# 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$$\begin{array}{cc}\hfill {R}_{\mathrm{d},\mathrm{bs}}^{2}& \triangleq \underset{0}{\overset{\infty}{\int}}\mathrm{d}uf\left(u\right){\left[{R}_{T}-\underset{0}{\overset{u}{\int}}\mathrm{d}R\left(s\right)\right]}^{2}\hfill \end{array}$$$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& =\underset{0}{\overset{\infty}{\int}}\mathrm{d}uf\left(u\right){\left[\underset{u}{\overset{\infty}{\int}}\mathrm{d}R\left(s\right)\right]}^{2}\hfill \end{array}$$$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& =2\underset{0}{\overset{\infty}{\int}}\mathrm{d}uF\left(u\right)\mathrm{d}R\left(u\right)\underset{u}{\overset{\infty}{\int}}\mathrm{d}R\left(s\right)\phantom{\rule{4pt}{0ex}},\hfill \end{array}$$

**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.