A Unified Scheme for TwoReceiver Broadcast Channels with Receiver Message Side Information
Abstract
This paper investigates the capacity regions of tworeceiver broadcast channels where each receiver (i) has both common and privatemessage requests, and (ii) knows part of the private message requested by the other receiver as side information. We first propose a transmission scheme and derive an inner bound for the tworeceiver memoryless broadcast channel. We next prove that this inner bound is tight for the deterministic channel and the more capable channel, thereby establishing their capacity regions. We show that this inner bound is also tight for all classes of tworeceiver broadcast channels whose capacity regions were known prior to this work. Our proposed scheme is consequently a unified capacityachieving scheme for these classes of broadcast channels.
Broadcast Channel, Capacity, Side Information \IEEEpeerreviewmaketitle
1 Introduction
We investigate the capacity regions of tworeceiver broadcast channels [1] with receiver message side information where each receiver may know some of the transmitted messages a priori. These channels are of interest due to applications such as multimedia broadcasting with packet loss, and the downlink phase of twoway relay channels [2]. The capacity regions of these channels are known for only the following special classes of the tworeceiver memoryless broadcast channel.

Specific message request and side information configuration (for all types of the channel):

Degraded message sets: one receiver needs to decode all the source messages, and the other one only a subset of the source messages [5]

Specific channel type (for all possible message requests and side information configurations):

Additive white Gaussian noise (AWGN) channel [6]

Less noisy channel

The capacity region for the less noisy case is obtained from the capacity region of the threereceiver less noisy broadcast channel where (i) only two receivers possess side information, and (ii) the request of the third receiver is only restricted to a common message demanded by all the receivers [7, Theorem 3].
The other results obtained from the existing capacity results for broadcast channels with three or more receivers, [4, 7, 8, 9, 10, 11], fall into the mentioned results for tworeceiver broadcast channels with complementary side information or degraded message sets.
(1)  
(2)  
(3)  
(4)  
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1.1 Contributions
We consider the message setup for tworeceiver broadcast channels that includes all possible message requests and side information configurations as special cases, i.e., each receiver (i) has both common and privatemessage requests, and (ii) knows part of the private message requested by the other receiver as side information. We propose a transmission scheme and derive an inner bound for the tworeceiver memoryless broadcast channel. We show that this inner bound (i) establishes the capacity regions for two new classes, namely the deterministic channel and the more capable channel, and (ii) is tight for all classes of tworeceiver broadcast channels with known capacity regions. A summary of the results is illustrated in Fig. 1.
2 System Model
We consider the tworeceiver discretetime memoryless broadcast channel , depicted in Fig. 2, with input , and outputs and . In this channel, is the transmitted codeword, and , is the channeloutput sequence at receiver . The transmitted codeword is a function of source messages, . The source messages are independent, and is uniformly distributed over the set , i.e., transmitted at rate bits per channel use.
We define two sets corresponding to each receiver. and are the set of messages requested by receivers 1 and 2 respectively. and are the set of messages known a priori to receivers 1 and 2 respectively. For receiver 1, is the part of the privatemessage request which is not known a priori to the other receiver, and is the part which is known. For receiver 2, these are and respectively.
A code for the channel consists of an encoding function
where denotes the Cartesian product, and denotes the fold Cartesian product of . It also consists of decoding functions
Average probability of error for this code is defined as
where , is the decoded at receiver 1, and , is the decoded at receiver 2.
Definition 1
A rate tuple is said to be achievable for the channel if there exists a sequence of codes with as .
Definition 2
The capacity region of the channel is the closure of the set of all achievable rate tuples .
Definition 3
A tworeceiver memoryless broadcast channel is said to be deterministic if the channel outputs are deterministic functions of the channel input, i.e., .
Definition 4
A tworeceiver memoryless broadcast channel is said to be more capable if for all input distributions .
3 Proposed Scheme and Inner Bound
In this section, we propose a transmission scheme and derive an inner bound for the tworeceiver memoryless broadcast channel with receiver message side information, stated as Theorem 1. The transmission scheme is constructed using Marton’s coding scheme [12, p. 205], superposition coding [1], and rate splitting.
Theorem 1
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(Codebook Construction) The codebook of the transmission scheme is formed from three subcodebooks which are constructed according to the distribution . Before subcodebook construction, using rate splitting, is divided into two independent messages at rate , and at rate such that .
The first subcodebook consists of i.i.d. codewords generated according to for each .
The second subcodebook consists of codewords generated according to where , i.e., for each , codewords are generated.
The third subcodebook consists of codewords generated according to where , i.e., for each , codewords are generated.
(Encoding) For the encoding, given , we first find a pair such that
where is the set of jointly typical sequences with respect to the considered distribution [12, p. 29]. If there is more than one pair, we arbitrary choose one of them, and if there does not exist one pair, we choose . We then construct the transmitted codeword as .
(Decoding) Receiver 1 decodes , if it is the unique tuple for which we have
otherwise the error is declared.
Receiver 2 similarly decodes , if it is the unique tuple for which we have
otherwise the error is declared.
We assume the transmitted messages are equal to one by the symmetry of code construction, and without loss of generality . Hence, the error events at receiver 1 are (6)–(9); note that there exist some other error events, but they yield redundant achievability conditions. The error events at receiver 2 are similarly written. Based on the error events, packing lemma [12, p. 45], and mutual covering lemma [12, p. 208], the achievability conditions are
We finally perform FourierMotzkin elimination to obtain the region in (1)–(5).
4 Capacity Results
In this section, using the derived inner bound in Theorem 1, we establish the capacity regions for two new classes, i.e., the deterministic channel, stated as Theorem 2, and the more capable channel, stated as Theorem 3. We also show that our inner bound is tight for all classes of tworeceiver broadcast channels whose capacity regions were known prior to this work.
Theorem 2
The capacity region of the tworeceiver deterministic broadcast channel with receiver message side information is the closure of the set of all rate tuples , each satisfying
(10)  
(11)  
(12)  
(13)  
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for some .
We present the achievability proof in the following, and the converse proof in Appendix A. {IEEEproof} (Achievability) The achievability part of Theorem 2 is proved by setting in (1)–(5).
Theorem 3
The capacity region of the tworeceiver more capable broadcast channel with receiver message side information is the closure of the set of all rate tuples , each satisfying
(15)  
(16)  
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for some .
We present the achievability proof in the following, and the converse proof in Appendix B. {IEEEproof} (Achievability) The achievability of Theorem 3 is proved by setting in (1)–(5). Note that implies that , and .
4.1 Discussion on Prior Known Results
In this subsection, we show that the derived inner bound in Theorem 1 is tight for all classes of tworeceiver broadcast channels with known capacity regions, as depicted in Fig. 1.
The capacity region of the tworeceiver memoryless broadcast channel with complementary side information is achieved by multiplexing all the requested messages in only one codebook [4, 3]. This scheme is a special case our scheme obtained by setting . Note that and are equal to zero in this message setup.
The capacity region of the tworeceiver memoryless broadcast channel with degraded message sets is achieved by superposition coding [5]. This scheme is a special case of our scheme obtained by setting or depending on whether receiver 1 or receiver 2 needs to decode the whole set of the source messages, respectively. Note that either or is equal to zero in this message setup.
The AWGN broadcast channel and the less noisy broadcast channel are a subset of the more capable broadcast channel [12], as depicted in Fig. 1, then our scheme can also achieve their capacity regions.
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(29) 
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5 Conclusion
We proposed a transmission scheme and derived an inner bound for the tworeceiver memoryless broadcast channel with receiver message side information. We considered the general message setup which includes all possible message requests and side information configurations as special cases. Our proposed scheme is a unified capacityachieving scheme for all classes of tworeceiver broadcast channels whose capacity regions had been previously established, and for two new classes, i.e., the deterministic channel and the more capable channel.
Appendix A
In this section, we present the converse proof for the tworeceiver deterministic broadcast channel with receiver message side information. The proof is based on the converse proof for the channel without receiver message side information [13]. {IEEEproof}(Converse Proof) By Fano’s inequality [12, p. 19], we have
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where as for . For the sake of simplicity, we use instead of for the remainder. The inequalities in (18)–(19) also lead to the following inequalities,
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Inequalities (23)–(27) yield conditions (10)–(14) respectively. To this end, we use the Csiszár sum identity [12, p. 25] based on which we have
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where and . We also need to define the auxiliary random variable as
We here show only how inequalities (23) and (26) yield conditions (10) and (13) respectively. We just need to follow similar steps for (24), (25), and (27).
In (23), we expand the mutual information term as follows
Then, since as , by using the standard timesharing argument [12, p. 114], we have
In (26), we first expand the mutual information terms as in (29). We then expand part 1 of (29) as in (30) where follows from (28). We also expand part 2 of (29) as in (31). Finally, since as , and
by using the standard timesharing argument, we have
(32)  
(33)  
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(37) 
Appendix B
In this section, we present the converse proof for the tworeceiver more capable broadcast channel with receiver message side information. The proof is based on the converse proof for the channel without receiver message side information [14].
(Converse Proof) Using (18) and (19), if a rate tuple is achievable, then it must satisfy (32)–(34).
In (32), we expand the mutual information term as follows