Applied Science, Faculty of
Engineering, School of (Okanagan)
DSpace
UBCO
Sajid, Adnan
Master of Applied Science - MASc
University of British Columbia
In modern communication networks, relays play a vital role in cooperative communication. Different relaying strategies like decode and forward, amplify and forward, and quantize and forward have been proposed for the relay networks. The use of relaying in communication networks improves spectral efficiency. For instance, in a two-way relay channel, two users can exchange data by using a transmission medium simultaneously. When these relay networks are used with advanced technologies like unmanned aerial vehicle (UAV) and mm-Wave networks, new limitations are faced like line of sight obstructions. Such networks become more susceptible to blockages leading to intermittent connectivity. However, the capacity of the intermittent two-way relay channel has not been studied in the literature. In this thesis, we calculate capacity bounds and achievable rates for the intermittent two-way relay channel. We classify our system model into three different cases depending on the knowledge of the channel state at different nodes in the networks. We consider an ideal case where channel states are available at all the nodes in the network and calculate the cut-set bound and achievable rates of different relaying schemes. We then extend our results to practical cases where limited channel state information is available. We compare the bounds and achievable rates, and discuss the capabilities and limitations of different relaying schemes in each of the three cases.
Transmission Schemes and AchievableRates for the Two-Way Relay Channelwith IntermittencybyAdnan SajidB.Sc. Hons., University of Engineering and Technology Taxila, 2019A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF APPLIED SCIENCEinTHE COLLEGE OF GRADUATE STUDIES(Electrical Engineering)THE UNIVERSITY OF BRITISH COLUMBIA(Okanagan)April 2022© Adnan Sajid, 2022The following individuals certify that they have read, and recommendto the College of Graduate Studies for acceptance, a thesis/dissertation en-titled:Transmission Schemes and Achievable Rates for the Two-Way RelayChannel with Intermittencysubmitted by Adnan Sajid in partial fulfilment of the requirements of thedegree of Master of Applied ScienceDr. Anas Chaaban, School of EngineeringSupervisorDr. Jahangir Hossain, School of EngineeringSupervisory Committee MemberDr. Lo¨ıc Markley, School of EngineeringSupervisory Committee MemberDr. Jian Liu, School of EngineeringUniversity ExamineriiAbstractIn modern communication networks, relays play a vital role in coopera-tive communication. Different relaying strategies like decode and forward,amplify and forward, and quantize and forward have been proposed for therelay networks. The use of relaying in communication networks improvesspectral efficiency. For instance, in a two-way relay channel, two users canexchange data by using a transmission medium simultaneously. When theserelay networks are used with advanced technologies like unmanned aerialvehicle (UAV) and mm-Wave networks, new limitations are faced like lineof sight obstructions. Such networks become more susceptible to blockagesleading to intermittent connectivity. However, the capacity of the intermit-tent two-way relay channel has not been studied in the literature. In thisthesis, we calculate capacity bounds and achievable rates for the intermittenttwo-way relay channel.We classify our system model into three different cases depending on theknowledge of the channel state at different nodes in the networks. We con-sider an ideal case where channel states are available at all the nodes in thenetwork and calculate the cut-set bound and achievable rates of different re-laying schemes. We then extend our results to practical cases where limitedchannel state information is available. We compare the bounds and achiev-able rates, and discuss the capabilities and limitations of different relayingschemes in each of the three cases.iiiLay SummaryConsider a network of water pipes in an irrigation system of a farm. Thisnetwork is deployed, to irrigate the whole farm efficiently. If some of thepipes get blocked by placing loaded crates on the pipes accidentally, waterwill not reach some areas of the farm and the efficiency of the network is com-promised. If we interpret these pipes as a transmission medium exchanginginformation between nodes in two-way relay channel which is blocked as well,then the quality of service and efficiency of the relay network is degraded.This necessitates developing and studying the performance of transmissionschemes for the relay channel with intermittency, which can provide per-formance guarantees and accurate information about the quality of servicethat can be achieved. In this thesis, we develop transmission schemes for theintermittent two-way relay channel and provide their performance analysisat different levels of intermittency.ivPrefaceThis thesis is conducted under the principal supervision of Dr. AnasChaaban. All the work of this thesis has been reviewed and improved byDr. Chaaban. He will also be a co-author with me in a research paperwhich is being prepared for submission in a well reputed Journal. This paperwill constitute of the system model explained in Chapter 2, which will beextended by using the results of a practical assumption from Chapter 4 ofthis thesis.vContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vContents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixList of Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . xiList of Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . xiiAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . xivDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvChapter 1: Introduction . . . . . . . . . . . . . . . . . . . . . . . 11.1 Background and Motivation . . . . . . . . . . . . . . . . . . . 11.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Thesis Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.3.1 Problem Statement . . . . . . . . . . . . . . . . . . . . 61.3.2 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . 71.3.3 Contributions and Outline of Thesis . . . . . . . . . . 7Chapter 2: System Model of the Two-Way Relay Channel(TWRC) and Preliminaries . . . . . . . . . . . . . 92.1 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2 Availability of Channel State Information (CSI) . . . . . . . 12viCONTENTS2.3 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.3.1 Information Measures and Useful Properties . . . . . . 142.3.2 Multiple Access Channel . . . . . . . . . . . . . . . . . 16Chapter 3: TWRC with Global Non Causal CSI Availibility 183.1 The Cut-Set Bound . . . . . . . . . . . . . . . . . . . . . . . 193.2 Transmission Schemes . . . . . . . . . . . . . . . . . . . . . . 243.2.1 Decode-and-Forward (DF) . . . . . . . . . . . . . . . . 243.2.2 Amplify-and-Forward (AF) . . . . . . . . . . . . . . . 263.2.3 Quantize-and-Forward (QF) . . . . . . . . . . . . . . . 293.3 Extension to Half Duplex . . . . . . . . . . . . . . . . . . . . 343.4 Numerical Illustration . . . . . . . . . . . . . . . . . . . . . . 37Chapter 4: TWRC with Practical CSI Assumptions . . . . . 444.1 TWRC with Non-Causal CSI at the Relay and Causal CSI atthe Users . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444.1.1 Cut-Set Bound . . . . . . . . . . . . . . . . . . . . . . 454.1.2 Transmission Schemes . . . . . . . . . . . . . . . . . . 454.1.3 Extension to Half Duplex . . . . . . . . . . . . . . . . 474.2 TWRC with Local CSI Available Causally at all Nodes . . . 534.2.1 Cut-Set Bound . . . . . . . . . . . . . . . . . . . . . . 534.2.2 Transmission Schemes . . . . . . . . . . . . . . . . . . 544.2.3 Extension to Half Duplex . . . . . . . . . . . . . . . . 564.3 Numerical Illustration . . . . . . . . . . . . . . . . . . . . . . 58Chapter 5: Conclusion and Discussion . . . . . . . . . . . . . . 645.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 645.2 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 665.3 Future Directions . . . . . . . . . . . . . . . . . . . . . . . . . 66Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75Appendix A: Rate of Transmission in a Gaussian Channel . . . . . 75viiList of TablesTable 2.1 CSI availability at different nodes in TWRC at timeinstant i . . . . . . . . . . . . . . . . . . . . . . . . . . 13viiiList of FiguresFigure 2.1 Gaussian Two-Way Relay Channel with Intermittency 10Figure 2.2 MAC System Model . . . . . . . . . . . . . . . . . . . 16Figure 3.1 Decode and Forward . . . . . . . . . . . . . . . . . . . 25Figure 3.2 Amplify and Forward . . . . . . . . . . . . . . . . . . 27Figure 3.3 Quantize-and-Forward . . . . . . . . . . . . . . . . . . 30Figure 3.4 Decoding at the users in Quantize and Forward . . . 31Figure 3.5 Transmission of Symbols in HD-TWRC when CSI isavailable Non-causally at all the nodes. . . . . . . . . 36Figure 3.6 Full-Duplex TWRC with intermittency . . . . . . . . 38Figure 3.7 Full-Duplex TWRC with intermittency at SNR1r =SNR2r = SNRr1 = SNRr2 = 25dB . . . . . . . . . 39Figure 3.8 Maximum sum rate of FD-TWRC at different valuesof intermittency, α = β . . . . . . . . . . . . . . . . . 40Figure 3.9 Achievable rates at SNR1r = 20dB, SNRr1 = 18dB,SNR2r = 22dB, and SNRr2 = 16dB. . . . . . . . . . 41Figure 3.10 Half-Duplex TWRC with intermittency at α = 0.45and β = 0.85 . . . . . . . . . . . . . . . . . . . . . . . 42Figure 3.11 Maximum sum rate of HD-TWRC at different valuesof intermittency at SNR1r = SNR2r = 10dB andSNRr1 = SNRr2 = 20dB . . . . . . . . . . . . . . . . 43Figure 4.1 Transmission of symbols in HD-TWRC when CSI isavailable non-causally at the relay and causally at theusers. . . . . . . . . . . . . . . . . . . . . . . . . . . . 48Figure 4.2 Full-Duplex TWRC with intermittency at SNR1r =SNR2r = SNRr1 = SNRr2 = 15dB . . . . . . . . . 59Figure 4.3 Maximum sum rate of the FD-TWRC with intermit-tency at SNR1r = SNR2r = 10dB and SNRr1 =SNRr2 = 15dB, β = α . . . . . . . . . . . . . . . . . 59ixLIST OF FIGURESFigure 4.4 Full-Duplex TWRC with intermittency at α = β =0.75 . . . . . . . . . . . . . . . . . . . . . . . . . . . 60Figure 4.5 Maximum sum rate of the HD-TWRC with intermit-tency, α = 1 . . . . . . . . . . . . . . . . . . . . . . . 61Figure 4.6 Maximum sum rate of the intermitent HD-TWRC fornon-causal global CSI at the relay and causal localCSI at the users . . . . . . . . . . . . . . . . . . . . . 62Figure 4.7 Maximum sum rate of the intermitent HD-TWRC forcausal local CSI at the users . . . . . . . . . . . . . . 63Figure A.1 Gaussian Channel . . . . . . . . . . . . . . . . . . . . 75xList of NotationsX,Y, Z Scalar random variablesx, y, z Deterministic quantity of random variable X,Y, ZXm Symbol at node m ϵ {1, 2, r}Xm,i Symbol transmitted at time instant i at node mXnm Vector of length n symbols of random variables(Xm,1,Xm,2, . . . ,Xm,n) at node mXim Vector of all symbols transmitted up till time instant i(Xm,1,Xm,2, . . . ,Xm,i) at node mW Set of alphabetsMj Message at node j ϵ {1, 2}hjr Channel constant from node j to the node rSj,i Channel state at node j at time instant iP{S} Probability of channel state S availabilityI(X;Y ) Mutual information between random variables X and YH(X) Entropy of random variable XRm Rate of transmission at node mp(y|x) Channel conditional probability distributionσ2 VarianceP Transmit poweri.i.d Independent and identically distributedN(0, σ2) Gaussian distribution with zero mean and variance σ2f Function defined using variables{f}ni=1 Sequence of function from i = 1 to nlog binary logarithm (logarithm to the base 2)α Uplink channel availabiltyβ Downlink channel availabiltyE ExpectationD DistortionxiList of AbbreviationsAF Amplify and ForwardAP Access PointCF Compute and ForwardCSI Channel State InformationDF Decode and ForwardFD Full DuplexFD-TWRC Full Duplex Two Way Relay ChannelHD Half DuplexHD-TWRC Half Duplex Two Way Relay ChannelIoT Internet of ThingsJBD Joint Blind-DifferentialLOS Line of SightmmWave milli-meter WaveMAC Multiple Access ChannelMIMO Multiple Input Multiple OutputpDF partial Decode and ForwardQF Quantize and ForwardSNR Signal to Noise RatioTHz Tera-HertzTWRC Two Way Relay ChannelxiiList of AbbreviationsUAV Unmanned Aerial VehiclexiiiAcknowledgementsI would like to express my sincere gratitude to Dr. Anas Chaaban, forhis guidance and endless support throughout the course of this work. Ithas been a great pleasure and privilege to work under his supervision andbenefit from his incredible knowledge and experience. What I learned fromhim- his dedication, hard work, and enthusiasm will be a lifelong asset.I am very grateful to have Dr. Jahangir Hossain and Dr. Lo¨ıc Markleyas supervisory committee members and course instructors, and to Dr. JianLiu as a university examiner. I am also grateful to my fellow team membersof the Communication Theory Lab.My wholehearted thanks to my mother for her prayers and all the efforts,she put into my upbringing. My siblings for always backing and cheeringme up during stressful times. Most importantly, my wife for her continuoussupport and belief in me, who has always stood by my side and kept moti-vating me to follow my goals. Last but not the least, Lisa and Ahmad whobecame a family to me and supported me whole this time.xivDedicationThis thesis is dedicated to my father, who passed away in October 2021.He supported me in all aspects throughout my degree, but unfortunately,he is not here to witness this moment of his son’s accomplishment. He isthe reason behind every success of my life.xvChapter 1IntroductionRelays have become an integral part of a number of communication net-works. With an increase in its application, it is important to investigatepossible limitations of communication networks using relays like loss of datadue to the blockage in a communication channel. Relays are being used withnew communication technologies, like mmWaves and Unmanned Aerial Ve-hicle (UAV) communication e.g. which are both susceptible to blockage,and hence it is important to study channels with intermittent connectiv-ity. Intermittency in the flow of data affects the quality of transmission.This effect can be portrayed through the transmission rates of such chan-nels. In this thesis, we study the capacity of the Two-Way Relay Channel(TWRC) under intermittent connectivity. We introduce the motivation andbackground of the research involved in the next section. Furthermore, wesummarize the research problems, objectives, and organization of the thesisat the end of this chapter.1.1 Background and MotivationWith the increase in the number of connected devices in the communi-cation networks, cooperative communication is attaining more significance,as it can help in overcoming coverage issues and improving the quality ofservice. Cooperative communication can be realized using different methodssuch as feedback and relaying. Relays can be used in different types of net-works, such as a point-to-point transmission from a source to a destination,where relays can be used to extend the coverage in a long-distance link. [1].Additionally, relays are used for multi-user cooperative networks to achievespatial diversity [2]. The relay can be thought of a static point, for examplean access point (AP) routing the signal from one device to another situatedat a long distance [3, 4]. Similarly, a mobile relay (like a UAV) can be usedto route messages from one node to another [5]. Relays are used in UAVcommunication to increase the coverage and improve the efficiency of thenetwork [6].11.2. Literature ReviewA relay can be used when the direct link between a source and a des-tination is missing where the relay channel acts as an alternative link [7].When two nodes, communicate via relay in both directions (both nodes aresources and destinations of information), the communication network canbe considered to be a TWRC. [8]. The TWRC has been studied in manyconfigurations such as Multiple Input Multiple Output (MIMO) networks[9], Internet of Things (IoT) [10], and vehicular communications [11].When a TWRC is used in wireless applications operating on higher fre-quencies, like mm Wave systems [12], the transmitted symbols are moresusceptible to blockage because of the reduced diffraction and higher prop-agation losses [13]. Even a human body can act as an obstacle and reducethe signal strength to 20dB [14]. Such a blockage based model is studiedin [15], where outage probability is analyzed by taking intermittent con-nectivity in consideration. Intermittency is also studied in [16] to designa throughput-optimal relay probing strategy for mm-Wave based cellularsystem. Similarly, when a TWRC is used in THz wireless networks [17],these networks are vulnerable to the channel attenuation blockages. This isbecause THz communication is highly dependent on the atmospheric con-ditions, for instance, water vapor can absorb the electromagnetic energy ofthe signals [18]. This results in coverage issues of THz networks. A coverageanalysis is provided in [19], where the effects of indoor blockages are stud-ied for THz networks. Another coverage analysis is provided in [20], whereblockage is assumed and a scheme is studied to to improve the coverage bychanging the antenna directivity at users and APs. Morever, since a TWRCalso arises in UAV [6] communication, which has benefits along with chal-lenges. The biggest challenge is to maintain the line of sight (LOS). Due tothe smaller wavelength, even a small object can be a hindrance for the LOSradio propagation [21, 22]. In short, when relays are used in these network,blockage may take place leading to intermittent connectivity. In this thesis,we aim to discuss the TWRC affected by the intermittency. Since two-wayrelaying is a special form of relaying, let us have glance at the history ofrelaying and bidirectional communication.1.2 Literature ReviewThe study of bidirectional communication from an information theoryperspective was first introduced by Claude Shannon in 1961 when he ex-plained the concept of two-way communication [23]. On the other hand, theconcept of relaying was first introduced in information theory by Van der21.2. Literature ReviewMeulen [24], where the connection between the source and the destinationis established with the help of the relay (called a one way relay channel orsingle relay channel). Different transmission and relaying schemes and theircapacities were calculated for the one way relay channel by Cover and ElGamal [25]. The one way relay channel can be extended to a TWRC and thetransmission schemes introduced in [25] can be generalized to the TWRC[26]. In this thesis, we will focus on various relaying schemes like decodeand forward (DF), amplify and forward (AF), and quantize (compress) andforward (QF). Next, we will review the mechanism and background of eachscheme used for the TWRC.Firstly, in DF, the relay decodes the message received in the uplink andre-encodes the messages for the downlink transmission. A relay acts like adigital regenerative repeater in DF as the received symbols of messages areregenerated using the codebook available at the relay. DF was introducedat first by Cover and El Gamal [25], in which block Markov superpositionencoding was used at the source, received symbols are decoded at the relay,and binning indexes are forwarded to the destination, which performs suc-cessive decoding. In [27], a backward decoding method is proposed wheredecoding is performed in reverse order after all the blocks are received at thedestination. It achieves the same rate for the discrete memory-less channelas [25]. DF has been used for the TWRC by Rankov and Wittneben in[26], where a partial Markov encoding is used by the two users for the trans-mission and the relay sends the superposition of the decoded signals. Incontrast, a different DF scheme was used by Xie in [28] where the users en-code their messages independently without using block-Markov coding andthe relay sends the binning indexes of the two decoded messages which canbe decoded again at the users’ by using the knowledge of their own mes-sages. Secondly, in AF, instead of decoding like in DF, the relay transmitsa scaled version of the received signals [29] which makes this scheme simplein terms of complexity. The use of AF scheme as a relaying scheme can betraced back to [30], where the achievable rate of the AF scheme was derivedwhen parallel independent observations of transmitted signals are amplifiedat the relay. This work is explained extensively in [31, p.61] as a forward-ing strategy without decoding at the relay. A numerical comparison of AFand DF at the relay is discussed in [31], where the achievable rate of AFwas shown to be better than DF for higher values of multi-access Signal toNoise Ratio (SNR) for a parallel relay channel acting as a single relay withtwo of the transmitted signals. In [32], AF (dubbed as scale and forward)was studied, where optimal achievable rates were calculated when a relay isdeployed between source and destination. As the relay acts like an analog31.2. Literature Reviewrepeater in AF, this AF scheme is also referred to as analog network coding[33]. Lastly, in QF (also known as compress and forward), the relay doesnot scale or decode the signal, rather, it quantizes (compresses) the receivedsignal and forwards it to the destination nodes, where decoding takes place.QF was originally proposed by Cover and El Gamal in [26]. An independentencoded message is received at the relay in each block. The relay compressesits noisy observation of the transmitted signal using Wyner-Ziv coding [34]and forwards the bin index of the compression. A 3-step sequential decod-ing is performed at the destination, in which the bin index is decoded first,followed by the compression index within that bin, followed by the messagereceived in the previous block which is then decoded using this compressionindex as side information. A 2-step decoding is also proposed in [35] whereinstead of decoding the compression index precisely after the bin index first,it decodes the message within that bin for some compressions index. It hasbeen shown that this 2-step decoding [35, 36] achieves the same rate as 3-stepdecoding [34] but uses a simpler method. A comparison of scalar and vectorquantization for the QF scheme is discussed in [37], where it is noticed thatusing scalar method leads to few quantization losses. The relaying schemesare not limited to DF, AF, and QF. Other relaying schemes like Computeand Forward (CF) using lattice coding [38], Joint Blind-Differential schemes(JBD) [39], and partial decode and forward (pDF) [40], etc. are also stud-ied in the literature. To achieve better performance, a combination of tworelaying schemes can be used, for example, splitting the source message intotwo parts and performing DF on one part and QF on the other part canenhance the performance of the standard QF scheme [41].Researchers have pooled their efforts to understand the capacity andtransmission schemes of the TWRC, like designing network coding sche-mes for the TWRC [42], and providing performance analysis and optimiza-tion of the TWRC using various relaying strategies [8]. Performance ofrelaying schemes in comparison with each other has always been a topicof interest. In [43], the achievable rates of pDF, a new scheme that usesa combination of CF and DF for TWRC are compared with the perfor-mance of each scheme separately. The results show that the superpositionof Gaussian and lattice coding outperforms each separate scheme. In [44],the sum-rates of relaying schemes like DF, QF and CF, etc. over a TWRCare compared in terms of their gap from the cut-set bound, which is a the-oretical upper bound that serves as a benchmark. It is noted that QF andCF have constant and unique gaps from the cut-set bound at high values ofSNR. Similarly, optimal power allocation parameters of the TWRC at therelay and communicating nodes are calculated for AF and DF in [45].41.2. Literature ReviewNowadays, relays are being used with advanced technologies of 5G (aswell as earlier generations), such as mmWave communication, UAV net-works, and MIMO networks [9, 46]. These networks are more susceptible tointermittency and result in intermittent relay channels. The sources of in-termittency have been discussed in Section. 1.1 in detail. Similar to [43, 44],the performance of relaying schemes for the TWRC needs to be investigatedas a function of parameters such as the SNR and the probability of blockage(intermittency).This thesis focuses on the TWRC with intermittent connectivity. In-termittency has been studied from an information-theoretic perspective inthe literature for various channels. The Multiple Access Channel (MAC), isimportant for describing DF schemes as we shall see in Section. 2.3.2, MAChas been investigated when connectivity is intermittent and the channel in-termittency state is available in the form of full Channel State Information(CSI) at the decoder and partial CSI at the encoders. The work in [47]derives inner and outer bounds for this channel by utilizing the knowledgeof channel states at different nodes. Similarly, in [48] achievable rates arecalculated for the intermittent MAC when CSI is only available at the re-ceiver. Achievable rates for the intermittent MAC are similar to the uplinkachievable rates of DF in an intermittent TWRC, where downlink achievablerates need to be investigated differently since the downlink in a TWRC re-sembles a broadcast channel with side information at the decoders. In [49],a network consisting of two users communicating through intermittent linkswith two receivers while interfering with each other was studied when CSIis available causally at the receiver. The capacity region of this network isformulated but the results have not been extended to the case where CSI isavailable non-causally at the receiver. The performance of this network canbe improved by deploying a relay between the two user-pairs, which turnsit into an interference relay channel [50] with intermittency. Moreover, ifthe user-pairs communicate in both directions, then the network becomes atwo-pair TWRC with intermittency. Such a two-pair TWRC with intermit-tency is studied in [50], where two user-pairs communicate with each otherin both directions via a relay, through intermittent links. In [50], it wasassumed that CSI is available in two forms: instantaneous CSI (knowledgeof channel state is available non-causally at all the nodes) and delayed CSI(knowledge of channel state is available causally at all the nodes). Outerand inner bounds on the achievable rates were derived, whereas the innerbounds were obtained by analyzing a CF that uses lattice coding and whichwas modified to utilize the knowledge of CSI.The aforementioned works focus on relay channels without intermittency,51.3. Thesis Workor channels with intermittency without a relay, with the exception of [50].Although [50] studies relay channels with intermittency, it only considersthe CF scheme at the relay by assuming equal SNRs at users in each pairand does not consider AF, DF, and QF. Similarly, it considers scenarioswhen instantaneous or delayed CSI is available at all the nodes in a full-duplex TWRC setup but the scenario where instantaneous CSI is availableat the relay and delayed CSI at the users has not been discussed. Thus,in this thesis, we will focus on the information-theoretic analysis of theTWRC with intermittency, considering a generic setup in terms of SNRs.We aim to analyze and/or modify relaying schemes like AF, DF and QF forthe TWRC with intermittency and derive their achievable rates. We theninvestigate the effect of intermittency on the TWRC by considering similarto [50] and different assumptions on CSI (detailed explanation follows in thenext chapter), firstly in the Full-Duplex (FD) case (where the symbols arereceived and transmitted in same time slot or frequency band), and then inthe Half-Duplex (HD) case (where the symbols are received and transmittedin different time slots or frequency bands). The main problem statement andthe objectives of this work are outlined next.1.3 Thesis WorkIn this section, we present the problem statement and summarize theobjectives, contributions and outline of the thesis.1.3.1 Problem StatementThe capacity region of TWRC has been investigated in the past [44].With the increase in the use of higher frequencies in wireless communica-tion, communication networks become more vulnerable to intermittency.Also, the use of aerial nodes (UAVs e.g.) for communications present in-termittent connectivity challenges since UAVs are susceptible to blockagein urban areas. Since two-way communication constitutes a prominent partof our daily communication activities, and since relaying is an importantfacilitator of communication over large networks, we need to investigatetransmission schemes and their performance in the TWRC under intermit-tent connectivity. Thus, the problem statement of this thesis isTo develop and study transmission schemes and their achievable rates in aTWRC with intermittent connectivity, and draw insights about the schemes’performance as a function of intermittency and SNR.61.3. Thesis WorkIn particular, we address the following questions− How is the capacity region of the TWRC affected by the intermittency?− How does intermittency affect the performance of different relayingschemes in the TWRC?− How does side information availability affect the performance of trans-mission schemes, including DF, AF, and QF?− Which relaying scheme has the best performance given a setting ofSNRs and intermittency ?We address these problems in this thesis by studying the capacity andinvestigating transmission schemes for the TWRC.1.3.2 ObjectivesIn order to answer the questions raised in the previous section, we aimto achieve the following objectives− Derive an outer bound on the capacity region of the intermittentTWRC.− Develop and analyze DF, AF, and QF schemes for the intermittentTWRC (some of which require modification to be suited for the in-termittent TWRC), and derive their achievable rates under causal ornoncausal (instantaneous) CSI availability at the users and/or the re-lay.− Study the performance of these relaying schemes under various settingsof SNRs, intermittency probability (in [0,1]), and CSI availability.1.3.3 Contributions and Outline of ThesisThe rest of the thesis has been structured as follows.− In Chapter 2, the system model is explained in detail, assumptionson CSI availability are listed which includes the combination of causaland/or non-causal CSI available at the nodes of intermittent TWRC,and preliminaries are summarized at the end of the chapter.71.3. Thesis Work− In Chapter 3, an ideal scenario is discussed in detail, where CSI isassumed to be available non-causally at all nodes in the network. Thecut-set bound and the achievable rates of different relaying schemesare derived for the Full-Duplex TWRC (FD-TWRC), and results areextended to the Half-Duplex TWRC (HD-TWRC). These results areillustrated numerically at the end of Chapter 3.− In Chapter 4, two practical assumptions on CSI (firstly, when causalCSI is available at the users and non-causal CSI at the relay andsecondly, when causal CSI is available at all the nodes) are consid-ered, and capacity bounds and achievable rates are derived for theFD-TWRC and extended to the HD-TWRC. The performance of therelaying schemes for these practical assumptions is analyzed using nu-merical evaluations.− In Chapter 5, we conclude this thesis by summarizing its contribution,and discussing its limitations and possible extensions.8Chapter 2System Model of theTwo-Way Relay Channel(TWRC) and PreliminariesIn this chapter, the system model of the TWRC is introduced. Further-more, the system model is classified into different scenarios based on theavailability of CSI or the lack thereof. Different preliminaries are discussedat the end of the chapter, which will be used in further chapters.2.1 System ModelThe system model of the TWRC with intermittent connectivity is shownin Fig. 2.1. This is a generic system model in which two users are commu-nicating with each other using a relay through intermittent channels, wherethe intermittency is represented by the binary variables S1,i, S2,i, T1,i, T2,i.Users want to exchange messages M1 and M2 with each other, which areencoded as Xn1 and Xn2 using functions {f1,i}ni=1 and {f2,i}ni=1. These func-tions encode the messages M1 and M2 using some knowledge of the channelstates U1,i and U2,i at the time instant i (to be specified later), and thepreviously received symbols Y i−11 and Yi−12 as follows .Xj,i = fj,i(Mj , Yi−1j , Uj,i), j = 1, 2, i = 1, 2, ..., n. (2.1)The information of the channel states Uj,i can include knowledge of all chan-nel states or some of them, as we illustrate later in Section. 2.2 These en-coded symbols are transmitted to the relay in n transmissions. The receivedsignal at the relay at time instant i is given byYr,i = h1rS1,iX1,i + h2rS2,iX2,i + Zr,i. (2.2)where X1,i and X2,i are the symbols transmitted from user 1 and user2 respectively, S1,i,S2,i ∈ {0, 1} are the channel states with probability92.1. System ModelRelayEncoder 2Decoder 1Encoder 1Decoder 2𝑋1,𝑖𝑌1,𝑖𝑋2,𝑖𝑌2,𝑖𝑆1,𝑖ℎ1𝑟 𝑆2,𝑖ℎ2𝑟𝑇2,𝑖ℎ𝑟2𝑇1,𝑖ℎ𝑟1𝑍𝑟,𝑖𝑍1,𝑖 𝑍2,𝑖𝑀1,𝑖 𝑀2,𝑖 𝑀2,𝑖User 1 User 2 𝑀1,𝑖Figure 2.1: Gaussian Two-Way Relay Channel with IntermittencyP{Sj,i = 1} = αj , j = 1, 2, i.e. (Bernoulli(αj)), h1r and h2r are the up-link channel coefficient from user 1 and user 2 respectively to the relay, andZr,i describes the additive Gaussian noise at the relay having zero mean andvariance σ2 represented as N(0,σ2). It is assumed without loss of generalitythatσ2 = 1,1nn∑i=1E[X2j,i] ≤ P, j = 1, 2. (2.3)Also, it is assumed that the channels h1r and h2r are static throughoutthe transmission, which can model block fading scenarios where the channelgains h1r and h2r remain constant during a coherence interval. Note thattogether h1r, h2r, S1,i, and S2,i model an intermittent channel where inter-mittency changes faster than fading, i.e., the intermittency states S1,i andS2,i change faster than the channel gains h1r and h2r during a coherenceinterval. The relay performs a set of functions {fr,i}ni=1 and transmits thesymbol Xr,i at time i given byXr,i = fr,i(Yi−1r , Ur,i), 1 ≤ i ≤ n,where Ur,i is the CSI at the relay, which we will also specify later in Section2.2. In general, the symbol Xr,i can be transmitted to the users at the sametime and same frequency band (i.e FD) or different time slot or frequency102.1. System Modelband (i.e. HD). The signal Y1,i received at user 1 at time instant i is givenbyY1,i = hr1T1,iXr,i + Z1,i, (2.4)where T1,i ∈ {0, 1} is the state of the channel between the relay and user 1 attime instant i, with distribution Bernoulli (β1), hr1 is the channel gain fromthe relay to user 1 which is assumed to be fixed during the transmissioninterval, and Z1,i is an additive Gaussian noise with zero mean and σ2variance, whereas it is also assumed that 1n∑ni=1 E[X2r,i] ≤ P .The signal received at user 2 at the time instant i is given asY2,i = hr2T2,iXr,i + Z2,i, (2.5)where T2,i, hr2, and Z2,i are defined similar to T1,i, hr1, and Z1,i, and T2,i isdistributed according to a Bernoulli(β2) distribution.On the receiver end, receiver j collects n instances of Yj,i, i.e., Ynj =(Yj,1, . . . , Yj,n). Then, it uses a decoding functions gj , in which the messageM̂k, k ̸= j, is decoded from the received signal Y nj the CSI Unj , and its ownmessage Mj , i.e.,M̂k = gj(Ynj , Unj ,Mj), k ̸= j (2.6)A (2nR1 ,2nR2 , n) code for this TWRC consists of set of integers (mes-sages) Mj = {1,2,...., 2nRj}, a set of encoder functionsfr,i and fj,i, i =1, ..., n, and decoding functions gj , for j = 1, 2. This code induces an errorprobability Pe(n) defined as the probability that (M1,M2) ̸= (M̂1, M̂2). Weaim to make this error probability small, which can be realized using channelcoding with n large, and by choosing Rj appropriately.We say that a rate pair (R1,R2) is achievable if it is possible (thereexists a sequence of codes) to make Pe(n) → 0 by increasing n. The set ofachievable the rate pairs (R1,R2) is the capacity region of the channel. Thegoal of this thesis is to study and bound this capacity region.We consider a symmetric setting for simplicity in this thesis, where S1,i =S2,i = Si and T1,i = T2,i = Ti. Thus, we have α1 = α2 = α and β1 =β2 = β. On the other hand, Si = Ti for the FD mode, since receptionand transmission take place during the same time slot (or frequency band)whereas this is not the case in the HD mode where Si ̸= Ti. Next, we willclassify different scenarios based on different assumptions on CSI availability.112.2. Availability of Channel State Information (CSI)2.2 Availability of Channel State Information(CSI)We study the capacity of the TWRC under different assumptions on theavailability of CSI at the communicating users and the relay, as mentionedbelow.Non-Causal Global CSI Availabile at the Relay and the UsersChannel states Si and Ti are known at both the users and relay atthe start of each transmission time slot, i.e. non causally which meansU1,i = U2,i = Ur,i = (Sn, Tn)This serves as the benchmark scenario for comparison, which constitutesan ideal case. This scenario helps in the understanding of the system modeland evaluating the transmission schemes for more practical scenarios.Non-Causal Global CSI at the Relay, and Causal Local CSI atthe UsersChannel states Si and Ti are known at the relay at the beginning of eachith transmission i.e. Ur,i = (Sn, Tn), whereas channel state Ti is known atthe user at the end of each ith transmission i.e U1,i = U2,i = Ti−1. Theusers knows the probability of uplink channel availability α.This scenario can arise when the relay can infer the blockage from itslocation e.g., or when the relay can estimate if blockage exists fast enoughwith high accuracy, such as when the relay is a UAV. This can also arisewhen the relay performs two tasks, a primary task and a secondary task.In the primary task it serves some other users in the network, and in thesecondary task it serves the TWRC communicating users. The two tasksmay be separated spatially using beamforming, to avoid interference betweenthe signals of the two tasks. In this case, the relay knows the states Si andTi because it knows which task is being performed at time i.Causal Local CSI at the Relay and the UsersChannel state Si is known at the relay at the end of transmission i inaddition to the probability of downlink channel availability β i.e. Ur,i =Si−1, and Ti is known at the users at the end of transmission i in additionto the probability of uplink channel availability α i.e U1,i = U2,i = Ti−1.This assumption is practical in the case of a blocked relay where neitherthe relay nor the users can know or predict blockage events. In this case, the122.2. Availability of Channel State Information (CSI)relay and the users can only estimate channel states Si and Ti, respectivelyat the end of the reception in time slot i at the relay and users respectively.The summary of these scenarios is shown in Table 2.1.Table 2.1: CSI availability at different nodes in TWRC at time instant iNon-Causal Global CSI Availability at the Relay and the UsersNodes UplinkChannelState (Si)DownlinkChannelState (Ti)Uplink Chan-nel Avail-abilty (α)DownlinkChannelAvailabilty(β)User 1 ✓ ✓ ✓ ✓Relay ✓ ✓ ✓ ✓User 2 ✓ ✓ ✓ ✓Non-Causal Global CSI at the Relay, and Causal Local CSI at the UsersNodes UplinkChannelState (Si)DownlinkChannelState (Ti)Uplink Chan-nel Avail-abilty (α)DownlinkChannelAvailabilty(β)User 1 ✓ ✓ ✓Relay ✓ ✓ ✓ ✓User 2 ✓ ✓ ✓Causal Local CSI at the Relay and the UsersNodes UplinkChannelState (Si)DownlinkChannelState (Ti)Uplink Chan-nel Avail-abilty (α)DownlinkChannelAvailabilty(β)User 1 ✓ ✓ ✓Relay ✓ ✓ ✓User 2 ✓ ✓ ✓In comparison to [50], the instantaneous CSI case in [50] resembles ourassumption when global non-causal CSI is available at all the nodes. Sim-ilarly, our delayed CSI case in [50] resembles our assumption when localcausal CSI is available at all the nodes. Unlike [50], we have also consideredanother practical assumption when non-causal CSI is available globally atthe relay and causal CSI available locally at the users.132.3. Preliminaries2.3 PreliminariesIn this section fundamental concepts from information and coding theoryare discussed which will be used for the derivation in future chapters.2.3.1 Information Measures and Useful PropertiesWe define the necessary preliminaries related to information measuresand their properties next.Definition 2.1. ([36, p. 17]) The entropy of random variable X with aprobability mass function p(x) is defined as:H(X) = −∑xp(x) log p(x), (2.7)which is the measure of average uncertainty in XDefinition 2.2. ([51, p. 33]) If (X,Y ) ∼ p(x, y), the conditional entropyH(X|Y ) is defined as:H(X|Y ) =∑xp(x)H(X|Y = y) = −∑x∑yp(x, y) log p(x|y). (2.8)Property : The joint entropy H(X,Y ) of a pair of random variables Xand Y satisfies the chain rule that is given by:H(X,Y ) = H(X) +H(Y |X). (2.9)Definition 2.3. ([36, p. 19]) The differential entropy of a continuousvariable X with probability density function f(x) is defined as:h(X) = −∫ ∞∞f(x) log f(x)dx. (2.10)The conditional differential entropy of Y given X is given as h(Y |X) =E[h(Y |X = x)] whereh(Y |X = x) = −∫ ∞∞f(y|x) log f(y|x)dy. (2.11)The mutual information is the measure of dependency between two ran-dom variables in terms of reduction in the uncertainty in one random variabledue to another random variable.142.3. PreliminariesDefinition 2.4. ([51, p. 139]) The mutual information for two randomvariables X and Y is given as:I(X;Y ) = H(X)−H(X|Y ). (2.12)Definition 2.5. ([51, p. 139]) The conditional mutual information ofrandom variables X and Y conditioned on Z is given by:I(X;Y |Z) = H(X|Z) +H(X|Y,Z). (2.13)Property : The mutual information between (X1, X2,· · · , Xn) and Y sat-isfies the chain rule given by:I(X1, X2, · · · , Xn;Y ) =n∑i=1I(Xi;Y |Xi−1, Xi−2, · · · , X1). (2.14)In addition to the aforementioned definitions, the following lemmas willbe useful in the sequel.Lemma 2.1Jensen’s Inequality ([51, p. 35]): If f is a convex function and X is arandom variable,f(X) ≥ f(EX), (2.15)where, E denotes the expectation.Lemma 2.2Fano’s Inequality ([36, p. 19]): For any estimator X̂ such that X →Y → X̂, with Pe = Pr(X ̸= X̂),H(Pe) + Pe log |X | ≥ H(Y |X̂) ≥ H(X|Y ). (2.16)This inequality can be weakened to1 + Pe log |X | ≥ H(X|Y ). (2.17)152.3. PreliminariesFigure 2.2: MAC System Model2.3.2 Multiple Access ChannelA thorough description of the discrete memoryless multiple-access chan-nel (MAC) can be found in ([52, p. 388]). Here, we provide a summaryof a result that will be needed in the thesis. A discrete memoryless chan-nel is defined by three sets (alphabets), V1, V2, and W, and a probabilitytransition matrix p(w|v1, v2). The channel is illustrated in Fig. 2.2A (2nR1 , 2nR2 ,n) code for the MAC consists of two sets of integersM1 = {1, 2, ..., 2nR1} and M2 = {1, 2, ..., 2nR2}, known as message sets, inaddition to encoding functions and decoding functions. Messages M1 ∈M1and M2 ∈M2 are encoded using functions,f :M1 → Vn1 , (2.18)and,f :M2 → Vn2 , (2.19)resulting in codewords V n1 and Vn2 which are transmitted using n uses ofthe channel. The received signal Wn after n uses of the channel is decodedusing the following function.g :Wn →M1 ×M2, (2.20)which maps the received Wn into a pair of messages (M ′1,M ′2). The ca-pacity region of MAC is the closure set of the set of achievable rate pairs(R1,R2), where a rate pair is said to be achievable if the error probabilityPr{(M1,M2) ̸= (M ′1,M ′2)} tends to zero as n tends to infinity. This capacityregion is described next.162.3. PreliminariesTheorem 2.1The capacity of a MAC (V1 × V2, p(w|v1, v2), W) is the closure of theconvex hull of all (R1,R2) satisfyingR1 < I(V1;W |V2), (2.21)R2 < I(V2;W |V1), (2.22)R1 +R2 < I(V1, V2;W ). (2.23)These preliminaries will be used in Chapter 3 and Chapter 4 to calculatethe cut-set bound and achievable rate regions of different relaying schemeslike DF, AF and QF. The MAC channel capacity will be used for DF to findthe achievable rates at the users. Next, we discuss the bench mark scenarioin detail.17Chapter 3TWRC with Global NonCausal CSI AvailibilityIn this chapter, we evaluate the transmission rate for different relayingschemes under global non causal CSI availability. The cut-set bound andachievable rates for relaying schemes are calculated under the assumptionmentioned in Section 2.2. The analysis of this benchmark scenario is per-formed for the FD and HD cases, where α = β for the FD and α ̸= β for theHD. Transmission rates are represented in the form of a function of (α, β)to accommodate both the FD and HD cases. The results are first expressedfor the FD case, and then the HD case is considered in Section 3.3.As discussed earlier in Section 1.2, the two user-pair TWRC with inter-mittency introduced in [50] only considers CF at the relay and assumes equalSNRs from each user to the relay in FD mode. In our system model, wehave considered a single user-pair TWRC with intermittent connectivity byassuming generic SNRs at all the nodes. We aim to to derive the achievablerates for AF, DF and QF transmission schemes of the relay for the FD andHD cases of intermittent-TWRC. The results for the two user-pair TWRCsetup developed in [50] can be abridged for our setup of single user-pairintermittent TWRC as follows. Any achievable rate pair (R1, R2) must liein the outer bound described by the following inequalitiesR1 ≤ min(α2log(1 +h21rPα),β2log(1 +h2r2Pβ)), (3.1)R2 ≤ min(α2log(1 +h22rPα),β2log(1 +h2r1Pβ)). (3.2)Also, rate pairs (R1, R2) satisfying the following constraints are achievableusing CFR1 ≤ min(α2log(1 +h21rPα)− 12,β2log(1 +h2r2Pβ(2− β))), (3.3)R2 ≤ min(α2log(1 +h22rPα)− 12,β2log(1 +h2r1Pβ(2− β))). (3.4)183.1. The Cut-Set BoundThe detailed derivation of the cut-set bound and achievable rates of DF,AF and QF schemes for our setup introduced in Section 2.1 is detailed next.3.1 The Cut-Set BoundIn this section, upper bounds on the transmission rates from user 1to 2 and user 2 to 1 are calculated. After that, these bounds are usedto characterize an outer bound on the capacity region of the TWRC withintermittency.In order to bound any achievable transmission rate R1 from user 1 touser 2, we derive the cut-set bound similar to [50], where instead of two-userpairs, we consider one user-pair and bound the rate from user 1 to the relayand user 2, and from user 1 and the relay to user 2. The bounds are derivedbelow for completeness. Note that while [46] considers a symmetric settingwhere h1r = h2r and hr1 = hr2, we consider the general setup here wherethese channels can be different.Following the fact that the message M1 is uniformly distributed over{1,2,· · · , 2nR1}, the number of bits in M1 which is nR1 will be the entropyof the message M1 i.e,nR1 = H(M1). (3.5)Using a similar method like the one used for multi-terminal networks in [52,p. 589], we rewrite (3.5) as followsnR1 = H(M1)−H(M1|M2, Y n2 , Y nr , Sn, Tn) +H(M1|M2, Y n2 , Y nr , Sn, Tn).(3.6)Using the definition of mutual information and using Fano’s inequality(since the messageM1 can be decoded fromM2 and Yn2 , Sn, Tn), see Lemma2.2. given the rate R1 is achievable, we can writenR1 ≤ I(M1;M2, Y n2 , Y nr , Sn, Tn) + nϵn. (3.7)where ϵn → 0 as n→∞.Applying the chain rule on the above equation, we obtainnR1 ≤ I(M1;M2, Sn, Tn) + I(M1;Y n2 , Y nr |M2, Sn, Tn) + nϵn. (3.8)The independence of M1, M2, Sn, Tn reduces the above equation tonR1 ≤ I(M1;Y n2 , Y nr |M2, Sn, Tn) + nϵn. (3.9)193.1. The Cut-Set BoundThis can be rewritten by applying chain rule as:nR1 ≤n∑i=1I(M1;Y2,i, Yr,i|M2, Sn, Tn, Y i−12 , Y i−1r ) + nϵn. (3.10)Since Xi2 can be constructed from M2, Yi−12 , Ti−1,Si−1 while on the otherhand Xir can be constructed from Yi−1r , Ti−1, Si−1, we can writenR1 ≤n∑i=1I(M1;Y2,i, Yr,i|M2, Sn, Tn, Y i−12 , Y i−1r , Xi2, Xir) + nϵn. (3.11)Applying chain rule on the above equation gives,nR1 ≤n∑i=1I(M1;Yr,i|M2, Sn, Tn, Y i−12 , Y i−1r , Xi2, Xir)+n∑i=1I(M1;Y2,i|M2, Sn, Tn, Y i−12 , Y i−1r , Xi2, Xir, Yr,i) + nϵn. (3.12)Since M1 is independent of Y2,i given the conditions in the second line of(3.12)n∑i=1I(M1;Y2,i|M2, Sn, Tn, Y i−12 , Y i−1r , Xi2, Xir, Yr,i) = 0.Therefore,nR1 ≤n∑i=1I(M1;Yr,i|M2, Sn, Tn, Y i−12 , Y i−1r , Xi2, Xir) + nϵn. (3.13)By the definition of mutual information, this can be re-written as:nR1 ≤n∑i=1h(Yr,i|M2, Sn, Tn, Y i−12 , Y i−1r , Xi2, Xir)−h(Yr,i|M2, Sn, Tn, Y i−12 , Y i−1r , Xi2, Xir,M1) + nϵn. (3.14)Since conditioning reduces entropy, we can writenR1 ≤n∑i=1h(Yr,i|Si, X2,i)− h(Zr,i) + nϵn. (3.15)203.1. The Cut-Set BoundPlugging the expression of the received signal Yr,i in above equation gives,nR1 ≤n∑i=1(h(h1rSiX1,i + Zr,i|Si)− h(Zr,i)) + nϵn,=n∑i=1p(si = 1)h(h1rX1,i + Zr,i) + p(si = 0)h(Zr,i)− h(Zr,i) + nϵn,=n∑i=1p(si = 1)(h(h1rX1,i + Zr,i)− h(Zr,i)) + nϵn.Since P{si = 1} = α, and since the above entropy is maximized whenX1,i follows a Gaussian distribution with zero mean and variance P1,i, whereP1,i is the transmission power when Si = 1 (see appendix A), we obtainnR1 ≤n∑i=1α2log(1 + h21rP1,i) + nϵn, (3.16)=n1nn∑i=1α2log(1 + h21rP1,i) + nϵn. (3.17)Using Jensen’s inequality (Lemma 2.1) we get,nR1 ≤ nα2log(1 + h21r1nn∑i=1P1,i) + nϵn. (3.18)Now, as the power constraint on X1 has to be satisfied, it is given by1nn∑i=1E[X21,i] ≤ P,By the law of large numbers, assuming n is large, we can write this constraintasαP1,i + (1− α)P ′1,i ≤ P,where P ′1,i is the transmission power when Si = 0. Note that the aboverate bound is maximized by choosing P ′1,i = 0 andP1,i =Pα. (3.19)Hence,nR1 ≤ nα2log(1 +h21rPα)+ nϵn. (3.20)213.1. The Cut-Set BoundDividing by n and letting n→∞ yields,R1 ≤ α2log(1 +h21rPα). (3.21)Similarly, an achievable transmission rate R2 from user 2 to the relay anduser 1 can be obtained as,R2 ≤ α2log(1 +h22rPα). (3.22)For simplicity, let us use the following notationsR1→r = α2log(1 +h21rPα), (3.23)andR2→r = α2log(1 +h22rPα). (3.24)which means the constraints R1 ≤ R1→r and R2 ≤ R2→r have to be satisfiedin order for (R1,R2) to be achievable.Now, we bound the transmission rate R1 of the other half of the com-munication path i.e, from user 1 and the relay to user 2. We start similarlyby writingnR1 = H(M1). (3.25)This can be re-written as,nR1 = H(M1)−H(M1|M2, Y n2 , Sn, Tn) +H(M1|M2, Y n2 , Sn, Tn). (3.26)By the definition of the mutual information and using Fano’s inequality(Lemma 2.2), we getnR1 ≤ I(M1;M2, Y n2 , Sn, Tn) + nϵn, (3.27)where ϵn → 0 as n→∞. Applying the chain rule, we obtainnR1 ≤ I(M1;M2, Sn, Tn) + I(M1;Y n2 |M2, Sn, Tn) + nϵn. (3.28)The independence of M1;M2, Sn, Tn leads tonR1 ≤ I(M1;Y n2 |M2, Sn, Tn) + nϵn. (3.29)223.1. The Cut-Set BoundBy applying chain rule, we getnR1 ≤n∑i=1I(M1;Y2,i|M2, Sn, Tn, Y i−12 ) + nϵn. (3.30)By the definition of mutual information,nR1 ≤n∑i=1h(Y2,i|M2, Sn, Tn, Y i−11 )− h(Y2,i|M2, Sn, Tn, Y i−11 ,M1) + nϵn.(3.31)After some simplification, we obtainnR1 ≤n∑i=1(h(Y2,i|Ti)− h(Z2,i)) + nϵn,=n∑i=1(p(ti = 1)h(hr2Xr,i + Z2,i) + p(ti = 0)h(Z2,i)− h(Z2,i)) + nϵn.Using Appendix A and since p(ti = 1) = β, this can be written asnR1 ≤n∑i=1β2log(1 + h2r2Pr,i) + nϵn, (3.32)= n1nn∑i=1β2log(1 + h2r2Pr,i) + nϵn. (3.33)Using Jensen’s inequality (Lemma 2.1),nR1 ≤ nβ2log(1 + h2r21nn∑i=1Pr,i) + nϵn. (3.34)Similar to the derivation of (3.19)Pr,i = P/β.Using this expression, dividing by n, and letting n→∞, we can obtainR1 ≤ β2log(1 +h2r2Pβ). (3.35)Using the same procedure for R2 yieldsR2 ≤ β2log(1 +h2r1Pβ). (3.36)233.2. Transmission SchemesAgain for simplicity, we define the following notationsRr→2 = β2log(1 +h2r2Pβ), (3.37)andRr→1 = β2log(1 +h2r1Pβ). (3.38)Combining all bounds, we obtain the following theorem for the cut-set boundof the intermittent TWRC with global CSI known at all the nodes.Theorem 3.1The capacity region C of the Gaussian FD-TWRC with intermittency isouter bounded by the region Co given byCo ={0 ≤ R1 ≤ min(R1→r,Rr→2),0 ≤ R2 ≤ min(R2→r,Rr→1).(3.39)Note that for h1r = h2r and hr1 = hr2, Theorem 3.1 coincides with (3.1)and (3.2).3.2 Transmission SchemesIn this section, we develop the achievable rates of various transmissionschemes that can be applied in this network. A relay can perform differentfunctions such as DF, AF and QF etc. These schemes are explained infollowing subsections.3.2.1 Decode-and-Forward (DF)We start with the decode and forward scheme in which X1 and X2 arethe codeword symbols being transmitted by the user 1 and user 2, respec-tively. Achievable rates of DF schemes for the conventional TWRC havebeen derived in [26]. We extend these derivation to the intermittent TWRCwith non-causal CSI available at all the nodes. In this case, since the uplinkchannel state S is available at the users, therefore, transmitted symbols arechosen to be Gaussian distributed with power P/α. Considering the as-sumptions of the benchmark scenario in which the channel state S is knownat the users, which means user 1 and user 2 only transmit when the chan-nel is available with transmit power of P/α, the DF scheme is explained inFig.3.1.243.2. Transmission Schemes𝑥11𝑥11𝑥11𝑋1𝑌𝑟 𝑋𝑟Decodeℎ1𝑟𝑆ℎ2𝑟𝑆𝑍𝑟User 1User 2 Relay𝑋2𝑋1Decode 𝑋2Decode 𝑋2Decode 𝑋1User 1User 2𝑋2𝑋1ℎ𝑟1𝑇ℎ𝑟2𝑇𝑌1𝑌2𝑍1𝑍2Figure 3.1: Decode and ForwardAt the relay, the messages are decoded jointly, leading to the rate con-straintsR1 ≤ I(X1;Yr|X2, S), (3.40)R2 ≤ I(X2;Yr|X1, S), (3.41)R1 +R2 ≤ I(X1, X2;Yr|S). (3.42)The above expressions are the extension of the MAC capacity describedin [48], but with non-causal CSI available at all the nodes instead of non-causal CSI available only at the receiver as in [48]. Evaluating these rateconstraints yields,R1 ≤ α2log(1 +h21rPα), (3.43)R2 ≤ α2log(1 +h22rPα), (3.44)R1 +R2 ≤ α2log(1 +(h21r + h22r)Pα). (3.45)After the successful decoding of the messages at the relay, the relay jointlyencodes the decoded messages into a signal Xr which is Gaussian distributedwith power P/β, which it transmits only when T = 1.253.2. Transmission SchemesThe rate constraints for the downlink transmission from the relay to eachuser given that each user has the knowledge of its own transmitted signaland the channel state, can be expressed asR1 ≤ I(X1;Y2|X2, T ), (3.46)R2 ≤ I(X2;Y1|X1, T ). (3.47)These rate constraints can be evaluated asR1 ≤ β2log(1 +h2r2Pβ), (3.48)R2 ≤ β2log(1 +h2r1Pβ). (3.49)The resulting rate constraints are summarized in the following proposition.Proposition 3.2.1The achievable rate of DF over the intermittent FD-TWRC with globalCSI available non-causally at all the nodes is given by the set of (R1,R2)pairs satisfying0 ≤ R1 ≤ min(α2log(1 +h21rPα),β2log(1 +h2r2Pβ)), (3.50)0 ≤ R2 ≤ min(α2log(1 +h22rPα),β2log(1 +h2r1Pβ)), (3.51)0 ≤ R1 +R2 ≤ α2log(1 +(h21r + h22r)Pα). (3.52)The disadvantage of DF is the need of high computation at the relay.Also, the MAC sum rate in the uplink phase limits the performance, themultiplexing loss cannot be avoided for high capacity downlink by makingthe DF sub-optimal [40, 53]. Therefore, we are going to study anotherscheme which requires no computation at the relay.3.2.2 Amplify-and-Forward (AF)For AF, the relay amplifies and forwards the received symbols to theusers. We use the derivation of the achievable rate of the AF scheme calcu-lated for the conventional TWRC [54] but adapted to our setting. When sig-nals are transmitted with non-causal CSI knowledge, the number of received263.2. Transmission Schemes𝑥11𝑥11𝑥11𝑋1𝑌𝑟 𝑋𝑟ℎ1𝑟𝑆ℎ2𝑟𝑆𝑍𝑟User 1User 2 Relay𝑋2Decode 𝑋2Decode 𝑋1User 1User 2𝑋2𝑋1ℎ𝑟1𝑇ℎ𝑟2𝑇𝜅Amplify𝑌1𝑌2𝑍1𝑍2Figure 3.2: Amplify and Forwardsymbols is the same as the number of the transmitted symbols. Thus, eachuser will send min(α, β)n symbols in n transmissions. Here it is assumedthat the users transmit at power P/min(α, β) using a random Gaussiancodebook.The relay has to perform a basic function of amplification as shown inFig.3.2. However, it only amplifies received signals when S = 1, and onlytransmits them when T = 1 where transmission takes place in the nexttransmission block (next frame). On the long term, when n is large, therewill be almost αn received symbols at the relay, and the relay can sendalmost βn symbols. So in general min(α, β)n symbols can be transmitted.Therefore the relay will scale the received signal Yr when S = 1 by a factorκ. Hence, its transmitted signal Xr will beXr = κYr, (3.53)which will be transmitted when T = 1, where κ is the power amplificationfactor chosen to ensure that the average power of signal Xr satisfies powerconstraint. i.e,E[X2r ] = κ2(h21rPα+h22rPα+ 1)≤ Pmin(α, β), (3.54)where signal with power Pα were transmitted from both the users and unit273.2. Transmission Schemesnoise power. From (2.2) in (3.53) yields toXr = κ(h1rSX1 + h2rSX2 + Zr). (3.55)The received signal at user 1 and user 2 will be exactly the same as (2.4)and (2.5) respectively.When the signals are received at their respective users, they can be easilydecoded when each user uses its own message as side information. First, eachuser will use its knowledge of Si and Ti to put the received symbols in theright order (note that the positions of the received symbols at the relay arechanged depending on Ti, since transmission takes place only when Ti = 1).Then, each user will decode its desired message using the knowledge of itsown message as side information. The rate of decoding X2 at user 1 can bewritten asR1 ≤ I(X2;Y1|X1, T ).Simplifying the above expression following the same procedure used in Sec-tion 3.1, we obtainR1 ≤ min(α, β)2log(1 +κ2h22rh2r1P(κ2h2r1 + 1)min(α, β)), (3.56)where the pre-log factor follows the fact that the relay has to transmit thesignal Xr when T = 1. Similarly, R2 can be calculated as:R1 ≤ I(X1;Y2, T |X2). (3.57)After simplification,R2 ≤ min(α, β)2log(1 +κ2h21rh2r1P(κ2h2r1 + 1)min(α, β)). (3.58)Thus the rate constraints for this scheme can be summarized as follows.Proposition 3.2.2The achievable rate of AF over the intermittent FD-TWRC with globalCSI available non-causally at all nodes is given by the set of (R1,R2) satis-fying0 ≤ R1 ≤ min(α, β)2log(1 +κ2h22rh2r1P(κ2h2r1 + 1)min(α, β)). (3.59)283.2. Transmission Schemes0 ≤ R2 ≤ min(α, β)2log(1 +κ2h21rh2r1P(κ2h2r1 + 1)min(α, β)). (3.60)In AF, no sophisticated processing is performed at the relay. It ampli-fies the received signal which means noise is also amplified and forwarded.However, no decoding of the uplink signal means relay will not incur anymultiplexing loss [40]. On the other hand, due to the accumulation of noiseand amplification, the AF scheme is generally sub-opitmal at low SNR [55]. Next, we are going to study another technique which uses a combinationof decode and forward and another scheme which is quantize and forward.3.2.3 Quantize-and-Forward (QF)We consider a hybrid QF scheme similar to the one in [44] which combinesDF and QF. However, the derivation of the achievable rate region of QFmust be modified using the constraints and assumption of the intermittentTWRC, when global CSI is available non-causally at the users and the relay.This scheme is explained next. User 1, assumed to have a stronger channelto the relay, sends a superposition of two signals at different powers, whileuser 2 sends one signal. When S = 1 the relay decodes one of the signals ofuser 1, subtracts it from the received signal, and quantizes the outcome. Itthen encodes the decoded signal and the quantization index, and forwardsthe resulting signals to the users. This is explained in details next.Transmit SignalsUser 1 splits its message M1 into two messages M11 and M12 with ratesR11 and R12 so that R11 + R12 = R1. Then it encodes M11 and M12into Xn11 and Xn12 with αn symbols which are independent and identicallydistributed (i.i.d.) N(0, P/α). User 2 encodesM2 into Xn12 with αn symbolswhich are i.i.d. N(0, P/α).The transmit signals from user 1 and user 2 canbe described as,X1 =√γ1X11 +√γ′1X12, (3.61)X2 =√γ2X21. (3.62)Where 0 ≤ γj ≤ 1 is a power allocation parameter,and γ′j = 1 − γj forj = 1, 2.As we shall see next, X11 is sent to user 2 using DF, while X12 and X21are delivered to their respective destinations using QF at the relay. Thesesignals are only transmitted when S = 1, which means αn symbols arereceived at the relay.293.2. Transmission Schemes𝑥11𝑋11𝝲1′𝝲1𝑋1𝑋21 𝑋2𝑌𝑟 𝑋𝑟DecodeQuantizeDelay−ℎ1 𝝲1ℎ1𝑟𝑆ℎ2𝑟𝑆𝑍𝑟User 1User 2 Relay𝑋12𝑀11𝑀𝑟𝝲2𝑀11𝑀12𝑀21EncoderFigure 3.3: Quantize-and-ForwardProcessing at RelayThe received signal at the relay is given by:Yr = h1rS(√γ1X11 +√γ′1X12) + h2rS√γ2X21 + Zr. (3.63)From Yr, we observe that the relay obtains a superposition of threesymbols from Gaussian codewords. The proposed scheme at the relay isas follows. The relay first decodes M11 by treating the remaining receivedsignalsX12 andX21 as noise. The relay discards erased symbols, and decodesonly from unerased symbols. This decoding is reliable as long asR11 ≤ I(X11;Yr|S). (3.64)Since user 1 transmits only when S = 1, we can rewrite the above constraintasR11 ≤αI(X11;Yr|S = 1),R11 ≤αI(X1;h1r(√γ1X11 +√γ′1X12) + h2r√γ2X21 + Zr|S = 1)Using Appendix A we can evaluate this as,R11 ≤ α2log(1 +γ1h21rPα+ γ′1h21rP + γ2h22rP). (3.65)303.2. Transmission SchemesDecode De-Quantize DecodeDecode De-Quantize Decode𝑋11 𝑋12𝑌2𝑋𝑟𝑋21𝑌1𝑋𝑟 𝑋12𝑋21User 1User 2𝑋11Figure 3.4: Decoding at the users in Quantize and ForwardIf (3.65) is fulfilled, the relay can decode M11 reliably and cancel it fromthe received signal in Yr. Thus, after subtracting the contribution of X11,we haveY ′r = h1rS√γ′1X12 + h2rS√γ2X21 + Zr. (3.66)Then quantization is performed on Y′r . The relay uses an optimal vectorquantizer of rate Rr to obtain the quantization index Mr. Then the relaycombines M11 and Mr into jointly encoded codeword Xnr with βn symbolsdistributed according to N(0, P/β), where signal Xr is only transmittedwhen channel state T is 1, which means βn symbols are transmitted in thissignal. Naturally, certain constraints of the distortion D in the quantizationprocess at relay needs to be considered using rate distortion theory [56]. Inorder to find these constraints we consider decoding.DecodingThe decoding process is explained in Fig.3.4. Note that user 1 has theadvantage that it knows M11. Thus user 1 can decode Xr using M11 asside information. The decoding of Xr at user 1 can be done with low error313.2. Transmission Schemesprobability as long asRr ≤ R(1)r = I(Xr;Y1|M11, T ),= I(Xr;hr1TXr + Z1|M11, T = 1),=β2log(1 +h2r1Pβ). (3.67)On the other hand, user 2 decodes M11 and Mr jointly, which can be donereliably if the following rate constraint is satisfiedR11 +Rr ≤ I(Xr;Y2|T ).This can be re-written asR11 ≤ I(Xr;Y2|T ),and,Rr ≤ I(Xr;Y2|T )−R11.These constraints can be evaluated toR11 ≤ β2log(1 +h2r2Pβ), (3.68)and,Rr ≤ R(2)r =β2log(1 +h2r2Pβ)−R11. (3.69)Thus, the decoding of Mr can be done with low error probability as longasRr ≤ min(R(1)r ,R(2)r ). (3.70)Now, user 1 and user 2 have decoded Mr which is the index of thequantized version Y ′r . After de-quantization [52], the users obtainYrQ = Y′r + ZQ, (3.71)where ZQ is the quantization noise with variance σ2Q which is independentof all other variables. By the rate-distortion theory [56], the rate and quan-tization noise are related throughRr > I(Y ′r ;YrQ) =12log(1 +σ2Y ′rσ2Q),323.2. Transmission Schemeswhere σ2Y ′r=(γ′1h21rPα +γ2h22rPα + 1). But since Rr ≤ min(R(1)r ,R(2)r ) from(3.70), then we must have1 +σ2Y ′rσ2Q≤ min{(1 +h2r1Pβ)β,(1 +h2r2Pβ)β2−R11}. (3.72)This leads toσ2Q ≥ max σ2Y ′r(β+h2r1Pβ)β − 1 ,σ2Y ′r(β+h2r2Pβ)β2−R11 − 1 , (3.73)which requires(β + h2r1Pβ)β> 1 and(β + h2r2Pβ)β2−R11 > 1. (3.74)If condition in (3.74) does not hold, then σ2Q can be assumed to be infinity.Otherwise, σ2Q is equal to the right-hand side of (3.73).If user 1 subtracts X12 from (3.71), we obtainYrQ = h2r√γ2X21 + Zr + ZQ. (3.75)The user then arranges the symbols of YrQ in the right order using its knowl-edge of Si, and then, it decodes M21, which can be done reliably if thefollowing constraint is satisfiedR21 ≤ α2log(1 +γ2h22rσ2Q + 1.Pα). (3.76)Similarly, when user 2 subtracts X21 from (3.71), YrQ will be reduced toYrQ = h1r√γ′1X12 + Zr + ZQ. (3.77)Then, it arranges the symbols using Si and decodes M21, leading to theconstraintR12 ≤ α2log(1 +γ′1h21rσ2Q + 1.Pα). (3.78)333.3. Extension to Half DuplexFrom (3.65) and (3.68)R11 ≤ min(α2log(1 +γ1h21rPα+ γ′1h21rP + γ2h22rP),β2log(1 +h2r2Pβ))(3.79)All the rate constraints for QF can be summarized in the form of the fol-lowing proposition.Proposition 3.2.3The rate of decoding M11,M12 and M21 in QF over the intermittentFD-TWRC, with global CSI non causally available at all the nodes can bewritten asR1 = R11 + α2log(1 +γ′1h21rσ2Q + 1.Pα), (3.80)R2 = α2log(1 +γ2h22rσ2Q + 1.Pα). (3.81)whereR11 ≤ min(α2log(1 +γ1h21rPα+ γ′1h21rP + γ2h22rP),β2log(1 +h2r2Pβ)),(3.82)σ2Y ′r =γ′1h21rPα+γ2h22rPα+ 1, (3.83)σ2Q = max σ2Y ′r(β+h2r1Pβ)β − 1 ,σ2Y ′r(β+h2r2Pβ)β2−R11 − 1, (3.84)if(β+h2r1Pβ)β> 1 and(β+h2r2Pβ)β2−R11 > 1, otherwise σ2Q =∞.We have evaluated the transmission rates of different relaying schemesfor FD-TWRC. These results can be extended to HD.3.3 Extension to Half DuplexIn this section, we express the cut-set bound and the achievable rates ofthe HD-TWRC with intermittency. Recall the system model of the TWRC343.3. Extension to Half Duplexexplained in Section 2.1, where channel states Si = Ti for the FD andSi ̸= Ti for the HD at the time instant i. This is because in the FD mode,symbols are transmitted and received at the same time and frequency bandwhereas in the HD mode, symbols are received in one time slot (or band)and transmitted in a different time slot (or band). Hence, signals from theusers are transmitted with power 2P/α in the HD mode as compare to P/αin the FD mode. Similarly, the relay transmits its signal with power 2P/β inthe HD mode instead of P/β in the FD mode. This is because we use half ofthe time (or band) in the HD mode as compared to the FD mode [57]. Theimpact of the HD constraint on the rate expressions is summarized below.Cut-Set Bound: The cut-set bound for the intermittent HD-TWRCwhen global CSI is known non-causally at all the nodes can be written as0 ≤ R1 ≤ min(α4log(1 +2h21rPα),β4log(1 +2h2r2Pβ)), (3.85)0 ≤ R2 ≤ min(α4log(1 +2h22rPα),β4log(1 +2h2r1Pβ)). (3.86)This is the same expression as for the FD case multiplied by 1/2 since halfthe time (or frequency band) is used for the uplink and the other half forthe downlink [58], and also with P/α and P/β are replaced with 2P/α and2P/β respectivelyDecode and Forward: The achievable rate of DF in the intermittentHD-TWRC can be re-written similarly as the set of (R1,R2) satisfying0 ≤ R1 ≤ min(α4log(1 +2h21rPα),β4log(1 +2h2r2Pβ)), (3.87)0 ≤ R2 ≤ min(α4log(1 +2h22rPα),β4log(1 +2h2r1Pβ)), (3.88)0 ≤ R1 +R2 ≤ α4log(1 +2(h21r + h22r)Pα). (3.89)Amplify and Forward: The achievable rate of AF over the intermit-tent HD-TWRC is given by the set of (R1,R2) satisfying0 ≤ R1 ≤ min(α, β)4log(1 +2κ2h22rh2r1P(κ2h2r1 + 1)min(α, β))), (3.90)0 ≤ R2 ≤ min(α, β)4log(1 +2κ2h21rh2r1P(κ2h2r1 + 1)min(α, β)). (3.91)353.3. Extension to Half DuplexFigure 3.5: Transmission of Symbols in HD-TWRC when CSI is availableNon-causally at all the nodes.where the amplification factor κ satisfies the following constraintκ2(h21rPα+h22rPα+12)≤ Pmin(α, β). (3.92)In (3.90) and (3.91), the factor min(α, β) represents the fraction of sym-bols received during the downlink transmission. Since channel state Sand T is known at the users, min(α, β)n symbols are received at the re-lay from the users with power 2P/min(α, β). The relay transmits onlymin(α, β)n symbols with power 2P/min(α, β). On the other hand, (3.92)is obtained because the power constraint at the relay becomes E[X2r ] ≤2P/min(a, b). Note that in this scheme, the position of the transmittedsymbols are changed by the relay as shown in Fig. 3.5 since the relay willtransmit the received symbols (first-in first-out) when the downlink channelis available (Ti = 1). However, the users can still re-arrange the receivedsymbols from the relay to the same positions as they were transmitted dueto their knowledge of Si and Ti. Thus, the users will rearrange the symbolsbefore decoding, leading the achievable rates in (3.90) and (3.91).Quantize and Forward: The achievable rate of QF over the intermit-tent HD-TWRC can be written asR1 = 12(R11 + α2log(1 +γ′1h21rσ2Q + 12Pα)), (3.93)R2 = α4log(1 +γ2h22rσ2Q + 12Pα), (3.94)363.4. Numerical IllustrationwhereR11 ≤ min(α2log(1 +2γ1h21rPα+ 2γ′1h21rP + 2γ2h22rP),β2log(1 +2h2r2Pβ)),(3.95)σ2Y ′r =2γ′1h21rPα+2γ2h22rPα+ 1, (3.96)σ2Q = max σ2Y ′r(β+h2r12Pβ)β − 1 ,σ2Y ′r(β+h2r22Pβ)β2−R11 − 1, (3.97)if(β+h2r12Pβ)β> 1 and(β+h2r22Pβ)β2−R11 > 1, otherwise σ2Q =∞.It is worth to note that the above rate expressions for QF in HD-TWRCare same as the FD case, except that the achievable rate is half of the FD-TWRC since half the time (or frequency band) is used for each phase, i.e.,the uplink and the downlink phases. The transmit power from the users is2P/α instead of P/α and the transmit power from the relay is 2P/β insteadof P/β. Similar to (3.74), the denominator in (3.97) should be greater thanzero otherwise σ2Q is assumed to be infinity. On the other hand, similar to AFthe QF process a the relay will change the order of the symbols transmittedfrom the users, because the relay will only compress the received symbolswhen Si = 1, and will only forward the compression index when Ti = 1(recall that Si ̸= Ti in this case). However, the users are still able toarrange the received symbols in the right position before decoding usingtheir knowledge of Si.Next we are going to preview and discuss numerical evaluations of theachievable rates over the intermittent TWRC with global CSI available noncausally at all the nodes.3.4 Numerical IllustrationIn order to analyze the numerical results of the TWRC, we considerSNR1r = h21rP , SNR2r = h22rP , SNRr1 = h2r1P , and SNRr2 = h2r2P . Wecompare between relaying schemes varying values of SNR. It is assumed thatusers are placed equidistant from the relay which means SNR1r = SNR2rand SNRr1 = SNRr2 for the sake of simplicity of presentation and toprovide a comparison of the CF scheme presented in [50] with our results373.4. Numerical Illustration(a) SNR1r = SNR2r = 5dB andSNRr1 = SNRr2 = 15dB at α =β = 0.85(b) SNR1r = SNR2r = 25dB andSNRr1 = SNRr2 = 15dB at α =β = 0.60Figure 3.6: Full-Duplex TWRC with intermittencyof DF, AF and QF schemes (noting that the results presented above aregeneral and apply for the general case).In Fig. 3.6 (a), the achievable rate regions of the FD-TWRC whereα = β are plotted when the uplink channel is weaker than the downlinkchannel. Here, we can see that AF has achieved rate close to the cut-setbound because the downlink SNR is high, which makes the relay capable offorwarding Yr to the users with high accuracy (relatively small noise poweras compare to the forwarded amplified received signal at the relay), whichenables users to achieve rates close to the uplink cut-set bounds.Similarly, QF came out to be the second most efficient scheme since thehigh SNR of the downlink channel allows quantization at higher rates atthe relay, which reduces distortion and allows users to achieve rates closeto the uplink cut-set bound. Note that the achievable region describedin Proposition 3.2.3 is denoted QF-1 in the Fig. 3.6. While the shapesof the achievable rate regions of DF and AF are symmetric with respectto the line R1 = R2, The shape of achievable rate region of QF is notsymmetrical because it is designed asymmetrically. Specifically, QF-1 favorsuser 1 whose message is split to two parts which grants additional flexibility(see Section. 3.2.3). Due to this, we also plot the achievable rate regionwhen user 2 is favored (denoted QF-2 in the figure) whose rate is similarto that in Proposition x with the subscripts 1 and 2 swapped, in additionto the convex hull of the union of the achievable regions of QF-1 and QF-2which is the overall achievable region of QF (denoted QF in the figure). On383.4. Numerical Illustration(a) α = β = 0.90 (b) α = β = 0.60Figure 3.7: Full-Duplex TWRC with intermittency at SNR1r = SNR2r =SNRr1 = SNRr2 = 25dBthe other hand, the achievable region of DF has a big gap from the cut-setbound as the sum rate constraint during the uplink transmission degradesits performance.However, in Fig. 3.6 (b) results are plotted when the uplink channel isstronger than the downlink channel. It is noted that DF attains achievablerates closer to the cut-set bound. Since, the uplink channel is strongerand the downlink channel is the bottleneck, the deteriorating effect of thesum rate constraint is minimized. For AF, the scaling factor κ is inverselyrelated to the uplink SNR which results in bigger gap from the cut-setbound. Similarly, the QF rate is low compared to the cutset bound becauseof the low downlink SNR which reduces the quantization rate leading tohigher distortion. We have compared the results of achievable rates of ourscheme with the achievable rates of the inner bound of CF summarized in(3.3) and (3.4) as plotted in Fig. 3.6, it shows that the performance of CFscheme designed for intermittent TWRC in [50] gets better by increasing theuplink SNR while intermittency is high in Fig. 3.6(b) as compare to Fig.3.6(a) as the lattice codewords are forwarded in uplink channel by utilizingthe knowledge of CSI and CF achieves a rate closer to cut-set bound byincreasing the SNR.Fig. 3.7 shows numerical results for the FD-TWRC when all SNRs areequal. Note that since the uplink and downlink SNR are equal, only thechannel intermittency probabilities α and β can affect the performance ofthe schemes. As significant difference is noted for the QF scheme which givesthe better performance when signals are transmitted on a less intermittent393.4. Numerical Illustration(a) SNR1r = SNR2r = 15dB andSNRr1 = SNRr2 = 25dB(b) SNR1r = SNR2r = 25dB andSNRr1 = SNRr2 = 15dBFigure 3.8: Maximum sum rate of FD-TWRC at different values of inter-mittency, α = βchannel. When α = β = 0.90, QF has a small gap from the cut-set boundand the gap increases with the increase in the level of intermittency i.eα = β = 0.60. This behaviour shows that the QF scheme is affected mostby the intermittency as the quantization noise increases with the increase inintermittency. Another significant difference is noticed for CF, in which thegap of the inner bound of CF from the cut-set bound increases significantlywith increase the intermittency (lower values of α = β) This shows thatthe performance of lattice encoding used in CF degrades with increasingintermittency. Though CF gives better performance than AF at α = β = 0.9but its performance decreases with increasing the intermittency α = β = 0.6.The sum-rate performance of the transmission schemes at different val-ues of intermittency i.e. α is shown in Fig. 3.8 for the FD TWRC. Whenthe downlink channel is stronger than the uplink channel, AF gives the bestperformance as compared to DF which is affected most by the intermittencydue to the sum rate constraint, and the performance of QF which is affectedby the large quantization noise decreases when α = β < 1. On the otherhand, CF gives the best performance for almost all α when the uplink chan-nel is stronger than the downlink channel as the lattice encoded symbols aretransmitted with high SNR during the uplink phase which makes the uplinkphase stronger as shown in (3.3) and (3.4) i.e., h21r/α > h2r2/β(2 − β). DFperform close to CF, as the deteriorating effect of the sum rate constraintduring the uplink transmission is minimized. AF still performs well in this403.4. Numerical Illustration(a) FD-TWRC α = β = 0.6 (b) HD-TWRC α = 0.75 and β =0.85Figure 3.9: Achievable rates at SNR1r = 20dB, SNRr1 = 18dB, SNR2r =22dB, and SNRr2 = 16dB.case when alpha is small (high intermittency).As we have discussed the cases where our results can be compared withthe results of CF from [50]. Now, we will discuss the limitations of the systemmodel of [50] which only considers SNR1r = SNR2r and SNRr1 = SNRr2for the FD-TWRC. We have derived the achievable rates of AF, DF andQF by assuming generic values of SNRs. We have also extended our resultsto the HD-TWRC. The results in [50] do not apply directly to the HD caseeven when SNR1r = SNR2r and SNRr1 = SNRr2 since in the HD case Sinot equal Ti, whereas [50] assumes they are equal. Thus, we do not comparewith CF in the HD case, bearing in mind that the AF scheme demonstratedsufficiently good performance close to CF in the FD case and it is expectedthat the same is true in the HD case. In Fig. 3.9, the results of achievablerates are plotted for different values of SNRs at each node. A comparisonof FD and HD is provided for unequal values of SNRs. The performance ofeach scheme is decreased almost equally for the HD-TWRC as the numberof symbols transmitted in HD mode are half of the symbols transmitted inFD mode.Similarly, we have also analyzed relaying schemes for the HD-TWRCwhen users are placed equidistant from the relay i.e. SNR1r = SNR2r andSNRr1 = SNRr2. In Fig. 3.10 (a), it is noticed that the achievable rateof AF is very close to the cut-set bound because the downlink SNR is largeand also β is large, which makes the uplink the bottleneck, whereas AF canforward the received signal at the relay to the users with low distortion.413.4. Numerical Illustration(a) SNR1r = SNR2r = 15dB andSNRr1 = SNRr2 = 25dB(b) SNR1r = SNR2r = 25dB andSNRr1 = SNRr2 = 15dBFigure 3.10: Half-Duplex TWRC with intermittency at α = 0.45 and β =0.85Fig. 3.10 (b) shows the results when the uplink channel is stronger than thedownlink channel. It means the negative affect of the sum rate constraintis minimized in DF, the performance of AF and QF is degraded when thedownlink channel is weak.In Fig. 3.11(a), the performance of transmission scheme is plotted byconsidering the uplink channel with intermitency i.e. at different values ofα and no intermittency during the downlink transmission i.e. β = 1. It isnoticed that with no intermittency and high SNR during the downlink com-munication makes the AF scheme, the best as the achievable rates of AFonly depend on the amplification factor during the downlink transmission.On the other hand, in Fig. 3.11(b), the performance is plotted under nointermittency in the uplink channel but with intermittency in the downlinkchannel. In this case, QF gives the best performance at low intermittency(large β) since the quantization rate becomes large leading to a small quan-tization distortion.In summary, in the FD-TWRC when global CSI is available non-causallyat all nodes, the AF scheme achieves the best performance for differentvalues of intermittency i.e. α = β, when the downlink SNR is equal orhigher than the uplink SNR. However, DF gives the best performance whenthe uplink SNR is higher than the downlink SNR. On the other hand, inthe HD-TWRC, DF and QF achieve the best performance at different levelsof downlink intermittency when the uplink channel is not intermittent andthe uplink SNR is larger or smaller then the downlink SNR. However, DF423.4. Numerical Illustration(a) No intermittency during thedownlink transmission, β = 1(b) No intermittency during theuplink transmission, α = 1Figure 3.11: Maximum sum rate of HD-TWRC at different values of inter-mittency at SNR1r = SNR2r = 10dB and SNRr1 = SNRr2 = 20dBand AF achieve the best performance when the downlink SNR is equalto the uplink SNR. AF gives the best performance for all high values ofSNR irrespective of the strength of the uplink and the downlink channel, atdifferent values of α.In next chapter, we study the intermittent TWRC when global CSI isknown noncausally at the relay and local CSI is known causally at the users,as well as when local CSI is known causally at all the nodes.43Chapter 4TWRC with Practical CSIAssumptionsIn this chapter, we study the intermittent TWRC under more practi-cal assumptions regarding the availability of CSI as defined in Section 2.2.Firstly, the cut-set bound and transmission rates are calculated under theassumption of non-casual global CSI knowledge at the relay and causal lo-cal CSI knowledge at the users. Secondly, the case with local CSI knowncausally at the relay and the users is considered. We focus on the FD-TWRCwhen α = β and then extend our study to the HD-TWRC when α ̸= β foreach scenario.Recall the [50], discussed in Section 1.2, studies on CSI assumptions:instantaneous CSI (discussed in Chapter 2) and delayed CSI (which shallbe discussed in this chapter’s Section 4.2). We have considered another dif-ferent and a practical assumption on CSI availability, which has not beeninvestigated for the TWRC before. We assume that non-causal CSI is avail-able at the relay and causal CSI is available at the users. The achievablerates for this scheme are provided next.4.1 TWRC with Non-Causal CSI at the Relayand Causal CSI at the UsersIn this scenario, it is assumed that global CSI is known non-causallyat the relay and causally at the users, which means channel state Si andTi are known non-causally at the relay and only Ti is known causally atthe users. This is the case when the relay can infer the blockage from itslocation such as when the relay is a UAV, or when the relay is performinga primary task of serving other nodes in the network and a secondary taskby acting as a two-way relay for the TWRC users. Though the results of[50] do not apply under this assumption, we will derive the cut-set boundand the achievable rates of the transmission schemes for the FD-TWRCby modifying the transmission schemes and derivation of achievable rates444.1. TWRC with Non-Causal CSI at the Relay and Causal CSI at the Usersprovided for benchmark scenario in Chapter 3 and extend our results laterto the HD mode.4.1.1 Cut-Set BoundRecall the derivation of cut-set bound for the benchmark scenario (non-causal global CSI known everywhere) in Section 3.1, where the rates oftransmission from user 1 to user 2 and vice versa are computed. It is quiteclear from derivation of the downlink cut-set bounds (from the relay to theusers) will be exactly the same as for the bench mark scenario. However, forthe uplink, the uplink channel state T is known at the relay only but not atthe users. The lack of the uplink CSI at the users results in the transmissionof symbols X1 and X2 with full power P all the time instead of power P/αduring a fraction alpha of the time, as in the benchmark scenario. Therefore,the cut-set bound derived in the previous chapter can be changed as givenin the following theorem.Theorem 4.1The capacity region C of the intermittent Gaussian FD-TWRC whenglobal CSI is known non-causally at the relay and local CSI known causallyat the user is outer bounded by the region Co given byCo =0 ≤ R1 ≤ min(α2 log(1 + h21rP), β2 log(1 +h2r2Pβ),0 ≤ R2 ≤ min(α2 log(1 + h22rP), β2 log(1 +h2r1Pβ).(4.1)4.1.2 Transmission SchemesThe achievable rates of different relaying schemes in the TWRC under ofnon-causal global CSI knowledge at the relay and causal local CSI knowledgeat the users is studied below.Decode and Forward: In DF similar to Section 3.2, under noncausalglobal CSI knowledge at the relay and causal local CSI knowledge at theusers, it is noticed that all the rate expressions will remain the same exceptfor the uplink transmit power which becomes P instead of P/α. Thesechanges in achievable rates can be summarized in the following proposition.Proposition 4.1.1The achievable rate of DF over the intermittent FD-TWRC with globalCSI available non-causally at the relay and local CSI known at the users is454.1. TWRC with Non-Causal CSI at the Relay and Causal CSI at the Usersgiven by the set of (R1,R2) pairs satisfying0 ≤ R1 ≤ min(α2log(1 + h21rP),β2log(1 +h2r2Pβ)), (4.2)0 ≤ R2 ≤ min(α2log(1 + h22rP),β2log(1 +h2r1Pβ)), (4.3)0 ≤ R1 +R2 ≤ α2log(1 + (h21r + h22r)P). (4.4)Amplify and Forward: In AF, users transmit n symbols with anN(0, P ) distribution but only αn symbols are received at the relay andthese symbols are amplified with a factor κ such that the power constraintat the relay is satisfied, i.e.,E[X2r ] = κ2(h21rP + h22rP + 1) ≤Pmin(α, β). (4.5)Note that since the number of the received and transmitted symbols shouldbe the same, min(α, β)n symbols are transmitted from the relay, whichenables the relay to transmit at power P/min(α, β). Also, note that sincein the FD mode Si = Ti, this means that the users know both Si and Ticausally which allows them to arrange the received symbols in the rightorder for decoding. Thus, the rate expressions will remain the same asfor the benchmark scenario except for the transmission power at the users,which will become P instead of P/α as shown in (4.5). Thus, the achievablerates of AF become as follows.Proposition 4.1.2The achievable rate of AF over the intermittent FD-TWRC with non-causal global CSI knowledge at the relay and causal local CSI knowledge atthe users is given by the set of (R1,R2) satisfying0 ≤ R1 ≤ min(α, β)2log(1 +κ2h22rh2r1Pκ2h2r1 + 1), (4.6)0 ≤ R2 ≤ min(α, β)2log(1 +κ2h21rh2r1Pκ2h2r1 + 1). (4.7)where k must satisfies the constraint in (4.5)Quantize and Forward: In QF, when non-causal global CSI is knownat the relay and causal local CSI is known at the users, the rate can be464.1. TWRC with Non-Causal CSI at the Relay and Causal CSI at the Usersanalyzed similar to the Section 3.2. It is noted that all the constraintswill remain the same except the uplink transmit power which becomes Pinstead of P/α due to the unavailability of channel state Si at the usersduring the uplink transmission. The users still need to arrange the symbolsof the decompressed signal from the relay in the right order before decoding,which they can do since Ti = Si in the full duplex case, and since users knowTi causally. The achievable rates can thus be expressed as follows.Proposition 4.1.3The achievable rate for QF in the intermittent FD-TWRC with non-causal global CSI at the relay and causal local CSI at the users can bewritten as a set of (R1, R2) satisfyingR1 = R11 + α2log(1 +γ′1h21rPσ2Q + 1), (4.8)R2 = α2log(1 +γ2h22rPσ2Q + 1). (4.9)whereR11 ≤min(α2log(1 +γ1h21rP1 + γ′1h21rP + γ2h22rP),β2log(1 +h2r2Pβ)),(4.10)σ2Y ′r =γ′1h21rP + γ2h22rP + 1, (4.11)σ2Q =max σ2Y ′r(β+h2r1Pβ)β − 1 ,σ2Y ′r(β+h2r2Pβ)β2−R11 − 1, (4.12)if(β+h2r1Pβ)β> 1 and(β+h2r2Pβ)β2−R11 > 1, otherwise σ2Q =∞. Next, weextend our results to HD case.4.1.3 Extension to Half DuplexIn this section, we extend the results evaluated in previous section to theHD-TWRC. In the HD-TWRC, the channel states Si ̸= Ti which impactsthe transmission rates significantly. Contrary to the benchmark scenario inthe FD mode where users know Si causally through their knowledge of Ti474.1. TWRC with Non-Causal CSI at the Relay and Causal CSI at the UsersFigure 4.1: Transmission of symbols in HD-TWRC when CSI is availablenon-causally at the relay and causally at the users.since Si = Ti, in the HD TWRC, this is not the case. Users do not knowSi, and cannot obtain it from Ti. In this case, symbols are transmitted fromeach user regardless of the channel state (all n symbols are transmitted,first-in first-out). On the other hand, the relay knows the downlink channelstate and it will only transmits when Ti = 1, which results in changing theindices of symbols as shown in Fig. 4.1 This will affect the AF and QFschemes significantly, where the users will not be able to arrange the receivedsymbols from the relay in the right order due to their lack of knowledge ofSi. Thus, another method has to be found to resolve this issue, or the relayshould not be allowed to change the order of symbols. These issues arediscussed next. But first, we present the cut-set bound for this case.Cut-Set Bound: The cut-set bound will have a similar expressions asin Section 3.3, except that the uplink transmission power will be 2P insteadof 2P/α as the channel state Si is unknown before each transmission at theusers. Therefore, the rate pair (R1,R2) for HD-TWRC can be re-written as:0 ≤ R1 ≤ min(α4log(1 + 2h21rP),β4log(1 +2h2r2Pβ)), (4.13)0 ≤ R2 ≤ min(α4log(1 + 2h22rP),β4log(1 +2h2r1Pβ)). (4.14)Decode and Forward: In DF, when global CSI is available at the relayand local CSI is available at the nodes, User 1 and User 2 transmit theirsymbols with power 2P , instead of P/α since they do not know the uplinkintermittency states Si and since they are active half of the time (or band).The relay transmits with power 2P/β since it knows Ti before transmission.The achievable rate of DF in the intermittent HD-TWRC can be re-written484.1. TWRC with Non-Causal CSI at the Relay and Causal CSI at the Usersas the set of (R1,R2) satisfying0 ≤ R1 ≤ min(α4log(1 + 2h21rP),β4log(1 +2h2r2Pβ)), (4.15)0 ≤ R2 ≤ min(α4log(1 + 2h22rP),β4log(1 +2h2r1Pβ)), (4.16)0 ≤ R1 +R2 ≤ α4log(1 + 2(h21r + h22r)P). (4.17)Amplify and Forward: In AF in this scenario, each user transmits nsymbols with power 2P , and only αn symbols are received at the relaywhich means 1− α symbols are lost during the uplink transmission. Thesereceived symbols are amplified at the relay and forwarded to the users whenTi = 1, which means min(α, β)n symbols are received at the users. However,since the users do not know Si, they have no way of putting these min(α, β)nsymbols in the right order as they were transmitted for decoding to work.Therefore, the channel state Si is transmitted by the relay to the usersin the form of an encoded signal with symbols Xh which is distributedaccording to N(0,2P/β). This signal will be added to the amplified symbolsand transmitted from the relay to the users. The rate of encoding {Si}ni=1to Xh is equal to the entropy of S, i.e.,Rh = H(S) = −α log(α)− (1− α) log(1− α). (4.18)The power constraint at the relay will be satisfied ifκ2(2h21rP + 2h22rP + 1) ≤2Pmin(α, β). (4.19)The relay will transmit Xr given asXr =√γrκYr +√1− γrXh. (4.20)where γr ∈ [0, 1] is the power allocation parameter. Note that only min(α, β)nsymbols Yr are forwarded, whereas the header message is encoded into βnsymbols. If α < β, then some symbols of Xr will not contain a contributionfrom Yr. The relay will transmit Xr when Ti = 1 since global CSI is knownnon-causally at the relay. Then, at the receiver side, user 1 will decode Xhfrom Y1 which leads to the rate constraintRh ≤ β2log(1 +2(1− γr)h2r1P(2γrκ2(h21r + h22r)h2r1P + γrκ2h2r1 + 1)β). (4.21)494.1. TWRC with Non-Causal CSI at the Relay and Causal CSI at the UsersAfter successfully decoding Xh, user 1 can arrange the symbols of Y1 in theright order as the symbols were transmitted by the users, and then decodeX2 using the knowledge of Xh and X1. The rate of decoding X2 can bewritten asR1 ≤ 12I(X2;Y1|X1, Xh, T ), (4.22)=min(α, β)4log(1 +2γrκ2h22rh2r1Pγrκ2h2r1 + 1), (4.23)where the factor 1/2 is introduced to the the HD operation. Similarly, therate of decoding Xh at user 2 can be written asRh ≤ β2log(1 +2(1− γr)h2r2P(2γrκ2(h21r + h22r)h2r2P + γrκ2h2r2 + 1)β), (4.24)and then X1 can be decoded from Y2 at user 2 after using Xh to arrangethe received signal Y2 in the right order leading to the achievable rateR2 ≤ 12I(X1;Y2|X2, Xh, T ), (4.25)=min(α, β)4log(1 +2γrκ2h22rh2r2Pγrκ2h2r2 + 1). (4.26)Note that these rates R1 and R2 are achievable only when Rh in (4.18)satisfies the constraints in (4.23) and (4.24) otherwise this scheme fails.Quantize and Forward: In QF for the HD-TWRC with causal localCSI at the users, users transmit their symbols X1 =√γ1X11+√γ′1X12 andX2 =√γ2X21, where X11, X12, and X21 follow an N(0, 2P ) distribution.M11 is decoded at the relay first, which can be done reliably if the ratesatisfiesR11 ≤ α2log(1 +2γ1h21rP1 + 2γ′1h21rP + 2γ2h22rP). (4.27)This rate is written in terms of bits/symbol of the uplink transmission,which will be divided by 2 later to represent the HD constraint. The relaysubtracts the contribution of X11, and quantizes the remaining part of thereceived signal is message Mr. This message will be sent to the users, whowill de-quantize it and obtain the αn received symbols at the relay withadditional distortion. However, the users also need to put these symbols inthe right order to be able to decode their desired messages. To do this, theywill need Si. Since, channel state Si is known at the relay, the relay encodes504.1. TWRC with Non-Causal CSI at the Relay and Causal CSI at the Users{Si}ni=1 in the form of a header message Mh with the rate of encoding Rhgiven byRh = H(S) = −α log(α)− (1− α) log(1− α). (4.28)The relay now jointly encodes M11, Mr and Mh into Xnr which follows ani.i.d. N(0, 2Pβ ) distribution, and transmits it to the users. User 1 will jointlydecode Mr and Mh using M11 as the side information, leading to the rateconstraintRr +Rh ≤ I(Xr;Y1|M11, T ). (4.29)This can be rewritten asRh ≤ I(Xr;Y1|M11, T ), (4.30)Rr ≤ I(Xr;Y1|M11, T )−Rh, (4.31)Using the Gaussian distribution of Xr, we obtainRh ≤ β2log(1 +2h2r1Pβ), (4.32)andRr ≤ β2log(1 +2h2r1Pβ)−Rh. (4.33)On the other hand, user 2 jointly decodes Mr, Mh and M11 , which can bedone reliably if the following rate constraint is satisfiedRr +Rh +R11 ≤ I(Xr;Y2|T ). (4.34)This can be re-written asRh ≤ I(Xr;Y2|T ), (4.35)R11 ≤ I(Xr;Y2|T )−Rh, (4.36)Rr ≤ I(Xr;Y2|T )−Rh −R11. (4.37)Evaluating the mutual information expression using the Gaussian distribu-tion of Xr yieldsRh ≤ β2log(1 +2h2r2Pβ), (4.38)R11 ≤ β2log(1 +2h2r2Pβ)−Rh, (4.39)514.1. TWRC with Non-Causal CSI at the Relay and Causal CSI at the UsersandRr ≤ β2log(1 +2h2r2Pβ)−Rh −R11. (4.40)Hence, the message Mh can be decoded with low error probability as longasRh ≤ min(β2log(1 +2h2r1Pβ),β2log(1 +2h2r2Pβ))(4.41)where Rh is as given in (4.28). Note that if the rate Rh defined in (4.28)does not satisfy (4.41) , the scheme fails.Finally, using the same procedure as in (3.71) - (3.78) but under the CSIassumption considered in this section, the achievable rates of QF over theintermittent HD-TWRC, with global CSI non causally available at the relayand local CSI known at the users can be written asR1 = R112+α4log(1 +2γ′1h21rPσ2Q + 1), (4.42)R2 = α4log(1 +2γ2h22rPσ2Q + 1). (4.43)whereR11 ≤min(α2log(1 +2γ1h21rP1 + 2γ′1h21rP + 2γ2h22rP),β2log(1 +2h2r2Pβ)−Rh),(4.44)Rh =− α log(α)− (1− α) log(1− α) (4.45)σ2Y ′r =2γ′1h21rP + 2γ2h22rP + 1, (4.46)σ2Q =max σ2Y ′r(β+h2r12Pβ)β2−Rh − 1,σ2Y ′r(β+h2r22Pβ)β2−R11−Rh − 1, (4.47)if(β+h2r12Pβ)β2−Rh > 1 and(β+h2r22Pβ)β2−R11−Rh > 1, otherwise σ2Q =∞.Next, we will study another practical assumption on the CSI availability inthe TWRC.524.2. TWRC with Local CSI Available Causally at all Nodes4.2 TWRC with Local CSI Available Causally atall NodesIn this section we will discussed a scenario, in which local CSI is knowncausally at all nodes, which means channel state Si is known at the relay andTi is known at the users at the end of transmission i. Using this assumption,we are going to evaluate cut-set bound and the achievable rates of the DF,AF and QF schemes. A similar assumption has been used for the two user-pair intermittent TWRC in [50], where inner and outer bound were derivedwhile assuming equal SNRs from each user to the relay and vice versa.We summarize the reults from [50] when re-written for the single user-pairsetting of the TWRC next. Any achievable rate pair (R1, R2) must lie inthe outer bound described by the constraintsR1 ≤ min(α2log(1 + h21rP),β2log(1 + h2r2P)), (4.48)R2 ≤ min(α2log(1 + h22rP),β2log(1 + h2r1P)). (4.49)Also, any rate pair satisfying the following inequalities is achievable usingCFR1 ≤ min(α2log(1 + h21rP)− △12,β2log(1 + h2r2P)− △22), (4.50)R2 ≤ min(α2log(1 + h22rP)− △12,β2log(1 + h2r1P)− △22), (4.51)where(△1,△2) =(1,β(1− β)3− β log 3 +β3− β log2πe12). (4.52)4.2.1 Cut-Set BoundUnder the assumptions of this section, channel states are unknown beforethe transmission of each signal. Therefore all the signals will be transmittedwith power P . The cut-set bound can be re-written as follows.Theorem 4.2The capacity region C of the intermittent Gaussian FD-TWRC whenlocal CSI is known at all the nodes is outer bounded by the region Co given534.2. TWRC with Local CSI Available Causally at all NodesbyCo ={0 ≤ R1 ≤ min(α2 log(1 + h21rP ), β2 log(1 + h2r2P ),0 ≤ R2 ≤ min(α2 log(1 + h22rP ), β2 log(1 + h2r1P ).(4.53)This outer bound exactly coincides with the one in (4.48) and (4.49).4.2.2 Transmission SchemesIn this section, we will discuss transmission schemes for the intermittentFD-TWRC.Decode and Forward: In DF when local CSI is known causally atall the nodes, it is obvious that the signals are transmitted with power Pfor both the uplink and the downlink transmissions instead of P/α andP/β respectively in the bench mark scenario. The effect of this change intransmission power is depicted in the following proposition.Proposition 4.2.1The achievable rate of DF over the intermittent FD-TWRC with localCSI available causally at all the nodes is given by the set of (R1,R2) pairssatisfying0 ≤ R1 ≤ min(α2log(1 + h21rP ),β2log(1 + h2r2P )), (4.54)0 ≤ R2 ≤ min(α2log(1 + h22rP ),β2log(1 + h2r1P )), (4.55)0 ≤ R1 +R2 ≤ α2log(1 + (h21r + h22r)P ). (4.56)Amplify and Forward: As compared to Section 3.2 (benchmark) andSection 4.1 (noncausal global CSI at the relay and causal local CSI at theusers), a significant difference in the rate expressions will be caused heredue to the limited CSI. When n symbols are transmitted from the userswith an N(0, P ) distribution, αn symbols are received at the relay due tothe intermittency. The relay transmits the received amplified symbols withpower P/α due to the unavailabilty Ti at the relay at time i. Note thatthe relay does not change the order of the symbols in this case, rather itamplifies each received symbol and forward it in the next time slot, whichdoes not change the order of symbols. Thus only αβn symbols are received544.2. TWRC with Local CSI Available Causally at all Nodesat the users during the downlink transmission. The amplification factor κat the relay must satisfyE[X2r ] = κ2(h21rP + h22rP + 1) ≤Pα. (4.57)Since no symbol re-ordering needs to be done at the users, each user de-codes the desired message while using its own message as side information.However, this decoding is done from a received signal that contains onlyαβn symbols.The achievable rates for this scheme are summarized in thefollowing proposition.Proposition 4.2.2The achievable rate of AF over the intermittent FD-TWRC with localCSI available causally at all the nodes is given by the set of (R1,R2) satis-fying0 ≤ R1 ≤ αβ2log(1 +κ2h22rh2r1Pκ2h2r1 + 1), (4.58)0 ≤ R2 ≤ αβ2log(1 +κ2h21rh2r1Pκ2h2r1 + 1), (4.59)where k must satisfy the power constraint in (4.57)Quantize and Forward: In QF, similar to the previous calculations ofthe achievable rates under the assumptions made in Section 4.1, the uplinktransmit power will remain the same (P ) but the downlink transmit powerP/β will change to P . The reason is the unavailability of Si at the usersand Ti at the relay before each transmission at time instant i. Note thatthe relay change the order of the symbols in this case, as Si = Ti in theFD mode which means that the users know both Si and Ti causally whichallows them to arrange the received symbols in the right order for decoding.We can summarize the achievable rates in the following proposition.Proposition 4.2.3The achievable rate for QF, when local CSI is known at all the nodes inFD-TWRC can be written as a set of (R1, R2) satisfyingR1 ≤ R11 + α2log(1 +γ′1h21rPσ2Q + 1), (4.60)R2 ≤ α2log(1 +γ2h22rPσ2Q + 1). (4.61)554.2. TWRC with Local CSI Available Causally at all NodeswhereR11 ≤min(α2log(1 +γ1h21rP1 + γ′1h21rP + γ2h22rP),β2log(1 + h2r2P)),(4.62)σ2Y ′r =γ′1h21rP + γ2h22rP + 1, (4.63)σ2Q =max{σ2Y ′r(1 + h2r1P )β − 1 ,σ2Y ′r(1 + h2r2P )β2−R11 − 1}, (4.64)if (1 + h2r2P )β2−R11 > 1 and (1 + h2r1P )β > 1, otherwise σ2Q =∞. Next, wewill extend these results to the HD case.4.2.3 Extension to Half DuplexIn this section, we will extend the results to the HD mode, when onlylocal CSI is known causally at all the nodes.Cut-Set Bound: In the intermittent HD-TWRC, channel state Si ̸= Tiat time instant i which means all the nodes receive the symbol at one instanti and transmit the symbol in next instant i+ 1, irrespective of the channelavailability as CSI is unknown before the transmission. Therefore, X1 andX2 will be transmitted with power 2P instead of 2P/α and Xr with power2P instead of 2P/β in comparison with the benchmark scenario. Otherwise,the rate expressions of the cut-set bound will be exactly same as describedin Section 3.3. Therefore, the rate pair (R1,R2) for HD-TWRC can bere-written as:0 ≤ R1 ≤ min(α4log(1 + 2h21rP ),β4log(1 + 2h2r2P )), (4.65)0 ≤ R2 ≤ min(α4log(1 + 2h22rP ),β4log(1 + 2h2r1P )). (4.66)Decode and Forward: In DF for HD-TWRC when local CSI is knownat all the nodes. The rate expression will be the same as Section 4.1.3,except the downlink transmit power will be 2P instead of 2P/β. Therefore,the achievable rate (R1,R2) for the DF in intermittent HD-TWRC can be564.2. TWRC with Local CSI Available Causally at all Nodeswritten as0 ≤ R1 ≤ min(α4log(1 + 2h21rP ),β4log(1 + 2h2r2P )), (4.67)0 ≤ R2 ≤ min(α4log(1 + 2h22rP ),β4log(1 + 2h2r1P )), (4.68)0 ≤ R1 +R2 ≤ α4log(1 + 2(h21r + h22r)P ). (4.69)Amplify and Forward: In HD-TWRC when local CSI is known causallyat all the nodes, each user transmits n symbols with power 2P and only αnsymbols are received at the relay. Since the relay does not know Ti beforetransmission in time slot i, and hence cannot rearrange the amplified re-ceived signals in a useful way so that they are sent when Ti = 1. Instead,the relay amplifies the received symbols and forwards them in the same or-der as they are received, without observing whether Ti = 1 or Ti = 0. Therelay transmits these symbols with transmit power 2P/α since it will onlytransmit αn symbols and is active only half of the time. Clearly, not all αnsymbols are received at the users, but rather only αβn. Thus the achievablerate of AF over the intermittent HD-TWRC is given by the set of (R1,R2)satisfying0 ≤ R1 ≤ αβ4log(1 +2κ2h22rh2r1Pκ2h2r1 + 1)(4.70)0 ≤ R2 ≤ αβ4log(1 +2κ2h21rh2r1Pκ2h2r1 + 1)(4.71)where the amplification factor κ should satisfy the following rate constraintκ2(h21rP + h22rP +12) ≤ Pα. (4.72)Quantize and Forward: Similar to the QF discussed in Section 4.1.3,we need a header message Mh and all the rate expressions will remain thesame except for the downlink transmit power which will be 2P instead of2P/β because relay is transmitting irrespective of the channel availability.The header message Mh will be decoded at the users with low error proba-bility as long asRh ≤ min(β2log(1 + 2h2r1P ),β2log(1 + 2h2r2P )), (4.73)574.3. Numerical IllustrationSimilarly, the change in the transmit power is depicted in the followingachievable rate expressions of QF in HD-TWRC.R1 ≤ R112+α4log(1 +2γ′1h21rPσ2Q + 1), (4.74)R2 ≤ α4log(1 +2γ2h22rPσ2Q + 1). (4.75)whereR11 ≤min(α2log(1 +2γ1h21rP1 + 2γ′1h21rP + 2γ2h22rP),β2log(1 + 2h2r2P )−Rh),(4.76)Rh =− α log(α)− (1− α) log(1− α), (4.77)σ2Y ′r =2γ′1h21rP + 2γ2h22rP + 1, (4.78)σ2Q =max{σ2Y ′r(1 + h2r12P )β2−R11 − 1 ,σ2Y ′r(1 + h2r22P )β2−R11−Rh − 1}, (4.79)if (1+h2r22P )β2−R11−Rh > 1 and (1+h2r12P )β2−Rh > 1, otherwise σ2Q =∞.However, Rh in (4.77) must satisfy (4.73), otherwise this scheme fails.Next, we will numerically analyze the behaviour of intermittent TWRCunder the assumptions of Section 4.1 and 4.2.4.3 Numerical IllustrationWhen FD-TWRC is analyzed using the results of Section 4.1 whereglobal CSI is available noncausally at the relay and local CSI is knownat the users causally, it is noticed that the behaviour of the transmissionschemes is the same as of the benchmark scenario except that the valuesof the rate pairs are degraded as the uplink channel power is P instead ofP/α. Similarly, further decrease in the values of achievable rates is noticedfor FD-TWRC when local CSI is known at all the nodes. This differencecan be noticed in Fig. 4.2.Thus, if CSI availability is limited at the nodes,the overall performance of the TWRC is degraded. In Fig. 4.2(b), resultsof QF, AF, and DF are compared with CF studied in [50] (see (4.50) and(4.51)). The plot shows that the performance of CF is similar to AF but thegap between CF and cut-set bound is lower as compared to the gap betweenAF and cut-set bound. It means computation using lattice coding enhancesthe performance.584.3. Numerical Illustration(a) Non-causal global CSI at therelay and causal local CSI at theusers with α = β = 0.85(b) Causal local CSI at all nodeswith α = β = 0.85Figure 4.2: Full-Duplex TWRC with intermittency at SNR1r = SNR2r =SNRr1 = SNRr2 = 15dB(a) Non-causal global CSI at therelay and causal local CSI at theusers(b) Causal local CSI at all nodesFigure 4.3: Maximum sum rate of the FD-TWRC with intermittency atSNR1r = SNR2r = 10dB and SNRr1 = SNRr2 = 15dB, β = αIn the comparison of two assumptions studied in this chapter, it is noticedthat different schemes behave differently as α changes. Fig. 4.3 shows theachievable sum-rate as a function of α for the FD-TWRC with limited CSIavailability. It is noticed that the AF scheme has the best performance overall values of α when noncausal CSI is available at the relay because downlinkchannel is available when uplink channel is available i.e. β = α, but QF and594.3. Numerical Illustration(a) Non-causal global CSI at therelay and causal local CSI atthe users when SNR1r = 12,SNR2r = 10dB, SNRr1 = 8, andSNRr2 = 15dB.(b) Causal local CSI at all nodeswhen SNR1r = 22,SNR2r = 20dBSNRr1 = 18, and SNRr2 = 25dBFigure 4.4: Full-Duplex TWRC with intermittency at α = β = 0.75DF have the best performance when only causal local CSI is available, withDF performing better for smaller α and QF performing better for larger α.This is because the full power in the rate expressions of QF, decreases thequantization noise at low intermittency i.e. the values of α closer to 1. DFgives better performance as the maximum sum-rate of AF and QF at highintermittency is decreased at α closer to zero. The results of CF are alsocompared when causal CSI is available at all the nodes. It is noticed thatthe performance of CF degrades with increasing the intermittency level. Infact, this scheme fails when the probability of channel availability α = β isvery small as shown in Fig. 4.3.As discussed in Section 1.2, the CF scheme used for intermittent TWRCin [50] is designed for a symmetrical setting of SNRs and a FD channel whereSi = Ti. We have designed AF, DF and QF for a generic setting of SNRs.We have derived the achievable rates of the relaying schemes for both theFD (i.e. Si = Ti) and the HD (i.e. Si ̸= Ti) cases. In Fig. 4.4(a), results forachievable rates of the transmission schemes are plotted for an asymmetricalsetting of SNRs for non-causal CSI availability at the relay and causal CSIavailability at the users. Similar performance is noted where DF performsbetter at low valued SNR combinations, whereas the performance of QFimproves for high values of SNRs when causal CSI is available at all thenodes as shown in Fig. 4.4(b).The CF scheme proposed in [50] does not applies for the HD case even604.3. Numerical Illustration(a) Non-causal global CSI at therelay and causal local CSI at theusers at SNR1r = SNR2r =SNRr1 = SNRr2 = 25dB(b) Causal local CSI at all nodesat SNR1r = SNR2r = 25dB andSNRr1 = SNRr2 = 15dBFigure 4.5: Maximum sum rate of the HD-TWRC with intermittency, α = 1for SNR1r = SNR2r and SNRr1 = SNRr2. Thus, we focus on DF, AF, andQF next, where we discuss the HD-TWRC when users are placed at equaldistances from the relay. In Fig. 4.5 (a), the AF/QF and DF achieve goodperformance for large and small values of β, respectively. This is becausethe uplink channel is not intermittent which makes the maximum sum-rateconstraint of DF redundant for lower values of β, which makes the downlinkbecome the bottleneck which is then achieved by DF. On the other hand,when beta is large, the sum-rate constraint of DF becomes relevant andreduces the rate compared to what can be achieved in the downlink which isnow large due to the larger beta. In this case, QF and AF perform well sincethe relay can forward its received signal to the users with low distortion.When the performance of the schemes is analyzed for HD-TWRC atdifferent values of β by keeping zero intermittency during the uplink trans-mission, when the uplink channel is equal or stronger than the downlinkchannel as shown in Fig. 4.5 (b) for local CSI available causally at all nodes.However, by keeping the uplink SNR higher than downlink, DF performsbest even at high values of β. As the affect of uplink sum-rate constraint isminimized by transmiting signals with power 2P at each node and keepingα = 1.Fig. 4.6 (a) shows the behaviour of the schemes for the intermittentHD-TWRC when global CSI is available non-causally at the relay and localCSI is available at the users, where it is assumed that the downlink channel614.3. Numerical Illustration(a) SNR1r = SNR2r = 10dB andSNRr1 = SNRr2 = 25dB(b) SNR1r = SNR2r = 20dB andSNRr1 = SNRr2 = 15dBFigure 4.6: Maximum sum rate of the intermitent HD-TWRC for non-causalglobal CSI at the relay and causal local CSI at the usersis available all the time with higher SNR than the uplink channel. QF givesthe best performance in this case as their is no intermitency in the downlinkchannel which means quantization noise will be minimal. On the otherhand, in Fig. 4.6 (b) the performance of the QF is compromised even at lowintermittency as the downlink SNR is low (quantization noise increases).When intermittent HD-TWRC with causal local CSI at all nodes is an-alyzed at the high SNR at the downlink channel as compare to the uplinkchannel by keeping β = 1. AF outperforms among all the schemes becausethe amplification factor κ is inversely related to the uplink channel whichmeans siganls are amplified with higher value of κ. On the other hand, QFbecomes the second effecient scheme as the distortion is minimized by in-creasing the downlink SNR. Whereas, the sum rate constraint is maximumat low SNR at the uplink channel for DF as shown in Fig. 4.7 (a). Behaviourcloser to the Fig. 4.7 (a) is noticed even by increasing the uplink SNR i.e20dB. However by decreasing the SNR of the downlink channel, distortionin QF is increased and efficiency of the scheme is compromised as shown inFig. 4.7 (b).In summary, for non-causal global CSI availability at the relay and causallocal CSI available at the user with no downlink intermittency, AF gives thebest performance when the downlink SNR is equal or greater than the up-link SNR in the HD and FD-TWRC. The DF schemes outperforms whenthe uplink SNR is greater than the downlink SNR in the FD-TWRC and QFachieves the best performance in the HD-TWRC. Similarly, for no uplink624.3. Numerical Illustration(a) SNR1r = SNR2r = 15dB andSNRr1 = SNRr2 = 20dB(b) SNR1r = SNR2r = 20dB andSNRr1 = SNRr2 = 15dBFigure 4.7: Maximum sum rate of the intermitent HD-TWRC for causallocal CSI at the usersintermittency, QF achieves the best performance at low downlink intermit-tency and DF achieves the best performance at high downlink intermittency.On the other hand, for causal CSI availability at all nodes, DF outperformswhen the uplink SNR is higher than the downlink SNR, and DF and QFachieves the best performance for different values of the uplink intermit-tency in the FD-TWRC. Similarly, AF achieves the best performance fordifferent combinations of high uplink and downlink SNR and for differentvalues of α in the HD-TWRC with no downlink intermittency. As almostsimilar performance (with some exemptions) to the previous case is noticedfor different values of β in the HD-TWRC with no uplink intermittency.In the next chapter, we will conclude and outline the limitations of ourthesis and give some future research directions.63Chapter 5Conclusion and DiscussionIn this section, we provide the conclusion of this thesis. We also discussthe limitations and future research directions.5.1 ConclusionIn this thesis, we evaluated the capacity of the TWRC with intermit-tency. We considered three different assumptions on the CSI availability,starting with an ideal scenario and leading to practical scenarios. We derivedthe achievable rates of different relaying schemes and compared them. Atfirst we derived achievable rates for an ideal assumption, where non-causalCSI is available at all the nodes. We noticed that the order of symbols trans-mitted in uplink and downlink phase is changed as shown in Fig. 3.5 butthe relay and users can re-arrange the symbols in the right order since CSIis available at all the nodes. When achievable rates of all the schemes areanalyzed numerically, it is noticed that AF outperforms the other schemesfor almost all combinations of high values of SNR in the FD-TWRC, and inthe HD-TWRC with uplink intermittency (with no downlink intermittency).However, DF and QF gives better performance for unequal SNRs when thedownlink phase is intermittent only. The performance of CF provided in [50]was also compared with DF, AF, and QF at different levels of intermittency.The achievable rate of CF is degraded by increasing the intermittency butthe performance of CF can be improved by increasing the uplink SNR. Weonly evaluated the performance of CF when users are placed equi-distantfrom the relay, which was the limitation of [50].In the second assumption, where non-causal CSI is available at the relayand causal CSI is available at the users. The order of transmitted symbolsis changed only during the downlink transmission. Unlike the benchmarkscenario, users can not re-arrange the symbols for decoding therefore a mod-ified scheme is designed for AF and QF which enable users to re-arrange thesymbol in the right order. No modification was required for DF since mes-sages are decoded and re-encoded at the relay . In the FD-TWRC, onlyAF and DF achieve sum rates close to the cut-set bound. However, in the645.1. ConclusionHD-TWRC, QF and AF achieve sum-rates close to the cut-set bound atlow downlink intermittency and DF outperforms at high values of downlinkintermittency.Lastly, when causal CSI is available at all the nodes, we use a modi-fied scheme for QF which enable users to re-arrange the symbols after de-quantization. This is done via a header signal transmitted from the relaywhich carries the uplink CSI to the users. The performance of DF is bettercompared to the previous scenarios at high values of SNR relative to thecut-set bound. However, the performance of QF and AF is degraded dueto the limited CSI. The results of CF in [50], were also evaluated under therestrictions of the setting in [50], where SNRs from each user to the relayare equal. It is noticed that the gap between the achievable rates of CF tothe cut-set bound increases from the cut-set bound with increasing inter-mittency. Infact, CF doesnot work at very high values of intermittency (αsmall).Overall, it is noticed that the QF scheme for the intermittent TWRC isaffected the most by intermittency as compared to the conventional TWRCsetups. In QF, the compression is performed at the relay, and if a smallnumber of symbols is received at the users, the compression index musthave low rate. Therefore, the performance of QF is compromised. However,better rates can be achieved for QF by decreasing the intermittency in theuplink phase and increasing the downlink SNR.We have achieved following objectives in this thesis:− We derived the cut-set bound and achievable rates of DF, AF, and QFfor the intermittent TWRC under three assumptions on CSI:– When non-causal CSI is available at all the nodes.– When non-causal CSI is available at the relay and causal CSI atthe users.– When causal CSI is available at all the nodes.− We analyzed the effect of CSI availability on the intermittent TWRCand provided modifications to the transmission schemes to solve prob-lems that may arise due to the lack of CSI.The results of this thesis will serve as a benchmark for different wirelessnetworks in which a relay is affected by blockage. It will also help in selectingthe best transmission schemes given some constraints on the CSI availabilityand SNRs. The limitations and future extension of this work are discussednext.655.2. Limitations5.2 LimitationsSome of the limitations of this work are listed below.− The use of a header signal in AF and QF adds an additional constrainton the achievable rate. Namely, the rate of the channel state (S)must be supported by the downlink channel, which reduces the rate ofinformation transmission in the downlink as a result. The computationcost of the scheme is also increased as an extra signal needs to betransmitted− We have focused on DF, AF, and QF. One limitation of this work isthat CF scheme similar to [50] was not considered. CF may need to bere-designed/modified for our settings, to enable asymmetry betweenSi and Ti, as well as between all channel’s SNRs. This can be aninteresting direction for future work.− Some CSI availability scenarios were considered in this thesis. How-ever, there are other CSI availability scenarios that we have not stud-ied, such as the extreme case where CSI is not available at any node,or available with error. In such a case, the schemes studied in thisthesis may require different modifications.All these limitations are interesting for future investigation. Additionallyfuture research directions are presented next.5.3 Future DirectionsThere are several ways to extend the research in this thesis. We willmention a few of them here. In this thesis, although we considered anasymmetric setting in terms of SNRs and in terms of uplink and downlinkstates, we still assumed symmetry in the uplink intermittency states (whichare assumed equal) and also in the downlink intermittency states (which areassumed equal). This models scenarios where blockages are close to the relay,which blocks both users’ channels at the same time. However, scenarioswhere this symmetry is not assumed are of interest too. Therefore, a possibleextension is to study fully asymmetric channels. We have also focused onDF, AF, and QF, but the literature shows that there are other relayingschemes like CF and JBD, etc., which achieve competitive achievable ratesfor conventional TWRCs but their performance in the intermittent TWRC665.3. Future Directionshas not been studied, and also the modifications that may need to be appliedto these schemes in the intermittent TWRC have not been investigated.Another extension is to consider a multi-user setting and evaluate achiev-able rates and compare the results with the capacity of the network, underchannel intermittency. Similarly, multi-relay networks with intermittentconnectivity can be studied.67Bibliography[1] Stefan Berger, Marc Kuhn, Armin Wittneben, Timo Unger, and AnjaKlein. Recent advances in amplify-and-forward two-hop relaying. IEEECommunications Magazine, 47(7):50–56, 2009. → pages 1[2] Kanchan Vardhe, Daryl Reynolds, and Brian D Woerner. Joint powerallocation and relay selection for multiuser cooperative communica-tion. 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IEEE, 2007. → pages 3574Appendix ARate of Transmission in aGaussian ChannelA Gaussian channel is a time-discrete channel with input X and outputY = X + Z, where Z models the noise and is normal distributed N (0, N)see Fig. A.1Figure A.1: Gaussian ChannelThe signal transmitted is limited by a power constraint E[X2] ≤ P .The transmission rate over this channel is given by the mutual informationbetween the transmitted and received signals.R ≤ I(X;Y ).We know that the mutual information can be rewritten as:I(X;Y ) = h(Y )− h(Y |X).Substituting Y by X + Z in the second term results inh(Y |X) = h(X + Z|X) = h(Z).75Appendix A. Rate of Transmission in a Gaussian ChannelThus,R ≤ h(Y )− h(Z).Since the distribution of noise is known to be a normal distribution withzero mean and variance N, we can geth(Z) =12log(2πeN).Since Y is a continuous random variable with zero mean and variance V .h(Y ) ≤ 12log(2πeV ).Since Y = X+Z, then its variance is V = P +N The rate can be re writtenas:R ≤12log(2πeV )− 12log(2πeN),=12log(2πe(P +N))− 12log(2πeN),=12log(2πe(P +N)2πeN),=12log(1 +PN), (A.1)where P is the transmitted signal power and N is the noise power.76
Thesis/Dissertation
eng
Electrical Engineering
Vancouver : University of British Columbia Library
Attribution-NonCommercial-NoDerivatives 4.0 International
Graduate
Chaaban, Anas
Transmission schemes and achievable rates for the two-way relay channel with intermittency
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