# The Broadcast Approach in Communication Networks

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## Abstract

**:**

Contents | ||

1 | Motivation and Overview | 4 |

1.1 What is the Broadcast Approach?................................................................................................................................................. | 4 | |

1.2 Degradedness and Superposition Coding........................................................................................................................................... | 5 | |

1.3 Application to Multimedia Communication......................................................................................................................................... | 6 | |

2 | Variable-to-Fixed Channel Coding | 7 |

2.1 Broadcast Approach in Wireless Channels......................................................................................................................................... | 7 | |

2.2 Relevance to the Broadcast Channel.............................................................................................................................................. | 8 | |

2.3 The SISO Broadcast Approach—Preliminaries..................................................................................................................................... | 9 | |

2.4 The MIMO Broadcast Approach..................................................................................................................................................... | 15 | |

2.4.1 Weak Supermajorization............................................................................................................................................ | 15 | |

2.4.2 Relation to Capacity.............................................................................................................................................. | 16 | |

2.4.3 The MIMO Broadcast Approach Derivation............................................................................................................................ | 17 | |

2.4.4 Degraded Message Sets............................................................................................................................................. | 19 | |

2.5 On Queuing and Multilayer Coding................................................................................................................................................ | 21 | |

2.5.1 Queue Model—Zero-Padding Queue.................................................................................................................................. | 22 | |

2.5.2 Delay Bounds for a Finite Level Code Layering..................................................................................................................... | 23 | |

2.5.3 Delay Bounds for Continuum Broadcasting........................................................................................................................... | 24 | |

2.6 Delay Constraints............................................................................................................................................................... | 27 | |

2.6.1 Mixed Delay Constraints........................................................................................................................................... | 27 | |

2.6.2 Broadcasting with Mixed Delay Constraints......................................................................................................................... | 28 | |

2.6.3 Parallel MIMO Two-State Fading Channel............................................................................................................................ | 30 | |

2.6.4 Capacity of Degraded Gaussian Broadcast Product Channels.......................................................................................................... | 31 | |

2.6.5 Extended Degraded Gaussian Broadcast Product Channels............................................................................................................. | 32 | |

2.6.6 Broadcast Encoding Scheme......................................................................................................................................... | 32 | |

2.7 Broadcast Approach via Dirty Paper Coding....................................................................................................................................... | 35 | |

3 | The Multiple Access Channel | 1 |

3.1 Overview........................................................................................................................................................................ | 35 | |

3.2 Network Model................................................................................................................................................................... | 35 | |

3.2.1 Discrete Channel Model............................................................................................................................................ | 37 | |

3.2.2 Continuous Channel Model.......................................................................................................................................... | 37 | |

3.3 Degradedness and Optimal Rate Spitting.......................................................................................................................................... | 38 | |

3.4 MAC without CSIT—Continuous Channels.......................................................................................................................................... | 38 | |

3.5 MAC without CSIT—Two-State Channels: Adapting Streams to the Single-User Channels............................................................................................. | 39 | |

3.6 MAC without CSIT—Two-State Channels: State-Dependent Layering................................................................................................................. | 40 | |

3.7 MAC without CSIT—Multi-State Channels: State-Dependent Layering............................................................................................................... | 45 | |

3.8 MAC with Local CSIT—Two-State Channels: Fixed Layering........................................................................................................................ | 47 | |

3.9 MAC with Local CSIT—Two-State Channels: State-Dependent Layering.............................................................................................................. | 48 | |

3.10 MAC with Local CSIT—Multi-State Channels: State-Dependent Layering............................................................................................................ | 51 | |

4 | The Interference Channel | 54 |

4.1 Overview........................................................................................................................................................................ | 54 | |

4.2 Broadcast Approach in the Interference Channel—Preliminaries.................................................................................................................. | 55 | |

4.3 Two-User Interference Channel without CSIT...................................................................................................................................... | 57 | |

4.3.1 Successive Decoding: Two-State Channel............................................................................................................................ | 58 | |

4.3.2 Successive Decoding: ℓ-State Channel............................................................................................................................ | 58 | |

4.3.3 Average Achievable Rate Region.................................................................................................................................... | 59 | |

4.3.4 Sum-Rate Gap Analysis............................................................................................................................................. | 61 | |

4.4 N-User Interference Channel without CSIT........................................................................................................................................ | 63 | |

4.5 Two-User Interference Channel with Partial CSIT................................................................................................................................. | 64 | |

4.5.1 Two-User Interference Channel with Partial CSIT—Scenario 1...................................................................................................... | 65 | |

4.5.2 Two-User Interference Channel with Partial CSIT—Scenario 2...................................................................................................... | 65 | |

5 | Relay Channels | 66 |

5.1 Overview........................................................................................................................................................................ | 66 | |

5.2 A Two-Hop Network............................................................................................................................................................... | 67 | |

5.2.1 Upper Bounds...................................................................................................................................................... | 68 | |

5.2.2 DF Strategies..................................................................................................................................................... | 70 | |

5.2.3 Continuous Broadcasting DF Strategies............................................................................................................................. | 71 | |

5.2.4 AF Relaying....................................................................................................................................................... | 75 | |

5.2.5 AQF Relay and Continuum Broadcasting.............................................................................................................................. | 76 | |

5.3 Cooperation Techniques of Two Co-Located Users.................................................................................................................................. | 78 | |

5.3.1 Lower and Upper Bounds............................................................................................................................................ | 80 | |

5.3.2 Naive AF Cooperation.............................................................................................................................................. | 82 | |

5.3.3 AF with Separate Preprocessing.................................................................................................................................... | 84 | |

5.3.4 Multi-Session AF with Separate Preprocessing...................................................................................................................... | 85 | |

5.3.5 Multi-Session Wyner–Ziv CF...................................................................................................................................... | 86 | |

5.4 Transmit Cooperation Techniques................................................................................................................................................. | 87 | |

5.4.1 Single-Layer Sequential Decode-and-Forward (SDF).................................................................................................................. | 88 | |

5.4.2 Continuous Broadcasting........................................................................................................................................... | 89 | |

5.4.3 Two Layer SDF—Successive Decoding............................................................................................................................... | 89 | |

5.5 Diamond Channel................................................................................................................................................................. | 92 | |

5.5.1 Decode-and-Forward................................................................................................................................................ | 92 | |

5.5.2 Amplify-and-Forward............................................................................................................................................... | 94 | |

5.6 Multi-Relay Networks............................................................................................................................................................ | 94 | |

5.6.1 Oblivious Relays.................................................................................................................................................. | 95 | |

5.6.2 Oblivious Agents.................................................................................................................................................. | 96 | |

5.7 Occasionally Available Relays................................................................................................................................................... | 97 | |

6 | Communications Networks | 98 |

6.1 Overview........................................................................................................................................................................ | 98 | |

6.2 Multi-User MAC Broadcasting with Linear Detection............................................................................................................................... | 98 | |

6.2.1 Channel Model..................................................................................................................................................... | 100 | |

6.2.2 Strongest Users Detection—Overview and Bounds................................................................................................................... | 100 | |

6.2.3 Broadcast Approach with Strongest Users Detection—(NO SIC)...................................................................................................... | 102 | |

6.2.4 SIC Broadcast Approach Upper Bound................................................................................................................................ | 103 | |

6.2.5 Broadcast Approach with Iterative SIC............................................................................................................................. | 104 | |

6.3 The Broadcast Approach for Source-Channel Coding................................................................................................................................ | 108 | |

6.3.1 SR with Finite Layer Coding....................................................................................................................................... | 109 | |

6.3.2 The Continuous SR-Broadcasting.................................................................................................................................... | 109 | |

6.4 The Information Bottleneck Channel.............................................................................................................................................. | 113 | |

6.4.1 Uncertainty of Bottleneck Capacity................................................................................................................................ | 115 | |

6.5 Transmitters with Energy Harvesting............................................................................................................................................. | 117 | |

6.5.1 Optimal Power Allocation Densities................................................................................................................................ | 119 | |

6.5.2 Optimal Power Allocation over Time................................................................................................................................ | 119 | |

6.5.3 Grouping the Constraints.......................................................................................................................................... | 120 | |

6.5.4 Dominant Constraints.............................................................................................................................................. | 121 | |

6.5.5 Optimality of Algorithm 1........................................................................................................................................ | 121 | |

7 | Outlook | 122 |

A | Constants of Theorem 7 | 126 |

B | Corner Points in Figure 16 | 127 |

References | 128 |

## 1. Motivation and Overview

#### 1.1. What is the Broadcast Approach?

#### 1.2. Degradedness and Superposition Coding

**1.****Degradedness in channel realizations:**The first step in specifying a broadcast approach for a given channel pertains to designating a notion of degradedness that facilitates rank-ordering different realizations of a channel based on their relative strengths. The premise for assigning such degradedness is that if communication is successful in a specific realization, it will also be successful in all realizations considered stronger. For instance, in a single-user single-antenna wireless channel that undergoes a flat-fading process, the fading gain can be a natural degradedness metric. In this channel, as the channel gain increases, the channel becomes stronger. Adopting a proper degradedness metric hinges on the channel model. While it can emerge naturally for some channels (e.g., single-user flat-fading), in general, selecting a degradedness metric is rather heuristic, if possible at all. For instance, in the multiple access channel, the sum-rate capacity can be used as a metric to designate degradedness, while in the interference channel, comparing different network realizations, in general, is not well-defined.**2.****Degradedness in message sets:**Parallel to degradedness in channel realization, in some systems, we might have a natural notion of degradedness in the message sets as well. Specifically, in some communication scenarios (e.g., video communication), the messages can be naturally divided into multiple ordered layers that incrementally specify the entire message. In such systems, the first layer conveys the baseline information (e.g., the lowest quality version of a video); the second layer provides additional information that incrementally refines the baseline information (e.g., refining video quality), and so on. Such a message structure specifies a natural way of ordering the information layers, which should also be used by the receiver to retrieve the messages successfully. Specifically, the receiver starts by decoding the baseline (lowest-ranked) layer, followed by the second layer, and so on. While some messages have inherent degradedness structures (e.g., audio/video signals), that is not the case in general. When facing messages without an inherent degradedness structure, a transmitter can still split a message into multiple, independently generated information layers. The decoders, which are not constrained by decoding the layers in any specific order, will decode as many layers as they afford based on the actual channel realization.

**Degraded message sets.**A message set with an inherent degradedness structure enforces a prescribed decoding order for the receiver.- -
- Degraded channels. When there is a natural notion of degradedness among channel realizations (e.g., in the single-user single-antenna flat-fading channel), we can designate one message to each channel realization such that the messages are rank-ordered in the same way that their associated channels are ordered. At the receiver side, based on the actual realization of the channel, the receiver decodes the messages designated to the weaker channels, e.g., in the weakest channel realization, the receiver decodes only the lowest-ranked message, and in the second weakest realization, it decodes the two lowest-ranked messages, and so on. Communication over a parallel Gaussian channel is an example in which one might face degradedness both in the channel and the message [20].
- -
- General channels. When lacking a natural notion of channel degradedness (e.g., in the single-user multi-antenna channel or the interference channel), we generally adopt an effective (even though imperfect) approach to rank order channel realizations. These orders will be used to prescribe an order according to which the messages will be decoded. The broadcast approach in such settings mimics the Körner–Marton coding approach for broadcast transmission with degraded message sets [21]. This approach is known to be optimal for a two-user broadcast channel with a degraded set of messages, while the optimal strategy for the general broadcast approach is an open problem despite the significant recent advances, e.g., [22].

**General message sets.**Without an inherent degradedness structure in the message, we have more freedom to generate the message set and associate the messages to different channel realizations. In general, each receiver has the freedom to decode any desired set of messages in any desired order. The single-user multi-antenna channel is an important example in which such an approach works effectively [23]. In this setting, while the channel is not degraded in general, different channel realizations are ordered based on the singular values of the channel matrix’s norm, which implies an order in channel capacities. In this setting, it is noteworthy that the specific choice of ordering the channels and assigning the set of messages decoded in each realization induces degradedness in the message set.

#### 1.3. Application to Multimedia Communication

## 2. Variable-to-Fixed Channel Coding

#### 2.1. Broadcast Approach in Wireless Channels

#### 2.2. Relevance to the Broadcast Channel

#### 2.3. The SISO Broadcast Approach—Preliminaries

#### 2.4. The MIMO Broadcast Approach

#### 2.4.1. Weak Supermajorization

#### 2.4.2. Relation to Capacity

#### 2.4.3. The MIMO Broadcast Approach Derivation

**minimal**eigenvalue ${\lambda}_{2}$ and the sum of eigenvalues ${\lambda}_{2}+{\lambda}_{1}$, respectively. Evidently, $u\ge 0,\phantom{\rule{0.166667em}{0ex}}v\ge 2u$. Say that ${\lambda}_{2},\phantom{\rule{0.166667em}{0ex}}{\lambda}_{1}$ take on the set of integer values $\{0,1,2,3,4\}$, then the layered system is described by $(u,v)$ in the order: $(0,0),\phantom{\rule{0.166667em}{0ex}}(0,1),\phantom{\rule{0.166667em}{0ex}}(0,2),\phantom{\rule{0.166667em}{0ex}}(0,3),\phantom{\rule{0.166667em}{0ex}}(0,4),\phantom{\rule{0.166667em}{0ex}}(1,2),\phantom{\rule{0.166667em}{0ex}}(1,3)$, $(1,4),\phantom{\rule{0.166667em}{0ex}}(2,4)$. The actual ordering of the layers is in fact immaterial, as will be shown, decoding is not done successively as in the SISO case [25], but rather according to what is decodable adhering to partial ordering.

#### 2.4.4. Degraded Message Sets

#### 2.5. On Queuing and Multilayer Coding

#### 2.5.1. Queue Model—Zero-Padding Queue

**waiting time**as the time measured from arrival until initially being served. The queue’s waiting time analysis can be done at embedded points: the beginning of every time slot. The random process of packet arrival random at each slot is a deterministic process denoted by $\lambda $ (bits/channel use).

#### 2.5.2. Delay Bounds for a Finite Level Code Layering

**Theorem**

**1**

**Corollary**

**1.**

#### 2.5.3. Delay Bounds for Continuum Broadcasting

- The number of layers is unlimited, that is $K\to \infty $.
- Since the layering is continuous, every layer i is associated with a fading gain parameter s. Every rate ${R}_{i}$ is associated with a differential rate $\mathrm{d}R\left(s\right)$ specified in (3).
- The cumulative rate ${\Re}_{K}$ should be replaced by$$\begin{array}{c}\hfill {R}_{T}=\underset{0}{\overset{\infty}{\int}}\mathrm{d}R\left(s\right)\phantom{\rule{4pt}{0ex}}.\end{array}$$
- The sum $\sum _{i=1}^{K}}{p}_{i}{\Re}_{K-i+1$ is actually the average rate and it turns to be ${R}_{\mathrm{bs}}$ (7) for the continuum case.
- Finally, in finite-level coding the expression $\sum _{i=1}^{K}}{p}_{i}{({\Re}_{K}-{\Re}_{K-i+1})}^{2}+\overline{p}{\Re}_{K}^{2$ turns out to be$$\begin{array}{cc}\hfill {R}_{\mathrm{d},\mathrm{bs}}^{2}& \triangleq \underset{0}{\overset{\infty}{\int}}\mathrm{d}uf\left(u\right){\left(\right)}^{{R}_{T}}2\hfill \end{array}$$$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& =\underset{0}{\overset{\infty}{\int}}\mathrm{d}uf\left(u\right){\left(\right)}^{\underset{u}{\overset{\infty}{\int}}}2\hfill \end{array}$$$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& =2\underset{0}{\overset{\infty}{\int}}\mathrm{d}uF\left(u\right)\mathrm{d}R\left(u\right)\underset{u}{\overset{\infty}{\int}}\mathrm{d}R\left(s\right)\phantom{\rule{4pt}{0ex}},\hfill \end{array}$$

**Corollary**

**2.**

**Corollary**

**3.**

#### 2.6. Delay Constraints

#### 2.6.1. Mixed Delay Constraints

#### 2.6.2. Broadcasting with Mixed Delay Constraints

**Lemma**

**1**

**Lemma**

**2**

**Proposition**

**1**

**Proposition**

**2**

#### 2.6.3. Parallel MIMO Two-State Fading Channel

#### 2.6.4. Capacity of Degraded Gaussian Broadcast Product Channels

#### 2.6.5. Extended Degraded Gaussian Broadcast Product Channels

#### 2.6.6. Broadcast Encoding Scheme

**Proposition**

**3.**

**Corollary**

**4.**

**Corollary**

**5.**

**Theorem**

**2.**

**independent broadcasting**. The broadcast approach for fading SISO channel relies on two main operations: superposition coding by layering at the transmitter; and successive interference cancellation at the receiver. The maximal expected sum rate of the symmetric two parallel two state channel under independent broadcasting is

**privately broadcasting**. This scheme is equivalent to setting $\alpha =0$ in Theorem 2, thus allocating encoding power from the common stream (${R}_{0}=0$) to the other streams ${R}_{\mathrm{AA}},{R}_{\mathrm{AB}},{R}_{\mathrm{BA}}$ and ${R}_{\mathrm{BB}}$, which achieves optimality for

#### 2.7. Broadcast Approach via Dirty Paper Coding

## 3. The Multiple Access Channel

#### 3.1. Overview

#### 3.2. Network Model

#### 3.2.1. Discrete Channel Model

#### 3.2.2. Continuous Channel Model

#### 3.3. Degradedness and Optimal Rate Spitting

#### 3.4. MAC without CSIT—Continuous Channels

#### 3.5. MAC without CSIT—Two-State Channels: Adapting Streams to the Single-User Channels

**Theorem**

**3**

#### 3.6. MAC without CSIT—Two-State Channels: State-Dependent Layering

- The second group of streams $\{{W}_{12}^{1},{W}_{21}^{2}\}$ are reserved to be decoded in addition to $\{{W}_{11}^{1},{W}_{11}^{2}\}$ when ${h}_{1}$ is strong, while ${h}_{2}$ is still weak.
- Alternatively, when ${h}_{1}$ is weak and ${h}_{2}$ is strong, instead the third group of streams, i.e., $\{{W}_{21}^{1},{W}_{12}^{2}\}$, are decoded.
- Finally, when both channels are strong, in addition to all the previous streams, the fourth group $\{{W}_{22}^{1},{W}_{22}^{2}\}$ is also decoded.

**Theorem**

**4**

**Proof.**

**Theorem**

**5**

**Proof.**

**Theorem**

**6**

#### 3.7. MAC without CSIT—Multi-State Channels: State-Dependent Layering

**Theorem**

**7**

#### 3.8. MAC with Local CSIT—Two-State Channels: Fixed Layering

**Theorem**

**8**

#### 3.9. MAC with Local CSIT—Two-State Channels: State-Dependent Layering

**State-dependent Layering.**In this approach, each transmitter, depending on the instantaneous state of the local CSI available to it, splits its message into independent information layers. Formally, when transmitter $i\in \{1,2\}$ is in the weak state, it encodes its message by only one layer, which we denote by ${U}_{11}^{i}$. On the other contrary, when transmitter $i\in \{1,2\}$ is in the strong state, it divides its message into two information layers, which we denote by ${U}_{12}^{i}$, and ${U}_{22}^{i}$. Hence, transmitter i adapts the codebook ${U}_{12}^{i}$ (or ${U}_{22}^{i}$) to the state in which the other transmitter experiences a weak (or strong) channel. A summary of the layering scheme and the assignment of the codebooks to different network states is provided in Figure 14. In this table, the cell associated with the state $({s}_{m},{s}_{n})$ for $m,n\in \{1,2\}$ specifies the codebook adapted to this state.**Decoding Scheme.**A successive decoding scheme is designed based on the premise that as the combined channel state becomes stronger, more layers are decoded. Based on this, the total number of codebooks decoded increases as one of the two channels becomes stronger. In this decoding scheme, the combination of codebooks decoded in different states is as follows (and it is summarizes in Table 5):

**State $({s}_{1},{s}_{1})$:**In this state, both transmitters experience weak states, and they generate codebooks $\{{U}_{11}^{1},{U}_{11}^{2}\}$ according to Figure 14. In this state, the receiver jointly decodes the baseline layers ${U}_{11}^{1}$ and ${U}_{11}^{2}$.**State $({s}_{2},{s}_{1})$:**When the channel of transmitter 1 is strong and the channel of transmitter 2 is weak, three codebooks $\{{U}_{12}^{1},{U}_{22}^{1},{U}_{11}^{2}\}$ are generated and transmitted. As specified by Table 5, the receiver jointly decodes $\{{U}_{12}^{1},{U}_{11}^{2}\}$. This is followed by decoding the remaining codebook, i.e., ${U}_{22}^{1}$.**State $({s}_{1},{s}_{2})$:**In this state, codebook generation and decoding are similar to those in the state $({s}_{2},{s}_{1})$, except that the roles of transmitters 1 and 2 are interchanged.**State $({s}_{2},{s}_{2})$:**Finally, when both transmitters experience strong channels, the receiver decodes four codebooks in the order specified by the last row of Table 5. Specifically, the receiver first jointly decodes the baseline layers $\{{U}_{12}^{1},{U}_{12}^{2}\}$, followed by jointly decoding the remaining codebooks $\{{U}_{22}^{1},{U}_{22}^{2}\}$.

**Achievable Rate Region.**Next, we provide an inner bound on the average capacity region. Recall that the average rate of transmitter i is denoted by ${\overline{R}}_{i}={\mathbb{E}}_{{h}_{1},{h}_{2}}\left[{R}_{i}({h}_{1},{h}_{2})\right]$, where the expectation is with respect to the random variables ${h}_{1}$ and ${h}_{2}$. Hence, the average capacity region is the convex hull of all simultaneously achievable average rates $({\overline{R}}_{1},{\overline{R}}_{2})$. Furthermore, we define ${\beta}_{ij}^{k}\in [0,1]$ as the ratio of the total power P assigned to information layer ${U}_{ij}^{k}$, where we have$$\sum _{i=1}^{j}{\beta}_{ij}^{k}=1$$

**Theorem**

**9**

**Outer Bound.**Next, we provide outer bounds on the average capacity region, and we compare them with the achievable rate region specified by Theorem 9.**Outer bound 2:**The second outer bound is the average capacity region of the two-user MAC with local CSI at transmitter 1 and full CSI at transmitter 2. Outer bound 2 is formally characterized in the following theorem.

**Theorem**

**10**

**Theorem**

**11**

**Theorem**

**12**

#### 3.10. MAC with Local CSIT—Multi-State Channels: State-Dependent Layering

- State $({s}_{1},{s}_{1})$: Start with the weakest channel combination $({s}_{1},{s}_{1})$, and reserve the baseline codebooks ${U}_{11}^{1},{U}_{11}^{2}$ to be the only codebooks to be decoded in this state. Define ${\mathcal{V}}_{11}^{i}=\left\{{U}_{11}^{i}\right\}$ as the set of codebooks that the receiver decodes from transmitter i when the channel state is $({s}_{1},{s}_{1})$.
- States $({s}_{1},{s}_{q})$ and $({s}_{q},{s}_{1})$: Next, construct the first row of the table. For this purpose, define ${\mathcal{V}}_{1q}^{2}$ as the set of the codebooks that the receiver decodes from transmitter 2, when the channel state is $({s}_{1},{s}_{q})$. Based on this, the set of codebooks in each state can be specified recursively. Specifically, in the state $({s}_{1},{s}_{q})$, decode what has been decoded in the preceding state $({s}_{1},{s}_{q-1})$, i.e., the set of codebooks ${\mathcal{V}}_{1(q-1)}^{2}$, plus new codebooks $\{{U}_{1q}^{1},\cdots ,{U}_{qq}^{1}\}$. Then, construct the first column of the table in a similar fashion, except that the roles of transmitter 1 and 2 are swapped.
- States $({s}_{q},{s}_{p})$ for $p,q>1$: By defining the set of codebooks that the receiver decodes from transmitter i in the state $({s}_{q},{s}_{p})$ by ${\mathcal{V}}_{qp}^{i}$, the codebooks decoded in this state are related to the ones decoded in two preceding states. Specifically, in state $({s}_{q},{s}_{p})$ decode codebooks ${\mathcal{V}}_{(p-1)q}^{1}$ and ${\mathcal{V}}_{p(q-1)}^{1}$. For example, for $\ell =3$, the codebooks decoded in $({s}_{2},{s}_{3})$ include those decoded for transmitter 1 in state $({s}_{2},{s}_{2})$ along with those decoded for transmitter 2 in channel state $({s}_{1},{s}_{3})$.

**Theorem**

**13**

## 4. The Interference Channel

#### 4.1. Overview

#### 4.2. Broadcast Approach in the Interference Channel—Preliminaries

#### 4.3. Two-User Interference Channel without CSIT

- Transmitter 1 (or 2) reserves the information layer ${V}_{1}^{k}$ (or ${V}_{2}^{k}$) for adapting it to the channel from transmitter 1 (or 2) to the unintended receiver ${y}_{2}^{k}$ (or ${y}_{1}^{k}$). Based on this designation, the intended receivers ${\left\{{y}_{1}^{k}\right\}}_{k=1}^{K}$ (or ${\left\{{y}_{2}^{k}\right\}}_{k=1}^{K}$) will decode all codebooks ${\left\{{V}_{1}^{k}\right\}}_{k=1}^{K}$ (or ${\left\{{V}_{2}^{k}\right\}}_{k=1}^{K}$), and the non-intended receivers ${\left\{{y}_{2}^{k}\right\}}_{k=1}^{K}$ (or ${\left\{{y}_{1}^{k}\right\}}_{k=1}^{K}$) will be decoding a subset of these codebooks. The selection of the subsets depends on on channel strengths of the receivers, such that the non-intended receiver ${y}_{2}^{k}$ (or ${y}_{1}^{k}$) decodes only codebooks ${\left\{{V}_{1}^{s}\right\}}_{s=1}^{k}$ (or ${\left\{{V}_{2}^{s}\right\}}_{s=1}^{k}$).
- Transmitter 1 (or 2) reserves the layer ${U}_{1}^{k}$ (or ${U}_{2}^{k}$) for adapting it to the channel from transmitter 2 (or 1) to the intended receiver ${y}_{1}^{k}$ (or ${y}_{2}^{k}$). Based on this designation, the unintended receivers ${\left\{{y}_{2}^{k}\right\}}_{k=1}^{K}$ (or ${\left\{{y}_{1}^{k}\right\}}_{k=1}^{K}$) will not decode any of the codebooks ${\left\{{U}_{1}^{k}\right\}}_{k=1}^{K}$ (or ${\left\{{U}_{2}^{k}\right\}}_{k=1}^{K}$), and the intended receivers ${\left\{{y}_{1}^{k}\right\}}_{k=1}^{K}$ (or ${\left\{{y}_{2}^{k}\right\}}_{k=1}^{K}$) will be decoding a subset of these codebooks. The selection of these subsets depends on channel strengths of the receives such that the intended receiver ${y}_{1}^{k}$ (or ${y}_{2}^{k}$) decodes only the codebooks ${\left\{{U}_{1}^{s}\right\}}_{s=1}^{k}$ (or ${\left\{{U}_{2}^{s}\right\}}_{s=1}^{k}$).

#### 4.3.1. Successive Decoding: Two-State Channel

#### 4.3.2. Successive Decoding: ℓ-State Channel

**Receiver 1—stage 1**(Codebooks ${\left\{{V}_{i}^{s}\right\}}_{s=1}^{q}$): Receiver 1 decodes one information layer from each transmitter in an alternating manner until all codebooks ${\left\{{V}_{1}^{s}\right\}}_{s=1}^{q}$ and ${\left\{{V}_{2}^{s}\right\}}_{s=1}^{q}$ are decoded. The first layer to be decoded in this stage depends on the state ${\beta}_{q}$. If ${\beta}_{q}<1$, the receiver starts by decoding codebook ${V}_{1}^{1}$ from transmitter 1, then it decodes the respective layer ${V}_{2}^{1}$ from transmitter 2, and continues alternating between the two transmitters. Otherwise, if ${\beta}_{q}>1$, receiver 1 first decodes ${V}_{2}^{1}$ from the interfering transmitter 2, followed by ${V}_{1}^{1}$ from transmitter 1, and continues alternating. By the end of stage 1, receiver 1 has decoded q codebooks from each transmitter.**Receiver 1—stage 2**(Codebooks ${\left\{{V}_{1}^{s}\right\}}_{s=q+1}^{K}$ and ${\left\{{U}_{1}^{s}\right\}}_{s=1}^{q}$): In stage 2, receiver 1 carries on decoding layers ${\left\{{V}_{1}^{s}\right\}}_{s=q+1}^{K}$ from transmitter 1, in an ascending order of the index s. Finally, receiver 1 decodes layers ${\left\{{U}_{1}^{s}\right\}}_{s=1}^{q}$ specially adapted to receivers ${\left\{{y}_{1}^{s}\right\}}_{s=1}^{q}$, in an ascending order of index s. Throughout stage 2, receiver 1 has additionally decoded K codebooks from its intended transmitter 1.

#### 4.3.3. Average Achievable Rate Region

**Theorem**

**14**

#### 4.3.4. Sum-Rate Gap Analysis

**Weak interference:**In the weak interference regime, the capacity with full CSIT is in unknown. In this regime, in order to quantify the gap of interest, we first evaluate the gap of the sum-rate achieved by the scheme of Section 4.3.2 to the sum-rate achieved by the HK scheme. By using this gap in conjunction with the known results on the gap between the sum-rate of HK and the sum-rate capacity, we provide an upper bound on the average sum-rate gap of interest.**Strong interference:**In the strong interference regime, the sum-rate capacity with full CSIT is known. It can be characterized by evaluating the sum-rate of the intersection of two capacity regions corresponding to two multiple access channels formed by the transmitters and each of the receivers [123].

**Theorem**

**15**

- (i)
- For $P\in {\mathrm{G}}_{1}=\left(\right)open="("\; close=")">0,\beta $ we have$$\begin{array}{cc}\hfill \Delta \left({\mathrm{G}}_{1}\right)& \le \frac{1}{3}\left(\right)open="["\; close="]">1+log\left(\right)open="("\; close=")">\frac{1+P(1+\beta )}{1+P(1+\frac{1}{\beta})}+\frac{1}{2}log(2+\beta )\hfill & \phantom{\rule{4pt}{0ex}}.\end{array}$$
- (ii)
- For $P\in {\mathrm{G}}_{2}=\left(\right)open="["\; close="]">\beta ,\beta ({\beta}^{2}+\beta -1)$ we have$$\begin{array}{c}\hfill \Delta \left({\mathrm{G}}_{2}\right)\le \frac{1}{3}\left(\right)open="["\; close="]">log\frac{4}{3}+log\left(\right)open="("\; close=")">\frac{1+P(1+\beta )}{1+P/\beta +\beta}+3log\frac{{(2+\beta )}^{2}}{1+2\beta}& \phantom{\rule{4pt}{0ex}}.\end{array}$$

**Theorem**

**16**

#### 4.4. N-User Interference Channel without CSIT

- Transmitter m adapts layer ${V}_{m}^{k}$ to the state of the channels linking all other transmitters to the unintended receivers $\{{y}_{1}^{k},\dots ,{y}_{{N}^{L-1}}^{k}\}\setminus \left\{{y}_{m}^{k}\right\}$: while the intended receivers ${\left\{{y}_{m}^{k}\right\}}_{k=1}^{S}$ will be decoding all codebooks ${\left\{{V}_{m}^{k}\right\}}_{k=1}^{S}$, the non-intended receivers $\{{y}_{1}^{k},\dots ,{y}_{S}^{k}\}\setminus \left\{{y}_{m}^{k}\right\}$ decode a subset of these codebooks depending on their channel strengths. More specifically, a non-intended receiver ${y}_{i}^{k}$ decodes only the codebooks ${\left\{{V}_{m}^{s}\right\}}_{s=1}^{k}$.
- Transmitter m adapts the layer ${U}_{m}^{k}$ to the state of the channels linking all other transmitters to the intended receiver ${y}_{m}^{k}$: while the unintended receivers $\{{y}_{1}^{k},\dots ,{y}_{S}^{k}\}\setminus \left\{{y}_{m}^{k}\right\}$ will not be decoding any of the codebooks ${\left\{{U}_{m}^{k}\right\}}_{k=1}^{S}$, the intended receivers ${\left\{{y}_{m}^{k}\right\}}_{k=1}^{S}$ decode a subset of these codebooks depending on their channel strengths. More specifically, the intended receiver ${y}_{m}^{k}$ decodes only the codebooks ${\left\{{U}_{m}^{s}\right\}}_{s=1}^{k}$.

#### 4.5. Two-User Interference Channel with Partial CSIT

#### 4.5.1. Two-User Interference Channel with Partial CSIT—Scenario 1

**State-dependent adaptive layering.**In this setting, each transmitter controls the interference that it imposes by leveraging the partially known CSI. Concurrently, each transmitter adapts one layer to every possible channel state at its intended receiver, overcoming the partial uncertainty about the other transmitter’s interfering link. Based on these observations, transmitter i splits its information stream into a certain set of codebooks depending on the state of its outgoing cross channel. We denote the set of codebooks transmitted by user i when ${a}_{j}=\sqrt{{\beta}_{k}}$ by ${\mathcal{C}}_{i}^{k}$. Each set ${\mathcal{C}}_{i}^{k}$ consists of $K+1$ codebooks given by

- layer ${V}_{1}^{k}$ (${V}_{2}^{k}$) is adapted to the cross channel state at the unintended receiver ${y}_{1}^{k}$ (${y}_{2}^{k}$); and
- layer ${U}_{1}^{k}\left(s\right)$ ( ${U}_{2}^{k}\left(s\right)$) is adapted to the cross channel state at the intended receiver ${y}_{2}^{s}$ (${y}_{1}^{s}$), for $s\in \{1,\cdots ,K\}$.

**Successive decoding.**Each codebook will be decoded by multiple receivers in the equivalent network formed by different receivers associated with different network states. Hence, each codebook rate will be constrained by its associated most degraded channel state. Furthermore, any undecoded layer at a particular receiver imposes interference, which degrades the achievable rate at that receiver. Motivated by these premises, a simple successive decoding scheme can be designed that specifies (i) the set of receivers at which each layer is decoded, and (ii) the order of successive decoding order at each receiver.

**Receiver ${y}_{1}^{s}$:**First, it decodes one layer from the unintended transmitter ${V}_{2}^{t}$ and remove it from the received signal. Secondly, it decodes the baseline layer from its intended transmitter ${V}_{1}^{s}$. Finally, depending on the network state $({\beta}_{s},{\beta}_{t})$, it successively decodes all the layers $\{{U}_{1}^{s}\left(1\right),\dots ,{U}_{1}^{s}\left(t\right)\}$.**Receiver ${y}_{2}^{t}$:**First, it decodes one layer from the unintended transmitter ${V}_{1}^{s}$ and remove it from the received signal. Secondly, it decodes the baseline layer from its intended transmitter ${V}_{2}^{t}$. Finally, depending on the network state $({\beta}_{s},{\beta}_{t})$, it successively decodes all the layers $\{{U}_{1}^{t}\left(1\right),\dots ,{U}_{1}^{t}\left(s\right)\}$.

#### 4.5.2. Two-User Interference Channel with Partial CSIT—Scenario 2

**State-dependent adaptive layering.**In contrast to Scenario 1, in this scenario, transmitter i knows ${a}_{i}$, and it is oblivious to the other channel. Lacking the extent of interference that each transmitter causes, transmitter i adapts multiple layers with different rates such that the unintended receiver opportunistically decodes and removes a part of the interfering according to the actual state of the channel. Simultaneously, transmitter i adapts the encoding rate of a single layer to be decoded only by its intended receiver based on the actual state of channel ${a}_{2}$. Based on this vision, transmitter i splits its information stream into a distinct set of codebooks corresponding to each state of the cross channel at its intended receiver. We denote the set of codebooks transmitted by user i when ${a}_{i}=\sqrt{{\beta}_{k}}$ by ${\mathcal{D}}_{i}^{k}$. Each set ${\mathcal{D}}_{i}^{k}$ consists of $K+1$ codebooks given by

- layer ${V}_{1}^{k}\left(s\right)$ (${V}_{2}^{k}\left(s\right)$) is adapted to the cross channel state at the unintended receiver ${y}_{2}^{s}$ (${y}_{1}^{s}$), for $s\in \{1,\dots ,K\}$; and
- layer ${U}_{1}^{k}$ (${U}_{2}^{k}$) is adapted to the cross channel state at the intended receiver ${y}_{1}^{k}$ (${y}_{2}^{k}$).

**Successive decoding.**Given that each codebook will opportunistically be decoded by multiple receivers, its maximum achievable rate is constrained by the most degraded network state in which it is decoded. Similarly to that of Scenario 1, a successive decoding scheme is devised that specifies (i) the set of receivers at which each layer is decoded, and (ii) the order of successive decoding order at each receiver.

**Receiver ${y}_{1}^{s}$:**First, it decodes one layer from the interfering signal ${V}_{2}^{t}\left(1\right)$. Afterwards, it decodes one layer from the intended signal ${V}_{1}^{s}\left(1\right)$. This receiver continues the decoding process in an alternating manner until codebooks ${\left\{{V}_{2}^{t}\left(j\right)\right\}}_{j=1}^{s}$ from transmitter 2 and codebooks ${\left\{{V}_{1}^{s}\left(j\right)\right\}}_{j=1}^{K}\}$ are decoded from the intended receiver 1 are successfully decoded. Finally, the last remaining layer from the intended message ${U}_{1}^{s}$ is decoded.**Receiver ${y}_{2}^{t}$:**First, it decodes one layer from the interfering signal ${V}_{1}^{s}\left(1\right)$. Afterwards, it decodes one layer from the intended signal ${V}_{2}^{t}\left(1\right)$. This receiver continues the decoding process in an alternating manner until codebooks ${\left\{{V}_{1}^{s}\left(j\right)\right\}}_{j=1}^{s}$ from transmitter 1 and codebooks ${\left\{{V}_{2}^{t}\left(j\right)\right\}}_{j=1}^{K}\}$ are decoded from the intended receiver 2 are successfully decoded. Lastly, the last remaining layer from the intended message ${U}_{2}^{t}$ is decoded.

## 5. Relay Channels

#### 5.1. Overview

#### 5.2. A Two-Hop Network

#### 5.2.1. Upper Bounds

#### 5.2.2. DF Strategies

#### 5.2.3. Continuous Broadcasting DF Strategies

**Coding Scheme I—Source: Outage and Relay: Continuum Broadcasting.**In this coding scheme, the source transmitter performs single-level coding. Whenever channel conditions allow decoding at the relay, it performs continuum broadcasting, as described in the previous subsection. Thus, the received rate at the destination depends on the instantaneous channel fading gain realization on the relay-destination link. Clearly, a necessary condition for receiving something at the destination is that channel conditions on the source-relay link will allow decoding. The source transmission rate is given by

**Proposition**

**4.**

**Coding Scheme II—Source: Continuum Broadcasting and Relay: Outage.**In this coding scheme, the source transmitter performs continuum broadcasting, as described in the previous subsection. The relay encodes the successfully decoded layers into a single-level block code. Thus, the rate of each transmission from the relay depends on the number of layers decoded. For a fading gain realization $\nu $ on the source-relay link, the decodable rate at the relay is

**Coding Scheme III—Source and Relay: Continuous Broadcasting.**In this scheme, both source and relay perform the optimal continuum broadcasting. The source transmitter encodes a continuum of layered codes. The relay decodes up to the maximal decodable layer. Then it retransmits the data in a continuum multi-layer code matched to the rate that has been decoded last. In this scheme, the source encoder has a single power distribution function, which depends only on a single fading gain parameter. The relay uses a power distribution that depends on the two fading gains on the source-relay and the relay-destination links.

#### 5.2.4. AF Relaying

**Proposition**

**5.**

#### 5.2.5. AQF Relay and Continuum Broadcasting

**Proposition**

**6.**

#### 5.3. Cooperation Techniques of Two Co-Located Users

- Naive AF—A helping node scales its input and relays it to the destined user, who jointly decodes the relay signal and the direct link signal.
- Separate preprocessing AF—A more efficient form of single-session AF is a separate preprocessing approach in which the co-located users exchange the values of the estimated fading gains, and each individually decodes the layers up to the smallest fading gain. The helping user removes the decoded common information from its received signal and performs AF on the residual signal to the destined user.
- Multi-session AF—Repeatedly separate preprocessing is followed by a transmitting cooperation information at both helper and destination nodes (on orthogonal channels). The preprocessing stage includes individual decoding of the received information from the direct link and previous cooperation sessions. Along the cooperation sessions, transmission of the next block already takes place. It means that multi-session cooperation introduces additional decoding delays
**without any impact on the throughput**. For this purpose, multiple parallel cooperation channels are assumed. For incorporating practical constraints on the multi-session approach, the total power of multi-session cooperation is restricted to ${P}_{r}$. This is identical to the power constraint in single-session cooperation.

- Naive CF—A helping node performs WZ-CF over the cooperation link. The destination informs the relay of its fading gain realization prior to the WZ compression. The destination performs optimal joint decoding of the WZ compressed signal forwarded over the cooperation link, and its own copy of the signal from the direct link.
- Separate preprocessing CF—Each user decodes independently up to the highest common decodable layer. Then WZ–CF cooperation takes place on the residual signal by WZ coding.
- Multi-session CF— Multi-session cooperation, as described for AF, is carried out in conjunction with successive refinement WZ [193] CF relaying.

#### 5.3.1. Lower and Upper Bounds

**Outage lower bound.**The single-layer coding expected rate is

**Broadcasting lower bound.**This bound is based on an SISO block fading channel, with receive CSI. The maximal expected broadcasting rate [23], for a Rayleigh fading channel is

**Outage upper bound.**Fully cooperating user bound is derived similarly to (311), with ${F}_{\mathrm{UB}}\left(u\right)$ as the fading gain distribution function.**Broadcasting upper bound.**The broadcasting upper bound is a two receive antenna block fading channel. The expected broadcasting rate for a Rayleigh fading channel [23] is

**Ergodic upper bound.**Ergodic bound for two receive antennas SIMO fading channel is ${C}_{\mathrm{erg}}\left(2\right)$ in (313),

**Single-session cut-set upper bound.**Another upper bound considered is the classical cut-set bound of the relay channel [40]. This bound may be useful for single-session cooperation, where the capacity of the cooperation link is rather small. Using the relay channel definitions in (306) and (307), and assuming a single cooperation session $K=1$, the cut-set bound for a Rayleigh fading channel is given by

#### 5.3.2. Naive AF Cooperation

**Proposition**

**7.**

#### 5.3.3. AF with Separate Preprocessing

**Proposition**

**8.**

#### 5.3.4. Multi-Session AF with Separate Preprocessing

**Proposition**

**9.**

#### 5.3.5. Multi-Session Wyner–Ziv CF

**Proposition**

**10.**

#### 5.4. Transmit Cooperation Techniques

#### 5.4.1. Single-Layer Sequential Decode-and-Forward (SDF)

#### 5.4.2. Continuous Broadcasting

#### 5.4.3. Two Layer SDF—Successive Decoding

**lower bound**for the achievable rate of oblivious relaying is considered here. In an oblivious setting, the maximal expected throughput without a helping relay is called a direct transmission rate. This rate serves as the lower bound to achievable rates for the relay channel.

**Proposition**

**11.**

**upper bounds**, reflecting full cooperation among transmitters. As the relay and source might have different power allocations, it is required to study the problem of MISO layering with individual power constraints per antenna. Consider first a sub-optimal approach where the same fractional power allocation per layer is used per antenna. In our setting this means $\alpha =\beta $ in (358) and (360), i.e., the first layer power allocation of the source and the relay is $\alpha {P}_{s}$ and $\alpha {P}_{r}$, respectively. The expected rate then, similarly to (361), becomes

**Proposition**

**12**