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Relay-Aided Communication Schemes for Wireless
Multiple Access and Multicast Channels
PhD Thesis
Jawwad Nasar Chattha
2011-06-0022
Advisor: Momin Ayub Uppal
Department of Electrical Engineering
Syed Babar Ali School of Science and Engineering
Lahore University of Management Sciences
Dedicated to my family, specially the 7 A’s (Ammi, Abbu,
Ayesha, Aima, Ahmad, Anam and Asma)
Acknowledgements
I am very thankful to Allah Almighty for providing me an opportunity to work under the supervision
of Dr. Momin Uppal. He is truly inspirational in every aspect and I strive everyday to follow his
footsteps. He is not only a patient supervisor but an excellent mentor and teacher. Thank you Dr.
Momin for all the support you have lent me during my stay in LUMS.
I am also very thankful to my committee members and faculty members in Electrical Engineering
Department, from whom I have learned a lot. I am specially thankful to all the teachers I have
taken courses from during my PhD. These include Dr. Shahid Masud, Dr. Zartash Afzal Uzmi,
Dr. Momin Uppal, Dr. Ijaz Haider Naqvi, Dr. Naveed-ul-Hassan, Dr. Ihsan Ayyub Qazi, Dr.
Muhammad Tahir, and Dr. Zubair Khalid. I am also thankful to all the support staff of the
Electrical Engineering Department for being very supportive.
I would also like to thank my colleagues from the ADCOM lab. It really has been an extraordinary
journey because of all the undergraduate, graduate students and research assistants that I interacted
with. They were all helpful either in relieving stress or making me regroup my thought towards
the problem at hand.
I would also like to acknowledge LUMS as an institution for striving for excellence and standing
out from the crowd of other institutions by providing quality education and research environment
to students. Thank you LUMS.
Contents
Acknowledgements iii
1 Introduction 1
1.1 Thesis Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1.1 Uplink Multiple Access Channel . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.1.2 Downlink Multicast Channel . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2 Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2 Preliminaries 8
2.1 Noisy Network Coding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.1.1 SNNC: Achievable Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2 Superposition versus Multiplexed Coding . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2.1 Superposition coding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2.2 Multiplexed Coding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3 NNC relaying for NOMA 19
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.2 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.3 NNC for Relay-Aided NOMA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.3.1 Achievable Rate Regions under Gaussian Signaling . . . . . . . . . . . . . . . 26
3.4 Outage performance analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
iv
3.5 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4 Joint NNC and DF for NOMA 36
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
4.2 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
4.3 The proposed J-NNC-DF Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4.3.1 Two-user case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4.3.2 General K-user case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.4 Outage performance analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.4.1 Achievable Rate Regions under Gaussian Signaling . . . . . . . . . . . . . . . 53
4.4.2 Outage analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.4.3 Comparison with benchmarks . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.4.4 Quantizer Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.5 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
5 Layered Multiplexed-Coded Relaying for Wireless Multicast 66
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
5.2 System Model and Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
5.3 Proposed LMDF Relaying . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
5.3.1 Two layers decoded at relay . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
5.3.2 One layer decoded at relay . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
5.3.3 No layer decoded at relay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
5.4 Simulations Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
5.5 Practical Implementations of LMDF . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
5.5.1 Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
5.5.2 Layering and superposition coding at the source . . . . . . . . . . . . . . . . 78
5.5.3 Decoding and multiplexed coding at the relay . . . . . . . . . . . . . . . . . . 79
v
5.5.4 LMDF System Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
5.5.5 Performance Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
5.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
6 Conclusions and Future Directions 90
6.1 Future Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
Appendix A Proof of Theorem 4.3.3 93
Appendix B Proof of Lemma 4.3.5 95
Appendix C Proof of Theorem 5.3.1 97
Appendix D Proof of Lemma 5.3.2 101
Bibliography 104
vi
List of Figures
1.1 The N-user uplink non-orthogonal multiple-access relay channel (NOMARC). . . . . 4
1.2 Downlink multicast channel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.1 Three node network with a dedicated relay. . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 2-user MARC with no side information . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.3 Multiplexed Codebook with side information at destination. . . . . . . . . . . . . . . 17
3.1 The two-user uplink non-orthogonal multiple-access relay channel (NOMARC). . . . 22
3.2 The probability of outage is evaluated by integrating the joint distribution of G1d
and G2d over the non-shaded region. . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.3 Probability of outage for Rs = 2 bits per channel use and β = 0.5. . . . . . . . . . . 34
3.4 Outage rate for an outage probability of 10−2 and β = 0.5. . . . . . . . . . . . . . . 35
4.1 The uplink non-orthogonal multiple-access relay channel. . . . . . . . . . . . . . . . 40
4.2 Capacity region of a two-user MAC along with a depiction of regions corresponding
to the events that (a) both users are decoded, (b) none of the users are decoded, and
(c) only one user is decoded. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4.3 Average probability of outage versus γ which is assumed to be the same on all links.
Bar chart in the background depicts the average probability of the relay successfully
decoding user messages. The sum-rate Rs = 4 b/s/Hz and fairness parameter β = 0.5. 62
4.4 Average outage rate versus γ for p∗ = 10−2 and β = 0.5. . . . . . . . . . . . . . . . . 64
4.5 Average outage-rate gain of J-NNC-DF over benchmarks at p∗ = 10−2 and β = 0.5. . 65
vii
5.1 Layered superposition coding with a 16-QAM constellation. . . . . . . . . . . . . . . 71
5.2 Network topology as well as the benefit of LMDF over SCDF and UDF at each user. 76
5.3 Network outage probability (for R = 1.5 b/s) and the network outage rate (for
p = 10−2) versus the transmission power. . . . . . . . . . . . . . . . . . . . . . . . . 77
5.4 Block diagram of the operations at the source. . . . . . . . . . . . . . . . . . . . . . 79
5.5 An illustration of how SM achieves SC with any α ∈ A8 through simple bit-connections
to a natural symbol mapper. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
5.6 Block diagram of the operations at the relay. . . . . . . . . . . . . . . . . . . . . . . 80
5.7 Simulated FER of LMDF versus THDF as a function of P = Ps = Pr at a particular
destination. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
5.8 Experimental setup with five USRPs placed in an indoor office environment. Alongside
each link, the average measured SNR is listed when the source and the relay USRP
use a transmit gain of 14 dB. Also shown at the individual nodes are the scatter
plots of the received constellations during T1 and T2. . . . . . . . . . . . . . . . . . . 87
5.9 Comparison of experimental FER results of LMDF and THDF at the three destinations
of Fig. 5.8. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
viii
List of Tables
2.1 SNNC coding for 3 node network with B=4 blocks . . . . . . . . . . . . . . . . . . . 13
3.1 NNC coding for 2-NOMARC with B=4 blocks. . . . . . . . . . . . . . . . . . . . . . 25
4.1 Joint NNC-DF coding with B=4 blocks. . . . . . . . . . . . . . . . . . . . . . . . . . 43
ix
Glossary
DF Decode-and-Forward
CF Compress-and-Forward
AF Amplify-and-Forward
BS Base Station
J-NNC-DF Joint-NNC-DF
NNC Noisy Network Coding
LNNC Long-message Noisy Network Coding
SNNC Short-message Noisy Network Coding
NOMARC Non-orthogonal Multiple Access Relay Channel
LMDF Layered Multiplexed Decode-and-Forward
SCDF Superposition Coded Decode-and-Forward
QAM Quadrature Amplitude Modulation
4G Fourth Generation
D2D Device-to-Device
OF Observe-and-Forward
HF Hash-and-Forward
QMF Quantize-Map-and-Forward
MAC Multiple Access Channel
MCC Multicast Channel
CSI Channel State Information
MC Multiplexed Coded
SC Superposition Coded
MIMO Multiple-Input and Multiple-Output
CoMP Coordinated Multi-Point
eBN E-UTRAN Node B
UE User Equipment
x
List of Publications
Below is a list of publications that have resulted directly from the research described in this thesis.
Journal articles
[J1] J. N. Chattha and M. Uppal, “Layered Multiplexed-Coded Relaying in Wireless Multicast
Using QAM Transmissions,” IEEE Communications Letters, vol. 20, no. 4, pp. 760-763,
2016
[J2] K. Mazher, U. B. Farooq, J. N. Chattha, and M. Uppal, “A Practical Layered Multiplexed-
Coded Relaying Scheme for Wireless Multicast, IEEE Transactions on Vehicular Technology,
vol. 67, no. 1, pp. 554-566, 2018
[J3] J. N. Chattha and M. Uppal, “Joint Noisy Network Coding and Decode-Forward Relaying
for Non-Orthogonal Multiple Access,”IEEE Transaction on Wireless Commun- ications, (Submitted
Feburary 2018)
Conference Papers
[C1] K. Mazher, F. Javed, U. B. Farooq, J. N. Chattha, and M. Uppal, “Layered Multiplexed-
Coded Relaying: Design and Experimental Evaluation,” in Proceedings of IEEE Global Communications
Conference (GLOBECOM), San Diego, CA, USA, 2015.
[C2] I. Ullah, F. U. Din, J. N. Chattha, and M. Uppal, “Compress-and-Forward Relaying:
Prototyping and Experimental Evaluation Using SDRs,” in Proceedings of 84th IEEE Vehicular
Technology Conference (VTC-Fall), Montreal, QC, Canada, 2016.
[C3] F. U. Din, J. N. Chattha, I. Ullah and M. Uppal, “A layered detect-compress-and-forward
coding scheme for the relay channel,” in Proceedings of 28th Annual International Symposium
on Personal, Indoor, and Mobile Radio Communications (PIMRC), Montreal, QC, Canada
2017.
xi
[C4] J. N. Chattha and M. Uppal, “Relay-aided non-orthogonal multiple access with noisy
network coding,” in Proceedings of IEEE International Conference on Communications (ICC),
Paris, France, 2017.
xii
Abstract
User cooperation through relaying is a powerful tool to combat fading and to increase robustness
of wireless networks. This thesis explores cooperative schemes for wireless multiple access and
multicast channels in the presence of a single dedicated relay. Novel cooperative schemes presented
here are based upon relay performing decode-and-forward (DF), noisy network coding (NNC) or a
combination of both DF and NNC.
The first half of this thesis presents cooperative schemes for a multiple access channel. It
considers an uplink non-orthogonal multiple access relay channel (NOMARC) in which multiple
users wish to communicate to a single base-station (BS) with the help of a single dedicated relay.
Firstly for a two- user setup, we derive the improved achievable rate region by employing NNC-only
relaying as opposed to conventional compress-and-forward (CF) relaying. Next, for the multiple
user setup, we propose a novel Joint NNC-DF (J-NNC-DF) scheme that utilizes DF cooperation
when messages from all user are successfully decoded at the relay and NNC when the relay is
unable to decode message of any one of the users. In the scenario when the relay is capable of
successfully decoding messages from only a subset of users, J-NNC-DF performs joint DF and NNC
encoding with DF applied to the set of messages that were decoded successfully, and NNC for the
set of messages that were not decoded successfully. After presenting the achievable rate regions,
we derive closed form expression for probability of outage for the proposed schemes. These outage
expressions permit selection of optimal quantizer noise variance selection to minimize probability of
outage. Both analysis and simulations confirm that the proposed J-NNC-DF scheme outperforms
other existing benchmarks such as DF-only, NNC-only and NNC-or-DF.
In the second part of this dissertation, we propose a cooperative scheme for a downlink multicast
network in which a BS wishes to communicate the same message to multiple users with the help of a
single dedicated relay. For this setup, we propose a layered multiplexed-coded decode-and-forward
(LMDF) relaying scheme. This scheme comprises of two major components: layering at the BS
and mulitplexed DF encoding at the relay. BS message is split into two layers, independently
encoded and mapped to a quadrature amplitude modulated (QAM) constellation, thus achieving
superposition. The benefit of superposition coding of the two layers is that it allows partial message
recovery at the relay and the users. On the other hand, multiplexed coding at the relay enables each
user to divert all channel resources towards decoding the layer(s) that remains unrecoverable from
the BS’s transmission. After deriving achievable rate regions, performance comparison is carried
out for the proposed schemes against superposition coded and unlayered BS transmissions.
In short, the dissertation proposes, analyzes and simulates J-NNC-DF and LMDF as viable
candidates for future generation wireless communication networks.
ii
Chapter 1
Introduction
The insatiable demand of higher data rates and greater reliability for wireless networks requires
multiple forms of diversity for user transmissions. One such type of diversity is known as cooperative
diversity. Cooperative diversity employs collaboration among multiple transmit and/or receive
antennas distributed across multiple users [1, 2]. Hence cooperative diversity emulates spatial
diversity gains through the use of antennas distributed across the network [3, 4]. Relay-aided
communication, considered a type of cooperative diversity, encompasses scenarios where user to
destination transmission is supported by a relay. Broadcast nature of the wireless medium enables
the relay to overhear and cooperate thus enhancing the source-to-destination transmissions. This
is achieved through various cooperative schemes deployed at the relay. Relays considered in this
thesis are altruistic in nature implying that the relay do not have any message of their own to
transmit and/or receive but they only help other user’s transmissions [5]. Relay nodes fulfilling
this altruistic behavior are also known as dedicated relays.
Cooperative communication has been a focus of research due to incorporation of relay-aided
cooperation schemes in the currently deployed fourth generation (4G) wireless networks standards
namely, the 3rd generation partnership project (3GPP) long term evolution-advanced (LTE-A)
[6, 7] and worldwide interoperability for microwave access (WiMAX) [8]. These standards discuss
scenarios for relaying in noise and interference limited regimes. These scenarios include relays
helping an edge user (located far away from a base station (BS)), access the BS, thus enhancing
1
signal range and/or network coverage area. On the other hand, relaying also boosts throughput
and reliability for users which are located within the coverage of BS, thus resulting in an increased
overall network capacity.
LTE-A, initiated in 2004 by 3GPP, targeted higher spectral efficiency and low-latency communica-
tion with achievable data rates as high as 300 Mb/s in the downlink and 75 Mb/s in the uplink, for
a 20MHz bandwidth. Apart from concepts including multiple-input and multiple output (MIMO)
transmission, self-organizing network operation and heterogeneous deployment, cooperative comm-
unication is also presented as a means to achieve these high data rates. Coordinated multi-point
(CoMP) [9, 10] transmission or reception is one such cooperative communication application. CoMP
is implemented by joint processing and transmission from multiple base stations (eNB) serving a
user equipment (UE) in the network. Relay nodes (RNs) can also be deployed in LTE-A systems
to assist the communication between the eNBs and the UEs. In comparison to eNBs, these RNs
transmits with lower power and hence have a smaller coverage area. Each UE can either be served
directly from a eNB or via the help of low power RNs. Hence CoMP transmission towards a UE
can be realized via two eNBs or between a RN and an eNB.
Similar to LTE-A, WiMAX also aims to extend the radio area coverage, increase the capacity
and to enhance the quality of service [11, 12]. However, to achieve these targets multi-antenna
processing schemes must be adapted. One of the ways to emulate multi-antenna behavior is
to deploy cooperative communication. Hence WiMAX too, like LTE-A, has proposed multiple
scenarios for cooperative communication. Apart from standards like LTE-A and WiMAX, which are
targeting cellular traffic, device-to-device (D2D), ad-hoc networks and sensor network communication
are some of the other applications where relaying is extensively deployed [13, 14, 15, 16, 17, 18] to
achieve better performance.
The idea of cooperative relaying was first presented in [19], where capacity bounds of a three
node network were discussed. Later in [20], authors investigated the capacity of the relay channel
and derived the achievable rates. Since then, a variety of cooperative scheme have been proposed in
[5, 21, 22, 23]. Most prevalent relaying schemes are decode-and-forward (DF), amplify-and-forward
(AF) and compress-and-forward (CF).
2
In the DF scheme, a relay first decodes then re-encodes and transmits the message. In the
scenario where relay is unable to decode, it does not cooperate. Hence cooperative gain in DF
diminishes when source-to-relay channel is weak or when source and relay are not physically close
together. As a result, it is well known that a decoding failure at the relay becomes a performance
bottleneck in DF cooperation [24]. DF scheme is beneficial when the source-to-relay channel is
strong, or when the source and relay are physically close. In comparison to DF, in AF the relay
simply amplifies without decoding the message received from the source. This amplified analog
retransmission has trivial implementation complexity compared to other schemes but results in
unwanted noise amplification. On the other hand in the CF scheme, the relay forwards a quantized
observation of the source message to the destination. CF turns out to be beneficial compared to
AF and DF whenever user-to-relay and user-to-destination channels are comparable with a strong
relay-to-destination link. CF comes in multiple flavors known by observe-and-forward (OF) [25],
estimate-and-forward (EF) [26] or hash-and-forward (HF) [27]. In addition to these flavors, other
variants of CF include Quantize-Map-and-Forward (QMF) [28] and noisy network coding (NNC)
[29]. All these CF based relaying schemes are employed when the relay is not able to decode user
message.
1.1 Thesis Contributions
As explained earlier, relay-aided cooperative communication is able to enhance throughput and
reliability of a wireless network. For this thesis, we concentrate on relay operation in the following
channels.
1. Uplink Multiple Access Channel (MAC)
2. Downlink Multicast Channel (MCC)
Considering these two channels, the objectives of the thesis are
• to develop cooperation schemes for these wireless channel models and to derive information
theoretic achievable rate regions for such cooperative schemes.
3
Base
StationRelay
hNd
User 2
User N
User 1
h4d
Figure 1.1: The N-user uplink non-orthogonal multiple-access relay channel (NOMARC).
• to analyze and compare these proposed cooperation schemes with various benchmark schemes
and to gain insights from these comparisons.
In the next subsection we introduce both uplink MAC and the downlink MCC which are
discussed throughout the thesis. We explain our contributions with regards to these channels. All
discussions are limited to brief explanations with details being discussed in subsequent chapters.
1.1.1 Uplink Multiple Access Channel
We first focus on uplink multiple access channel where multiple users transmit simultaneously
towards the BS via a relay. Such a channel is also known as non-orthogonal multiple access relay
channel (NOMARC). An example of N -user NOMARC is depicted in Fig. 1.1.
The cooperative schemes we devise for the NOMARC depicted in Fig. 1.1 are based on NNC
rather than traditional CF. NNC relaying is preferred as CF faces additional constraint due to
the necessity of decoding quantization indices before user message decoding. Since NNC does not
enforce intermediate recovery of the quantization indices and attempts to recover the user message
directly, NNC is beneficial over CF relay in fading type of scenarios prevalant in wireless channels
[30].
4
For the NOMARC, our contribution is two-fold. First, we propose a cooperation scheme for
the NOMARC based solely on NNC. The NNC encoding at the relay involves quantizing the signal
received from the two-users before re-encoding and transmission to the BS. The BS thus receives
simultaneous transmissions, not only from both users, but also from the relay, and attempts to
recover the users’ messages by performing a one-shot joint decoding operation. For the NNC-only
based relaying scheme, we first provide the rate region for the two-user NOMARC before comparing
with conventional CF for different channel state information (CSI) regimes. This comparison shows
that the NNC-only scheme outperforms the CF-only scheme even when the relay has access to global
CSI. This is opposed to the three-node relay network, for which it is known that NNC and CF
result in the same information theoretic performance under global CSI at the relay [31]. We further
derive closed-form expressions for probability of outage for the case when relay has access to both
the transmit and receive side CSI, as well as for the case when the relay has knowledge of only the
receive side CSI.
Our second contribution in the context of a NOMARC is to propose a novel joint-NNC-DF
(J-NNC-DF) scheme for multiple users that are aided through a dedicated relay which may be able
to decode all, none, or only a subset of the users. This disparity in number of user messages being
successfully decoded at the relay is due to fading experienced by the wireless channels because
of which each user experiences a different link quality to the relay. When all user messages are
decoded at the relay, we show that the optimal choice for the relay is to utilize DF. On the other end
when none of the user messages is successfully recovered at the relay, NNC becomes the natural
choice. Lastly when only a subset of user messages are successfully recovered at the relay, the
proposed scheme lets the relay utilize a single codebook that jointly encodes using (a) DF for the
successfully decoded user messages and (b) NNC for users which were not successfully decoded. We
derive the information theoretic conditions necessary for successful decoding at the BS. Using these
information theoretic analysis, we then present closed-form expression for probability of outage for
J-NNC-DF under the assumption that the relay has access to local CSI only. Based on these closed-
form outage expression, the optimal quantization noise variance selection criteria is presented for
a two-user NOMARC. Simulation results indicate SNR gains of up to 2.1 dB, 1.7 dB, 0.75 dB over
5
Base
StationRelay
hNd
User 2
User N
User 1
h4d
Figure 1.2: Downlink multicast channel.
NNC-only, DF-only, and DF-or -NNC, respectively, while the respective maximum average outage
rate gains are observed to be up to 1 b/s/Hz, 0.8 b/s/Hz, and 0.4 b/s/Hz.
1.1.2 Downlink Multicast Channel
In the second part of this thesis, we consider a downlink multicast channel (MCC), where a base
station (BS) is aided through a relay to communicate the same message to multiple users. Each
user tries to completely decode the BS message. A system diagram for such a channel model is
shown below in Fig.1.2.
We propose a novel cooperative scheme based on layered transmission from the BS. BS transm-
issions employ a layered DF scheme based on multi-level coding which partitions the source message
into different layers. Theses layers are superposition coded (SC) before transmission. The amount of
source message or the number of superposition coded layers decoded by the relay and destination
depends on the quality of the respective channels. SC at the source allows both the relay and
the destination to partially decode the source information, which results in an enhancement of
throughput. However, employing SC at the relay is not an optimal choice. SC by nature splits
the transmission power into the message layers, hence SC at the relay causes wastage of the power
6
given that the partial message (a subset of total number of layers) has already been decoded at
the destination. In order to avoid this power penalty, we propose a multiplexed-coded (MC) [27]
transmission approach. MC based transmission from the relay, allows the destination to utilize
all channel resources solely towards decoding the layers that remained unrecoverable from the
source’s initial transmission. We propose LMDF relaying with quadrature amplitude modulation
(QAM) transmissions for a multicast network. For a setup with 15 users uniformly spread around
the relay, simulation results indicate that LMDF outperforms conventional unlayered DF by 1.7 dB
and superposition coding-based DF by 1.1 dB, while the respective gains in outage rate at individual
users are up to 48.6% and 31.4%. A practical implementation of LMDF relaying strategy for a
wide-band wireless multicast network is also presented. LMDF is implemented using convolutional
codes, with the relays code optimized specifically for MC transmission. We conduct over-the-air
experiments, utilizing software defined radios, in an indoor office environment using an OFDM-
based physical layer, and illustrate the considerable performance benefits of LMDF over benchmarks
such as conventional two-hop DF.
1.2 Organization
Chapter 2 of the thesis discusses CF and its variant NNC along with user message encoding
techniques, namely superposition and multiplexed coding. We present an NNC based two-user
NOMARC scheme in Chapter 3 while in Chapter 4 a joint NNC and DF based relaying scheme
known as J-NNC-DF for NOMARC is presented. This is followed by Chapter 5 that presents the
layered multiplexed DF scheme known as LMDF based on QAM transmissions. Chapter 6 presents
thesis conclusions and possible future extensions.
7
Chapter 2
Preliminaries
In this chapter, we discuss key concepts that form the basis of this thesis. We formally introduce
NNC and highlight key differences between NNC and CF. Two variants of NNC (long and short-
message) are discussed along with their comparison. NNC achievability rate region proof for a
three-node relay network is also presented. In addition to NNC, we also discuss user message
encoding techniques including superposition coding and multiplexed coding. We elaborate upon the
idea of multiplexed codebook for a relay channel and its application to MCC and MAC. Key ideas
discussed in this chapter build the necessary background to understand contributions presented in
subsequent chapters.
2.1 Noisy Network Coding
In order to understand and appreciate the subtle difference between NNC and CF relaying, we
consider it important to first have a discussion regarding CF. CF was first presented as a relaying
scheme in [20] for a three-node network involving a single dedicated relay as shown in Fig. 2.1. CF
is beneficial when the relay is unable to decode the user message yet it can still aid the message
transmission by compressing and forwarding its observation to the destination. This compressed
message from the relay helps the destination in decoding the user message.
Consider a full duplex three-node Discrete Memoryless Relay Channel (DM-RC) where transmissions
from a user and relay are denoted by X1 and Xr respectively, while messages received at the relay
8
and BS are given by Yr and Yd respectively. If Yr corresponds to the quantized version of Yr, the
achievable rate for CF presented by authors in [20] is given as
RCF ≤ I(X1; Yr, Yd|Xr) (2.1)
subject to I(Xr;Yd) ≥ I(Yr; Yr|Xr, Yd) (2.2)
Above bound is a result of block Markov coding with the user transmission divided into multiple
Base
Station
Relay
User
Figure 2.1: Three node network with a dedicated relay.
blocks and a new message being encoded in each block. Such a user encoding technique is referred
to as cumulative encoding. After each block of transmission, the relay compresses its received signal
and forwards the bin index of the compression using Wyner-Ziv coding [32]. At the BS, decoding
is performed in a sequential manner for each block. The BS receiver decodes the bin index followed
by relay compression index decoding. Finally the BS uses this compression index to decode the user
message sent in the previous block. The above described CF scheme is referred to as the original
or successive CF. This name originates from the fact that BS performs block-wise decoding which
is successive in nature with the relays compression index being decoded prior to user message
decoding at the destination. This is clearly indicated by the information theoretic constraints
(2.1)-(2.2). In particular, the right-hand side of (2.2) represents the Wyner-Ziv compression rate,
while the left-hand side is the channel capacity on the relay-destination link (while treating the
9
source transmissions as interference). Clearly, (2.2) is indicative of the condition necessary for
successfully decoding the compression index. Under the constraint that compression index is
successfully recovered, (2.1), represents the information that the quantization variable Yr and the
relay reception Yd conveys about the source transmissions X1 (given that the relay transmissions
Xr have been successfully recovered).
An equivalent bound for CF relaying was presented in [33, 34], where the achievable transmission
rate is given by
RCF ≤ min{I(X1; Yr, Yd|Xr), I(X1, Xr;Yd)− I(Yr; Yr|X1, Xr, Yd))} (2.3)
Both these bounds are equivalent with difference in the way decoding is carried out at the destination
[35]. The former bound enforces the destination to decode compression indices whereas the latter
bypasses this explicit compression index decoding. After decoding the bin indices, message is
decoded for some compression indices within that bin instead of decoding the compression index. It
has been shown in [36, 37] that under the availability of global CSI at the relay, the achievable rates
presented in (2.1)-(2.2) and (2.3) are equivalent with the latter one having a simpler representation.
Some useful insights with reference to (2.1)-(2.2) and (2.3) include,
• Any rate satisfying (2.1)-(2.2) also satisfies (2.3).
• (2.1)-(2.2) enforce the need for knowledge of global CSI at the relay since in successive CF,
quantization indices need to be decoded prior to user message decoding. In particular, the
quantization variable Yr needs to be chosen such that the Wyner-Ziv constraint (2.2) is
satisfied. On the other hand, no channel knowledge is required in the case of (2.3) where
relay can choose any Yr that is close to Yr .
• If the constraint of (2.2) is not satisfied, quantization indices are not decodable at the BS. In
this case, the relay transmission starts acting like interference for the source-to-destination
link thus reducing the achievable rate to be less than (2.1). In fact, in such a situation, the
CF achievable rate reduces to that below of direct transmission.
Over the years, multiple variants of CF have been developed, with differences in user encoding,
10
relay operation and BS decoding strategies. The most prominent schemes among these variants
are the two NNC types namely long-message NNC (LNNC) [29] and short-message NNC (SNNC)
[38].
LNNC has three new ideas compared to original CF. The first is message repetition or repetitive
encoding, in which user transmits the same message over multiple consecutive blocks using independent
codebooks. Secondly, the relay does not perform Wyner-Ziv encoding, but rather forwards the
compression index directly using a joint source-channel codebook. Thirdly, the destination performs
joint decoding over all blocks without explicit decoding of compression indices. Compared to the
original version of CF, NNC has the same rate region for a single relay channel but achieves a larger
rate region when applied to multi-source networks. This gain in achievable rate region comes at
the price of delay in decoding of user message because of inherit message repetition in each block.
The delay occurs since the destination only starts decoding after transmission of all blocks. For a
three node network the achievable rate region for LNNC is proven to be
RLNNC ≤ min{I(X1; Yr, Yd|Xr), I(X1, Xr;Yd)− I(Yr; Yr|X1, Xr, Yd))} (2.4)
SNNC, which is also known as Quantize-Map-and-Forward (QMF) [28] or CF without binning
[35] employs cumulative encoding for each user message. Apart from cumulative rather than
repetitive encoding, QMF/SNNC is similar to LNNC with relay performing compression on the
user message through quantization and forwarding the quantized message which is jointly decoded
at the BS. This scheme results in a similar achievable rate as that of the CF bound in (2.3), but
with an additional constraint. More specifically, the SNNC achievable rate, by employing joint
decoding over all the transmission blocks, is proven in [39] to be
RSNNC ≤ min{I(X1; Yr, Yd|Xr), I(X1, Xr;Yd)− I(Yr; Yr|X1, Xr, Yd))} (2.5)
subject to I(Xr;Yd) + I(Yr;X1, Yd|Xr) ≥ I(Yr;Yr|Xr) (2.6)
An equivalent form of SNNC achievable rate region (2.5) using the equivalence of (2.1)-(2.2) and
11
(2.3) is given by
RSNNC ≤ I(X1; Yr, Yd|Xr) (2.7)
subject to I(Xr;Yd) ≥ I(Yr; Yr|Xr, Yd) (2.8)
Comparing (2.6) and (2.8), we can prove (2.6) is redundant, if the following is true
I(Yr; Yr|Xr, Yd) ≥ I(Yr;Yr|Xr)− I(Yr;X1, Yd|Xr) (2.9)
(2.9) is true since
I(Yr; Yr|Xr, Yd) = I(Yr; Yr, X1|Xr, Yd)
= I(Yr;X1|Xr, Yd) + I(Yr; Yr|X1, Xr, Yd)
≥ I(Yr; Yr|X1, Xr, Yd)
= I(Yr;X1, Xr, Yd, Yr|Xr)− I(Yr;X1, Yd|Xr)
= I(Yr;Yr|Xr)− I(Yr;X1, Yd|Xr)
This proof is also mentioned in [35]. This redundancy implies that both LNNC and SNNC
achievable rate regions converge when joint decoding is employed at the BS. Rather than joint
decoding the BS can also employ block-wise backward decoding which gives the relay flexibility
to deploy hybrid schemes as discussed in the later chapters. In the scenario when BS employs
block-wise backward decoding rather than joint decoding over all blocks, an additional constraint
appears, which is given by
I(Xr;Yd|X1) ≥ I(Yr;Yr|X1, Xr, Yd) (2.10)
This additional constraint (2.10) ensures successful quantization index decoding in the last block
while performing block-wise decoding. A comparison of (2.8) and (2.10) highlights key differences
between the two constraints. (2.10) is a constraint between the channel capacity of relay-destination
link and the Wyner-Ziv compression rate (while assuming the source transmissions are already
12
known). (2.10) requires only local CSI knowledge at the relay. In the scenario when (2.10) is not
satisfied relay transmission start acting like interference for the user-destination link. In order to
provide more insights into the operations involved in SNNC, we next provide a proof of achievable
rate given by (2.5) and (2.10).
2.1.1 SNNC: Achievable Rates
In the following, we provide an overview, from an information theoretic perspective for SNNC-
based three node network. Code construction and encoding at the user and the relay are discussed
followed by an explanation of the decoding process at the relay and the BS. The discussion concludes
with the probability of error analysis resulting in the achievable rate region of SNNC for the three
node network.
Code Construction: In cumulative encoding, transmission is performed in B + 1 blocks,
and we generate a different codebook for each block as shown in Table 2.1. User message is split
into B equal parts, where wb represents the message to be transmitted in the bth block. For each
block b ∈ (1, 2, . . . , B + 1), 2nR1 codewords x1b(wb), wb = 1, 2, . . . , 2nR1 of length-n are generated,
independently using P (X1) (with the assumption that x1(B+1)(w(B+1)) = 1). At the relay two
codebooks are generated, one for quantizing the relay received message and the other for relay’s
transmissions. Quantization codebook denoted by yr(vb−1, vb) where vb = 1, 2, . . . , 2nRr is generated
as a two-dimensional length-n codebook of 2nRr rows and 2nRr columns for each block b by drawing
independently and randomly from the distribution P (Yr). For relay transmissions, 2nRr length-n
sequences are generated by choosing 2nRr codewords xr(v1) independently using P (Xr).
Table 2.1: SNNC coding for 3 node network with B=4 blocks
Blocks 1 2 3 4 5
User x11(w1) x12(w2) x13(w3) x14(w4) x15(1)
Relay xr1(1) xr2(v1) xr3(v2) xr4(v3) xr5(v4)
Quantization yr1(1, v1) yr2(v1, v2) yr3(v2, v3) yr4(v3, v4) yr4(v4, 1)
Relay and BS decoding Relay performs block-wise forward decoding. For a received length-n
13
sequence yrb at the end of block-b, the relay performs joint source-channel encoding by finding an
index vb such that xrb(v(b−1)), yrb, and yrb(v(b−1), vb) are jointly typical, i.e.,
{xrb(v(b−1)), yrb, yrb(v(b−1), vb)
}∈ T (n)
ε .
Here, v(b−1) is the known encoding index from the previous block (with the assumption that v0 = 1).
If there is no such index vb that satisfies the typicality condition, the relay selects an index from the
set{
1, . . . , 2nRr}
uniformly at random. For the selected index vb, the relay transmits the sequence
xr(b+1)(vb) in block-(b+ 1).
At the BS, block-wise backward decoding is performed. Thus, having received the length-n
sequence ydb at the end of block-b and having recovered the indices wb and vb from earlier decoding
rounds, the BS tries to find a pair of unique indices (w1(b−1) and v(b−1)) such that
{x1b(wb), yrb(v(b−1), vb), xr(v(b−1)),ydb
}∈ T (n)
ε
If no such unique pair is found, an error is declared. The procedure is repeated for each block from
b = B + 1, all the way back to b = 2.
Probability of error analysis By symmetry of code construction, the conditional probability
of error does not depend on which message index (wb) is sent from the user. Similarly the conditional
probability of error does not depend on the quantization index vb chosen by the relay. Thus, without
loss of generality, the probability of error can be evaluated by assuming that the message index
and the quantization index are all equal to 1 for all blocks b ∈ {1, . . . , B + 1}. At the BS, an error
happens if for some b, one or more of the following events occur
ER(vb) ,{(
yrb, yrb(1, vb), xrb(1))6∈ T (n)
ε
}for all vb, and
ED(w(b−1), v(b−1)) ,{(
x1b(w(b−1), 1), yrb(v(b−1), 1), xrb(v(b−1)
),ydb
)∈ T (n)
ε
}
for some (w1(b−1), v(b−1)) 6= (1, 1).
14
By the covering lemma and the union of events bound (over B blocks), the probability of the
first event tends to zero as n→∞ if Rr > I(Yr;Yr|Xr) + δ(ε′). For the second error event, we have
P (ED) = P
( ⋃(w1(b−1),v(b−1))6=(1,1)
ED(w1(b−1), v(b−1))
)
≤∑
w1(b−1) 6=1
P
(ED(w1(b−1), 1)
)+
∑v(b−1) 6=1
P
(ED(1, v(b−1))
)+
∑(w1(b−1),v(b−1)) 6=(1,1)
P
(ED(w1(b−1), v(b−1))
)
Using standard typicality arguments, it can be shown that the probability of error approaches zero
as n→∞ if
R1 < I(X1; Yr, Yd|Xr) (2.11)
Rr < I(Xr;Yd|X1) + I(Yr;X1, Yd|Xr) (2.12)
R1 +Rr < I(X1, Xr;Yd) + I(Yr;X1, Yd|Xr) (2.13)
By eliminating Rr through the use of the constraint Rr > I(Yr; Yr|XR), the bounds in (2.11)–(2.13)
yield the rate region presented above in (2.5) and (2.10).
2.2 Superposition versus Multiplexed Coding
Two types of user message coding are employed at the users and/or relay in this thesis. The purpose
of these codebooks is to recombine the multi-level messages (layered transmission) from a single
user decoded at the relay or to recombine the decoded received signal from multiple users at the
relay. In this section we describe both these coding schemes in detail, highlighting the need for
multiplexed coding. We start our discusssion by describing superposition coding (SC).
2.2.1 Superposition coding
We show the difference between superposition and multiplexed coding for a two-user NOMARC as
shown in Fig. 2.2. User messages w1 ∈ {1, . . . , 2nR1} and w2 ∈ {1, . . . , 2nR2} from each user are
encoded individually using independently generated codebooks at each user. Here R1 and R2 are
15
individual rate of transmission for the two users, respectively. Both relay and destination attempt
to decode the messages transmitted from the users. Under the assumption that messages from both
the users are successfully decoded at the relay, SC of the two users is possible. SC at the relay is
carried out by re-encoding both messages in a similar manner as done by the two users initially.
The difference lies in splitting of the total transmit power of relay. In SC total transmit power of
the relay is split between the two users through a factor of α ∈ [0, 1]. This results in a transmit
signal given by
Xr(w1, w2) =√αX1(w1) +
√1− αX2(w2)
where X1(w1) and X2(w2) are the encoded message from user 1 and user 2, respectively. SC at
relay is the natural choice in scenarios when no side information (no prior knowledge of two user
message) is present at the BS. SC does not hold well in scenarios when either of the two user
message is known at the BS. In such a scenario, a fraction of the total power dedicated for the user
message already known at the BS is wasted.
Base
StationRelay
h2d
User 2
User 1
Figure 2.2: 2-user MARC with no side information
Now suppose a BS has some side information, for instance the BS has already decoded one
of the user message namely X1(w1). NOMARC is one such scenario where prior knowledge of
16
user messages at the BS is possible due to presence of direct link between users and destination.
Assuming the availability of such a side information in the form of already decoded message of user
1 at the BS, then α fraction of the transmission power is wasted. This power penalty incurred by
the realy due to prior knowledge can be overcome through a coding scheme known as multiplexed
coding.
2.2.2 Multiplexed Coding
Multiplexed coding (MC) was first proposed in [27, 40]. In MC, the two users messages at the
relay are encoded using a single codebook. For ease of explanation, let us only focus on the two-
dimensional case, although multiplexing can be achieved in higher dimensions as well. The two
dimensional codebook is constructed by randomly generating 2n(R1+R2) independent and identically
distributed (i.i.d.) length-n sequences. The sequences are then arranged in a table with 2nR1
rows and 2nR2 columns with entries denoted by Xr(w1, w2), where indices w1 ∈ {1, . . . , 2nR1} and
w2 ∈ {1, . . . , 2nR2} correspond to the two user messages. Given both the messages are decoded, the
relay transmits length-n codeword residing at row-w1 and column-w2 of the multiplexed codebook
as shown in the Fig. 2.3. As for the decoding, if no user is decoded at the BS beforehand, then the
......
... ...
...
Xr(2,2nR2)
Xr(2nR1,2nR2)Xr(2nR1, 2)Xr(2nR1, 1)
Xr(2, 1)
Xr(1,2nR2)Xr(1, 1) Xr(1, 2) ...
Codeword Length (n)
Figure 2.3: Multiplexed Codebook with side information at destination.
17
BS decoder searches for a jointly typical codeword over all 2n(R1+R2) entries of the two dimensional
codebook, with successful recovery of both layers achieved if R1 + R2 is less than the channel
capacity. On the other hand, if a destination had recovered one users message beforehand, then
the decoding operation would involve a search for the jointly typical codeword only over the 2nR2
entries of the row indexed by the user 2 information. Successful decoding will occur if R2 is less
than the channel capacity, thus allowing all channel resources to be diverted towards decoding the
unknown layer. For instance, as shown in the figure below, if the BS already knows the user 1’s
index as w1 = 1, it would have searched over the 2nR2 length-n entries of the first row of the
codebook (illustrated with a darker shade in the figure).
Multiplexed coding for multicast channel
Above description highlights the use of MC in MAC. For multicast channel multiplexed coding can
be employed by introducing layered transmission from the BS to users. These layers or multi-level
message from the BS can be multiplexed coded at the relay to avoid power penalty associated with
SC. Detailed description of the use of MC for MCC is present in Chapter 5.
2.3 Conclusions
In this Chapter, we highlight the key differences between CF and NNC. We also present two
variants of NNC namely LNNC and SNNC with their information theoretic achievable rates for a
three node network. Finally the idea of superposition and multiplexed coding are discussed. In the
end, multiplexed coding benefits are presented for the MAC and MCC scenario.
18
Chapter 3
NNC relaying for NOMA
In this chapter we consider two-user relay-aided uplink non-orthogonal multiple access (NOMA)
with noisy network coding (NNC) at the relay. The NNC-based cooperation strategy at the relay
relies on noisy quantization followed by joint source-channel coding, while at the destination it
utilizes joint decoding using the non-orthogonal signals received from the sources and the relay.
After presenting the information theoretic achievable rates for the NNC-based cooperation scheme,
we discuss how it outperforms conventional compress-and-forward (CF) cooperation even when the
relay has access to global channel state information (CSI). For the NNC-based approach, we also
derive closed-form expressions for probability of outage when the relay has access to local CSI,
as well as when the relay has knowledge of the receive CSI only. These closed-form expressions
help in appropriately choosing the quantizer so as to optimize the outage performance. Simulation
indicate that NNC-based relaying outperforms CF by 2.2 dB when global CSI is available at the
relay. Additionally, the availability of global CSI results in an improvement of 3.8 dB and 5.4 dB
over the case when relay has access only to local CSI and the receive CSI, respectively.
3.1 Introduction
Non-orthogonal multiple access (NOMA) is one of the candidate techniques for addressing the
envisioned high-throughput massive-connectivity requirements of 5G networks [41]. NOMA schemes
have advantage over orthogonal multiple access schemes (OMA) such as time division multiple
19
access (TDMA) and frequency-division multiple access (FDMA) for both single and multiple
antenna systems due to NOMAs larger achievable capacity regions [42, 43]. In this chapter,
we consider uplink NOMA with a pair of users transmitting their respective data to a BS via
a relaying node as depicted in Fig. 3.1. We refer to this NOMA architecture as the NOMA relay
channel (NOMARC), a subclass of multiple access relay channels (MARC) initially introduced
in [44]. Amongst the cooperation schemes that can be deployed in the NOMARC, the three
prevalent strategies, originially proposed for the three-node relay network, are amplify-and-forward
(AF), decode-and-forward (DF) and compress-and-forward (CF). In AF, an amplified version of the
signal received at the relay is forwarded to the BS, while in DF, the relay attempts to decode before
transmitting the re-encoded message to the BS. On the other hand, in the CF scheme, the relay
quantizes its received signal, and applies distributed compression before onward transmission to the
BS. In addition to these conventional strategies, recent cooperation schemes include quantize-map-
and-forward (QMF) [45] as well as the NNC scheme [29]. The QMF strategy in essence stems from
the analysis of a wireless network as a deterministic model. NNC on the other hand extends QMF
so as to obtain a single letter expression of achievable rate through vector quantization at the relay.
NNC bounds are within a tighter gap to the cut-set bound than the QMF scheme employing scalar
quantization. NNC is also closely related to CF, with the major difference lying in the decoding
operation at the BS. In particular, while CF relies on multi-stage decoding with the recovery of
relay’s quantization indices followed by decoding of the user message, NNC does not necessitate
intermediate recovery of the quantization indices and attempts to recover the user message in a one-
shot decoding process. Since NNC does not require explicit decoding of the quantization indices,
it is beneficial over CF in fading type of scenarios [30].
Relay-aided NOMA for 5G networks has recently attracted considerable research attention.
This includes work where NOMA is used for efficient broadcasting, by which the throughput of a
network is improved. An example of one such scheme is [46], where a two-step relaying strategy
based on NOMA is introduced for improving the rates. NOMA outage probability is studied in
[47] and [48]. Additional studies for instance, [49] proposes a hybrid scheme for the NOMARC
by combining AF and DF relaying, depicting performance gains in a half-duplex scenario with
20
multiple relays. Similarly, [50] considers multi-user uplink access with successive interference
cancellation for 5G networks. Here, a suboptimal approach with user pairing is proposed to reduce
complexity of decoding at the BS. In [51], authors highlight the benefits of joint network channel
coding in NOMARC over orthogonal MARC with selective DF relaying. Similarly, [52] presents
outage achievable rate for various cooperative schemes. Information theoretic analysis for downlink
NOMA by forming relaying broadcast channels is considered in [53], while additional work showing
performance gains of cooperative NOMA include [54] and [55]. For a downlink NOMA setting, [56]
studies simultaneous information and power transfer, presenting closed-form expression for outage
probability and system throughput. Some additional works on cooperative NOMA can also be
found in [54, 57, 55, 56, 58, 59, 60]. Apart from these work in [31], the quantization distortion level
for QMF relaying in a three-node network is optimized to achieve the best outage performance
under various levels of known channel conditions.
In this chapter, we focus on uplink NOMA with two users transmitting to the BS over the same
time-frequency slot with the help of a dedicated full-duplex relay. The relay performs NNC, which
involves quantizing the signal received from the two users before re-encoding and transmission
to the BS. The BS thus receives simultaneous transmissions, not only from both users, but also
from the relay, and attempts to recover the users’ messages by performing a one-shot joint decoding
operation. For the NNC-based relaying scheme, we first provide the information theoretic achievable
rate region for the two-user NOMARC. Comparing NNC relaying with conventional CF, we show
that the former outperforms the latter even when the relay has access to global channel state
information (CSI). This is opposed to the three-node relay network, for which it is known that
NNC and CF result in the same information theoretic performance under global CSI at the relay
[31]. Since access to global CSI at the relay may be impractical in certain situations, we also carry
out performance analysis of the NNC-based relaying under limited CSI availability at the relay. In
particular, we derive closed-form expressions for probability of outage for the case when relay has
access to both the transmit and receive CSI, as well as for the case when the relay has knowledge
of only the receive CSI. Using these closed-form expressions, we numerically determine the optimal
quantizer distortion level so as to maximize outage performance. Simulation results indicate that
21
Base
Station
User 1
User 2
Relay
Figure 3.1: The two-user uplink non-orthogonal multiple-access relay channel (NOMARC).
the NNC-based relaying under global CSI outperforms CF by a margin of 2.2 dB. In addition, the
availability of global CSI in NNC-based relaying results in a performance gain of 3.8 dB and 5.4
dB over the cases when relay has access to local CSI, and the receive CSI, respectively.
The remainder of this chapter is organized as follows. Section 3.2 describes the system model,
while the NNC-based relaying scheme for NOMARC along with the associated achievable rate-
region is presented in Section 3.3. Outage performance evaluations and the corresponding simulation
results are presented in Sections 3.4 and 3.5, respectively, while Section 3.6 concludes the chapter.
3.2 System Model
We consider uplink NOMA transmissions, in which two users send their respective messages to a BS
with the help of a dedicated relaying node, as depicted in Fig. 3.1. The relay is assumed to operate
in the full-duplex mode and is therefore capable of transmitting to the BS while simultaneously
receiving from the users. The baseband signal received at the relay under a frequency-flat fading
model is given by
Yr = h1rX1 + h2rX2 + Zr,
22
where Xi is the transmission from user-i (i = 1, 2), hir is the channel gain between user-i and the
relay, whereas Zr is complex circularly symmetric zero-mean Gaussian noise. Similarly, the signal
received at the BS is denoted by
Yd = h1dX1 + h2dX2 + hrdXr + Zd,
where Xr is the relay’s transmission, hid is the channel gain between user-i and the BS, hrd is
the channel coefficient on the relay-BS link, and Zd is zero-mean Gaussian noise. Without loss in
generality, we assume that the noise terms Zr and Zd are of unit-variance, and that E[|X1|2
]=
E[|X2|2
]= E
[|Xr|2
]= 1. Moreover, the channel coefficients hjk are assumed to be circularly
symmetric Gaussian with zero-mean and variance γjk. Since the noise terms are assumed to be
unit-variance, and the transmissions are assumed to be of unit-energy, γjk characterizes the average
signal-to-noise ratio (SNR) between node j ∈ {1, 2, r}, k ∈ {r, d}, j 6= k.
3.3 NNC for Relay-Aided NOMA
We consider an NNC-based cooperation strategy for the NOMARC. The NNC scheme, as originally
proposed in [29] for the three-node relay channel relies on quantizing the received signal Yr at
the relay to obtain Yr, and then encoding it with a joint source-channel code before onward
transmissions to the BS. The BS applies joint decoding on the signals received from the source and
the relay to recover the original source message. The NNC-based cooperation for the considered
two-user NOMARC follows the same approach, except that the relay quantizes the signal received
from the two users over non-orthogonal channels, and that the BS attempts to jointly recover the
two users’ messages from what it receives from the users as well as the relay. In the following,
we describe the NNC-based relaying strategy for the NOMARC by providing an overview, from
an information theoretic perspective, of the (a) code construction at the two users and the relay,
(b) the corresponding encoding process, and (c) the decoding process at the BS. We then provide
the achievable rate region for this strategy when all nodes utilize transmissions from the Gaussian
alphabet and the relay employs Gaussian quantization.
23
Codebook generation and encoding at the users: Since the relay operates in the full-duplex
mode, the NNC-based cooperation scheme relies on a block-processing approach. In particular,
the transmissions from the two users are divided into B blocks, with an independent codebook
utilized in each block. Thus, if each block spans a total of n symbols, a total of 2nBR1 length-
n sequences are generated at user-1 for each block b ∈ {1, . . . , B}. The sequences, denoted by
x1b (w1) are indexed by nBR1-bit message of user-1 denoted by w1 ∈{
1, . . . , 2nBR1}
. Moreover,
the sequences are assumed to be generated in a random independent identically distributed (i.i.d.)
fashion according to the distribution P (X1). Similar codebook generation takes place at user-2
with the 2nBR2 length-n sequences for each block-b denoted by x2b (w2) with w2 ∈{
1, . . . , 2nBR2}
.
These sequences are assumed to be generated, once again in a random i.i.d. fashion, according to
the distribution P (X2). Given message w1 and w2, the two users simultaneously transmit x1b(w1)
and x2b(w2) in block-b. It is worth noting that the original message sequences at the two users are
not split into smaller blocks and encoded independently; rather a single long message of length-
nBRi bits is used to index the coded sequences across multiple blocks. NNC variant described
above and considered throughout this chapter is LNNC.
Codebook generation and encoding at the relay: At the relay, two codebooks are generated;
one for quantizing the relay received signal and the other for re-encoding the quantized output
before onward transmission to the BS. For the transmissions, a length-n codebook of size 2nRr is
generated, where Rr is an auxiliary relay encoding rate that can ultimately be eliminated from the
achievable rate region evaluations. The codebook is formed by randomly generating 2nRr length-n
i.i.d. sequences xrb (vb) according to the distribution P (Xr) with vb ∈{
1, . . . , 2nRr}
. On the other
hand, the quantization codebook is two-dimensional and is indexed by the transmission index vb−1
of block b−1 as well the index vb of the current block. For each vb−1 ∈{
1, . . . , 2nRr}
, the codebook
consists of a total of 2nRr length-n sequences (indexed by vb) denoted by yr(vb|vb−1), that are
randomly and conditionally independently generated with symbol-i (i = 1, . . . , n) following the
distribution P (Yr|Xr).
24
For a received sequence yr(j) at the end of block-j, the relay performs quantization by finding
an index vb such that (by convention, we assume v0 = 1)
{yr(b), yr(vb|vb−1),xr(vb−1)
}∈ T (n)
ε ,
where T (n)ε denotes the set of length-n typical sequences for an arbitrary small ε > 0. If there
is more than one such index, the relay selects one of them uniformly at random. If there is no
such index, it selects an index from the set{
1, . . . , 2nRr}
uniformly at random. For the selected
index vb, the relay transmits xr(vb) in block b+ 1. Both user and relay encoding operation are also
summarized in Table 3.1.
Decoding at the BS: Decoding at the BS takes place after all B blocks have been received, at
Table 3.1: NNC coding for 2-NOMARC with B=4 blocks.
Blocks 1 2 3 4
User 1 x11(w1) x12(w1) x13(w1) x14(w1)
User 2 x21(w2) x22(w2) x23(w2) x24(w2)
Relay xr1(1) xr2(v1) xr3(v2) xr4(v3)
Quantization yr1(v1|1) yr2(v2|v1) yr3(v3|v2) yr4(v4|v3)
which point it finds a pair of unique indices (w1b, w2b) such that
{x1b(w1b),x2b(w2b), yr(vb−1|vb),xrb(vb−1), ydb
}∈ T (n)
ε
for all b ∈ {1, . . . , B} and for some v1, v2, ..., vB.
Using joint typicality and random coding argument, it can be shown that the aforementioned
scheme can achieve an arbitrarily small probability of error as long as the transmission rates of the
two users lie in the rate region defined by Theorem 3.3.1.
Theorem 3.3.1. The achievable rate region R for two user NOMARC with NNC-based cooperation
is given by
R ,{
(R1, R2)∣∣∣R1 ≤ C1, R2 ≤ C2, R1 +R2 ≤ Cs
}(3.1)
25
where,
C1 = min{I(X1; Yr, Yd|X2, Xr), I(X1, Xr;Yd|X2)− I(Yr;Yr|X1, X2, Xr, Yd)} (3.2)
C2 = min{I(X2; Yr, Yd|X1, Xr), I(X2, Xr;Yd|X1)− I(Yr;Yr|X2, X1, Xr, Yd)} (3.3)
Cs = min{I(X1, X2; Yr, Yd|Xr), I(X1, X2, Xr;Yd)− I(Yr;Yr|X1, X2, Xr, Yd)} (3.4)
Proof. Proof is an extension of the achievable rate of a single user presented in (2.4) to two
user scenario. The same NOMARC rate region can also be deduced from the achievability proof
presented in [29] for the multi-source multi-destination network.
3.3.1 Achievable Rate Regions under Gaussian Signaling
Under Gaussian transmissions Xj ∼ CN (0, 1), j ∈ {1, 2, r}, we employ a Gaussian quantization
model, under which the quantized version of Yr is given as Yr = Yr + Zq. Here Zq represents
zero-mean Gaussian noise of variance q; a quantity that parameterizes the quantizer fidelity. Under
this model, the rate-region boundaries in (3.2)–(3.4) evaluate to
C1 = min{
log(1 +G1r
1 + q+G1d),
⌈log(1 +G1d +Grd)− log
1 + q
q
⌉+}(3.5)
C2 = min{
log(1 +G2r
1 + q+G2d),
⌈log(1 +G2d +Grd)− log
1 + q
q
⌉+}(3.6)
Cs = min{
log(1 +G1r +G2r
1 + q+G1d +G2d),
⌈log(1 +G1d +G2d +Grd)−log
1 + q
q
⌉+}(3.7)
where dxe+ , max(0, x) and Gij = |hid|2 are the (real) channel gains with i ∈ {1, 2, r}, j ∈ {r, d}
and i 6= j.
3.4 Outage performance analysis
In this section, we analyze outage performance of the NNC-based relaying scheme for the two-user
NOMARC described in Section 3.3. Let the transmission rates of the two users be denoted by R1
and R2, and the sum-rate by Rs = R1 + R2. We assume an arbitrary fairness constraint between
the two users. In particular, for a given β ∈ [0, 1], we assume that the transmission rate of user-1
26
and user-2 must be fractions β and βc , 1 − β of the sum-rate, respectively, i.e., R1 = βRs and
R2 = βcRs. For performance analysis, we consider an outage event to occur when either of the two
users’ message is not recoverable at the BS. Thus, the probability of outage for a fixed sum-rate R
(and a fairness parameter β) is given as
po (R) , Pr
(R > min
(C1
β,C2
βc, Cs
)), (3.8)
where C1, C2, and Cs are given in (3.5)–(3.7). We also consider outage rate as a performance metric
which is defined as the maximum transmission sum-rate (under a fixed β) which ensures that the
probability of outage does not exceed a threshold ε. Mathematically,
Ro(ε) , max{R∣∣∣po (R) ≤ ε
}(3.9)
It is clear that the probability of outage as well as the outage rate is a function of the quantization
noise variance q. Thus, depending on the level of available CSI, the relay can choose q so as to attain
the best possible outage performance. In the following, we first consider the outage performance of
the NNC-based cooperation scheme when the relay has access to global CSI. We then analyze the
outage probability of the proposed scheme for the case when relay has access to its local transmit
and receive CSI. This is followed by performance analysis for the most practical situation in which
the relay has access only to its receive CSI.
Global CSI at the relay
For this case, the relay is assumed to have perfect knowledge of the instantaneous channel realizations
on all five links; users to the relay, relay to the BS, as well as users to the BS. Under this situation,
it is clear from (3.8) and (3.9) that the optimum q that minimizes the probability of outage (or
maximizes the outage rate) is given by
q∗g = arg maxq>0
min
(C1
β,C2
βc, Cs
). (3.10)
27
Each of the capacity terms (3.5)-(3.7) is a minima of two terms, first a non-increasing term and the
second non-decreasing term. Hence these capacity terms have their respective individual maximas
at the intersection of the non-increasing and the non-decreasing terms given as
q∗1 , arg maxq>0
C1 =1 +G1r +G1d
Grd
q∗2 , arg maxq>0
C2 =1 +G2r +G2d
Grd
q∗s , arg maxq>0
Cs =1 +G1r +G2r +G1d +G2d
Grd
Since the real channel gains Gij where i ∈ {1, 2, r}, j ∈ {r, d} and i 6= j are non-negative hence
both q∗1 and q∗2 are always less than q∗s . Using this observation we can reduce the search space
for optimal quantizer variance to q∗g ∈ [min (q∗1, q∗2) , q∗s ]. Moreover, due to non-increasing and
non-decreasing behavior of terms within the minima operator of (3.5)-(3.7) it is evident that the
optimum solution in fact lies either on the boundaries of this interval, or at one of those intersections
of the terms C1β , C2
βc , and Cs that lie within this interval.
At this point, it is instructive to compare NNC-based relaying with conventional CF relaying
for the NOMARC under global CSI at the relay. Recall that while the relaying operation in CF
stays the same as that with NNC, the latter, as opposed to the former, does not necessarily require
the quantized signal Yr to be perfectly recoverable at the BS. For further analysis we first present
the information theoretic achievable rate region for CF for NOMARC
Theorem 3.4.1. The achievable rate region RCF for two user NOMARC with CF-based cooperation
is given by
RCF ,{
(R1, R2)∣∣∣R1 ≤ C1c, R2 ≤ C2c, R1 +R2 ≤ Csc
}(3.11)
where,
C1c = min{I(X1; Yr, Yd|X2, Xr), I(X1, X2, Xr;Yd)− I(Yr;Yr|X1, X2, Xr, Yd)} (3.12)
C2c = min{I(X2; Yr, Yd|X1, Xr), I(X1, X2, Xr;Yd)− I(Yr;Yr|X1, X2, Xr, Yd)} (3.13)
Csc = min{I(X1, X2; Yr, Yd|Xr), I(X1, X2, Xr;Yd)− I(Yr;Yr|X1, X2, Xr, Yd)} (3.14)
28
Proof. Proof is an extension of the achievable rate of a single user presented in (2.3) to two user
scenario.
From the equivalence of (3.4) and (3.14) it can be deduced that when q∗g = q∗s , the quantized
signal can be perfectly recovered at the BS, and as a result the achievable rate-region of NNC-based
relaying is the same as that of CF. On the other hand, if q∗g < q∗s , then BS in the CF scheme considers
relay transmission as interference for the direct link as the BS is unable to recover the quantized
signal. Due to this inability of the BS in CF the achievable rate-region of NNC-based relaying for
NOMARC turns out to be strictly greater than that with CF relaying. Hence NNC outperforms
CF which is constraint to decode the quantized transmission received via the relay. This behavior
is opposed to a three-node relay network in which, under global CSI at the relay, NNC is known
to have the same achievable rate as that of CF [31]. This is verified by our simulation results, in
which NNC-based relaying is shown to achieve better performance than its CF counterpart.
Local CSI at the relay
In this case, we assume that the relay has knowledge of instantaneous channel realizations for
both its incoming and outgoing links, i.e., the relay has perfect knowledge of h1r, h2r, and hrd,
while it does not have access to the realizations of h1d and h2d. Thus, the relay has the luxury of
adjusting the quantization noise variance q as a function of the known local channel realizations
so as to minimize the probability of outage. The probability of outage, given the known channel
realizations can be computed as
pl0(R) , Pr
(R > min
(C1
β,C2
βc, Cs
) ∣∣∣h1r, h2r, hrd
)(3.15)
=1− Pr({G1d ≥ µ}
⋂{G2d ≥ ν}
⋂{G1d +G2d ≥ η}
),
29
Figure 3.2: The probability of outage is evaluated by integrating the joint distribution of G1d andG2d over the non-shaded region.
where µ , dmax (a, b−Grd)e+, ν , dmax (c, d−Grd)e+, η , dmax (e, f −Grd)e+, and
a , 2βR − G1r1+q − 1, b , 2βR
(1 + q
q
)− 1,
c , 2βcR − G2r
1+q − 1, d , 2βcR
(1 + q
q
)− 1,
e , 2R − G1r+G2r1+q − 1, f , 2R
(1 + q
q
)− 1
Since h1d and h2d are independent and exponentially distributed, the probability of outage can be
computed by integrating a product of two exponential density functions over the non-shaded region
shown in Fig. 3.2. Considering a symmetric case with γ1d = γ2d = γ, (3.15) can be evaluated as
pl0(R) = 1− 1
γ2
∫ ∞ν
∫ ∞µ
e−xγ e− yγ dxdy +
1
γ2
∫ max(η−µ,ν)
ν
∫ η−y
µe−xγ e− yγ dxdy (3.16)
= 1− e−max(η,µ+ν)
γ
(1 +dη − (µ+ ν)e+
γ
)(3.17)
30
which is a closed-form expression for probability of outage as a function of the local CSI realizations.
Thus, for given local channel realizations, the optimum q is given by
q∗l = arg maxq>0
e−max(η,µ+ν)
γ
(1 +dη − (µ+ ν)e+
γ
)(3.18)
The choice q∗l for given local channel realizations ensures that the overall probability of outage is
minimized as well. Since the optimization problem in (3.18) appears to be highly non-linear, we
rely on numerical search when generating simulation results in Section 3.5.
Receiver CSI at the relay
Receiver CSI at the relay applies to the more practical scenario where the relay has instantaneous
knowledge of h1r and h2r only. The conditional probability of outage given these realizations is
given by
pr0(R) = Pr
(R > min
(C1
β,C2
βc, Cs
) ∣∣∣h1r, h2r
)(3.19)
Considering once again a symmetric case with γ1d = γ2d = γrd = γ, the probability of outage for
the case of receiver side CSI at the relay can be computed by marginalizing (3.17) over Grd. Thus,
using z , Grd as the marginalization variable, the probability of outage in this case is given as
pr0(R) = 1− 1
γ
∫ ∞0e− z+max(η,µ+ν)
γ
(1 +dη − (µ+ ν)e+
γ
)dz (3.20)
To evaluate (3.20) in closed-form, consider the set S , {b− a, d− c, f − e} for given channel realizations
h1r and h2r. Let m3 = max S, m1 = min S and m2 = S \ {m3 ∪m1}, then (3.20) can be written
using piecewise integration as
pr0(R) = p1 + p2 + p3 + p4 (3.21)
where pi, 1γ
∫m1
mi−1e− zγ pl0dz, i ∈ {1, 2, 3, 4}, m0 = 0, and m4 = ∞. The terms pi can be computed
using straight-forward, albeit tedious algebra; their exact forms are provided below. Given this
closed-form for the outage probability, the relay can choose q as a function of h1r and h2r so as to
31
minimize the conditional outage probability. For this case as well, we resort to numerical search
over q for optimizing the outage performance.
p1 =
1− e
−m1γ −
(2m1(γ+f−b−d)+m2
12γ2
)e
−fγ if f ≤ (b+ d−m1)
1− e−fγ + e
−(b+d)γ − e
−m1γ −
(m1−b−d+f
γ
)(1 + f−3d−2a−b−m1
2γ
)e
−fγ else if b+ d− f > 0
0 otherwise
p2 =
0 if m1 = m2
e−m1γ − e
−m2γ − e
−(b+d−m2)γ + e
−(b+d−m1)γ if m1 >
b+d−e2
(1−
(γ+e−b−d
γ
)e
−eγ
)(e
−(f−e)γ − e
−m2γ
)+2e
−eγ
(e
−(m2)γ m2+γ
γ − e−m1γ m1+γ
γ
)else if m2 <
b+d−e2
e−(b+d+m1)
γ + e−m1γ − e
−m2γ
−e−(b+d+e)
2γ − e−eγ
(γ+e−b−d
γ
)(e
−(b+d−e)2γ − e
−m2γ
)+2e
−eγ
((m2+γγ
)e
−m2γ −
(b+d−e+2γ
2γ
)e− b+d−e
2γ
)otherwise
else ifm1 =f−e
e−m1γ − e
−m2γ − (m2−m1)
γ2(γ + df − α1e+) e
−max(f,α1)γ otherwise
32
p3 =
0 if m2 = m3
e−m1γ − e
−m3γ if m3 < f − a− c
−(m3 −m1)(
2(γ+f−a−c)−m3−m1
γ2
)e
−fγ
(e−m2γ − e
−m3γ )
(1− e(
−(a+c)γ
))
else if m2 > f − a− c
e−m2γ − e
−m3γ − e
−fγ + e
−(a+c+m3)γ
(a+ c+m2 − f)(
2(γ+f−a−c)−m3−m1
γ2
)e
−fγ otherwise
else ifm3 =f−e
e−m2γ − e
−m3γ −
(m3−m2
γ
)e
−αγ if m3 < α2 − e
e−m2γ − e
−m3γ
(1−
(γ+e−α2
γ
)e
−eγ
)−e
−eγ
((m3+γ
γ )e−m3γ − (m2+γ
γ )e−m2γ
)else if m2 > α2 − e
e−m2γ − e
−m3γ −
(α2−e−m2
γ
)e
−(b+c)γ
−(γ+e−α2
γ
)(e
−α2γ − e
−(m3+e)γ
)+(m3+γγ
)e
−(m3−e)γ −
(α2−e+γ
γ
)e
−α2γ otherwise
otherwise
p4 =
(1− e
−max(e,a+c)γ
(1 +de− (a+ c)e+
γ
))e
−m3γ
where α1 =
a+ d if m1 = b− a
b+ c otherwise
, α2 =
b+ c if m3 = b− a
a+ d otherwise
3.5 Simulation Results
We consider the outage performance for the symmetric case in which the average SNRs on all links
is equal to γ. In Fig. 3.3, we plot the probability of outage versus γ when the sum-rate is fixed to
R = 2 bits per channel use, and the fairness parameter is β = 0.5. For comparison, we consider
NNC relaying for the three cases discussed in Section 4.4. With global CSI and optimal q∗g choice
33
SNR(dB)0 2 4 6 8 10 12 14 16
Pro
babi
lity
of o
utag
e
10-2
10-1
100
NNC-Global CSICF-Global CSINNC-Local CSINNC-Receive CSI
Figure 3.3: Probability of outage for Rs = 2 bits per channel use and β = 0.5.
of the quantization distortion, we observe that NNC-based relaying outperforms CF (with global
CSI) by a margin of 2.2 dB at an outage probability of 10−2. Thus, as opposed to the three-node
relay network, it is clear that NNC-based relaying in NOMARC outperforms CF even if global
CSI is available at the relay. When the relay has access only to its local channel realizations, the
performance at outage probability of 10−2 degrades by 3.8 dB, while under only receiver side CSI,
the performance gap increases to 5.4 dB.
In Fig. 3.4, we plot the outage rate for various schemes for a fixed outage probability of 10−2
and a fairness parameter of β = 0.5. At an outage rate of 3 bits per channel use, the performance
gap of NNC-relaying under global CSI from CF is 2.2 dB, from NNC-relaying with local CSI at the
relay is 4.4 dB, and from NNC-relaying with receiver-side CSI at the relay is 6 dB.
34
SNR(dB)8 10 12 14 16 18 20
Out
age
rate
(bi
ts p
er c
hann
el u
se)
0
1
2
3
4
5
6
NNC-Global CSICF-Global CSINNC-Local CSINNC-Receive CSI
Figure 3.4: Outage rate for an outage probability of 10−2 and β = 0.5.
3.6 Conclusions
We have considered two-user uplink NOMA in the presence of a single dedicated relay. For such
a NOMARC, we consider an NNC-based relaying strategy and provide the associated information
theoretic achievable rate region. Using this rate-region, as well as fairness rate-constraint between
the two users, we have derived closed-form expressions for probability of outage under various
levels of CSI availability at the relay. These closed-form expressions help the relay in appropriately
choosing the quantization noise variance so as to maximize the outage performance. We also show
that under global CSI availability at the relay, NNC-based NOMARC scheme always performs
better than CF-based relaying. SNR benefits gained under various degree of CSI knowledge at the
relay are shown through simulation results.
35
Chapter 4
Joint NNC and DF for NOMA
In this chapter, we consider a non-orthogonal multiple access channel in which multiple users
communicate with a base-station through a single dedicated relay. For this setup, we propose
a novel Joint-NNC-DF cooperation scheme that employs noisy network coded (NNC) relaying
with opportunistic decode-and-forward (DF). The proposed scheme utilizes DF cooperation when
messages from all user are successfully decoded at the relay and NNC when the relay is unable to
decode any one of the users. In the scenario when the relay decodes only a subset of users, the
proposed scheme utilizes joint DF and NNC encoding with DF applied to the set of successfully
recovered messages, and NNC to the rest. This chapter builds on top of contributions made in the
previous chapter in which we had presented an NNC-only scheme for two-user relay-aided NOMA.
NNC scheme variant with cummulative encoding is presented to enable joint encoding of multiple
users with NNC and DF. We derive the information theoretic achievable rates for the proposed
scheme and show that it has a strictly better outage performance than either of the conventional DF-
only, NNC-only, or DF-or-NNC schemes. For the two-user case, we derive closed-form expressions
for the outage performance under Rayleigh fading while assuming access only to the local channel
state information at the relay; expressions that let us choose the NNC quantization parameter so
as to attain optimum outage performance. Simulation results validate the theoretical findings and
depict that the proposed Joint-NNC-DF scheme achieves a performance gain of 0.75 dB, 1.7 dB
and 2.1 dB compared to the DF-or-NNC only, DF-only and NNC-only strategies respectively.
36
4.1 Introduction
In this chapter, we again consider uplink NOMAR where multiple users transmit their respective
data to a single base-station (BS) via a dedicated relaying node. As mentioned previously in
Chapter 3 the three prevalent strategies, originially proposed for the three-node relay network but
also applicable to NOMAR, are amplify-and-forward (AF), decode-and-forward (DF) and compress-
and-forward (CF). In AF, an amplified version of the signal received at the relay is forwarded to
the BS, while in DF, the relay attempts to decode before transmitting the re-encoded message to
the BS. On the other hand, in the CF scheme, the relay quantizes its received signal, and applies
distributed compression before onward transmission to the BS.
In NOMAR setup, each user experiences a different channel quality to the relay. As a result,
the relay may be able to decode all, none or only a subset of the users. When all user messages are
decoded at the relay, the most natural choice for the relay is to utilize DF cooperation. On the other
end is the case when none of the messages is successfully recovered, for which NNC is a reasonable
choice as was employed in the previous chapter. A more likely and important scenario though is
when only a subset of user messages are successfully recovered at the relay. Keeping this scenario in
mind, we propose a novel Joint-NNC-DF (J-NNC-DF) scheme for the NOMAR, in which the relay
utilizes a single codebook that jointly (a) encodes the successfully decoded messages and (b) applies
NNC to the noisy signal corresponding to the users whose messages remained unrecoverable. The
proposed J-NNC-DF scheme falls under the category of hybrid or mixed relaying schemes.
Relevant works falling under the category of mixed or hybrid scheme involving AF and DF
include [61, 62, 63, 64, 65, 66, 67]. In general, these hybrid schemes have better performance
compared to DF-only and AF-only schemes. Specifically, in [61] a hybrid decode-amplify-forward
scheme is proposed where the relay exploits merits of both DF-only and AF-only scheme. Similarly
a hybrid fixed decode-forward and amplify-forward (HDAF) relaying scheme is introduced in [62, 63]
which outperforms adaptive decode forward and AF in terms of symbol error performance. It is also
shown in [62, 63] that the performance gain of the proposed scheme is dependent on the location
of the relay. Keeping in mind the relay location dependency of hybrid schemes, [64] proposes a
hybrid decode-amplify-forward protocol in which the relays close to the source perform AF while
37
the relays further away from the source perform decode-and-forward. Another study [65] conducts
the outage performance of a hybrid decode-amplify-forward protocol resulting in the best-relay
selection scheme. Similarly in another work [66] authors propose an incremental hybrid decode AF
protocol in which a relay either keeps silent or transmits message based on DF and/or AF. Hence
the transmissions from the relay are dependent upon the channel qualities between user and relay.
For a two-way relay network, [67] proposes a hybrid DF and AF with network coding (HDAF-NC)
scheme where the relay nodes can still forward the network coded information when one of the
packets from the two sources cannot be decoded.
Additional works focusing on combination of DF and CF include [68], where a hybrid DF
and CF based scheme is presented. Similarly [69], combines DF and CF while trying to optimize
power allocation. Studies relevant to mixed DF and NNC strategies include [70] that presents a
combination of DF and NNC for a three node network consisting of a transmitter-receiver pair aided
by a dedicated relay. Authors consider splitting user message into a common and a private message
which are superposition coded before transmission. The common message is different in each block
and is decoded at the relay and BS as in DF, while the private message is the same for all blocks
and is decoded only at the destination as in NNC. Similarly, [71] proposes superposition coding for
the rate-split message but presents the generalized achievable rates for multiple-source multicast
networks. This work also provides a proof depicting higher achievable rates of the proposed scheme
compared to NNC-only or CF-only relaying. In [72], the authors propose the use of NNC with
DF through rate-splitting. This proposed relaying scheme highlights the interference limitation of
a mixed scheme which is mitigated through the use of short-NNC relaying. A simplistic two-relay
network is considered and the achievable rates are derived. Another relevant work [73] combines
NNC with partial DF (PDF) scheme for a single-source multicast network. Each relay in the
network simultaneously performs NNC and PDF. A superposed message based on NNC and PDF
is forwarded towards the destination. After the end of all block transmissions, joint decoding
is carried out to recover the message. Another work combining NNC with DF is [74] for the
cooperative unicast scenario, with a single source and a single destination being aided by multiple
relays. Here, each relay is allowed to try DF relaying but all of the relays also transmit the
38
noisy descriptions of their received messages. These noisy description of the messages help the BS
in decoding the DF relayed message. Another recent work on hybrid cooperative scheme which
involves multiple source transmission is discussed in [30]. Here, short-NNC is proposed with DF
option for a more generic relaying scenario, with multiple sources, relays and destinations. However,
as opposed to the J-NNC-DF scheme we propose in this chapter, the relaying scheme in [30] employs
either DF-or -NNC depending upon the channel conditions.
Works highlighted thus far include cooperative schemes that require global CSI for performing
rate-splitting in case of a single source. Since global CSI implies that the relay has a prior knowledge
of all channels, it limits practical implementation of such cooperative schemes. In addition, in the
absence of global CSI, claimed performance gains of these cooperative schemes are not realizable
using naive partitioning at the source. Moreover, the existing hybrid schemes rely on NNC-only, DF-
only, or DF-or -NNC, missing out on the possibility of joint encoding of NNC and DF. As opposed
to existing work, our proposed J-NNC-DF strategy involves joint NNC and DF encoding in the
absence of global CSI knowledge at the sources and the relay. For the proposed scheme, we present
the encoding and decoding procedures and derive the associated information theoretical conditions
for successful message recovery. More importantly, with outage behavior as a performance metric,
we show that the proposed scheme strictly outperforms the conventional NNC-only, DF-only, as
well as the DF-or -NNC schemes that exist in the literature. Considering Rayleigh fading channels
under the assumption that the sources do not have access to any CSI while the relay has access
to local CSI only, we then derive a closed-form expression for the probability of outage for a two-
user scenario. The closed-form result serves as a useful tool for optimizing the relevant NNC
quantization parameters so as to attain the best outage performance. Additionally, using these
closed-form results, we are able to reduce the search space for optimizing quantization parameter.
For a 2-user case, simulations depict that the J-NNC-DF scheme achieves a performance gain of
0.75 dB, 1.7 dB and 2.1 dB compared to the DF-or-NNC only, DF-only and NNC-only schemes,
respectively. We consider it important to note that our focus of outage evaluations specifically
on the two-user case is driven by the well-established argument that pair-wise NOMA is the most
practically viable solution for future-generation networks [75]. This is in part due to the complexity
39
Base
StationRelay
hNd
User 2
User N
User 1
h4d
Figure 4.1: The uplink non-orthogonal multiple-access relay channel.
of multi-user detection increasing exponentially with increasing number of users [76].
The remainder of this chapter is organized as follows. Section 4.2 describes the system model,
while the proposed J-NNC-DF scheme along with the associated information theoretic achievability
analysis is presented in Section 4.3. Outage analysis and discussions on quantizer design are
presented in Section 4.4, while simulation results comparing proposed J-NNC-DF with other
benchmarks are presented in Section 4.5. Lastly, Section 4.6 concludes this chapter.
4.2 System Model
We consider uplink NOMA transmissions, depicted in Fig. 4.1, in which K users send their
respective messages to a BS with the help of a single dedicated relay. The relay is assumed to operate
in the full-duplex mode and is therefore capable of transmitting to the BS while simultaneously
receiving from the users. The K users access the channel in a non-orthogonal fashion, with the
baseband signal received at the relay under a frequency-flat fading model given by
Yr =K∑i=1
hirXi + Zr.
40
Here Xi is the baseband transmission from user-i (i = 1, 2, ...,K), hir is the complex channel gain
between the relay and user-i, and Zr is complex circularly symmetric zero-mean Gaussian noise.
Similarly, the signal received at the BS is denoted by
Yd =K∑i=1
hidXi + hrdXr + Zd,
where Xr is the relay’s transmission, hid is the complex channel gain between user-i and the BS, hrd
is the channel coefficient on the relay-BS link, and Zd is zero-mean Gaussian noise. Without loss in
generality, we assume that the noise terms Zr and Zd are of unit-variance. Moreover, all the channels
are assumed to experience Rayleigh fading with hjk, j ∈ {1, . . . ,K, r}, k ∈ {1, . . . ,K, r, d}, j 6= k,
being zero-mean circularly symmetric Gaussian and E[|hjk|2
]= γjk. We also assume, once again
without loss in generality, that all transmissions are of unit-energy, i.e., E[|Xj |2
]= 1 for all
j ∈ {1, . . . ,K, r}. As a result, the variable γjk parameterizes the average signal-to-noise ratio
(SNR) between node j and k.
4.3 The proposed J-NNC-DF Scheme
For clarity of exposition and ease of understanding, the following first presents the proposed
J-NNC-DF scheme for the two-user case through a detailed description of the encoding and decoding
processes, as well as the associated information theoretic constraints that must be satisfied for
successful decoding. This will be followed by a generalization to an arbitrary K-user case.
4.3.1 Two-user case
For the proposed J-NNC-DF scheme, the relay utilizes a single codebook to jointly encode the user
messages that were successfully decoded while at the same time providing NNC to the ones that
were not. However, since the users are assumed to have no CSI, the encoding process at the users
stay the same regardless of what messages are successfully decoded at the relay. In the following,
we first describe the encoding process at the users, followed by the decoding and encoding process
at the relay that opportunistically switches between different operations depending on what user
41
messages have been decoded. For each case, a description of the decoding strategy at the BS along
with the associated information theoretic conditions for successful decoding are also presented.
Encoding at the users
The message wi from user-i, i ∈ {1, 2}, is encoded as follows. For a transmission rate of Ri bits per
sample (b/s), and a codeword length of n samples per block, the message wi consisting of nBRi bits
is first split into B blocks wi1, wi2, ..., wiB of nRi bits each. Through this splitting of the message
sequence, independent encoding is enforced in each block with the transmission completed using a
total of B+1 blocks. For each block b ∈ {1, . . . , B+1}, a two dimensional lengh-n codebook of 2nRi
rows and 2nRi columns is generated by drawing independently and randomly from the distribution
P (Xi). During block b, the transmitted sequence from user-i is xib(wi(b−1), wib
)as shown in the
first row of Table 4.1. Here xib (·, ·) refers to the two dimensional encoding function during block
b with wi0 = wi(B+1) = 1, a fact that is known at the relay and the destination. The reason for
employing the two-dimensional codebook is to aid the encoding and decoding at the relay and the
BS, as will be explained in the next subsection. It is also noteworthy that this procedure of splitting
the message into B blocks and encoding the split messages in each block is akin to the so-called
short-message NNC strategy [30]. This is in contrast to the long-message NNC scheme [29] in which
the same message wi is encoded with independent codebooks over all blocks. As was pointed out
in [39], both strategies result in same performance even for the multiple source multicast network
under the availability of global CSI at the relay ; the additional constraint present in short-message
NNC strategy becomes redundant and the two scheme become equivalent in terms of achievable
rates. However, as opposed to short-message NNC, the DF option at the relay with long-message
NNC becomes infeasible [73] as the effective coding rate per block is quite high. It is to exploit the
DF option that we utilize the short-message NNC in the proposed J-NNC-DF scheme.
Encoding at the relay and decoding at the BS
For the proposed J-NNC-DF scheme, the relay first attempts to recover the users’ information
using forward decoding. Since the relaying operation is dependent upon the number of users it can
42
Table 4.1: Joint NNC-DF coding with B=4 blocks.
Blocks 1 2 3 4 5
User 1 x11(1, w11) x12(w11, w12) x13(w12, w13) x14(w13, w14) x15(w14, 1)
User 2 x21(1, w21) x22(w21, w22) x23(w22, w23) x24(w23, w24) x25(w24, 1)E2 occurs xr1(1, 1) xr2(w11, w21) xr3(w12, w22) xr4(w13, w23) xr5(w14, w24)
Relay E0 occurs xr1(1) xr2(v1) xr3(v2) xr4(v3) xr5(v4)
E1 occurs xr1(1, 1) xr2(v1, w11) xr3(v2, w12) xr4(v3, w13) xr5(v4, w14)
Quantization yr1(1, v1) yr2(v1, v2) yr3(v2, v3) yr4(v3, v4) yr4(v4, 1)
successfully decode, the encoding at the relay and the decoding at the BS will vary according to
three disjoint events: (a) When both user messages are decoded at the relay, (b) when none of the
user messages are decoded at the relay, and (c) when only one user is decoded at the relay. The
following describes the operations corresponding to these distinct events.
Both users decoded at the relay When both users are successfully decoded at the relay,
the natural relaying choice is DF. For DF cooperation in each block, the relay forms a two-
dimensional length-n codebook of size 2nR1 × 2nR2 by drawing randomly and independently from
the distribution P (Xr); for block-b, the encoding function is denoted by xrb (·, ·). The forward
decoding and encoding operation at the relay for an arbitrary block b, 1 ≤ b ≤ B, operates as
follows. Given that the message sequences w1(b−1) and w2(b−1) are successfully decoded, the relay
utilizes the received sequence during block-b to recover the message sequences w1b and w2b of the
two users. In particular, the relay attempts to jointly recover w1b and w2b by only searching in
the rows of the two-dimensional source codebooks that correspond to the known sequences w1(b−1)
and w2(b−1). Thus, the relay effectively sees a conventional multiple-access channel (MAC) from
the two users. As B →∞, we can conclude that the relay is capable of successfully recovering the
user messages in each block if the transmission rates R1 and R2 lie in the two-user MAC capacity
region [77] shown in Fig. 4.2. In other words, both user messages are successfully decoded if the
43
No user decoded
One user decoded
Both user decoded
Figure 4.2: Capacity region of a two-user MAC along with a depiction of regions corresponding tothe events that (a) both users are decoded, (b) none of the users are decoded, and (c) only one useris decoded.
following event is satisfied
E2 ,{R1 ≤ I(X1;Yr|X2)
}⋂{R2 ≤ I(X2;Yr|X1)
}⋂{R1 +R2 ≤ I(X1, X2;Yr)
}. (4.1)
Given that the event E2 is satisfied, the relay encodes w1b and w2b during block-(b + 1) using the
encoding function xr(b+1) (w1b, w2b) before onward transmission to the BS, as indicated in Table 4.1.
The BS utilizes backward decoding. In particular, having perfectly recovered the message sequences
w1b and w2b already, the BS attempts to recover the message sequences w1(b−1) and w2(b−1) from
the received sequence during block-b. The successful decoding conditions for this case when B →∞
is specified below.
Theorem 4.3.1. Given that the relay decodes both users (i.e. the event E2 in (4.1) is satisfied),
the proposed J-NNC-DF scheme allows the BS to successfully recover both users as long as the
44
transmission rates of the two users lie in a rate-region defined by
R2 ,{
(R1, R2)∣∣∣R1 ≤ C1,2, R2 ≤ C2,2, R1 +R2 ≤ Cs,2
}(4.2)
where,
C1,2 = I(X1, Xr;Yd|X2) (4.3)
C2,2 = I(X2, Xr;Yd|X1) (4.4)
Cs,2 = I(X1, X2, Xr;Yd) (4.5)
The achievable rate region can be derived using standard joint typicality decoding arguments.
The exact proof is very similar to the one discussed in [78], and will therefore not be reproduced
here. Having said this, an insightful observation is that the region in (4.2) is similar to that of a
conventional two-user MAC, except that the single user information bounds C1 and C2 as well as
the sum-rate bound Cs includes an additional variable Xr. This makes intuitive sense; Xr in the
mutual information terms is representative of the fact that the relay forwards the encoded version
of exactly the same information that is being transmitted from the users over a cooperative MAC.
No user decoded at the relay Although the relay attempts to decode both users, it is quite
possible that it is unable to successfully recover any one of the user messages. As seen in Fig. 4.2,
this happens when the following event is satisfied
E0 ,{R1 > I(X1;Yr)
}⋂{R2 > I(X2;Yr)
}⋂{R1 +R2 > I(X1, X2;Yr)
}, (4.6)
i.e., the relay cannot decode either of the message even by treating the other as interference. For
this particular case, the proposed J-NNC-DF scheme employs (short-message) NNC relaying. In
particular, during each block, the relay quantizes the received signal Yr and generates the encoded
version Xr for onward transmission towards the BS. For the purpose, two distinct codebooks are
generated; one for quantizing the relay received signal and the other for relay’s transmissions. For
45
each block, let Yr be the quantization random variable that follows the conditional distribution
P (Yr|Yr), with the associated marginal distribution given by P (Yr). Also, let Rr be an auxiliary
relay encoding rate (that will ultimately be eliminated from the achievable rate region evaluations).
For quantization, a two-dimensional length-n codebook of 2nRr rows and 2nRr columns is generated
for each block-b by drawing independently and randomly from the distribution P (Yr); the corresponding
encoding function is denoted by yrb(·, ·). At the same time, for each block-b, a codebook containing
2nRr length-n sequences is also generated by drawing independently and randomly from the distribution
P (Xr); the encoding function for this codebook is denoted by xrb(·). The NNC encoding at the relay
proceeds as follows. For a received length-n sequence yrb at the end of block-b, the relay performs
joint source-channel encoding by finding an index vb such that xrb(v(b−1)), yrb, and yrb(v(b−1), vb)
are jointly typical, i.e., {xrb(v(b−1)), yrb, yrb(v(b−1), vb)
}∈ T (n)
ε .
Here, v(b−1) is the known encoding index from the previous block (with the assumption that v0 = 1).
If there is no such index vb that satisfies the typicality condition, the relay selects an index from the
set{
1, . . . , 2nRr}
uniformly at random. For the selected index vb, the relay transmits the sequence
xr(b+1)(vb) in block-(b+ 1).
At the BS, block-wise backward decoding is performed. Thus, having received the length-n
sequence ydb at the end of block-b and having recovered the indices w1b, w2b, and vb from earlier
decoding rounds, the BS tries to find a triplet of unique indices (w1(b−1), w2(b−1), v(b−1)) such that
{x1b(w1(b−1), w1b), x2b(w2(b−1), w2b), yrb(v(b−1), vb), xr(v(b−1)),ydb
}∈ T (n)
ε
If no such unique triplet is found, an error is declared. The procedure is repeated for each block
from b = B + 1, all the way back to b = 2. As B → ∞, the conditions necessary for successful
decoding are given below.
Theorem 4.3.2. Given that the relay decodes no users (i.e. the event E0 in (4.6) is satisfied),
the BS can successfully decode both users with the proposed J-NNC-DF scheme if the transmission
46
rates lie inside the following region.
R0 ,{
(R1, R2)∣∣∣R1 ≤ C1,0, R2 ≤ C2,0, R1 +R2 ≤ Cs,0
}(4.7)
where,
C1,0 = min{I(X1; Yr, Yd|X2, Xr), I(X1, Xr;Yd|X2)− I(Yr;Yr|X1, X2, Xr, Yd)
}(4.8)
C2,0 = min{I(X2; Yr, Yd|X1, Xr), I(X2, Xr;Yd|X1)− I(Yr;Yr|X2, X1, Xr, Yd)
}(4.9)
Cs,0 = min{I(X1, X2; Yr, Yd|Xr), I(X1, X2, Xr;Yd)− I(Yr;Yr|X1, X2, Xr, Yd)
}(4.10)
under the condition that the following inequality is satisfied
I(Xr;Yd|X1, X2) > I(Y ;Yr|X1, X2, Xr, Yd) (4.11)
Detailed proof regarding this achievable rate, albeit for a three-node relay network, region is
presented in Section 2.1.1 of this thesis. An extension of that proof to this two-user NOMAR case
is relatively straightforward. A similar proof, albeit using a slightly different coding scheme, can
also be seen in [39].
Exactly one user is decoded at the relay Finally, the most interesting case is when the relay
is capable of successfully decoding only one of the two users. We assume without loss in generality,
that during the forward decoding process, the relay is capable of recovering user-1 message w1b
during block-b, while that of user-2 remains unrecoverable. As seen in Fig. 4.2, this corresponds
to the event
E1 ,{R1 ≤ I(X1;Yr)
}⋂{R2 > I(X2;Yr|X1)
}. (4.12)
Under such a scenario, the proposed J-NNC-DF scheme employs a transmission with the joint
NNC and DF encoding strategy outlined in the following. Once again, two codebooks are generated
at the relay; one for quantization and the second for the relay’s transmissions. For the latter,
47
given a quantization rate Rr, the relay forms a two-dimensional length-n codebook of 2nR1 rows
and 2nRr columns by drawing randomly and independently from the distribution P (Xr). On
the other hand, the quantization codebook is similar to the one described earlier except that the
quantization is carried out on the relay received signal from which the user-1 contribution has
been removed. More specifically, the quantization during block-b is carried out on the sequence
yrb = yrb − h1rx1b
(w1(b−1), w1b
), which can be formed due to (a) the fact that the relay has already
been able to perfectly recover w1b and w1(b−1), and (b) the assumption that the relay has perfect
knowledge of the channel coefficient h1r. Thus the quantization codebook is formed by random
drawings of Yr that has a conditional distribution P (Yr|Yr) and the associated marginal distribution
given by P (Yr). The quantization during block-b is performed by finding an index vb such that
{yrb, yrb(v(b−1), vb), xrb(v(b−1), w1(b−1))
}∈ T (n)
ε
where yrb(·, ·) and xrb(·, ·) are the block-b encoding functions corresponding to the quantization
and the relay transmission codebook, respectively. If there is more than one such index, or if there
exists no such index, the relay selects an index from the set{
1, . . . , 2nRr}
uniformly at random.
For the selected index vb, the relay transmits the sequence xr(b+1)(vb, w1b) in block-(b + 1). As
before, v0 = vB+1 = 1, a fact that is known at the BS for decoding purposes.
At the BS, backward block-wise decoding is employed. In particular, for each block-b, the
decoding operation involves finding a triplet of unique indices (w1(b−1), w2(b−1), v(b−1)) such that
{x1b(w1(b−1), w1b),x2b(w2(b−1), w2b), yrb(v(b−1), vb), xrb
(v(b−1), w1(b−1)
), ydb
}∈ T (n)
ε ,
where it is assumed that the indices w1b, w2b, and vb have been perfectly recovered during the
previous decoding step. As B →∞, the achievable rate-region due to the joint encoding scheme is
given in the following.
Theorem 4.3.3. Given that the relay decodes user-1 only (i.e. the event E1 in (4.12) is satisfied),
the proposed J-NNC-DF scheme allows the BS to successfully recover messages of both users if their
48
transmission rates lie inside the rate region defined by
R1 ,{
(R1, R2)∣∣∣R1 ≤ C1,1, R2 ≤ C2,1, R1 +R2 ≤ Cs,1
}(4.13)
where
C1,1 = I(X1, Xr;Yd|X2)− I(Yr; Yr|X1, X2, Xr, Yd) (4.14)
C2,1 = min{I(X2; Yr, Yd|X1, Xr), I(X2, Xr;Yd|X1)− I(Yr; Yr|X1, X2, Xr, Yd)
}(4.15)
Cs,1 = min{I(X1, X2, Xr; Yr, Yd), I(X1, X2, Xr;Yd)− I(Yr; Yr|X1, X2, Xr, Yd)
}(4.16)
under the condition that the following inequality be satisfied
I(Xr;Yd|X1, X2) > I(Yr; Yr|X1, X2, Xr, Yd). (4.17)
Proof. See Appendix A.
4.3.2 General K-user case
For the general case, let K , {1, 2, ...,K} be the set of all users. At each user, encoding is carried
out using a two-dimensional codebook, exactly in the same manner as highlighted for the two-user
case in Table 4.1. Without loss in generality, we assume that the relay is able to successfully decode
the messages of M ≤ K users with D denoting the set of such users; the set of users whose messages
remain unrecoverable at the relay is denoted by Dc , K − D. Using the achievable rate-region of
a standard multiple-access channel between the users and the relay, it can be shown that only the
messages of users in D are successfully recovered if the following event is satisfied
ED ,⋂V⊆D
{∑j∈V
Rj ≤ I(XV ;Yr|XVc)} ⋂T ⊆Dc
{∑i∈T
Ri > I(XT ;Yr|XD)}
(4.18)
49
where Vc , D − V. In addition, XV denotes the collection of the transmit random variables
indexed by the elements in the set V. In particular, if V = {v1, . . . , vd} is a set of users, then
XV , {Xv1 , . . . , Xvd}.
If the relay decodes all users (i.e., M = K), there is no need for quantization, and the relay
transmits from a K-dimensional codebook of size 2nR1× . . .×2nRK . On the other hand, if M < K,
the relay removes the contributions of the users in D from its received signal before quantization.
In particular, for each block b, it forms yrb = yrb−∑
j∈D hjrxjb(wj(b−1), wjb
). The relay encoding
once again makes use of two codebooks; one for quantization and the other for transmission. For
each block, the quantization codebook is two dimensional of size 2nRr×2nRr constructed by random
and independent drawings from the distribution P (Yr). On the other hand, if d1, . . . , dM denote
the M elements of the set D, an M + 1-dimensional transmission codebook, of size 2nRr × 2nRd1 ×
2nRd2 × . . .× 2nRdM is constructed by independently drawing from P (Xr). For encoding in block-b,
the relay attempts to find an index vb such that
{yrb, yrb(v(b−1), vb),xrb
(v(b−1), wd1(b−1), . . . , wdM (b−1)
)}∈ T (n)
ε .
For the selected index, the relay transmits the sequence xr(b+1)
(vb, wd1(b), . . . , wdM (b)
)in block-
(b + 1) with v0 = vB+1 = 1. Decoding at BS for block b is carried out in a backward fashion by
finding a set of unique indices {w1(b−1), . . . , wK(b−1), v(b−1)} such that
{x1b(w1(b−1), w1b), . . . , xKb(wK(b−1), wKb), yrb(v(b−1), vb)
,xrb(v(b−1), wd1(b−1), . . . , wdM (b−1)
),ydb
}∈ T (n)
ε .
It is important to point out here that when D = K, i.e., all users are decoded, the relay does not
perform the quantization typicality test, and instead selects vb = 1 for all b = 0, . . . , B + 1; a fact
that we assume is known to the BS. In addition, since no quantization actually takes place in this
case, we let the variable Yr = 0, an assumption that leads to the following generalized rate region.
Theorem 4.3.4. For the general K-user NOMAR, if the relay is capable of decoding users in
the set D ⊆ K, the destination can successfully recover the messages of all users as long as the
50
transmission rates lie in the region
RG ,
{(R1, R2, ..., RK)
∣∣∣∑i∈V
Ri ≤ CDF (V),∑j∈T
Rj ≤ CNNC(T ),
∑i∈V
Ri +∑j∈T
Rj ≤ CNNC−DF (V, T ) ∀ V ⊆ D, T ⊆ Dc}
(4.19)
where,
CDF (V) = I(XV , Xr;Yd|XVc)− I(Yr; Yr|XK, Xr, Yd) (4.20)
CNNC(T ) = min{I(XT ; Yr, Yd|XT c , Xr),
I(XT , Xr;Yd|XT c)− I(Yr; Yr|XK, Xr, Yd)} (4.21)
CNNC−DF (V, T ) = min{I(XV , XT , Xr; Yr, Yd|X{V ∪ T }c),
I(XV , XT , Xr;Yd|X{V ∪ T }c)− I(Yr; Yr|XK, Xr, Yd)}, (4.22)
(with Vc = D − V, T c , Dc − T , and {V ∪ T }c , K − {V ∪ T }
)under the constraint that the
following inequality be satisfied
I(Xr;Yd|XK) > I(Yr; Yr|XK, Xr, Yd). (4.23)
The achievability of the region above can be proved through a straightforward, albeit tedious
extension of the presentation in Appendix A. In order to gain insights into the achievable region
presented in Theorem 4.3.4, we make the following observations depending upon the number of
users decoded at the relay.
1. When all users are decoded
First, it can be verified that for the case when all users were decoded at the relay, (4.19)
does indeed reduce to the DF-only rate-region. In particular, since Dc is an empty set, the
constraint involving CNNC (T ) becomes irrelevant. Moreover, as noted above, Yr = 0 when
D = K, because of which CNNC−DF (V, T ) is the same as CDF (V). Consequently, the only
51
constraint governing achievability is the standard DF constraint given by
∑i∈V
Ri ≤ I (XV , Xr;Yd|XVc) , ∀V ⊆ K
2. When no user message is decoded
Since in this case V is always a null-set, the DF constraint of (4.20) is irrelevant. Moreover,
using the chain rule of mutual information, it can be verified that (4.21) is no greater than
(4.22), because of which the latter also becomes redundant. As a result, the achievable region
is governed, as it should be, only by the constraints
∑i∈T
Ri ≤ CNNC (T ) ,∀T ⊆ K
3. When only a subset of users are decoded
In this scenario, all bounds of Theorem 4.3.4 contribute to form a rate region. It is worth
noting that CDF (V) in (4.20) of the J-NNC-DF scheme consists of two terms. The first
corresponds to the conventional DF bound (for the decoded users) one would have obtained
had the quantized indices not been encoded in the relay’s transmission. The second term
therefore is a rate penalty that the decoded users suffer due to the their messages being
jointly encoded with the quantization index. The expression (4.21) is a straight forward
extension of the rates evaluated in (4.10) for NNC-only relaying. Lastly, (4.22) presents the
sum rate highlighting the benefit of joint encoding of decoded and quantized message at the
relay.
In the following lemma, we prove that this bound is strictly greater than a bound achieved by
employing NNC-only relaying scheme. From (4.20)–(4.22), we also observe how the choice of Yr
provides competing benefits to the users in D and to those in Dc. On the one hand, choosing Yr
to be more correlated to Yr (in other words, choosing the quantization noise to be small) helps the
users in Dc but causes a larger penalty to the users in D, and vice-versa. Finally, we point out
that the constraint (4.23) appears because the BS decoding requires that the quantization index vB
52
be recovered perfectly at the destination, using which backward decoding is performed. We note
though that it possible to get rid of this constraint if one assumes that the quantization index vB
is somehow communicated to the BS perfectly over a highly reliable channel, as was assumed in
[30] for short-message NNC relaying.
Next, in order to analytically depict the performance superiority of the proposed J-NNC-DF
strategy over NNC-only and DF-only schemes in the next section, we will make use of the following
result.
Lemma 4.3.5. For any D, V ⊆ D and T ⊆ Dc, we have
CDF (V) ≥ CNNC(V) (4.24)
CNNC−DF (V, T ) ≥ CNNC(V ∪ T ) (4.25)
Proof. See Appendix B.
4.4 Outage performance analysis
In this section, we examine outage performance of the proposed J-NNC-DF scheme for the two-user
case under Gaussian signaling from all nodes. The performance metrics include outage probability
and outage rate, exact specifications of which follow in subsequent subsections. We point out that
we limit our attention to the two-user case not only due to its analytical tractability, but also
because of its potential significance in future generation networks, as outlined in the introduction.
4.4.1 Achievable Rate Regions under Gaussian Signaling
For analysing the outage performance of the proposed J-NNC-DF scheme, we assume that all node
transmissions are Gaussian, given by Xk ∼ CN (0, 1), k ∈ {1, 2, r}. With Gaussian signaling, an
appropriate choice of the quantization model, as employed for example in [30], is Yr = Yr + Zq.
Here Zq denotes the zero-mean complex Gaussian noise with variance q, with the parameter q
determining the quantizer fidelity. The achievable rate region for the proposed J-NNC-DF scheme
is governed by the occurrence of events E2, E0, and E1 defined in (4.1), (4.6) and (4.12), respectively.
53
Given that E2 in (4.1) is satisfied, i.e., both users are decoded at the relay, the rate bounds (4.3)–
(4.5) evaluate to
C1,2 = log (1 +G1d +Grd) , (4.26)
C2,2 = log (1 +G2d +Grd) , and (4.27)
Cs,2 = log (1 +G1d +G2d +Grd) , (4.28)
where Gid = |hid|2 for all i ∈ {1, 2, r} are the (real) channel gains. Similarly, given that no user
is decoded at the relay, i.e., the event E0 in (4.6) is satisfied, the rate bounds (4.8)-(4.10) under
Gaussian signaling can be evaluated as
C1,0 = min{
log(1 +G1r
1 + q+G1d),
⌈log(1 +G1d +Grd)− log
1 + q
q
⌉+}, (4.29)
C2,0 = min{
log(1 +G2r
1 + q+G2d),
⌈log(1 +G2d +Grd)− log
1 + q
q
⌉+}, and (4.30)
Cs,0 = min{
log(1+G1r +G2r
1 + q+G1d+G2d),⌈
log(1+G1d+G2d+Grd)− log1 + q
q
⌉+}, (4.31)
where G1r = |h1r|2, G2r = |h2r|2, and dxe+ , max(0, x). Moreover, since the quantization indices
need to be recovered perfectly at the BS, the above rate-region is valid as long as (4.11) is satisfied.
This constraint evaluated under Gaussian signaling is given by
log(1 +Grd) > log(1 + q
q) (4.32)
54
Finally, given the case when only user-1 was decoded at the relay, i.e., the event E1 in (4.12) is
satisfied, the rate bounds (4.14)–(4.16) under Gaussian signaling become
C1,1 =⌈log(1 +G1d +Grd)− log
1 + q
q
⌉+(4.33)
C2,1 = min{
log(1 +G2r
1 + q+G2d),
⌈log(1 +G2d +Grd)− log
1 + q
q
⌉+}(4.34)
Cs,1 = min{
log(1+G1r +G2r
1 + q+G1d+G2d),⌈
log(1+G1d+G2d+Grd)− log1 + q
q
⌉+}(4.35)
under the constraint (4.17) which again translates to (4.32).
4.4.2 Outage analysis
For the outage analysis, we assume an arbitrary fairness constraint parameter β ∈ [0, 1] between
the two users. In particular, if Rs , R1 +R2 is the sum-rate, we assume that the transmission rates
of user-1 and user-2 must be fractions β and βc , 1−β of the sum-rate, respectively, i.e., R1 = βRs
and R2 = βcRs. In addition, as mentioned earlier, we assume availability of local CSI at the relay;
a more realistic scenario compared to the usual assumption of global CSI availability in existing
literature. More specifically, we assume that the relay has perfect knowledge of the magnitude and
phase of receive coefficients h1r and h2r. As for the transmit CSI, we assume that the relay has
perfect knowledge only of the channel gain Grd and not the phase. These assumptions once again
are in line with real-world systems where it is relatively straightforward to acquire the magnitude
and phase of the receive CSI. On the other hand, while the magnitude of the transmit CSI can be
acquired through channel feedback and/or channel reciprocity, acquiring the corresponding phase
is a lot more challenging due to phase drifts between the transmitter and receiver oscillators.
For outage analysis, we consider an outage event to occur when either of the two users’ messages
is not decoded at the BS. Before presenting the outage analysis, we note that for the achievable
rate bounds (4.29)–(4.31) and (4.33)–(4.35) of the proposed J-NNC-DF scheme to be valid, the
backward decoding constraint (4.32) must be satisfied. Fortunately, we observe that the constraint
is only a function of Grd (and not of the corresponding phase) that is perfectly known at the relay.
55
Thus, the outage performance analysis carried out in the following implicitly assumes that q is
always chosen such that q > 1Grd
, thus ensuring that (4.32) is always satisfied. Moreover, for the
known G1r and G2r, only one of the events E0, E1, and E2 are true. Thus, if po (Rs|Ei) is the outage
probability as a function of the sum-rate Rs conditioned on the event Ei, i = 0, 1, 2, the overall
probability of outage can be evaluated as1
po (Rs) , po (Rs|E2) .1 (E2) + po (Rs|E0) .1 (E0) + po (Rs|E1) .1 (E1) , (4.36)
where 1 (Ei) is an indicator function equal to one when the event Ei is satisfied and zero otherwise.
Next, we analyze each one of the conditional outage probabilities. Given E2, an outage occurs when
either R1 = βRs > C1,2, R2 = βcRs > C2,2, or Rs > Cs,2. Thus, an outage occurs when Rs is
greater than the minimum ofC1,2
β ,C2,2
βc , and Cs,2. As a result, the probability of outage given E2 is
given by
p0(Rs|E2) = P
(Rs > min
(C1,2
β,C2,2
βc, Cs,2
) ∣∣∣G1r, G2r, Grd
)(4.37)
= 1− P({G1d ≥ µ2}
⋂{G2d ≥ ν2}
⋂{G1d +G2d ≥ η2}
),
where the probability is computed over the random variables G1d and G2d, and where µ2, ν2, and
η2 are functions of the known deterministic parameters and are given by
µ2 ,⌈2βRs − 1−Grd
⌉+, ν2 ,
⌈2β
cRs − 1−Grd⌉+, η2 ,
⌈2Rs − 1−Grd
⌉+.
Using similar arguments, the probability of outage given the event E0 can be evaluated as
p0(Rs|E0) = P
(Rs > min
(C1,0
β,C2,0
βc, Cs,0
) ∣∣∣G1r, G2r, Grd
)(4.38)
= 1− P({G1d ≥ µ0}
⋂{G2d ≥ ν0}
⋂{G1d +G2d ≥ η0}
),
1We note that in addition to Rs, the probabilities are also parameterized by the fairness parameter β as well as theknown local CSI gains G1r, G2r, and Grd. However, for ease of notation, we do not explicitly denote this dependency.
56
where µ0 , dmax (a0, b0)e+, ν0 , dmax (c0, d0)e+, η0 , dmax (e0, f0)e+, and
a0 , 2βRs − G1r
1 + q− 1, b0 , 2βRs
(1 + q
q
)− 1−Grd,
c0 , 2βcRs − G2r
1 + q− 1, d0 , 2β
cRs
(1 + q
q
)− 1−Grd,
e0 , 2Rs − G1r +G2r
1 + q− 1, f0 , 2Rs
(1 + q
q
)− 1−Grd
Finally, the probability of outage when only one user is decoded at the relay is given by
p0(Rs|E1) = 1− P({G1d ≥ µ1}
⋂{G2d ≥ ν1}
⋂{G1d +G2d ≥ η1}
), (4.39)
with µ1 , dmax (a1, b1)e+, ν1 , dmax (c1, d1)e+, η1 , dmax (e1, f1)e+, and
a1 ,
a0 User 2 decoded,
0 User 1 decoded,
b1 = b0, c1 ,
c0 User 1 decoded ,
0 User 2 decoded ,
d1 = d0, e1 = e0, f1 = f0
The probabilities in (4.37), (4.38), and (4.39) can be evaluated by integrating the joint probability
density function of G1d and G2d over the non-shaded region as previously depicted in chapter 3 Fig.
3.2. Since G1d and G2d are independent and exponentially distributed, this probability, assuming
a symmetric case with γ1d = γ2d = γ, can be evaluated as
p0(Rs|Ei) = 1− 1
γ2
∫ ∞νi
∫ ∞µi
e− xγ e− yγ dxdy +
1
γ2
∫ max(ηi−µi,νi)
νi
∫ ηi−y
µi
e− xγ e− yγ dxdy
= 1− e−max(ηi,νi+µi)
γ
(1 +dηi − (νi + µi)e+
γ
)(4.40)
for all i = 0, 1, 2, and where µi, νi, and ηi for all cases are specified in the preceding discussions.
57
4.4.3 Comparison with benchmarks
At this point, it is instructive to compare the proposed J-NNC-DF scheme with the following
conventional benchmarks existing in the literature: (a) DF-only, (b) NNC-only, and (c) DF-or -
NNC. As the name implies, in the DF-only approach, the relay cooperates only if it is capable of
successfully recovering both user messages; otherwise the relay stays silent. Thus, the probability
of outage with the DF-only approach is given by
pDF0 (Rs) = p0(Rs|E2)1(E2) + p0(Rs|E2, Grd = 0) [1− 1(E2)] , (4.41)
where p0(Rs|E2, Grd = 0) is the probability of outage when only the direct transmissions are utilized,
and can be obtained by substituting Grd = 0 in the expression for p0(Rs|E2). On the other hand,
with the NNC-only approach, the relay does not attempt to decode the user messages and always
performs NNC. As a result, the associated probability of outage is given by
pNNC0 (Rs) = p0(Rs|E0). (4.42)
Finally, with the DF-or -NNC approach, the relay performs DF cooperation if it is capable of
successfully recovering both messages, and carries out NNC cooperation otherwise. Thus, the
probability of outage associated with this benchmark is given by
pDForNNC0 (Rs) = p0(Rs|E2)1(E2) + p0(Rs|E0) [1− 1(E2)] . (4.43)
The following lemma compares the outage performance of the J-NNC-DF scheme with these
benchmarks.
Lemma 4.4.1. The outage performance of the proposed J-NNC-DF scheme for the two-user case
is better than the DF-only, NNC-only, as well as the DF-or-NNC approach.
Proof. Making use of Lemma 4.3.5, as well as analysing the corresponding two-user bounds directly,
we can easily conclude that C1,2 ≥ C1,1 ≥ C1,0, C2,2 ≥ C2,1 ≥ C2,0, and Cs,2 ≥ Cs,1 ≥ Cs,0. As a
result, for the same parameters q and β, as well as the same channel gains G1d, G2d, and Grd, we
58
have
p0(Rs|E0) ≥ p0(Rs|E1) ≥ p0(Rs|E2). (4.44)
Moreover, it can also be easily verified that had only the direct paths between the users and the
BS utilized, the corresponding probability of outage is always worse than that achieved with NNC
cooperation, i.e.,
p0(Rs|E2, Grd = 0) ≥ p0(Rs|E0). (4.45)
Using (4.44) and (4.45), and comparing (4.36) with (4.41)–(4.43), it can be easily seen that the
outage probability corresponding to J-NNC-DF scheme satisfies
p0(Rs) ≤ pDForNNC0 (Rs) ≤ pDF0 (Rs), and
p0(Rs) ≤ pDForNNC0 (Rs) ≤ pNNC0 (Rs).
Thus, Lemma 4.4.1 points to the intuitive fact that if the relay is capable of successfully
recovering a user’s message, it is always better off applying DF cooperation to it. We also consider
it important to point out that while Lemma 4.4.1 proves the outage performance superiority of J-
NNC-DF scheme for the two-user case only, the same ideas can be extended to the general K-user
case as well. Thus, if all users satisfy fairness constraints that reduce to one the overall degree of
freedom in choosing the user rates, the result in Lemma 4.3.5 can be utilized to demonstrate that
the J-NNC-DF scheme outperforms the given benchmarks for any number of users.
4.4.4 Quantizer Design
As mentioned in the preceding subsections, we restrict our attention to a Gaussian quantizer
according to which Yr = Yr + Zq; here Zq is the zero-mean Gaussian quantization noise with
variance q. Thus, design of an appropriate quantizer entails choosing the variance q that achieves
59
the best outage performance. More specifically, the optimum quantization noise variance is
q∗ = arg minq> 1
Grd
p0(Rs), (4.46)
where the outage probability p0(Rs) is given by (4.36). We note that p0(Rs) is a function of the
local CSI at the relay, and thus the optimum quantization variance will also turn out to be a
function of these channel conditions. In addition, for given local CSI at the relay, only one of the
three events E2, E1, and E0 is satisfied, with quantization involved only in the latter two. As a
result, q can be optimized independently for the two events. Thus, the optimum quantization noise
variance corresponding to the event i ∈ {0, 1} is
q∗i = arg minq> 1
Grd
p0(Rs|Ei). (4.47)
Given the outage probability p0(Rs|Ei) in (4.40), deriving a closed-form expression for q∗i appears to
be a daunting task. Consequently, we resort to numerical methods of searching over the quantization
variance. However, instead of exhaustively looking for all q ∈ ( 1Grd
,∞), we analyze the behavior of
the parameters µi, νi and ηi (i = 0, 1) to reduce the search space. First, from the descriptions of
the parameters (ai, bi, ci, di, ei, fi) following (4.38) and (4.39), we note they are all monotones in q,
with (ai, ci, ei) being non-decreasing and (bi, di, fi) being non-increasing functions of q. Let qxy be
the (unique) positive q at which the parameters x and y intersect; x, y ∈ {ai, bi, ci, di, ei, fi}, and
x 6= y. Moreover, let qmax , max(qab, qcd, qef ). For q > qmax, we have µi = ai, νi = ci and ηi = ei,
which are all non-decreasing functions of q. Thus, for q > qmax, we have the following piecewise
definition of the probability of outage
p0(Rs|Ei)∣∣∣q>qmax
=
1− e−
eiγ
(1 + ei−ai−ci
γ
)ei ≥ ai + ci
1− e−ai+ciγ otherwise
(4.48)
Next, we observe that for i = 0, i.e., the case when no user is decoded, the term e0−a0−c0 becomes
independent of q. As a result, for q > qmax, the probability p0(Rs|E0) is increasing in q, which in
60
turn implies that q∗0 < qmax. On the other hand, when i = 1, i.e., when only one of the users has
been decoded, the behavior is not necessarily monotone for q > qmax. In particular, we observe
that for i = 1, while the second term in (4.48) is increasing in q, the first term may not be. Thus
to find an upper bound on q∗1, we differentiate the first term with respect to q and set it to zero
yielding the solution
q+ =G1r
2Rs − 2βcRs + γ(1− G1rG1r+G2r
)− 1, (4.49)
where without loss in generality, we have assumed that user-1 message was decoded at the relay.
By carrying out an analysis of the slope of the first term of (4.48), it can be easily shown that it is
increasing in q for all q > q+. Thus, we conclude that the optimum quantization noise for the case
when only one user is decoded at the relay satisfies q∗1 < max (qmax, q+).
For providing a lower bound on the optimum quantization noise, we define
qmin , min(qab, qcd, qef ).
Next, we observe that for all 0 < q < qmin, we have µi = bi, νi = ei, and ηi = fi, which are all
non-increasing functions of q. Thus, for q < qmin, we have
p0(Rs|Ei)∣∣∣0<q<qmin
=
1− e−
fiγ
(1 + fi−bi−di
γ
)iffi > bi + di
1− e−bi+diγ otherwise
(4.50)
First, we note that since b1 = b0, d1 = d0, and f1 = f0, the probability of outage in the interval
0 < q < qmin is the same regardless of the number of users successfully decoded at the relay. Second,
we note that the term in the second case of (4.50) is monotonically decreasing in q, but the first
term may not be so. Thus, to find a lower bound on q∗i , we differentiate the first term with respect
to q and set it to zero yielding the solution
q− =(2Rs − 2βRs − 2β
cRs)2Rs
(γ − 2Rs)(2Rs − 2βRs − 2βcRs)− (1 +Grd + γ)2Rs(4.51)
Once again, an analysis of the slope of the first term in (4.50) indicates that it is decreasing in q for
61
all q < qmin. Thus, we conclude that the optimum quantization noise variance satisfies the lower
bound q∗i > max(
1Grd
,min (qmin, q−))
.
In short, the optimum quantization noise variance always lie in the following intervals
q∗0 ∈(ql , qmax
)(4.52)
q∗1 ∈(ql , max
(qmax, q
+) ), (4.53)
with the lower bound for both cases being ql = max(
1Grd
,min (qmin, q−))
.
Average SNR γ (dB)
4 5 6 7 8 9 10 11 12 13 14
Pro
babili
ty o
f decodin
g a
t re
lay
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Pr(ǫ2)
Pr(ǫ1)
Pr(ǫ0)
Pro
babili
ty o
f outa
ge a
t B
S
10-3
10-2
10-1
100
NNC-only
DF-only
DF-or-NNC
J-NNC-DF
100100100
Figure 4.3: Average probability of outage versus γ which is assumed to be the same on all links.Bar chart in the background depicts the average probability of the relay successfully decoding usermessages. The sum-rate Rs = 4 b/s/Hz and fairness parameter β = 0.5.
4.5 Simulation Results
In this section, we present outage performance results of the proposed J-NNC-DF strategy for the
two-user case. As mentioned earlier, we use the benchmarks (a) NNC-only [39], (b) DF-only, in
62
which the relay cooperates only when all users are successfully decoded, and (c) the DF-or -NNC
scheme of [30] in which the relay performs DF when all messages are decoded, otherwise it falls
back to NNC relaying.
In Fig. 4.3, we present the outage probability performance of the proposed J-NNC-DF scheme
along with those of the benchmarks for a fixed sum-rate of Rs = 4 bits/seconds/Hz (b/s/Hz) and a
fairness parameter β = 0.5. In addition, we assume that all links experience the same average SNR
γ. Moreover, the curves correspond to the outage probability that has been averaged over local CSI
realizations. The curves have been evaluated after numerically optimizing over the quantization
noise variance q (within the bounds derived in Section 4.4.4) for each realization of the local CSI.
As can be seen from the figure, at a probability of outage of 10−2, the average SNR gain achieved by
deploying a DF-only relaying scheme in place of NNC-only is approximately 0.4 dB. On the other
hand, DF-or -NNC achieves an SNR gain of approximately 0.95 dB over a DF-only approach. As
expected from the theoretical analysis in Section 4.4, the proposed J-NNC-DF scheme outperforms
all benchmarks, with an SNR gain of approximately 0.75 dB compared to the DF-or -NNC scheme.
In addition to the overall probability of outage at the BS, the bar-chart in the background of Fig.
4.3 depicts the average probability (averaged over the local CSI realizations) with which the relay
successfully decodes user messages. In particular, the heights of the fattest cyan bars, the medium
green bars, and the slimmest blue bars indicate the average probability with which the relay decodes
both users, one user, and none of the users, respectively.
In order to gauge the performance analysis at different sum-rates, we also consider outage rate
as a performance metric. More specifically, the outage rate is defined as the maximum transmission
sum-rate (under a fixed β) which ensures that the probability of outage does not exceed a threshold
p∗. Mathematically, (for given realizations of the local CSI) it is given as
Ro(p∗) , max
{Rs
∣∣∣po (Rs) ≤ p∗}. (4.54)
The average outage rate (averaged over local CSI realizations) versus average SNR γ (once again
assumed to be the same on all links) at p∗ = 10−2 and β = 0.5 is depicted in Fig. 4.4. At an
outage rate of 4 b/s/Hz, it can be verified that the SNR gaps between different benchmarks are, as
63
Average SNR γ (dB)
5 6 7 8 9 10 11 12 13
Ave
rag
e O
uta
ge
Ra
te (
bits/H
z)
2
2.5
3
3.5
4
4.5
5
J-NNC-DF
DF-or-NNC
DF-only
NNC-only
Figure 4.4: Average outage rate versus γ for p∗ = 10−2 and β = 0.5.
they should be, exactly the same as the ones we observed in Fig. 4.3 at outage probability of 10−2.
We observe that the average outage rate gain between J-NNC-DF and DF-or -NNC decreases with
increasing SNR. For instance, the gain decreases from 1.4 dB at an SNR of 6.4 dB down to a gap of
0.2 dB at an SNR of 13 dB. This decrease in gain is due to the fact that the relay is more likely to
decode both users at the relay as the SNR increases, thus causing the performance of DF-or -NNC
to be closer to that of J-NNC-DF.
While Figs. 4.3–4.4 illustrate the performance of the proposed J-NNC-DF scheme for the
symmetric SNR scenario, in Fig. 4.5, we illustrate the performance benefit of the proposed strategy
for the asymmetric SNR case. In particular, assuming γ1d = γ2d, and γ1r = γ2r = γrd, we depict
the average outage rate gain of J-NNC-DF over other benchmarks at p∗ = 10−2 and β = 0.5.
We observe that J-NNC-DF achieves a maximum outage rate gain of 1 b/s/Hz over NNC-only
in the vicinity of γ1d = γ2d = 1 dB, and γ1r = γ2r = γrd = 4 dB. This highlights a visible and
intuitive trend that higher gains are achieved when the relay links are stronger compared to the
direct links. Similar trends are visible for performance gain over DF-only and DF-or -NNC schemes
with respective maximas of 0.8 b/s/Hz and 0.4 b/s/Hz.
64
Figure 4.5: Average outage-rate gain of J-NNC-DF over benchmarks at p∗ = 10−2 and β = 0.5.
4.6 Conclusions
We have considered uplink NOMA in the presence of a single dedicated relay. For such a NOMAR,
we propose a Joint-NNC-DF relaying scheme that utilizes joint DF and NNC encoding at the relay.
We carry out information theoretical analysis and derive conditions required for successful decoding
at the BS. Using this analysis, we derive closed-form expressions for the probability of outage under
the assumption of availability of local CSI at the relay. These expressions help us in appropriately
choosing the quantization noise variance so as to achieve the best outage performance. We also
analytically show that the proposed scheme is guaranteed to outperform existing benchmarks in
the literature. Simulation results confirm our analysis, indicating SNR gains of up to 2.1 dB, 1.7
dB, 0.75 dB over NNC-only, DF-only, and DF-or -NNC, respectively, while the respective maximum
average outage rate gains are observed to be up to 1 b/s/Hz, 0.8 b/s/Hz, and 0.4 b/s/Hz.
65
Chapter 5
Layered Multiplexed-Coded Relaying
for Wireless Multicast
In this chapter, we propose a layered multiplexed-coded decode-and-forward (LMDF) relaying
scheme for a wireless multicast network. Splitting the message into two layers, the source encodes
them separately and maps them to a quadrature amplitude modulation (QAM) constellation to
achieve superposition, thus allowing partial message recovery at the relay and the destinations. In
addition to the layering at the source, a key component of LMDF is multiplexed coding at the relay
that enables each destination to divert all channel resources towards decoding the layer(s) that
remains unrecoverable from the source’s transmission. We derive information theoretic achievable
rates for LMDF and consider network outage as a performance measure. For a setup with 15
users uniformly spread around the relay, simulations indicate that LMDF outperforms conventional
unlayered DF by 1.7 dB and superposition coding-based DF by 1.1 dB, while the respective gains in
outage rate at individual users are up to 48.6% and 31.4%. A practical design and implementation
of LMDF relaying strategy for a wide-band wireless multicast network is also presented. LMDF
is implemented using convolutional codes, with the relays code optimized specifically keeping its
multiplexed nature in mind. We conduct over-the-air experiments, utilizing National Instruments
USRPs 2921, in an indoor office environment using an OFDM-based physical layer, and illustrate
the considerable performance benefits of LMDF over benchmarks such as conventional two-hop DF.
66
5.1 Introduction
It is well known that a decoding failure at the relay becomes a performance bottleneck in decode-
forward (DF) cooperation [24]. For mitigating this bottleneck, layering coupled with superposition
coding (SC) can be utilized to enable partial message recovery at the relay [79, 46, 80, 81, 82].
Most existing works in this domain focus on the optimization of rate and/or power allocation while
assuming availability of global channel state information (CSI). For instance, [79] employs two-
level SC, optimizing both rate and power allocation to increase throughput whereas [46] uses SC
to improve spectral efficiency based on known CSI. Similarly, [80] improves distortion exponent
through SC of source layers. The work in [81] evaluates the achievable rates for half duplex multi-
level source transmission, whereas [82] optimizes the expected user satisfaction through power
allocation of the SC layers.
In this chapter, we propose layered multiplexed-coded decode-and-forward (LMDF) relaying
for a wireless multicast network in which all transmissions are mapped to quadrature amplitude
modulation (QAM) constellations. QAM is well suited from practical implementation prespective
where higher-order QAM constellation can be thought of as superposition of two lower-order
QAMs. For instance, a 16-QAM constellation can be thought of as a superposition of two 4-
QAM constellations, one of which has twice the amplitude of the other. The message at the source
is partitioned into two layers, each of which is encoded with an independent codebook. Exploiting
the inherent unequal power (UEP) allocation accorded to the bit-planes of a naturally mapped
QAM constellation, LMDF achieves superposition through a simple mapping operation. This
enables partial message recovery at the relay, enabling cooperation even if the complete message
is not decoded at the relay. The majority of existing works on layered cooperation consider the
three-node orthogonal DF network, while assuming ideal feedback channel and/or the availability
of global CSI. At the same time, the prevalent strategy is to utilize SC at the relay as well. In
this work, we consider a multicast network under a practical scenario in which the source and
relay have no knowledge of the instantaneous CSI. Under such a model, superposition coding at
the relay causes wastage of the power allocated to layers already decoded at the destination. In
order to avoid this power penalty, we propose a multiplexed-coded [40] approach. Under a single
67
broadcast from the relay, multiplexed coding enables each destination to utilize all channel resources
solely towards decoding the layers that remained unrecoverable from the source’s transmission. We
derive the information theoretic rates for LMDF, specifically with QAM transmissions from both
the source and the relay. Simulations results in a Rayleigh fading environment with 15 users
indicate performance gains of up to 1.7 and 1.1 dB over conventional unlayered DF (UDF) and
superposition-coded layered DF (SCDF), respectively. In addition, the outage rates with LMDF at
individual users exhibit a gain of up to 48.6% and 31.4% compared to UDF and SCDF, respectively.
The remainder of this chapter is organized as follows. Section 5.2 describes the system model
after building some preliminary background knowledge. The proposed LMDF scheme along with the
associated achievable rate-region is presented in Section-5.3. Outage performance evaluations and
the corresponding simulation results are presented in Section-5.4. Section 5.5 describes practical
implementation of LMDF while Section-5.6 concludes this chapter.
5.2 System Model and Preliminaries
We consider a multicast network consisting of a single source transmitting the same information to
N destinations through a single relay. The relay is assumed to operate in the half-duplex mode with
the overall transmission divided into two time-slots; the durations of the first and second time-slots
are fractions γ ∈ (0, 1) and γc , 1 − γ of the total transmit duration, respectively. During the
first time-slot, only the source transmits, with the signals received at the relay and destination
d ∈ {1, . . . , N} given as
Yr = hsrXs + Zr, and Yd1 = hsdXs + Zd1,
respectively. Here Xs is the source’s QAM transmission with E[|Xs|2
]= Ps, hsr and hsd are
the channel coefficients on the source-relay and the source-destination links, respectively, whereas
Zr and Zd1 ∼ CN (0, 1) are complex circularly symmetric Gaussian noise, that without loss of
generality are assumed to be of unit variance. During the second time slot, both source and relay
transmit, with the signal at destination d represented by Yd2 = hsdXs+hrdXr+Zd2, where Xr is the
relay’s QAM transmission with E[|Xr|2
]= Pr, hrd is the channel coefficient on the relay-destination
68
link, and Zd2 ∼ CN (0, 1). The channel gains on all links are assumed to be circularly symmetric
Gaussian, changing independently from one transmission block to another. In addition, the channel
gains are assumed to follow a simple path-loss model, normalized by the source-relay link. Under
this model, we have dsr = 1, E[|hsr|2
]= 1, E
[|hrd|2
]= (drd)
−ρ, and E[|hsd|2
]= (dsd)
−ρ. Here
dsd and drd are the respective distances normalized with dsr and ρ is the path-loss coefficient. We
assume that the transmitters have no knowledge of the instantaneous CSI, but are aware of its
statistics. In order to describe the information theoretic achievable rates, we define the following
notation. If Y = Γ1X1 + Γ2X2 + Z with Z ∼ CN (0, 1), X1, X2 both drawn from unit-energy M -
QAM constellations, and Γ1, Γ2 the complex channel coefficients, we denote the channel’s sum-rate
as
CS (Γ1,Γ2) , I(X1, X2;Y ) = h(Y )− h(Z) (5.1)
= −∞∫−∞
f(y) log2 f (y) dy − log2(πe)
where f (Y ) = 1M2
M∑i=1
M∑j=1
1πe|y−Γ1xi1−Γ2xj2|2 is the probability density function of the complex-valued
received signal, {xi1}, {xj2} are the constellation points of the unit-energy M -QAM constellations,
and the integral is over the entire complex plane. Similarly, if Y = ΓX + Z with Z ∼ CN (0, 1), X
drawn from an unit energy M -QAM constellation, and Γ the complex channel coefficient, we will
denote the constrained capacity of this channel as
CM (Γ) , I(X;Y ), (5.2)
which can be obtained by substituting Γ1 = Γ and Γ2 = 0 in (5.1). In addition, we denote the
effective channels on the three links by Γsr , hsr√Ps, Γsd , hsd
√Ps, Γrd , hrd
√Pr, the set of
channels by Γd , {Γsd,Γsr,Γrd}, and the average magnitudes by Γd = E [{|Γsd|, |Γsr|, |Γrd|}].
69
5.3 Proposed LMDF Relaying
For the proposed LMDF scheme, the source message of k bits is split into two parts of lengths βk
and βck bits, with β ∈ (0, 1) and βc , 1− β. The two parts are encoded independently to produce
equal-length codewords. Thus if the overall encoding rate is R bits/symbol (b/s), the rates for the
two layers are R1 = βR and R2 = βcR. If Xs1 and Xs2 are the unit-energy coded symbols of the
two layers, the source transmits
Xs =√αPsXs1 +
√αcPsXs2, (5.3)
where α ∈ (0, 1) is the power allocation parameter and αc , 1−α. Thus the unequal protection is
provided in two ways. Through unequal rate allocation, characterized the parameter β, and through
unequal power allocation which is characterized the parameter α. Keeping practical considerations
in mind, we restrict our attention to the case where the transmissions from the source as well as
the relay are constrained to be drawn from M -QAM constellations. In addition, we rely on the
observation that an M -QAM constellation can be broken down into a superposition of an M1-QAM
and an M2-QAM constellation (such that M = M1M2). Thus, we will assume that Xs1 and Xs2 are
drawn from M1-QAM and M2-QAM constellations respectively, with α chosen appropriately so as
to ensure that the superposition results in Xs belonging to the required M -QAM constellation. We
also note that such superposition could be practically induced by a simple bit-mapping operation.
An illustration of this for M = 16 and M1=M2=4 is shown in Fig. 5.1. The two portions of the
original message are encoded separately using a 4-QAM alphabet (each symbol represented by two
bits). If Xs1 and Xs2 are each drawn from a 4-QAM alphabet, it can be easily shown that their
superposition with α = 45 results in a 16-QAM constellation. Moreover, we see that connecting the
encoder outputs of the first layer to the most-significant bits (shown with a boldface blue font in
Fig. 5.1) and the output of the second layer to the two least significant bits, one can induce the
superposition operation of (5.3) with M1 = M2 = 4 and α = 45 . We point out that the idea is
similar to the so-called hierarchial modulation concept proposed for broadcasting applications [83],
which can be applied to other QAM constellations by defining appropriate mapping operations.
70
16
-QA
M M
ap
pe
r
dLdL
Encoder R1
QPSK Mapper
× dL
Encoder R2
QPSK Mapper
×
∑
1000
01 11
00 10
01 11
0000 0010 1000 1010
0001 0011 1001 1011
0100 0110 1100 1110
0101 0111 1101 1111
MSBLSB
Figure 5.1: Layered superposition coding with a 16-QAM constellation.
In the following, we provide a description of LMDF and its achievable rates by separately
discussing the cases where the relay either decodes both layers, one layer, or no source layer.
5.3.1 Two layers decoded at relay
The relay first attempts to decode both source layers. Since the layers are encoded separately, they
can be viewed as independent users transmitting over a multiple access channel (MAC). The relay
is capable of decoding both layers if R1 and R2 lie inside the two-user MAC capacity region [40],
i.e.,
R1 = βR ≤ γI(Xs1;Yr|Xs2) = γCM1
(√αΓsr
)R2 = βcR ≤ γI(Xs2;Yr|Xs1) = γCM2
(√αcΓsr
)(5.4)
R1 +R2 = R ≤ γI(Xs1, Xs2;Yr) = γCM (Γsr)
71
Combining the three constraints, the maximum transmission rate allowed for successful recovery of
the two layers at the relay can be conveniently represented as1
Cr2 ,γmin
(CM1 (
√αΓsr)
β,CM2
(√αcΓsr
)βc
,CM (Γsr)
)(5.5)
Given that the relay decodes both layers, the most straight-forward strategy, as is prevalent in
the literature on layered relaying, is for the relay to re-encode the two layers with independent
codebooks to obtain Xr1 and Xr2, and to use a power allocation factor αr ∈ (0, 1) to obtain a
superposed transmission Xr =√αrPrXr1 +
√αcrPrXr2. Notice that if the destination was able to
recover Layer-1 from the source’s transmission during the first time-slot, a fraction αr of the relay
power is wasted in transmitting information that is already known. In order to avoid this penalty, we
propose a multiplexed-coded approach at the relay. A two-dimensional multiplexed codebook [40]
is constructed by randomly generating 2n(R1+R2) independent and identically distributed length-n
sequences drawn from an M -QAM alphabet. The sequences are then arranged in a table with 2nR1
rows and 2nR2 columns. Given indices m1 ∈ {1, . . . , 2nR1} and m2 ∈ {1, . . . , 2nR2} corresponding to
the two layers, the encoder transmits the length-n codeword residing at row-m1 and column-m2 of
the multiplexed codebook. If a given destination had recovered Layer-1 beforehand, the decoding
operation would involve a search for the jointly typical codeword only over the 2nR2 entries of the
row indexed by the Layer-1 information. Successful decoding will occur if R2 is less than the channel
capacity, thus allowing all channel resources to be diverted towards decoding the unknown layer.
At the same time, had any layer not been known, the decoder searches over the two dimensional
codebook, allowing successful recovery of both layers if R1 +R2 is less than the channel capacity.
Thus, LMDF employs a multiplexed codebook at the relay during the second time-slot. In
addition, the source also transmits simultaneously using a multiplexed codebook.
Theorem 5.3.1. Given that both layers are decoded at the relay, using similar arguments as that
in [40], the constraints that govern correct decoding of both layers at the destination can be given
1For notational ease, we omit the variables (γ, β,Γd) as arguments from the left side of the equality in (5.5), and(5.9)–(5.12).
72
as
βR ≤ γI (Xs1;Yd1|Xs2) + γcI (Xs, Xr;Yd2) (5.6)
βcR ≤ γI (Xs2;Yd1|Xs1) + γcI (Xs, Xr;Yd2) (5.7)
R ≤ γI (Xs1, Xs2;Yd1) + γcI (Xs, Xr;Yd2) (5.8)
Proof. See Appendix C for proof
Had superposition coding been used at the relay (and the source) during the second time-slot,
the analogous constraints to (5.6) and (5.7) would have been
βR ≤ γI (Xs1;Yd1|Xs2) + γcI (Xs1, Xr1;Yd2|Xs2, Xr2)
βcR ≤ γI (Xs2;Yd1|Xs1) + γcI (Xs2, Xr2;Yd2|Xs1, Xr1) ,
respectively, while the sum-constraint (5.8) would have remained the same. One immediately
observes that these rates are strictly smaller than the LMDF counterparts. This is because of the
fact that during the second time-slot, had Layer-1 been decoded already, the power allocated to
Xr1 and Xs1 during the second time-slot would have been wasted. This is opposed to LMDF in
which all transmit power from the source and the relay can be utilized for the recovery of Layer-2
information. Combining constraints (5.6)-(5.8) and writing the mutual information in terms of the
constrained channel capacities defined in Section 5.2, the maximum achievable rate Cd2 (γ, β,Γd) at
destination d under the condition that two layers be decoded at the relay is given in (5.9).
73
Cd2 = min
(γCM1 (
√αΓsd)+γcCS (Γsd,Γrd)
β,γCM2
(√αcΓsd
)+γcCS (Γsd,Γrd)
βc,
γCM (Γsd)+γcCS (Γsd,Γrd),Cr2
)(5.9)
Cd1 = min
(γCM1 (
√αΓsd)+γcCS (Γsd,Γrd)
β,γCM2
(√αcΓsd
)+γcCM (Γsd)
βc,
γCM (Γsd)+γcCS (Γsd,Γrd),Cr1
)(5.10)
Cd0 = min
(γCM1 (
√αΓsd)+γcCM (Γsd)
β,γCM2
(√αcΓsd
)+γcCM (Γsd)
βc, CM (Γsd)
)(5.11)
5.3.2 One layer decoded at relay
In case the relay is unable to decode both layers, it attempts to recover Layer-1 by treating Layer-2
as interference. Relay is able to decode one layer if the transmission rate is less than
C1r,γ
βI(Xs1;Yr)=
γ
β
[CM (Γsd)− CM2
(√αcΓsd
)](5.12)
When only Layer-1 is decoded, LMDF employs a single-user codebook to map its information to a
codeword drawn from a M -ary QAM alphabet. Meanwhile the source switches from superposition
coding to multiplexed coding in the second time slot. Under this scenario, we once again obtain
MAC-type capacity constraints required for successful decoding of both layers. Combining these
constraints with (5.12), the maximum achievable rate Cd1 (γ, β,Γd) under the condition that one
layer is decoded at the relay is given in (5.10).
5.3.3 No layer decoded at relay
In case the relay is unable to successfully decode any layer, the relay remains silent. However, the
source switches to a multiplexed-coded transmission during the second time-slot. Using the same
arguments as before, the maximum achievable rate in this situation can be computed as (5.11).
Lemma 5.3.2.
74
For given parameters γ, β, Γd, the destination is capable of successfully recovering both layers
if the transmission rate R is no greater than
CLMDF (γ, β,Γd) = max
(Cd2 , C
d1 , C
d0
). (5.13)
Proof. See Appendix D for proof
5.4 Simulations Results
For performance comparisons, we consider the network outage behavior. The outage probability
at destination d for a transmission rate of R b/s is defined as
Pd(γ, β,Γd, R) = Pr (CLMDF (γ, β,Γd) < R) ,
whereas we define the network outage probability as the probability of outage averaged over the N
users, i.e.,
P (γ, β,Γ, R) =1
N
N∑d=1
Pd(γ, β,Γd, R), (5.14)
where we have used the notation Γ , {Γ1, . . . ,ΓN}. In addition, we also evaluate network outage
rate, defined as the maximum transmission rate such that the network outage probability does not
exceed p. Mathematically,
R(γ, β,Γ, p) = max{R∣∣P (γ, β,Γ, R) ≤ p}. (5.15)
For performance evaluation, we consider the two dimensional network topology with N = 15
users shown in Fig. 5.2. The source is located at the origin (not shown in the figure) with the
relay situated unit distance away at an (X,Y ) location of (1, 0). Due to the geometrical symmetry
around the X-axis, we consider the users to be spread only in the positive Y direction. In addition,
75
Figure 5.2: Network topology as well as the benefit of LMDF over SCDF and UDF at each user.
we set ρ = 3 which is representative of an urban micro-cell environment. In the simulations for
LMDF as well as SCDF, we set M1 = M2 = 4. Through a simple software mapping shown in Fig.
5.1, we achieve superposition with α = 45 to produce 16-QAM transmissions for both cases. For a
fixed transmission power of P = Ps = Pr = 6.3 dB (for which the optimized network outage rate
of LMDF at p = 10−2 is 1.5 b/s), we compute the outage rate at each individual user for LMDF
as well as the benchmarks SCDF and UDF. The transmissions for UDF are also assumed to be
drawn from a 16-QAM constellation. At any given user, the figure illustrates the percentage gain in
outage rate of LMDF over the benchmarks, with a maximum gain of 31.4 % and 48.6% over SCDF
and UDF, respectively. The optimal β for both LMDF and SCDF is 0.625, whereas the optimal γ
for LMDF, SCDF, and UDF is 0.675, 0.65, and 0.625, respectively.
76
5 6 7 8 9 1010
−3
10−2
10−1
Ave
rage
Net
wor
k O
utag
e P
roba
bilit
y
5 6 7 8 9 100
0.5
1
1.5
2
2.5
3
Ave
rage
Net
wor
k O
utag
e R
ate
Transmission Power (dB)
LMDFSCDFUDF
Figure 5.3: Network outage probability (for R = 1.5 b/s) and the network outage rate (for p = 10−2)versus the transmission power.
In Fig. 5.3, we show the network outage rate versus the transmission power for p = 10−2.
Each curve is obtained by numerically searching for the β and γ that maximizes (5.15). It can be
seen that at an outage rate of 1.5 b/s, LMDF outperforms SCDF and UDF by 1.1 dB and 1.7 dB,
respectively. Similarly, at an outage rate of 2.0 b/s, the performance gains of LMDF over SCDF and
UDF are 1.0 dB and 1.9 dB, respectively. We point out that these performance gains are observed
after having independently optimized the SCDF and UDF curves over the parameters β and γ. On
the same figure, we also provide a comparison of outage probability versus the transmission power
for R = 1.5 b/s. These curves were also obtained after optimizing (5.14) over β and γ.
5.5 Practical Implementations of LMDF
In this section, we present a practical implementation of the proposed LMDF scheme. Discussion
focuses on the design and implementation of a practical LMDF relaying strategy [84] for a wide-band
wireless multicast network. A system-level implementation of LMDF catering for imperfections of
77
real-world RF equipment such as carrier and timing offsets is presented. The relaying takes place
over two orthogonal time-slots of equal duration. During the first time-slot, denoted by T1, only
the source transmits. During the second time-slot, denoted by T2, the relay transmits while the
source stays silent.
5.5.1 Related Work
The work closest to practical implementation of LMDF scheme is the so-called network modulation
strategy of [85], in which a software mapping scheme for achieving UEP was developed for QAM
constellations. However, the strategy in [85] requires channel feedback from participating nodes,
based on which a constellation is chosen to service only one particular user. On the other hand,
LMDF is capable of simultaneously servicing many diverse destinations without requiring global
CSI. Finally, although experimental prototyping of relaying has been carried out before [86, 87, 88,
89], to the best of our knowledge, this work is the first to experimentally evaluate the benefit of
layering in a cooperative multicast environment.
5.5.2 Layering and superposition coding at the source
As discussed in LMDF source transmission are layered in nature. Layering is accomplished by
splitting the message sequence into multiple portions and encoding them with independent error
correction codes. The UEP can be provided through unequal rate and/or unequal power allocation
for different layers. We consider a bi-layer strategy in which UEP is achieved through the latter.
Thus, the message m is split into two equal portions m1 and m2, each of which is encoded with
an independent rate-R binary code to generate codeword sequences c1 and c2, as shown in Fig.
5.4. Next, each of the codewords are superposition coded. In order to avoid significant changes
required for SC, we propose a methodology, subsequently referred to as superposition mapping
(SM), that is capable of inducing superposition, albeit for a finite set of power allocations, through a
simple bit-mapping operation. Given that the original M -PAM constellation uses natural mapping,
superposition with a required α can simply be achieved by appropriately connecting c1 and c2 to
the log2M bits of the M -PAM mapper. As an example, the respective connections for each α ∈ A8
78
MSB Layer m2
Superposition
Mapper
CRC LSB Layer m1 CRC
Conv CodeΠConv Code Π
OFDM
Modulator
High Protected
Layer
Low Protected
Layer
C2 C1
Figure 5.4: Block diagram of the operations at the source.
α = 1/37
α = 1/17
α = 4/29
α = 1/5
α = 4/13
α = 9/25
High Protected
Layer
Low Protected
Layer
Mapper
8 PAM
Natural
Symbol
Mapping
MSB
LSB
High Protected
Layer
Low Protected
Layer
Mapper
8 PAM
Natural
Symbol
Mapping
MSB
LSB
High Protected
Layer
Low Protected
Layer
Mapper
8 PAM
Natural
Symbol
Mapping
MSB
LSB
High Protected
Layer
Low Protected
Layer
Mapper
8 PAM
Natural
Symbol
Mapping
MSB
LSB
High Protected
Layer
Low Protected
Layer
Mapper
8 PAM
Natural
Symbol
Mapping
MSB
LSB
High Protected
Layer
Low Protected
Layer
Mapper
8 PAM
Natural
Symbol
Mapping
MSB
LSB
Transmitted Symbols Unused Symbols NOT Operation
000 001 010 011 100 101 110 111
000 001 010 011 100 101 110 111
000 001 010 011 100 101 110 111
000 001 010 011 100 101 110 111 000 001 010 011 100 101 110 111
000 001 010 011 100 101 110 111
Figure 5.5: An illustration of how SM achieves SC with any α ∈ A8 through simple bit-connectionsto a natural symbol mapper.
are illustrated in Fig. 5.5, where
A8 =
{1
37,
1
17,
4
29,1
5,
4
13,
9
25
}, (5.16)
The resulting symbol stream from SM, as shown in Fig. 5.4, is then fed to the OFDM modulator
before being broadcast over the air to the relay as well as to the destinations.
5.5.3 Decoding and multiplexed coding at the relay
Keeping the required low computational complexity in mind, we use a suboptimum but an implementation-
friendly approach that relies on parallel recovery of the layers as shown in Fig. 5.6. Given the
received symbols, hard-decisions are formed on each sub-carrier to generate bit-streams c1 and c2
79
that are then serially collected, de-interleaved, and fed into the layers’ individual binary decoders.
At the relay, LMDF utilizes a multiplexed-coded approach that, with a single transmission, caters
MD Symbol
DecodingConv
Decoder
Conv
Decoder
OFDM
Demodulator
CRC Check
CRC Check
Multiplexed
Coding &
Transmission
Õ-1
Õ-1
Figure 5.6: Block diagram of the operations at the relay.
for the disparity in the number of layers decoded at different destinations. A two-layer multiplexed
codebook is essentially arranged in a two-dimensional table with the rows indexed by m1 and
the columns by m2. If the destination has not been able to decode any layer from the source’s
transmission, it searches the two-dimensional codebook for the appropriate codeword. On the other
hand, if the destination had been able to decode m2, it searches for potential codewords only in
the column indexed by m2. Thus, prior knowledge of m2 reduces the search space, aiding recovery
of m1. In other words, multiplexed coding will induce all resources on the relay-destination link to
be allocated to m1 without the relay ever knowing that the destination had already decoded m2.
The biggest advantage of the proposed approach is in a multicast setting. Given the heterogeneous
channel conditions, the destinations could have recovered a disparate number of layers during T1.
Multiplexed coding in this scenario serves as a universal transmission from the relay that not only
services those destinations that have no prior knowledge of any layer, but also those that were able
to recover m2 during T1.
5.5.4 LMDF System Design
In order to evaluate the feasibility of the proposed LMDF scheme under practical constraints such
as noisy channel estimates, carrier frequency offsets (CFO), imperfect timing estimation etc., we
develop a system-level implementation using an IEEE 802.11-type baseline OFDM modem. As the
block diagram in Fig. 5.4 depicts, a message within a given packet is split into two equal portions
80
with a separate cyclic redundancy check (CRC) appended to each layer. Each layer (along with
the CRC bits) is encoded with an identical off-the-shelf (n, k) convolutional code. The encoded
bit-streams are interleaved before using SM to obtain symbols drawn from a subset of an M2-QAM
constellation. The practical choice of α∗ for SM is based on a feedback from the relay. The resulting
symbols are interleaved and fed into the OFDM modulator, the output of which is broadcast to the
relay and the destinations. At the relay, the received signal is processed by the OFDM demodulator
that performs the required synchronization, channel estimation, and equalization operations. The
demodulator output symbols are fed into an MD symbol decoder, yielding estimated bit-streams c1
and c2. These are de-interleaved and decoded independently using standard hard-decision Viterbi
decoding. The CRC bits help identifying whether a given layer was decoded successfully. If the relay
successfully recovers both layers, it re-encodes the data using a convolutional code, the generator
polynomials of which are specially designed to account for its multiplexed nature. The encoded bit-
stream from this multiplexed code is interleaved, mapped to a regular 16-QAM constellation, and
transmitted to the destinations through the baseline OFDM modulator. As for the destinations,
the decoding during T1 is identical to that at the relay. At the end of T1, the destinations only
save the layers that were decoded successfully while discarding everything else. During T2, the
signal received from the relay is processed with the OFDM 16-QAM demodulator to obtain a
single bit-stream. This bit-stream along with any layer decoded during T1 is fed into a multiplexed
hard-decision Viterbi decoder.
In the following, we provide a brief description of the baseline OFDM modem followed by a
discussion on multiplexed encoding and decoding using convolutional codes.
Baseline OFDM modem
We only provide a brief overview of the modem operations. The total bandwidth is divided into
N = 64 sub-carriers, leaving the DC as well as the six and five sub-carriers at each end empty. From
the remaining 52 sub-carriers, four carry the predefined pilot symbols while 48 carry data. This
is followed by a standard inverse Fast Fourier Transform (IFFT) operation to achieve modulation
onto orthogonal frequency bands. In addition, the payload of OFDM symbols are preceded by
81
ten ‘Short’ and two ‘Long’ preamble OFDM symbols that are used for frame synchronization,
channel estimation, and CFO compensation at the receiver. At the receiver, the start-of-packet
is detected through an autocorrelation-based approach [90] which also provides an estimate of
the CFO between the transmitter and the receiver. The long preamble sequences are then used
to estimate the channel, the effect of which is then equalized. The cross-correlation of the long
preambles are also used to refine the estimate of the start-of-packet position. Moreover, the pilot
symbols embedded in the data frames are used to estimate any residual CFO or a change in channel
phase and amplitude that may occur during the transmission of the payload [91].
Multiplexed Coding through Convolutional Codes
In this section, we describe the multiplexed encoding at the relay using an (n, k) convolutional code
followed by details of the code design as well as a description of the decoding methodology at the
destination. Since the convolutional code has k distinct inputs, the relay splits a given layer into
k equal portions. If mi,l = [mi,l(1), . . . ,mi,l(L)], l = 1, . . . , k are the k portions corresponding to
Layer-i, i = 1, 2, their equivalent polynomial representations are
mi,l(D) = mi,l(1) +mi,l(2)D + . . .+mi,l(L)DL−1. (5.17)
For multiplexed encoding, the information bit-sequences corresponding to the two layers are
interleaved such that the bits from m1 and m2 are placed at even and odd indices, respectively. It
is this even-odd interleaving that induces the multiplexed nature in the encoding process. Indeed,
the set of codeword sequences of the convolutional code can be rearranged to form a two-dimensional
(square) table with the rows indexed by m1 and the columns by m2. If the two-dimensional code
corresponds to a rate- kn code, a single column indexed by the known information m2 corresponds
to a rate- k2n code, which under a good code design should have a better threshold than the original
two-dimensional structure.
Multiplexed Code Design: The polynomial representation of the even-odd interleaved sequence
82
on input l of the convolutional encoder is given by
ml(D) = m1,l
(D2)
+Dm2,l
(D2). (5.18)
If gl,j(D) represents the generator polynomial from input l to output j (j = 1, . . . , n), the polynomial
corresponding to output sequence j is given by
xj(D) =
k∑l=1
ml(D)gl,j(D)
=
k∑l=1
m1,l(D2)gl,j(D) +D
k∑l=1
m2,l(D2)gl,j(D) (5.19)
where all algebraic operations are over the binary field. The generator polynomials are assumed to
be of rational form, i.e., gl,j(D) =gfl,j(D)
gbl (D), with gfl,j(D) and gbl (D) being the feed-forward and the
feed-back polynomial, respectively, and gl,j(D) being the corresponding series expansion. The coded
bit-streams corresponding to x1(D), . . . , xn(D) are interleaved and mapped to 16-QAM symbols
before onward transmission to the destinations, as described earlier. The code described by (5.19)
will be referred to as the mother code in subsequent discussions.
The objective of the code design is to search for a structure that achieves good performance
not only for the case when the destination tries to recover both layers, but also for the case
when it attempts to recover m1 using m2 as side information. For both cases, we use the free
distance of the convolution code as a performance metric. It is clear that in the absence of any side
information, recovery of m1 and m2 corresponds to standard Viterbi decoding on the mother code’s
trellis. Good generator polynomials under this conventional setup are the ones that maximize the
free distance df,m of the code, and can be found through an exhaustive search [92]. Optimized
generator polynomials for this conventional setup can be found in Table 12.1 of [92].
We next look at the case when the destination attempts to recover m1 in the presence of m2 as
known side information. We observe that for m2,l(D) 6= 0, the set of codeword sequences defined
by (5.19) forms a coset of the linear binary code corresponding to m2,l(D) = 0. It is known that
the set of Hamming distances between pairs of codewords in all cosets remains the same, and is
83
equivalent to the weight profile of the linear code corresponding to m2,l(D) = 0, which we will refer
to as the induced code. Thus, maximizing the free distance of the cosets is equivalent to maximizing
the free distance of the induced code. For the induced code, we observe that every k input bits
(of m1) produce 2n coded bits; n coded bits at the even time index corresponding to m1 and n
bits at the odd time index corresponding to the zeroed sequence m2 (which however do depend on
m1 due to the encoder memory). As a result, given the side information, the rate of the induced
code is halved. Let xj(D) = x(0)j
(D2)
+ Dx(1)j
(D2), with x
(0)j (D) and x
(1)j (D) representing the
polynomials corresponding to the coded bits at the even and odd indices, respectively. Let gf(0)l,j (D),
gf(1)l,j (D), g
b(0)l (D), and g
b(1)l (D) be similarly defined.
The 2n coded bit-streams of the induced code are given by (j = 1, . . . , n, i = 0, 1)
x(i)j (D) =
k∑l=1
m1,l(D)g(i)l,j (D) (5.20)
with
g(0)l,j (D) =
gf(0)l,j (D)g
b(0)l (D) +Dg
f(1)l,j (D)g
b(1)l (D)
gbl (D)(5.21)
g(1)l,j (D) =
gf(0)l,j (D)g
b(1)l (D) + g
f(1)l,j (D)g
b(0)l (D)
gbl (D)(5.22)
For the code design, we carry out an exhaustive search over the polynomials gfl,j(D) and gbl (D)
without constraining gfl,l(D) to be equal to gbl (D). If df,m and df,i represent the free distance of the
mother code and the induced code, respectively, we identify a subset of polynomials whose df,m is
the same as that corresponding to the codes optimized under a conventional setup. Among this
subset, we pick the polynomials that maximize df,i. The motivation behind this approach is an
attempt to optimize the induced code under the constraint that the performance (in the sense of
free distance) at any user is no worse than benchmark schemes. It can be seen that the optimized
polynomials can achieve a significantly higher df,i than that achieved with the conventional codes.
84
5.5.5 Performance Evaluation
For performance evaluation of the proposed practical LMDF system, we not only rely on simulations
but also utilize a prototype implementation using SDRs. While not discounting the importance
of simulations, the purpose of this prototyping is to experimentally demonstrate the performance
benefits of LMDF in real radio environment without relying on simplistic models or assumptions
that are typically employed in simulations.
Benchmark scheme considered for performance evaluation is the two-hop decode-and-forward
(THDF). In THDF, the transmission is split into two orthogonal timeslots. The source transmits
in the first and the relay tries to decode the source message. Then in the next timeslot relay
transmits the decoded message. After the end of second timeslot the destination attempts to
recover the source message solely from the relay’s transmissions.
Simulation Results
In this section, we present the performance of LMDF with convolutional codes and the baseline
OFDM modem described in Section 5.5.4. In Fig. 5.7, we compare the frame-error rate (FER) of
LMDF versus that of THDF at a particular destination in a multicast scenario. At the source, we
utilize 64-QAM as the mother constellation for SM, and a standard 16-state R = 12 convolutional
code. At the relay, we utilize a 16-state R = 12 multiplexed convolutional code. Moreover, we
assume that the source and the relay utilize the same transmit power. In order to estimate the
effect of imperfect synchronization and channel estimation, we also include the FER curves for
simulations with these offsets. We observe that under practical synchronization and estimation,
and a FER of 2%, LMDF can provide a gain of more than 3.3 dB over THDF.
Experimental Setup and Evaluation
For experimental prototyping and performance evaluation, we utilize the USRP 2921 from National
Instruments [93] interfaced with Mathworks Simulink. All transmissions from the source as well
as the relay nodes utilize a bandwidth of 10 MHz centered at a carrier frequency of 5.5 GHz;
with a cyclic prefix of 20% and the usage of 48 out of 64 sub-carriers for data transmission, this
85
Figure 5.7: Simulated FER of LMDF versus THDF as a function of P = Ps = Pr at a particulardestination.
amounts to a symbol rate of 6×106 symbols per second. For performance evaluation in a multicast
setting, we utilize a total of five USRP devices; the source, relay, and three destinations placed in
an indoor office environment. The relative channel conditions are shown in Fig. 5.8 which lists the
average SNR measured on the individual links (averaged across the sub-carriers and across multiple
measurements) at a USRP transmit gain of 14 dB (from both the source and the relay). The figure
also depicts scatter plots (after synchronization and one-tap channel equalization) of the received
constellations corresponding to an arbitrary packet transmission on the links. It can be seen that at
the given transmit gain, the SNR on the source-relay link is 29.3 dB; strong enough for the source
to use SM with α = 137 . This can be seen by the relatively clean scatter plot at the relay in which
the 16 constellation points corresponding to α = 137 are received as distinct clusters. On the other
hand, the SNR on the Source-Destination 3 (S-D3) link is 14.8 dB which allows D3 to successfully
recover m2 with a high probability, while recovery of m1 fails most of the time. This is also visible
in the scatter plot, in which the four points within a given quadrant are not distinguishable whereas
86
Source
D2
D1
D3
29.3 dB
14.8 dB
14.8 dB
8.4 dB
T1 T2 T1 T2
T1 T2
16 dB
19.8 dB
15.7 dB
Relay
Figure 5.8: Experimental setup with five USRPs placed in an indoor office environment. Alongsideeach link, the average measured SNR is listed when the source and the relay USRP use a transmitgain of 14 dB. Also shown at the individual nodes are the scatter plots of the received constellationsduring T1 and T2.
the points in the four quadrants appear as distinct clusters. The same holds true on the S-D1 link
which has the same average SNR of 14.8 dB. The difference between the two destinations lies during
T2 in which the R-D3 link has an SNR of 19.8 dB, while R-D1 has a lower SNR of 16 dB. Thus, one
should expect LMDF to outperform THDF at D1 by a considerable margin since the knowledge of
m2 during T1 will aid the decoding process during T2, and thus compensate for the relatively low
SNR on the R-D1 link. On the other hand, one expects the margin to be smaller at D3 since the
R-D3 link is strong enough for THDF to recover both layers solely from the relay’s transmission
most of the time. In contrast to D1 and D3, the SNR on the S-D2 link is only 8.4 dB, due to which
even the decoding of m2 fails most of the time. Thus, for D2, we can expect the performance of
LMDF to be more or less the same as that with THDF.
For the setup described above, Fig. 5.9 shows the FER at the three destinations as a function
of the received SNR at the relay. The SNR at the relay is varied by changing the transmit gain
of the source, which in turn also causes SNR variation at the destinations. The transmit gain at
87
the relay is varied by exactly the same amount as that at the source, due to which the relative
channel conditions on all links remain approximately the same as those specified in Fig. 5.8. As
expected, the biggest advantage of LMDF over THDF is at D1, requiring approximately 3 dB less
SNR to achieve an FER of 2%. One also observes a gain at D3 of approximately 1.5 dB, which is
not as significant as that at D1 due to the relatively stronger relay-destination link. On the other
hand, due to the weak source-destination link, the performance of LMDF at D2 is approximately
the same but no worse than THDF.
18 20 22 24 26 2810
−2
10−1
100
Receive SNR at Relay (dB)
Fra
me E
rror
Rate
(F
ER
)
Two−Hop D1
Two−Hop D2
Two−Hop D3
LMDF D1
LMDF D2
LMDF D3
Figure 5.9: Comparison of experimental FER results of LMDF and THDF at the three destinationsof Fig. 5.8.
5.6 Conclusions
We propose LMDF relaying with QAM transmissions for a multicast network. The two main
components of the scheme are layering at the source and multiplexed coding at the relay. While
layering at the source allows partial message recovery at the relay, multiplexed coding allows all
88
channel resources to be utilized for decoding the layer(s) not recoverable through the source’s
transmission. Considering network outage as a performance measure, LMDF exhibits considerable
gains over conventional UDF as well as SCDF.
We also present a system-level implementation using an OFDM-based modem, and convolutional
codes that are specifically optimized to cater for their multiplexed nature. In addition to simulating
the system design over IEEE 802.11 TGn indoor channel models, we also conduct experimental
performance evaluations using SDRs. Both simulations and experimental results indicate considerable
performance benefits of LMDF over the conventional THDF strategy.
89
Chapter 6
Conclusions and Future Directions
In this chapter, we conclude by highlighting the main contributions of the thesis. The objective of
the thesis is to develop novel relaying strategies for a multiple access and a multicast network. In
this context the first setup we consider is the two-user uplink NOMARC. We consider an NNC-
only relaying scheme and provide the associated information theoretic achievable rate region for the
two-user NOMARC. Based on this rate region we derive closed-form expressions for probability of
outage under various levels of CSI availability at the relay. These closed-form expressions contribute
towards optimal choice of the quantization noise variance so as to maximize the outage performance.
For a two-user NOMARC setting it is also shown that under global CSI availability at the relay,
NNC-based NOMARC scheme always performs better than CF-based relaying.
Next, we consider an uplink NOMAR with multiple users transmitting in the presence of a
single dedicated relay. For such a NOMAR, we propose a Joint-NNC-DF relaying scheme that
utilizes joint DF and NNC encoding of the messages that are decoded at the relay, and those
that are not. Information theoretical analysis is carried out to yield achievable rate region and
conditions necessary for successful decoding at the BS. We also derive closed-form expressions for
the probability of outage under the assumption of availability of local CSI at the relay. These
expressions help us in appropriately choosing the quantization noise variance so as to achieve the
best outage performance. We also analytically show that the proposed scheme is guaranteed to
outperform existing benchmarks presented in the literature. Simulation results confirm our analysis,
90
indicating SNR gains of up to 2.1 dB, 1.7 dB, 0.75 dB over NNC-only, DF-only, and DF-or-NNC,
respectively, while the maximum average outage rate gains are observed to be up to 1 b/s/Hz, 0.8
b/s/Hz, and 0.4 b/s/Hz over NNC-only, DF-only, and DF-or-NNC, respectively.
Second setup we consider is the multicast network, where we propose LMDF relaying with QAM
transmissions. The two main components of the proposed scheme are layering at the source and
multiplexed coding at the relay. While layering at the source allows partial message recovery at
the relay, multiplexed coding allows all channel resources to be utilized for decoding the layer(s)
not recoverable through the source’s transmission. Achievable rate region for LMDF is presented
before network outage probability and network outage rate for finite constellation are simulated
in comparison to conventional UDF as well as SCDF. We also present LMDF implementation
on software defined radios using convolutional codes modified for MC transmission. Performance
comparison of the implementation work also highlights significant gains of LMDF in comparison
to two-hop DF.
6.1 Future Directions
In this section, we highlight the possible future contributions stemming from the research discussed
in this thesis. There are multiple avenues that seem worth exploring. Some of these future directions
are listed below.
1. One possible direction is the practical implementation of J-NNC-DF scheme proposed in
Chapter 4. This work could leverage lessons learned during practical design and implementation
of LMDF for finite constellation mentioned in section 5.5.
2. In Chapter 3 and 4, we presented J-NNC-DF for an uplink NOMARC scenario. A possible
future direction is to consider the application of J-NNC-DF scheme for a downlink multicast
channel discussed in Chapter 5. LMDF is presented under the assumption that all layers from
a layered source are successfully decoded at the relay. While pursuing J-NNC-DF, instead of
LMDF, we can explore the scenario when a subset of the total layers are decoded at the relay.
Intuitively joint encoding of the DF layers with the quantized message of relay should be
91
beneficial. Thus another direction of future work could be to quantify joint encoding benefit
for multicast scenarios.
3. The possibility of exploring other interesting relaying scenario like the two-way relay channel
[94], the diamond relay channel [31, 95] or the multi-source multi-relay channel model is
another future direction. These channel models are well explored with either NNC or DF
scheme but a joint encoding scheme such as J-NNC-DF would further improve the achievable
rates region and outage performance. It would be interesting to examine if closed-form
probability of outage can be formulated for these said channel models. Also another interesting
challenge would be to reduce the quantizer variance search space similar to reduced search
space for J-NNC-DF discussed in section 4.4.4.
4. Another plausable direction is to jointly optimize relay selection and resource allocations for
a cooperative network setting. This research direction could be similar to studies conducted
in [96, 97, 98, 99], where joint optimization of resource allocation is conducted with either
relay selection or relay strategy. The result of this direction could fit very well in real life
cellular networks where a BS transmission are relayed towards users.
5. Since J-NNC-DF involves block-wise backward decoding of the user and relay transmissions
at the BS, so an interesting research focus could be to analyze and quantify these decoding
latencies incurred in J-NNC-DF compared to other relaying strategies.
6. Lastly we can pursue proposed cooperative schemes under the limitation of finite block length
models. This again would be an interesting and challenging problem since the achievability
proof for both J-NNC-DF and LMDF assume infinite block length models. A practical
implementation using off-the-shelf codebook like convolutional or polar code is another future
direction. Works such as [100, 101] can be utilized, where polar codes are implemented in
cooperative communication setup.
Above discussion points to some of the possible future direction which can extend the research
work discussed in this thesis.
92
Appendix A
Proof of Theorem 4.3.3
To prove the achievability region for Joint-NNC-DF, we follow same line of arguments as stated in
[77] for a multiple access channel. By symmetry of code construction, the conditional probability of
error does not depend on which pair of message indices (w1b, w2b) is sent from the users. Similarly
the conditional probability of error does not depend on quantization index vb chosen by the relay.
Thus, without loss of generality, the probability of error can be evaluated by assuming that the
message indices and the quantization index are all equal to 1 for all blocks b ∈ {1, . . . , B + 1}. At
the BS an error happens if for some b, one or more of the following events occur
ER(vb) ,{(
yrb, yrb(1, vb), xrb(1, 1))6∈ T (n)
ε
}for all vb, and
ED(w1(b−1), w2(b−1), v(b−1)) ,{(x1b(w1(b−1), 1),x2b(w2(b−1), 1), yrb(v(b−1), 1), xrb
(v(b−1), w1(b−1)
),ydb
)∈ T (n)
ε
}
for some (w1(b−1), w2(b−1), v(b−1)) 6= (1, 1, 1). By the covering lemma and the union of events bound
(over B blocks), the probability of the first event tends to zero as n→∞ if Rr > I(Yr; Yr|Xr)+δ(ε′).
93
For the second error event, we have
P (ED) = P
( ⋃(w1(b−1),w2(b−1),v(b−1)) 6=(1,1,1)
ED(w1(b−1), w2(b−1), v(b−1))
)
≤∑
w1(b−1) 6=1
P
(ED(w1(b−1), 1, 1)
)+
∑w2(b−1) 6=1
P
(ED(1, w2(b−1), 1)
)+
∑v(b−1) 6=1
P
(ED(1, 1, v(b−1))
)
+∑
(w1(b−1),w1(b−1)) 6=(1,1)
P
(ED(w1(b−1), w2(b−1), 1)
)+
∑(w1(b−1),v(b−1))6=(1,1)
P
(ED(w1(b−1), 1, v(b−1))
)
+∑
(w2(b−1),v(b−1)) 6=(1,1)
P
(ED(1, w2(b−1), v(b−1))
)+
∑(w1(b−1),w2(b−1),v(b−1))6=(1,1,1)
P
(ED(w1(b−1), w2(b−1), v(b−1))
)
Using standard typicality arguments, it can be shown that the probability of error approaches zero
as n→∞ if
R1 < I(X1, Xr; Yr, Yd|X2) (A.1)
R2 < I(X2; Yr, Yd|X1, Xr) (A.2)
Rr < I(Xr;Yd|X1, X2) + I(Yr;X1, X2, Yd|Xr) (A.3)
R1 +R2 < I(X1, X2, Xr; Yr, Yd) (A.4)
R1 +Rr < I(X1, Xr;Yd|X2) + I(Yr;X1, X2, Yd|Xr) (A.5)
R2 +Rr < I(X2, Xr;Yd|X1) + I(Yr;X1, X2, Yd|Xr) (A.6)
R1 +R2 +Rr < I(X1, X2, Xr;Yd) + I(Yr;X1, X2, Yd|Xr) (A.7)
By eliminating Rr through the use of the constraint Rr > I(Yr; Yr|XR), the bounds in (A.1)–(A.7)
yield the rate region presented in Theorem 4.3.3.
94
Appendix B
Proof of Lemma 4.3.5
Analysing (4.20) and (4.21), we note that
CNNC(V) = min{I(XV , Xr; Yr, Yd|XVc), CDF (V)}. (B.1)
This immediately establishes the first statement (4.24) of the lemma. For proving the second
statement (4.25) of the lemma, we use (4.21) and (4.22) to obtain
CNNC−DF (V, T ) = min{I(XV , XT , Xr; Yr, Yd|X{V∪T }c),
I(XV , XT , Xr;Yd|X{V∪T }c)− I(Yr; Yr|XK, Xr, Yd)} (B.2)
CNNC(V ∪ T ) = min{I(XV , XT ; Yr, Yd|X{V∪T }c , Xr),
I(XV , XT , Xr;Yd|X{V∪T }c)− I(Yr; Yr|XK, Xr, Yd)} (B.3)
where for (B.3), we have used the fact that V and T are disjoint because of which X{V∪T } =
{XV , XT }. We see the second terms within the minimum operators of (B.2) and (B.3) are the
same. Therefore, we only compare the first terms of the two expressions. Using the chain rule of
mutual information, we have
I(XV , XT , Xr; Yr, Yd|X{V∪T }c) = I(Xr; Yr, Yd|X{V∪T }c) + I(XV , XT ; Yr, Yd|X{V∪T }c , Xr)
> I(XV , XT ; Yr, Yd|X{V∪T }c , Xr).
95
This proves the inequality in (4.25).
96
Appendix C
Proof of Theorem 5.3.1
To prove the achievability, we follow same line of argument as in [77] for the multiple access channel.
We restrict our attention to discrete alphabets; it can be extended to continuous alphabets in a
straight-forward manner.
Encoding
During the first time-slot, the source splits the message, and encodes the two layers separately
with rate R1 = βR and R2 = βcR, respectively. The two codebooks are constructed randomly
with symbols drawn from the M1- and M2-PAM alphabet, respectively.. Given index i and j,
i ∈(1, 2, ..., 2nR1
), j ∈
(1, 2, ..., 2nR2
), the source transmits a symbol-by-symbol superposition of
the sequences Xs1(i) and Xs2(j), each of length γn. The received sequence at the destination is
Yd1. Given both layers are recovered at the relay after the first time slot, the source and the relay
both use independent multiplexed codebooks (each constructed randomly with symbols drawn from
the M -PAM alphabet) and transmit Xs(i, j) and Xr(i, j), respectively. The received sequence at
the destination is Yd2 and is of length γcn symbols. All codebooks are revealed to the receivers.
97
Decoding
Let A(γn)ε denote the set of jointly typical {Xs1,Xs2,Yd1} sequences and A
(γcn)ε denote the set of
jointly typical {Xs,Xr,Yd2} sequences. The receiver chooses the pair (i, j) such that
{Xs1(i),Xs2(j),Yd1} ∈ A(γn)ε and {Xs(i, j),Xr(i, j),Yd2} ∈ A(γcn)
ε
If such a pair does not exist, or is not unique, an error is declared.
Analysis of the probability of error
By the symmetry of the random code construction, the conditional probability of error does not
depend on which pair of indices is sent. Thus, the conditional probability of error is the same as the
unconditional probability of error. So, without loss of generality, we can assume that (i, j) = (1, 1)
was sent. We have an error if either the correct codewords are not typical with the received sequence
or there is a pair of incorrect codewords that are typical with the received sequence. Defining the
events
Eij = {Xs1(i),Xs2(j),Yd1, } ∈ A(γn)ε and {Xs(i, j),Xr(i, j),Yd2} ∈ A(γcn)
ε ) (C.1)
Then by the union of events bound, the error probability given the pair (1,1) was sent is given by,
Pne = P (Ec11
⋃∪(i,j)6=(1,1)Eij)
≤ P (Ec11) +∑
i 6=1,j=1
P (Ei1) +∑
i=1,j 6=1
P (E1j) +∑
i 6=1,j 6=1
P (Eij) (C.2)
From AEP, we have P (Ec11)→ 0. On the other hand, using the same line of arguments as in [77],
we have for i 6= 1,
98
P (Ei1) =∑
{Xs1(i),Xs2(j),Yd1,}∈A(γn)ε ,{Xs(i,j),Xr(i,j),Yd2}∈A
(γcn)ε
p(Xs1)p(Xs2,Yd1)p(Xs)p(Xr)p(Yd2)
=∑
{Xs1,Xs2,Yd1)∈A(γn)ε
p(Xs1)p(Xs2,Yd1)∑
(Xs,Xr,Yd2)∈A(γcn)ε
p(Xs)p(Xr)p(Yd2)
≤ |A(γn)ε |2−γn(H(Xs1)+H(Xs2,Yd1)−2ε)|A(γcn)
ε |2−γcn(H(Xs)+H(Xr)+H(Yd2)−3ε)
≤ 2−γn(H(Xs1)+H(Xs2,Yd1)−H(Xs1,Xs2,Yd1)−3ε)2−γcn(H(Xs)+H(Xr)+H(Yd2)−H(Xs,Xr,Yd2)−4ε)
≤ 2−γn(I(Xs1;Yd1|Xs2)−3ε)2−γcn(I(Xs,Xr;Yd2)−4ε)
where through independence ofXs1 andXs2 we conclude I(Xs1;Yd1, Xs2) = I(Xs1;Xs2)+I(Xs1;Yd1|Xs2) =
I(Xs1;Yd1|Xs2). Using the same line of arguments, we can easily conclude that for j 6= 1
P (E1j) ≤ 2−γn(I(Xs2;Yd1|Xs1)−3ε)2−γcn(I(Xs,Xr;Yd2)−4ε)
Finally for i 6= 1, j 6= 1
P (Eij) =∑
{Xs1(i),Xs2(j),Yd1,}∈A(γn)ε ,{Xs(i,j),Xr(i,j),Yd2}∈A
(γcn)ε
p(Xs1)p(Xs2)p(Yd1)p(Xs)p(Xr)p(Yd2)
=∑
(Xs1,Xs2,Yd1)∈A(γn)ε
p(Xs1)p(Xs2)p(Yd1)∑
(Xs,Xr,Yd2)∈A(γcn)ε
p(Xs)p(Xr)p(Yd2)
≤ |A(γn)ε |2−γn(H(Xs1)+H(Xs2)+H(Yd1)−2ε)|A(γcn)
ε |2−γcn(H(Xs)+H(Xr)+H(Yd2)−H(Xs,Xr,Yd2)−3ε)
≤ 2−γn(H(Xs1)+H(Xs2)+H(Yd1)−H(Xs1,Xs2,Yd1)−3ε)2−γcn(H(Xs)+H(Xr)+H(Yd2)−H(Xs,Xr,Yd2)−4ε)
≤ 2−γn(I(Xs1,Xs2;Yd1)−3ε)2−γcn(I(Xs,Xr;Yd2)−4ε)
Since there are a total of 2nR1 and 2nR2 codewords of layer-1 and layer-2, respectively, (C.2)
becomes
Pne ≤ P (Ec11) + 2nR12−γn(I(Xs1;Yd1|Xs2)−4ε)2−γcn(I(Xs,Xr;Yd2)−4ε)
+2nR22−n(I(Xs2;Yd1|Xs1)−4ε)2−γcn(I(Xs,Xr;Yd2)−4ε)
+2n(R1+R2)2−γn(I(Xs1,Xs2;Yd1)−4ε)2−γcn(I(Xs,Xr;Yd2)−4ε)
99
For an arbitrary ε > 0, the probability of error tends to 0 as n → 0 if the following conditions
are satisfied.
R1 = βR ≤ γI (Xs1;Yd1|Xs2) + γcI (Xs, Xr;Yd2)
R2 = βcR ≤ γI (Xs2;Yd1|Xs1) + γcI (Xs, Xr;Yd2)
R1 +R2 = R ≤ γI (Xs1, Xs2;Yd1) + γcI (Xs, Xr;Yd2)
This completes the proof that equations (5.6), (5.7) and (5.8) are indeed the correct constraints for
correct recovery of both layers at the destination (given that both layers are decoded at the relay).
100
Appendix D
Proof of Lemma 5.3.2
We define following notations in order to prove this argument.
Cd2 = min
(γCM1 (αSsd) + γcCS (Ssd, Srd)
β,γCM2 (αcSsd) + γcCS (Ssd, Srd)
βc,
γCM (Ssd) + γcCS (Ssd, Srd)
)Cd1 = min
(γCM1 (αSsd) + γcCS (Ssd, Srd)
β,γCM2 (αcSsd) + γcCM (Ssd)
βc,
γCM (Ssd) + γcCS (Ssd, Srd)
)Cd2 = min
(Cd2 , C
r2
)Cd1 = min
(Cd1 , C
r1
)
Thus, Cd2 and Cd1 are the maximum achievable rates given the number of layers decoded at the relay
were two and one, respectively. For a fixed transmission rate R, an outage event will be declared
as per the following psuedo code:
if (R1 < Cr2) % if both layers decoded at relay
Outage event if (R > Cd2 )
else if (R1 < Cr1) & (R > Cr2) % if only one layer decoded at relay
Outage event if (R > Cd1 )
101
else if (R > Cr1) % if no layer decoded at relay
Outage event if (R > Cd0 )
For further analysis, we define the following events: A ,{R < Cr2
},B ,
{R > Cd2
},C ,
{R >
Cd1},D ,
{R > Cr1
}and E ,
{R > Cd0
}. It is easy to see that these events satisfy the following
properties:
B ⊆ C ⊆ E,
D ⊆ A.
Let the complement of an event be denoted as {.}. The outage event can be written in terms of
the sets A,B,C,D and E as follows.
{Outage Event} = (A ∩B) ∪ (D ∩ A ∩C) ∪ (D ∩E)
=[(A ∩B) ∪ (D ∩ A)
]∩[(A ∩B) ∪C
]∪ (D ∩E)
=
[[(A ∩B) ∪ (D ∩ A)
]∩C
]∪ (D ∩E) (D.1)
=
[{[(A ∩B) ∪ (D ∩ A)
]∩C
}∪D
]∩[{[
(A ∩B) ∪ (D ∩ A)]∩C
}∪E
](D.2)
102
{Outage Event} =
[{[(A ∩B) ∪ (D ∩ A)
]∩C
}∪D)
]∩E (D.3)
=
[{[(A ∩B) ∪ (D ∩ A)
]∪D
}∩{
C ∪D
}]∩E
=
[((A ∩B) ∪
[(D ∩ A) ∪D
]]∩ (C ∪D) ∩E
=
[((A ∩B) ∪
[(D ∪D) ∩ (A ∪D)
]]∩ (C ∪D) ∩E
=
[(A ∩B) ∪ (A ∪D)
]∩ (C ∪D) ∩E
=
[(A ∩B) ∪ A
]∩ (C ∪D) ∩E (D.4)
=
[(A ∪ A) ∩ (B ∪ A)
]∩ (C ∪D) ∩E
= (B ∪ A) ∩ (C ∪D) ∩E
where the equality (1) uses B ⊆ C, hence (A ∩B) ⊆ C, thus implying[(A ∩B) ∪C
]= C. The
equality (2) follows since
{[(A∩B)∪(D∩A)
]∩C
}⊆ C ⊆ E =⇒
[{[(A ∩B) ∪ (D ∩ A)
]∩C
}∪E
]=
E and equality (3) relies on the fact that D ⊆ A =⇒ (A ∪ D) = A. Most of the remaining
equalities follow from the distributive law of sets.
Next, we note that the event (B ∪ A) ={R > min
(Cd2 , C
r2
)= Cd2
}and similarly the event
(C ∪ D) ={R > min
(Cd1 , C
r1
)= Cd1
}. Hence writing equation (4) in terms of these conditions,
we have the outage event equal to the event
{R > max
(Cd2 , C
d1 , C
d0
)}which can be written as{
R > CLMDF
}with CLMDF = max
(Cd2 , C
d1 , C
d0
)being the maximum transmission rate possible
for the destination to achieve successful recovery of data.
103
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