Information Theoretic Limits of State-dependent Networks

Information Theoretic Limits of State-dependent Networks

Yunhao Sun
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INFORMATION THEORETIC LIMITS OF STATE-DEPENDENT NETWORKS

by
Yunhao Sun

B.E.(EE), Beijing Institute of Technology, 2011

M.S.(EE), NYU Tandon School of Engineering, 2013

DISSERTATION

Submitted in partial fulfillment of the requirements for the degree of

Doctor of Philosophy in Electrical Engineering

Syracuse University

July 2017

Advised by:

Prof. Yingbin Liang

Copyright © 2017 Yunhao Sun
All rights reserved

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Abstract

We investigate the information theoretic limits of two types of state-dependent models in this dissertation. These models capture a wide range of wireless communication scenarios where there are interference cognition among transmitters. Hence, information theoretic studies of these models provide useful guidelines for designing new interference cancellation schemes in practical wireless networks.

In particular, we first study the two-user state-dependent Gaussian multiple access channel (MAC) with a helper. The channel is corrupted by an additive Gaussian state sequence known to neither the transmitters nor the receiver, but to a helper noncausally, which assists state cancellation at the receiver. Inner and outer bounds on the capacity region are first derived, which improve the state-of-the-art bounds given in the literature. Further comparison of these bounds yields either segments on the capacity region boundary or the full capacity region by considering various regimes of channel parameters.

We then study the two-user Gaussian state-dependent Z-interference channel (Z-IC), in which two receivers are corrupted respectively by two correlated states that are noncausally known to transmitters, but unknown to receivers. Three interference regimes are studied, and the capacity region or the sum capacity boundary is characterized either fully or partially under various channel parameters. The impact of the correlation between the states on the cancellation of state and interference as well as the achievability of the capacity is demonstrated via numerical analysis.

Finally, we extend our results on the state-dependent Z-IC to the state-dependent regular IC. As both receivers in the regular IC are interfered, more sophisticated achievable schemes are designed. For the very strong regime, the capacity region is achieved by a scheme where the two transmitters implement a cooperative dirty paper coding. For the strong but not very strong regime, the sum-rate capacity is characterized by rate splitting, layered dirty paper coding and successive cancellation. For the weak regime, the sum-rate capacity is achieved via dirty paper coding individually at two receivers as well as treating interference as noise. Numerical investigation indicates that for the regular IC, the correlation between states impacts the achievability of the channel capacity in a different way from that of the Z-IC.

Chapter 1 Introduction

Since the early 1940s, when Claude E. Shannon established the maximum amount of information that can be sent over a noisy channel in his classic papers [1] and [2], information theory has driven the evolution of communication systems from one generation to another. Shannon’s original work focused on the discrete memoryless channel, in which the transition probability distribution (i.e., the noise characteristics of the channel) is perfectly known to both the transmitter and the receiver. In this scenario, he proved theoretically the existence of coding and decoding schemes to achieve any rate below the channel capacity, and proved that such capacity is the reliable transmission limit via a converse argument. In particular, Shannon came up with the idea of random coding (see [3]) to show the achievability of the rate, the ingenious tool that is still widely used in information theory today and is used throughout the rest of this dissertation.

Then, Shannon’s basic approach was extended by both mathematicians and engineers to more general models with respect to information sources, coding structures, and performance measures. The fundamental theorem for entropy was extended to the same generality as the ordinary ergodic theorems by McMillan [4] and Breiman [5] and the result is now known as the Shannon-McMillan-Breiman theorem (the asymptotic equipartition theorem or AEP, the ergodic theorem of information theory, and the entropy theorem). A variety of detailed proofs of the basic coding theorems and stronger versions of the theorems for memoryless, Markov, and other special cases of random processes were developed.

In [6], Robert M. Gray pointed out that there are two primary goals of information theory: The first is the development of the fundamental theoretical limit on the achievable performance when communicating a given information source over a given communication channel using optimal (but only theoretical) coding schemes from within a prescribed class. The second goal is the development of practical coding schemes, e.g., structured encoder(s) and decoder(s), which provide the performance that is reasonably good in comparison with the optimal performance given by the theory.

During the development of practical communication systems, both of these two goals need to be fulfilled. In this dissertation, we mainly investigate the first aspect of information theory. In particular, we focus on a type of state-dependent channels, and explore the dirty paper coding as a useful tool to understand the fundamental performance limit of such a type of channels. The remainder of this chapter provides background materials and outlines the contributions of this dissertation.

1.1 Motivation

Interference management is one of important issues that determine the spectral efficiency of wireless networks. Techniques to deal with interference in all up-to-date cellular networks follow the basic principle of orthogonalizing transmissions in time, frequency, code, and space, which yields TDMA, FDMA, CDMA and more advanced OFDM technologies. However, orthogonal schemes typically do not reach the best spectral efficiency and are not information theoretically optimal in general. On the other hand, various advanced non-orthogonal interference cancellation schemes are proposed, motivated by information theoretic designs. For example, the Han-Kobayashi [7] scheme is based on the idea that the receiver decodes interference partially, and then subtracts it from the output to reduce the interference. Such a scheme is recently exploited in a down-link non-orthogonal multiple access (NOMA) scheme for interference management in [8, 9] and [10]. However, such successive interference cancellation requires users to share codebooks and hence can be very complex to implement in practice, especially when transmissions are not within the same network domain.

My dissertation aims at investigating a new framework for interference control in wireless networks based on the following key perspectives of the interference. In fact, interference signals in nature contain coded information sent to certain intended nodes, and hence such signals as codeword sequences are typically noncausally known by various nodes in the network. For example, an interferer clearly knows the interference signal that it causes to other users noncausally, because such interference is the codeword that this interferer transmits to its intended receivers. As another example, if the interferer is a base station, it can easily inform other base stations about its interference via the backhaul network or inform access points in the cell via wired links. The major observation here is that interference cognition (i.e., the knowledge of interference being informed to various nodes) naturally exists or can be established at very low costs in networks. Thus, nodes that possess noncausal interference information should be able to exploit it to assist cancellation of such interference.

One major advantage of such an idea is that the design of interference cancellation is handled mainly at the interferer or the helper side, which are typically powerful nodes (such as base stations and access points) in networks and can hence easily take the extra load of interference cancellation. Since the design is on one side, it is more efficient and does not require sharing codebooks as in successive interference cancellation in the Han-Kobayashi scheme. Moreover, the design can be made transparent to nodes being interfered with. This is very useful in cognitive networks (i.e., [11, 12, 13]) and internet of things (IoT) networks [14, 15, 16]. With the interference to primary nodes being canceled by the interferer itself or helper nodes, the access of primary channels can be made simultaneous and transparent from primary networks.

In this dissertation, we explore two types of information theoretic models, which capture the impact of interference cognition in wireless networks, and set the goal of best exploiting such information for interference cancellation. These two types of models respectively represent two angles to treat the interference cognition and thus focus on two performance objectives to accomplish. We next explain the essence of these two types of models by their basic versions.

  • The first type of models are state-dependent channels with the state known at a helper, where neither the transmitter nor the receiver knows the information of channel interference. Instead, there is a dedicate helper who has the information of channel interference and assists the receiver to cancel the channel state. This model can be illustrated via an example (see Fig. 1.1). In this model, we consider a device-to-device (D2D) communication in a picocell located inside a macrocell of a cellular network. The D2D communication inside the picocell is corrupted by the interference . The helper (i.e., wireless router) in the picocell, which knows the information of channel state through the wire cable, sends its signal to help the D2D communication. Although the help signal may also cause interference to the macrocell base station, as long as the power of is much less than the power of , there is still significant gain in throughput. In fact, our results demonstrate that the helper can use a relatively small amount of power to completely cancel the interference that the base station causes to D2D users (e.g., the picocell users in our example) even if the interference is as large as infinite.

    Fig. 1.1: state-dependent channels with states known at helper
  • The second type of models are state-dependent channels with the state known at transmitters, so that the transmitters can help the receiver to cancel the state interference. This class of models can be illustrated via a simple example as follows. Consider a multi-cell network, where base station 1 sends information to a cellular user. It is typical that base station 2 in cell 2 can cause interference to the cellular user in cell 1. In this case, base station 2 can send such interference to base station 1 through the backhaul network, so that base station 1 can use its noncausal knowledge about the interference to cancel the interference efficiently.

    Fig. 1.2: state-dependent channels with states known at transmitter

The above two basic scenarios can be broadened to a wide range of network models to capture various scenarios arising in practical networks: multiple receivers can be interfered with by the same or different interfering sources, multiple helpers can cooperatively assist interference cancellation, and interference from different interfering sources can be known distributively by different helpers. All of these models have their representative characteristics which give rise to unique designs and technical challenges, and comprehensive exploration of these models will yield a new framework for interference management based on interference cognition.

Thus, the goal of this dissertation is to conduct extensive investigations of the two types of models in order to develop comprehensive understandings of the impact of cognitive interference on interference management in various network environments and devise a set of analysis techniques for characterizing the information theoretic performance of these models. Our studies will provide useful guidelines to significantly improve interference management technologies in practical wireless networks. In particular, we design new interference/state cancellation schemes that maximize the performance of these systems, and characterize the fundamental communication limits of the basic models. Then, we can understand the impact of different channel parameters on the achievability of the channel capacity through our numerical analysis.

1.2 Related Work

In this section, we introduce studies in the literature that are related to the results in this dissertation. The study of state-dependent channels were initiated by Shannon in [17], in which the channel model with causal knowledge of the channel state at the transmitter was studied. In [18], the point-to-point channel with the state known noncausally at the transmitter was studied, and the capacity was obtained for the discrete memoryless channel via Gel’fand-Pinsker binning. Based on this result, in [19], the capacity for the state-dependent point-to-point Gaussian channel was obtained, and it was shown that the state can be perfectly canceled as if there is no state interference. The achievable scheme was referred to as "dirty paper coding".

Various state-dependent network models have been studied, including the state-dependent multiple-access channel (MAC) in [20, 21, 22, 23, 24], the state-dependent broadcast channel in [25, 26, 27, 28], the state-dependent relay channel in [29, 26, 30, 31], and the state-dependent interference channel (IC) as we discuss below.

More closely to our work on the state-dependent ICs in Chapter 3 and Chapter 4 are a few studies on the state-dependent ICs as follows. A state-dependent IC model was studied in [32, 33] with two receivers corrupted by the same state, and in [34] with two receivers corrupted by independent states. More recently, in[35], both the state-dependent regular IC and Z-IC were studied, where the receivers are corrupted by the same but differently scaled state. Furthermore, in [36, 37], a type of the state-dependent Z-IC was studied, in which only one receiver is corrupted by the state and the state information is known only to the other transmitter. In [38], a type of the state-dependent Z-IC with two states was studied, where each transmitter knows only the state that corrupts its corresponding receiver. In [39], a state-dependent Z-interference broadcast channel was studied, in which one transmitter has only one message for its corresponding receiver, and the other transmitter has two messages respectively for two receivers. Both receivers are corrupted by the same state, which is known to both transmitters.

In [40] and [41], a model of the cognitive state-dependent IC was also studied, in which both transmitters (i.e., the primary and cognitive transmitters) jointly send one message to receiver 1, and the cognitive transmitter sends an additional message separately to receiver 2. The state is noncausally known at the cognitive transmitter only. In [42, 43], two state-dependent cognitive IC models were studied, where one transmitter knows both messages, and the two receivers are corrupted by two states which are know to both the two transmitters.

In all the previous work on the state-dependent IC and Z-IC, the states at two receivers are either assumed to be independent, or to be the same but differently scaled, with the exception of [38] that allows correlation between states. However, [38] assumes that each transmitter knows only one state at its corresponding receiver, and hence two transmitters cannot cooperate to cancel the states. In this dissertation, we investigate the state-dependent IC and Z-IC with the two receivers being corrupted respectively by two correlated states and with both transmitters knowing both states in order for them to cooperate. The state sequences are assumed to be known at both transmitters. The main focus of this dissertation is on the Gaussian state-dependent IC and Z-IC, where the receivers are corrupted by additive interference, state, and noise. The aim is to design encoding and decoding schemes to handle interference as well as to cancel the state at the receivers. In particular, we are interested in answering the following two fundamental questions: (1) whether or under what conditions both states can be simultaneously fully canceled so that the capacity for the IC and Z-IC without state can be achieved; and (2) what is the impact that the correlation between two states make towards state cancellation and capacity achievability.

A common nature that the above models share is that for each message to be transmitted, at least one transmitter in the system knows both the message and the state, and can incorporate the state information in encoding of the message so that state interference at the corresponding receiver can be canceled However, in practice, it is often the case that transmitters that have messages intended for receivers do not know the state, whereas some third-party nodes know the state, but do not know the message. In such a mismatched case, a helper user can assist all the interfered users to cancel state, though state information cannot be exploited in encoding of messages. A number of previously studied models capture such mismatched property. In [44], the point-to-point channel with a helper was studied, in which a transmitter sends a message to a state-dependent receiver, and a helper knows the state noncausally and can help the transmission. Lattice coding was designed in [44] for the helper to assist state cancellation at the receiver, and was shown to be optimal under certain channel conditions. A number of more general models were then further studied, which include the point-to-point channel as a special case. More specifically, in [45, 46], the state-dependent MAC was studied, which can be viewed as the point-to-point model with the helper also having its own message to the receiver. Two more general state-dependent MACs were studied in [47] and [48], which can be viewed as the MAC model in [45, 46] respectively with the helper further knowing the transmitter’s message and with one more state corruption known at the transmitter. In [49], the state-dependent Z-interference channel was studied, which can be viewed as the point-to-point model with the helper also having a message to its own receiver. In [29, 30], the state-dependent relay channel was studied, which can be viewed as the point-to-point model with the helper also receiving information from the transmitter and serving as a relay. When these models reduce to the point-to-point model here, the results in [46, 48, 30, 49] characterize the capacity of the Gaussian channel as the state power goes to infinity as in [44]. In particular, the achievable scheme in [48] is based on lattice coding similar to [44], and the scheme in [46, 30, 49] can be viewed as single-bin dirty paper coding (i.e., a special case of dirty paper coding [18, 19] with only one bin). The channel capacity of channels with helper remained unknown until, in [50], a new achievable scheme was introduced. under which the channel capacity for point-to-point channel with helper is characterized when the power of channel state is finite and the power of helper is small.

In this dissertation, we are interested in the state-dependent MAC with a helper. Various state-dependent MAC models were studied previously, which are related but different from the MAC model with a helper studied in this dissertation. State-dependent MAC models with state causally or strictly causally known at the transmitter were studied in [20, 21, 22, 23, 24], whereas our model assumes that the state is noncausally known at the helper. The two-user MAC with state noncausally known at the transmitters has been previously studied in various cases. [51, 52] studied the MAC model with state noncausally known at both transmitters, while [45, 46] assumed that the state is known only to one transmitter. [47] studied the cognitive MAC model in which one transmitter also knows the other transmitter’s message in addition to the noncausal state information. Furthermore, [53, 48] studied the model with the receiver being corrupted by two independent states and each state is known noncausally to one transmitter. In all these two-user MAC models with noncausal state information, at least one transmitter knows the state information, and can hence encode messages by incorporating the state information. Our MAC model is different in that only an additional helper knows the state information and assists to cancel the state. Our goal is to characterize the capacity region either fully or partially for such a model.

1.3 Contributions and Organization of Dissertation

The rest of the dissertation is organized as follows. In Chapter 2, we study the two-user state-dependent MAC with a helper. In this model, transmitters 1 and 2 respectively send two messages to one receiver, which is corrupted by an independent and identically distributed (i.i.d.) state sequence. The state sequence is known to neither the transmitters nor the receiver, but is known to a helper noncausally, which thus assists state interference cancellation at the receiver. Our focus is on the Gaussian channel with additive state. An outer bound on the capacity region is first derived, and an inner bound is then obtained based on a scheme that integrates direct state cancellation and single-bin dirty paper coding. By comparing the inner and outer bounds, the channel parameters are partitioned into appropriate cases, and for each case, either segments on the capacity region boundary or the full capacity region are characterized.

In Chapter 3 and Chapter 4, we investigate the Gaussian state-dependent IC and Z-IC, in which two receivers are corrupted respectively by two different but correlated states that are noncausally known to two transmitters and but are unknown to the receivers. Three interference regimes are studied, and the capacity region or sum capacity boundary is characterized either fully or partially under various channel parameters. For the very strong regime, the capacity region is achieved by a scheme where the two transmitters implement a cooperative dirty paper coding. For the strong but not very strong regime, the sum-rate capacity is characterized by rate splitting, layered dirty paper coding and successive cancellation. For the weak regime, the sum-rate capacity is achieved via dirty paper coding individually at two receivers as well as treating interference as noise. Furthermore, the impact of the correlation between states on cancellation of state and interference as well as achievability of capacity is explored with numerical illustrations.

This dissertation leads to the following two conference publications [54, 55], one journal paper publication [50] and one journal paper to be submitted [56].

Chapter 2 Helper-Assisted State Cancellation for Multiple Access Channels

In this chapter, we study the state-dependent MAC channel with a helper, where two transmitters wish to send the messages to a receiver over a state-corrupted channel, and a helper knows the state information noncausally and wishes to assist the receiver to cancel the state interference.

The rest of this chapter is organized as follows. We first describe the channel model. Then, we provide lower and upper bounds on the capacity. By analyzing the lower bounds to compare them with our upper bounds, we characterize the capacity region either fully or partially in various cases.

2.1 Channel Model

Fig. 2.1: The state-dependent MAC with a helper

We consider the state-dependent MAC with a helper (as shown in Fig. 2.1), in which transmitter 1 sends a message , and transmitter 2 sends a message to the receiver. The encoder at transmitter maps a message to a codeword for . The two inputs and are transmitted over the MAC to a receiver, which is corrupted by an i.i.d. state sequence . The state sequence is known to neither the transmitters nor the receiver, but is known to a helper noncausally. Hence, the helper assists the receiver to cancel the state interference. The encoder at the helper maps the state sequence into a codeword . The channel transition probability is given by . The decoder at the receiver maps the received sequence into two messages for .

The average probability of error for a length- code is defined as

(2.1)

A rate pair is achievable if there exist a sequence of message sets with for , and encoder-decoder tuples such that the average error probability as . We define the capacity region to be the closure of the set of all achievable rate pairs .

We focus on the state-dependent Gaussian channel with the output at the receiver for one channel use given by

(2.2)

where the noise variables , and . Both the noise variables and the state variable are i.i.d. over channel uses. The channel inputs , and are subject to the average power constraints , and .

Our goal is to characterize the capacity region of the Gaussian channel under various channel parameters .

2.2 Outer and Inner Bounds

We first provide an outer bound on the capacity region as follows, in which the first terms in the "min" improve the corresponding bounds give in [57].

Proposition 1.

An outer bound on the capacity region of the state-dependent Gaussian MAC with a helper consists of rate pairs satisfying:

(2.3a)
(2.3b)
(2.3c)

for some that satisfies .

Proof.

See Section 2.4.1. ∎

The second terms in the "min" in (2.3a)-(2.3c) capture the capacity region of the Gaussian MAC without state. If these bounds dominate the outer bound, then it is possible to design achievable schemes to fully cancel the state. Otherwise, if the first terms in the "min" in (2.3a)-(2.3c) dominate the outer bound, then the state cannot be fully canceled by any scheme, and the capacity region of the state-dependent MAC is smaller than that of the MAC without state.

We next derive an achievable region for the channel based on an achievable scheme that integrates direct state cancellation and single-bin dirty paper coding. In particular, since the helper does not know the messages, dirty paper coding naturally involves only one bin. More specifically, an auxiliary random variable (represented by in Proposition 2) is generated to incorporate the state information so that the receiver decodes such variable first to cancel the state and then decode the transmitters’ information. Based on such an achievable scheme, we derive the following inner bound on the capacity region.

Proposition 2.

For the discrete memoryless state-dependent MAC with a helper, an inner bound on the capacity region consists of rate pairs satisfying:

(2.4a)
(2.4b)
(2.4c)

for some distribution .

Proof.

See Section 2.4.2. ∎

Based on the above inner bound, we derive the following inner bound for the Gaussian channel.

Proposition 3.

For the state-dependent Gaussian MAC with a helper, an inner bound on the capacity region consists of rate pairs satisfying:

(2.5a)
(2.5b)
(2.5c)

for some real constants and satisfying . In the above bounds,

(2.6)
(2.7)

where .

Proof.

The region follows from Proposition 2 by choosing the joint Gaussian distribution for random variables as follows:

where are independent. The constraint on follows due to the power constraint on . ∎

We note that the above construction of the input of the helper reflects two state cancelation schemes: the term represents direct cancelation of some state power in the output of the receiver; and the variable is used for dirty paper coding via generation of the state-correlated auxiliary variable . Hence, the parameter controls the balance of two schemes in the integrated scheme, and can be optimized to achieve the best performance. This scheme is also equivalent to the one with , where and are correlated. While such approaches have been considered in the literature (see e.g., [45]), we believe that selecting and successively provides a more operational meaning to the correlation structure.

2.3 Capacity Characterization

By comparing the inner and outer bounds provided in Section 2.2, we characterize the capacity region or segments on the capacity boundary in various channel cases. Our idea is to separately analyze the bounds (2.5a)-(2.5c) in the inner bound and characterize conditions on the channel parameters under which these bounds respectively meet the bounds (2.3a)-(2.3c) in the outer bound. In such cases, the corresponding segment on the capacity region is characterized.

We first consider the bound on in (2.5a). Let

(2.8)
(2.9)

Then takes the following form

(2.10)

where maximizes

In fact, is set to maximize for fixed , and is set to maximize the function with being replaced by . If , then is achievable, and this meets the outer bound (the first term in "min" in (2.3a)). Thus, one segment of the capacity region is specified by .

Furthermore, we set and then obtain:

(2.11)

If , i.e., where holds for some , then is achievable, and this meets the outer bound (the second term in "min" in (2.3a)). This also equals the maximum rate for when the channel is not corrupted by state. Thus, one segment of the capacity region is specified by .

we demonstrate our characterization of the capacity via numerical plots.

Fig. 2.2: Lower and upper bounds on the capacity for the state-dependent channel with a helper

In Fig. 2.2, we fix , and , and plot the lower bounds

and the upper bounds in Proposition 1 as functions of the helper’s power . It can be seen that the lower bound matches the upper bound 1 () when , which characterize the capacity, and the lower bound matches the upper bound 2 () when , which corresponds to the capacity . The numerical result also suggests that when is small, the channel capacity is limited by the helper’s power and increases as the helper’s power increases. However, as becomes large enough, the channel capacity is determined only by the transmitter’s power , in which case the state is perfectly canceled. We further note that the channel capacity without state can even be achieved when (e.g., ). This implies that for these cases, the state is fully canceled not only by state subtraction, but also by precoding the state via single-bin dirty paper coding. We finally note that a better achievable rate can be achieved by the convex envelop of the two lower bounds, which does not yield further capacity result and is not shown in Fig. 2.2.

Fig. 2.3: Ranges of parameters for which the capacity is characterized

In Fig. 2.3, we fix , and plot the range of the channel parameters for which we characterize the capacity. Each point in the figure corresponds to one parameter pair . The upper shaded area corresponds to channel parameters that satisfy (4.7b), i.e., is large enough compared to , and hence the capacity of the channel without state can be achieved. The lower shaded area corresponds to channel parameters that satisfy (4.7a), and hence the capacity is characterized by a function of not only , but also and .

Similarly, following the above arguments, segments on the capacity region boundary corresponding to bounds on and can be characterized.

Summarizing the above analysis, we obtain the following characterization of segments of the capacity region boundary.

Theorem 1.

The channel parameters can be partitioned into the sets , where

If , then captures one segment of the capacity region boundary, where the state cannot be fully canceled. If , then captures one segment of the capacity region boundary, where the state is fully canceled. If , segment of the capacity region boundary is not characterized.

The channel parameters can alternatively be partitioned into the sets , where

where are defined similarly to (2.8) with being replaced by . If , then captures one segment of the capacity region boundary, where the state cannot be fully canceled. If , then captures one segment of the capacity region boundary, where the state is fully canceled.

Furthermore, the channel parameters can also be partitioned into the sets , where

where are defined similarly to (2.8) with being replaced by . If , then captures one segment of the sum capacity, where the state cannot be fully canceled. If , then captures one segment of the sum capacity, where the state is fully canceled.

The above theorem describes three partitions of the channel parameters respectively characterizing segments on the capacity region corresponding to , and . Then intersection of three sets (with each from one partition) collectively characterizes all segments on the capacity region boundary. For example, if a given channel parameter tuple satisfies , then following Theorem 1, line segments characterized by , , and are on the capacity region boundary. Since parameters and that achieve these segments are not the same, the intersection of these segments are not on the capacity region boundary.

Fig. 2.4: Segments of the capacity region for all cases of channel parameters

Fig. 2.4 lists all possible intersections of sets that the channel parameters can belong to. In principle, there should be cases. We further note that if , they must belong to and . Hence, the total number of cases becomes . For each case in Fig. 2.4, we use the solid red lines to represent the segments on the capacity region that are characterized in Theorem 1, and we also mark the value of the capacity that each segment corresponds to as characterized in Theorem 1.

We note that for several cases, segments on the capacity region boundary are characterized to be strictly inside the capacity region of the MAC without the state, i.e., the state cannot be fully canceled. For example, for cases with and , sum capacity segments are characterized to be smaller than the sum capacity of the MAC without state. These cases include mostly channel parameters with finite , and thus contain much larger sets of channel parameters than [57] that characterizes such sum capacity segment only for infinite .

We further note an interesting case (the last case in Fig. 2.4), for which the capacity region is fully characterized. We state this result in the following theorem.

Theorem 2.

If , i.e.,

(2.12)

where for some, then the capacity region of the state-dependent Gaussian MAC contains satisfying

which achieves the capacity region of the Gaussian MAC without state.

Theorem 2 implies that the state is fully canceled if the channel parameters satisfy the condition (2.12). We further note two special sets of channel parameters in this case. First, if , then and the condition clearly holds. This is not surprising because the helper has enough power to directly cancel the state. Secondly, if , then the condition holds for for arbitrarily large . This implies that if the helper’s power is above a certain threshold, then the state can always be canceled for arbitrary state power (even for infinite ).

2.4 Technical Proofs

2.4.1 Proof of Proposition 1

The second bounds in "min" in (2.3a)-(2.3c) follow from the capacity of the Gaussian MAC without state. The remaining bounds arise due to capability of the helper for assisting state cancelation and are derived as follows.

Consider a code with an average error probability . The probability distribution on is given by

(2.13)

By Fano’s inequality, we have

(2.14)

where as .

We first bound based on Fano’s inequality as follows:

(2.15)

where (a) follows because and are independent, and (b) follows because is an i.i.d. sequence.

We bound the first term in the above equation as

(2.16)
(2.17)

where .

It is easy to obtain bounds on the second and third terms in (2.15) as follows.

(2.18)
(2.19)

We next bound the last term in (2.15) as follows.

(2.20)

where the last equation follows by setting

so that and are uncorrelated.

Combining the above four bounds, we obtain the following upper bound on .