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Capacity of the LDA and a simple achievable strategy for the Gaussian noise channel
In the previous section we showed that PDF achieves the gDoF of the Gaus- sian HD relay channel. PDF is based on block Markov encoding and joint decoding [87], which can be too complex to realize in practical systems. For this reason we seek now to design schemes that are simpler than PDF and that are still gDoF optimal. In order to do so, we consider the LDA in (2.2). Based on the many recent success stories, such as [19], we first determine the capacity achieving scheme for the LDA and we then try to ‘translate’ it into a gDoF-optimal scheme for the Gaussian HD relay channel. The rational is the “folk’s theorem” that the capacity of the LDA gives the gDoF of the corresponding Gaussian noise channel.
Analytical gaps
In Sections 2.3 and 2.4 we described upper and lower bounds to determine the gDoF of the Gaussian HD relay channel. In Section 2.4 we proposed a scheme inspired by the analysis of the LDA channel that also achieves the optimal gDoF. We now show that the same upper and lower bounds are to within a constant gap of one another thereby concluding the proof of Theorem 1. We consider both the case of random switch and of deterministic switch for the relay. For completeness we also consider the CF lower bound. Proposition 7. PDF with random switch is optimal to within 1 bit. Proof. The proof can be found in Appendix 2.F. Proposition 8. PDF with deterministic switch is optimal to within 1 bit. Proof. The proof can be found in Appendix 2.G. The intuition of why the gap does not improve with random switch is that there exist channel parameters for which direct transmission is approx- imately optimal (when min{C, I} S); in the case of direct transmission there are no benefits to use the relay at all and silencing the relay is a case of deterministic switch.
Gaussian HD relay channel with direct link
Figure 2.6(b) and Figure 2.6(c) show the rates achieved by using the dif- ferent achievable schemes presented in the previous sections for a channel with S > 0. In Figure 2.6(b) the channel conditions are such that PDF outperforms CF, while in Figure 2.6(c) the opposite holds. In Figure 2.6(b) the PDF strategy with random switch (red curve with maximum rate 11.66 bits/ch.use) outperforms both the CF with random switch (cyan curve with maximum rate 11.11 bits/ch.use) and the PDF with deterministic switch (blue curve with maximum rate 11.4 bits/ch.use); then the PDF with deterministic switch outperforms the CF with deterministic switch (magenta curve with maximum rate 10.94 bits/ch.use), which is also encompassed by the CF with random switch. Differently from the case without direct link, we observe that the maximum CF rates both in Figure 2.6(b) and in Fig- ure 2.6(c) are achieved with the choice Q = ;, i.e., the time-sharing random variable Q is a constant. This is due to the fact that the source is always heard by the destination even when the relay transmits so there is no need for the source to remain silent when the relay sends. Figure 2.6(d) shows, as a function of SNR and for « sd = 1, (« rd, « sr) 2 [0, 2.4], the maximum gap between the cut-set upper bound r(CS−HD) in (2.7b) and the following lower bounds with deterministic switch: the PDF lower bound obtained from r(PDF−HD) in (2.17) with I(PDF) 0 = 0, the CF lower bound in Remark 5 in Appendix 2.I, and the LDAi lower bound in (2.39).
Table of contents :
Acknowledgements
Abstract
Contents
List of Figures
Acronyms
Notations
1 Introduction
1.1 Motivation
1.2 Background
1.2.1 Half-Duplex Relay Networks
1.2.2 The Interference Channel with Source Cooperation
1.3 Contributions of this dissertation
1.3.1 Part I
1.3.2 Part II
I Half-Duplex Relay Networks
2 Half-Duplex Relay Channel
2.1 System model
2.1.1 General memoryless channel
2.1.2 The Gaussian noise channel
2.1.3 The deterministic / noiseless channel
2.2 Overview of the main results
2.3 The gDoF for the Gaussian HD relay channel
2.3.1 Cut-set upper bounds
2.3.2 PDF lower bounds
2.4 Capacity of the LDA and a simple achievable strategy for the Gaussian noise channel
2.4.1 Capacity of the LDA
2.4.2 LDAi: an achievable strategy for the Gaussian HD relay channel inspired by the LDA
2.5 Analytical gaps
2.6 Numerical gaps
2.6.1 Gaussian HD relay channel without a source-destination link (single-relay line network)
2.6.2 Gaussian HD relay channel with direct link
2.7 Conclusions and future directions
2.A Proof of Proposition 1
2.B Proof of Proposition 2
2.C Proof of Proposition 3
2.D Proof of Proposition 4
2.E Proof of Proposition 6
2.F Proof of Proposition 7
2.G Proof of Proposition 8
2.H Proof of Proposition 9
2.I Achievable rate with CF
2.J Proof of Proposition 10
3 The Half-Duplex Multi-Relay Network
3.1 System model
3.2 Background and overview of the main results
3.3 Capacity to within a constant gap
3.3.1 Channel Model
3.3.2 Inner Bound
3.3.3 Outer Bound
3.3.4 Gap
3.4 Simple schedules for a class of HD multi-relay networks
3.4.1 Proof Step 1
3.4.2 Proof Step 2
3.4.3 Proof Step 3
3.5 The gDoF and its relation to the MWBM problem.
3.6 Network examples
3.6.1 Example 1: HD relay network with N = 2 relays
3.6.2 Example 2: HD relay network with N = 1relay equipped with mr = 2 antennas
3.7 Applications of Theorem 6
3.7.1 The MIMO point-to-point channel
3.7.2 The relay-aided BC
3.7.3 The MISO K-user BC
3.8 Conclusions and future directions
3.A Proof that I(fix)
A in (3.9) is submodular
3.B (Approximately) Optimal simple schedule for N = 2.
3.C Proof of Theorem 6
3.D Upper and lower bounds for I(fix) ; in (3.54) and I(fix) {1} in (3.55) 120
3.E Water filling power allocation for I(fix) ; in (3.54) and I(fix) {1} in (3.55)
II The Causal Cognitive Interference Channel, or the Interference Channel with Unilateral Source Cooperation
4 Case I: Full-Duplex CTx
4.1 System Model
4.1.1 General memoryless channel
4.1.2 ISD channel
4.1.3 The Gaussian noise channel
4.2 Overview of the main results
4.3 Outer bounds on the capacity region for the CCIC
4.3.1 Known outer bounds and some generalizations
4.3.2 Novel outer bounds
4.3.3 Outer bounds evaluated for the Gaussian CCIC
4.4 The capacity region to within a constant gap for the symmet- ric Gaussian CCIC
4.4.1 Regime 1 (strong interference I)
4.4.2 Regime 2 (strong interference II)
4.4.3 Regime 3 (strong interference III)
4.4.4 Regime 4 (weak interference I)
4.4.5 Regime 5 (weak interference II)
4.4.6 Regime 6 (weak interference III)
4.4.7 Implication of the gap result
4.5 The capacity region to within a constant gap for the Gaussian Z-channel
4.5.1 Case C Sp: when unilateral cooperation might not be useful
4.5.2 Case C > Sp, Sc Ic (i.e., strong interference at PRx): when unilateral
