (D1 ∗ D2 ∗ D3 ) ≤ R12 C( 2N0 ),GS ,D P1
GS ,D P2

1 + H2 (D1 ∗ D2 ∗ D3 ) ≤ R11 C( 1N0 ) + R12 C( 2N0 ).

4.4

Power Minimization with Rate Constraints

In this section, a power minimization problem with the conditions for reliable transmission is derived. Then, SCA
is applied for solving the non-convex problem. The power minimization problem with constraints shown in (4.9)
is expressed asminimize P1 + P2 + Pr
P,Dsubject to

GS1 ,R P1
N0 ), (i)GS2 ,R P2
1
1 − H2 (D2 ) ≤ R2 C( N0 ), (ii)G Pr
1 − H2 (D3 ) ≤ R1r C( R,DN0 ), (iii)
GS ,D P1
1
H2 (D1 ∗ D2 ∗ D3 ) ≤ R1 C( 1N0 ), (iv)
GS ,D P2
H2 (D1 ∗ D2 ∗ D3 ) ≤ R12 C( 2N0 ), (v)1 − H2 (D1 ) ≤ R11 C((4.10)

1 + H2 (D1 ∗ D2 ∗ D3 ) ≤GS1 ,D P1
GS2 ,D P2
11
R1 C( N0 ) + R2 C( N0 ) (vi),where P = [P1 , P2 , Pr ] and D = [D1 , D2 , D3 ].
4.4.1

Successive Convex Approximation

In the following, we derive an SCA for the non-convex power minimization problem shown in (4.10). It is wellknown that binary entropy function H2 : [0, 0.5] → [0, 1] is concave and increasing when its argument is in [0, 0.5].
Furthermore, logarithm is also a concave function and therefore, constraints (i-iii) are convex constraints. Letf (D1 , D2 , D3 ) : [0, 0.5]3 → [0, 0.5] denote the binary convolution D1 ∗ D2 ∗ D3 . Calculating the Hessian of f , it canbe shown that f is a concave function. Using a vector composition result from [BV04, Chpt. 3.], it is directly seen

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that H2 ( f (D1 , D2 , D3 )) is a concave function. Therefore, it needs to be approximated by its convex upper bound,i.e., linear function. A local linear approximation of f (D1 , D2 , D3 ) at a point (D̂1 , D̂2 , D̂3 ) is derived asfˆ(D1 , D2 , D3 ) =∂ f (D̂1 , D̂2 , D̂3 )(Dk − D̂k ).
∂ Dk
k=1
3

f (D̂1 , D̂2 , D̂3 ) + ∑(4.11)

Due to the concavity of f , it holds that fˆ(D1 , D2 , D3 ) ≥ f (D1 , D2 , D3 ). Let fˆ(D1 , D2 , D3 ) = D̃. Similarly, we canapproximate H2 (D̃) at a point D̂ by Ĥ2 (D̃) asĤ2 (D̃) = H2 (D̂) + log2 (1 − D̂
)(D̃ − D̂).
D̂

(4.12)

Now it holds that Ĥ2 (D̃) ≥ H2 (D̃). Using the property that H2 (D̃) is an increasing function and using the original
notation of fˆ(D1 , D2 , D3 ), it also holds thatĤ2 ( fˆ(D1 , D2 , D3 )) ≥ H2 ( f (D1 , D2 , D3 )).

(4.13)

The approximated optimization problem is written as
minimizeP,Dsubject toP1 + P2 + Pr
GS1 ,R P1
N0 ),
G
S
,R P2
1 − H2 (D2 ) ≤ R12 C( N2 0 ),G Pr
1 − H2 (D3 ) ≤ R1r C( R,D
N0 ),
GS ,D P1Ĥ2 ( fˆ(D1 , D2 , D3 )) ≤ R11 C( 1N0 ),GS ,D P2Ĥ2 ( fˆ(D1 , D2 , D3 )) ≤ R12 C( 2N0 ),1 − H2 (D1 ) ≤ R11 C((4.14)

1 + Ĥ2 ( fˆ(D1 , D2 , D3 )) ≤GS1 ,D P1
GS2 ,D P2
11
R1 C( N0 ) + R2 C( N0 ),which is a convex problem and can be efficiently solved using standard optimization tools, e.g., interior-point
methods [BV04]. It is worth noticing that due to the inequality (4.13), the solution of (4.14) satisfies the constraintsof the original problem (4.10).
(0)

The SCA algorithm starts by a feasible initialization D̂k = D̂k , ∀k. After this, (4.14) is solved yielding a solution(∗)
Dk which is used as a new point for the linear approximation. The procedure is repeated until convergence. Algo-rithm 2 provides the description for the SCA algorithm. The inequality (4.13) states that the linear approximationis always above the approximated function. Therefore, this algorithm is guaranteed to monotonically converge toa local optimum of the original problem [MW78].
Algorithm 2: Successive convex approximation algorithm.
(0)

1: Set D̂k = D̂k, ∀k.

2: repeat3:
Solve Eq. (4.14).
(∗)

4:
Update D̂k = D̂k
5: until Convergence.

4.5

, ∀k.

Simulation Results

In this section, we present the numerical results to demonstrate the impact of power allocation in RESCUE system
and illustrate its superiority compared with lossless DF in terms of total power consumption. The pathloss exponent
l is empirically set at 3.52 according to [YG11]. The simulation setup is defined in Fig. 4.3. We will place thesource and destination nodes into the Euclidean space and move the relay along the horizontal axis.

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S1

0.5
0.4

dS ,D

0.3

1

dS ,R

0.2

1

0.1

D

dR,D

R

0

d

−0.1

S ,R
2

−0.2

dS ,D

−0.3

2

S2

−0.4
−0.5

0

0.2

0.4

0.6

0.8

1

Figure 4.3: Simulation setup.

2
DF
RESCUE
1.8

sum power

1.6

1.4

1.2

1

0.8

0

0.2

0.4

0.6

0.8

1

dR,D

Figure 4.4: Sum power versus relay location.
As a first simulation example, we consider symmetric scenario where the nodes are placed at the points S1 =(0, 0.5), S2 = (0, −0.5) and D = (1, 0), i.e., dS1,R = dS2,R and dS1,D = dS2,D . Sum power versus dR , D is plotted inFig. 4.4. It can be seen that the RESCUE scheme performs better in terms of the total power consumption when
the relay is close to the destination.
In the second example, the nodes are placed at the points S1 = (0, 1), S2 = (0, −1) and D = (1, 0). The results areshown in Fig. 4.5. The range where the RESCUE scheme is superior is larger compared to the case presented inFig. 4.4 because the sources are further away from the relay and the destination.
The third example is an asymmetric case where the nodes are at the points S1 = (0, 1), S2 = (0, −2) and D = (1, 0).
The results are shown in Fig. 4.6. It can be seen that the RESCUE scheme is superior compared to the losslesscase through the whole range of relay positions considered.

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2
DF
RESCUE
1.8

sum power

1.6

1.4

1.2

1

0.8

0

0.2

0.4

0.6

0.8

1

dR,D

Figure 4.5: Sum power versus relay location.

6
DF
RESCUE
5.5

sum power

5

4.5

4

3.5

3

2.5

0

0.2

0.4

0.6

0.8

1

dR,D

Figure 4.6: Sum power versus relay location.

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4.6

D2.2.1 Version 1.0

Summary

Power allocation for TS4 with RESCUE functionality was investigated in this section. This work was about toprovide the ultimate limits for the lossy MARC system and show how it performs compared to lossless DF. It
was shown that the RESCUE scheme can achieve superior performance in terms of the total power consumption
compared to the conventional DF strategy. It was shown that when the signal-to-noise power ratios of the links
decrease, the performance gap between the lossy and lossless scheme increases. The RESCUE system presented inthis section is especially beneficial in unpredictable environment, where the link budget is not enough to guarantee
the lossless transmission.

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D2.2.1 Version 1.0

Conclusion

In this deliverable, the intermediate results regarding the optimal power allocation and rate distortion minimization
for different toy scenarios have been presented. With the concept of “links-on-the-fly”, the objectives are mainly
focusing on the outage probability minimization subject to the total power constraint and its dual of the transmit
power minimization subject to the fixed outage probability constraint, where only statistic channel information areneeded at the source/relay nodes for the power allocation. Apart from this, assuming the system with the local
or global instantaneous channel information, some algorithms to solve the MIMO based sum-rate maximization
problem and the power minimization problem under the lossy based rate region constraint have also been provided.
Although the instantaneous channel information is difficult to obtain, such kind of analysis could be used as thebaselines for the further power allocation research comparison.
In detail, the optimal power allocation algorithms for the basic erroneous DF relaying network (i.e., TS1) has
been firstly investigated. Here, the optimal power allocation for the LosFoR based outage probability has been
proposed, where the closed-form of the approximated outage probability has been provided, and its convexity has
been proved. As shown in the simulation results, the approximation can ensure sufficient accuracy, and comparing
with the equal power allocation, the proposed optimization can achieve much lower outage probability especiallywhen the relay node is near to the destination node. Following, the system model has been extended to MIMO
case in order to enjoy the increased multiplexing gain. In this case, for its simplicity, ZF based precoding/decoding
matrices have been proposed for the interference cancellation. Then, the relay-oriented source power allocation
algorithm for MIMO S-DF relaying has been developed to maximize the system sum-rate subject to the total source
power constraint, and the simulation results have shown the improved system performance by comparing with theconventional relay assisted schemes.
The second consideration is the power allocation algorithms for solving the CEO problem. As the way of analysing
the outage probability for TS1, the optimal power allocation algorithms to minimize the outage probability subject
to the total power constraint and the total transmit power subject to the fixed outage probability for TS2 have also
been investigated. As expected, comparing with the equal power allocation, the proposed algorithm has shown its
improved performance in terms of the outage probability and energy saving. In addition, the distributed transmit
power allocation among all relays for the CEO problem has also been investigated, which aims to equalize the
SNR limits for each relay decoder. Such process can remove or reduce the ramp region of the joint decoder. The
proposed algorithm has shown significant improvements in the SNR limits for the relay decoders, and the BER
curve of the joint decoder qualitatively follows as expected.
Moreover, the optimal power allocation for orthogonal MARC allowing intra-link errors has also been investigated.
In this case, in order to guarantee the reliable transmission of the lossy based system, the rate constraints have been
provided, and the power optimization problem has been formulated as minimizing the total transmit power subject
to simultaneously satisfy the provided lossy rate constraints. Such problem can be solved based on SCA algorithm,
and convergence of the algorithm has been justified. It has been shown that the proposed scheme outperforms theconventional DF strategy in terms of the total power consumption. Furthermore, when SNRs of the links decrease,the performance gap between the lossy and lossless scheme increases.
To sum up, the intermediate results of optimal distributed power allocation algorithms have been provided, andthe performances of proposed algorithms have been evaluated using the RESCUE provided toy scenarios whichwere presented in D1.2.1. For the next step, in order to further analyse the benefits of the “links-on-the-fly”
concept, more comprehensive and practically used power allocation algorithms will be exploited through computer
simulations and SDR platform test, and all of these will be documented in D2.2.2.

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6.

D2.2.1 Version 1.0

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