Cooperative Transmission and Radio Resources Management for Wireless Networks
Broadcast Relay Channel
Achievable Rates for the Non-Degraded Gaussian Broadcast Relay Channel
Adrian Agustin de Dios
Dept. Teoria de la Señal y Comunicaciones (TSC) Grupo de Procesado de la Señal(GPS) Universidad Politecnica de Catalunya (UPC)
Barcelona, SPAIN [email protected]
Research Stage at the Multimedia Communications Laboratory (MCL) University of Texas at Dallas (UTD)
Richardson, TX, USA BE 2005 (Sept-Dec 2005)
1 SUPPORT ... 3
2 OBJECTIVES AND WORK PLAN DEVELOPED IN THE RESEARCH STAGE... 3
3 INTRODUCTION TO THE BROADCAST RELAY CHANNEL... 4
4 FULL DUPLEX CASE ... 6
4.1 OUTER BOUNDS... 7
4.2 ACHIEVABLE REGION... 7
4.2.1 EASYCASEFORBROADCAST-RELAYCHANNEL ... 8
4.2.2 RELAYCHANNELWITHSIDEINFORMATION(SI)... 13
4.2.3 NON-DEGRADEDBROADCASTRELAYCHANNELWITHSI ... 15
5 PRELIMINARY CONCLUSIONS ... 15
6 REFERENCES... 16
7 ANNEX A: PAPER SUBMITTED TO ISIT 2006 ... 16
1 Support
This work was financed by the Departament d'Universitats, Recerca i Societat de la Informació de la Generalitat de Catalunya (DURSI) with the scholarships: BE (Beques per a estades de recerca fora de Catalunya) and FI ( Formacio de Personal Investigador 2005FIR-00260)
2 Objectives and work plan developed in the research stage
This research stage has carried out at the Multimedia Communication Laboratory (MCL) at the University of Texas at Dallas (UTD) under the supervision of Professor Aria Nosratinia. It started on September, 19th and has finished on December, 21st of 2005 (3 months).
This research stage has been a good opportunity to complement and extend the work developed in the Ph.D thesis named “Cooperative Transmission and Radio Resources Management for wireless networks” and whose Thesis Project was presented in July 2005 at Universitat Politécnica de Catalunya (UPC). Professor Aria Nosratinia has been working in the last years in the same area. Additionally he is author of some relevant references considered in the thesis project.
The main objective of the research stage was to improve the knowledge over the cooperative transmission and the radio resources management. Initially, as I pointed in the scholarship application form, there were some possible points to study during this research period.
However, after a presentation of all my work to the Professor and his Ph.D. students and also taking into account his recent work and his experience on Information Theory, he suggested to study the cooperative transmission from an Information Theory point of view deeper than I had been done until that moment. That way will provide a strong theoretic basis to all my previous work. However, taking into account the duration of the research stage (3 months), the work done will be an excellent starting point. For this reason we have analyzed the scenario in which I have been working and we have identified the main aspects to be considered. It can be considered as a broadcast relay channel (BRC).
The work that I had done assumes a scenario with one source and multiple destinations.
Moreover, in order to improve the performance in terms of mutual information, it has been considered that for each destination there is another nearby terminal (called relay) which will help to improve the performance of the destination. Basically, up to now, I have been considering that the system works in TDMA (Time Division Multiple Access), providing to each user one period of time (slot) and avoiding the interference between users. Moreover, all the relays should transmit in the same slot in order to compensate the extra use of resources. In this last slot there is a interfering scenario. However, this scheme is not optimal from the information theory point of view. The main idea is to know how far is from the optimal one.
For this reason we have considered a scenario where the source and the relays transmit simultaneously. Basically, the scenario has one source, two destinations and two relays. This scheme combines different types of channels considered in the information theory, such as the relay channel, broadcast channel (BC) and interference channel (IFC) (if the relays transmit information intended only to its associated destination). For instance the capacity of the relay channel is only known when the channel is degraded. The Capacity Region (CR) (the optimal way to encode the signals for the different users, i.e. related to radio resources management) of the MIMO (Multiple Input Multiple Output) BC has been proved recently using the Side Information at the source (also known as Dirty Paper Coding) and for the IFC, the CR is only known for some interference values, as long as for the other values, the CR is just bounded.
Recently, some papers have appeared talking about the Broadcast-Relay channel providing the
Capacity Regions when the channel is degraded and there is only one relay (it is not the same as in my scenario).
The work plan has consisted on studying the relevant papers of the different Information Theory areas present in my scenario with the help of Professor A.Nosratinia, learning the common tools to deal with the different problems. Basically, in this report I will present how to tackle my scenario (also a Broadcast Relay Channel - BRC) and will show some results. Our scenario is non-degraded in general, for this reason we believe that one way to obtain one achievable region (and maybe the capacity region (maximum achievable region)) is the use of Side Information in the BRC in a similar way as it was done for the non-degraded BC. However, our BRC is more complex than the BC. Therefore, during this research stage we have started to study how to apply the Side Information to the relay channel, when it is known by both source and relay, and when only is known to the source. This first step has to be considered if we want to apply Side Information knowledge to the Broadcast Relay channel.
Moreover, it must be stood out that as a result of the work done during this research stage a paper has been produced to be submitted to the 2006 International Symposium on Information Theory (ISIT), named “Achievable Rates for the Relay-Assisted Broadcast Channel”. Finally, it is expected to continue the joint research started in this stage at UTD.
3 Introduction to the Broadcast Relay Channel
In this work we will study the Capacity Region (CR) of the Broadcast Relay Channel (BRC).
This type of channel was firstly presented in [13], showing the benefits of relaying over the broadcast channel (BC) for two users. In that environment, it was assumed two different types of relaying. The former one is the user-relaying, where one user can receive data and help the other user to improve its performance. The second one is the dumb-relay, where there is one dumb relay helping to both users. The achievable region for degraded channel is obtained.
Moreover in [14] an extension to multiple users/relays is performed, providing achievable regions and deriving new outer bounds to the CR also for the degraded channel. In this case, it is assumed that each user receives its own information and also helps the remaining users to receive their intended data.
In this work we have considered a scenario similar to ones given in [13] and [14], but in this case we have assumed that there are nodes devoted only to help the other users (act as relays), see Figure 1. This scheme tries to model the scenario where a source sends independent messages for two users, and additionally, these users can have access to at least one relay to enhance their performance. Therefore these messages will also be received by the relays. This scenario combines multiple-relays and multiple-users with broadcast, [8]-[11] and relay transmissions, [7]. Moreover, depending on the role played by the relays, i.e. in case that each RS only try to help to its associated user, then this scenario is an interference channel (IFC), [1]- [5]. However because of all RSs are in receiving mode they might have side information (SI) at the transmitter about the signals transmitted by the remaining RSs. In that case a coding following the work of Slepian-Wolf and Gel’fand-Pinsker should be considered. Some references related to this work are [15],[16],[17].
A practical implementation of this scenario could be applied to downlink transmissions of a centralized cellular system with relays as all my Ph.D work done up to now. This work starts to study the optimal transmission scheme for this scenario. In the following we are going to describe some of the most interesting configurations to analyze, which will help as a starting point.
S
U2
U1
R1
R2 b a c
Figure 1.- Broadcast Relay channel with 2 users
This scenario presents a large number of situations to analyze. In order to study them easily we will consider the most relevant in the full and half duplex mode, respectively:
1. Full Duplex Mode
1.1. All the transmitters synchronized (c=0 and relays-cooperation).- In this case both relays (RSs) can transmit a combination of the messages devoted to both users.
1.2. The relay’s transmitter is only synchronized with its associated user (c=0 and relays-cooperation).- Now both RSs also transmit a combination of messages for both users, but for example user 1 will receive the signal intended to user 2 asynchronously.
1.3. The relay’s transmitter is only synchronized with its associated user (c=0 and no relays-cooperation).- Each RS transmits a message intended for its associated user. Note that in this case, the relay transmissions are performed over a channel similar to an Interference Channel (IFC).
1.4. All the transmitters synchronized (c≠0 and relays-cooperation).- This case is different from (1.1) because both relays can help one to each other.
1.5. The relay’s transmitter is only synchronized with its associated user (c≠0 and relays-cooperation).- In this case we assume that the signal received by the relays from the other relays is asynchronous with respect the signal received from the source.
2. Half Duplex Mode.- In this case there is a period of time where the source is transmitting and the relays and users are receiving data, and a second period of time where the source and relays are transmitting and the users are in receiving mode.
Therefore, in this case the relays cannot help one another, i.e c=0.
2.1. The relay’s transmitter is only synchronized with its associated user and there is relay-cooperation (if it is possible)
2.2. The relay’s transmitter is only synchronized with its associated user and there is no-relay-cooperation
It seems that the most interesting situations to analyze are: {1.2, 1.4, and 1.5} and {2.1 and 2.2}.
The objectives of the work will be to obtain the achievability region given a receiver structure and outer bounds of our scenario, analyzing the differences between achievability regions and outer bounds for the different cases. Needless to say that all this work cannot be done during the research stage (3 months). Basically, we have started analyzing the full duplex case and during my Ph.D work I will continue developing the remaining points. Additionally, in order to extend
the analysis of this scenario, the study of the multiplexing-diversity tradeoff, impact of multiple antennas and use of other techniques such as Amplify and Forward (A&F) are aspects that can be considered for the future.
In the following, the work developed during the research stage is presented. Section §4 presents the BRC for the full duplex case, taking into account outer bounds obtained with “max-flown min cut theorem” [12] and the achievable region taking into account some configurations of degradedness between the different elements of the scenario. However, as it will be seen in
§4.2.1, this scenario is not degraded in general and some of the common tools considered cannot be applied now. For that reason, in section §4.2.2 the Side Information (or State Information) at the source and the relays has been considered. In that section the main scenarios are presented and moreover the paper submitted to the 2006 International Symposium on Information Theory (ISIT) will be added in annex A.
4 Full duplex case
Figure 2 presents the Broadcast Relay Channel (BRC) given 2 final users with 2 associated relays (RSs) for the full duplex case. We assume that RSs act as dumb relays, i.e. they are devoted only to help the end destination users to enhance their total throughput.
b a
2: 2
Y X
1: 1
Y X X
Y1
Y2
c
Figure 2.- Broadcast Relay channel with 2 users for the full duplex case
This BRC consist of a channel input X , relay input X , X1 2, relay receive values Y ,Y1 2 and two user receive values Y ,Y1 2. The probability transition function is given by
(
1, , ,1 2 2| , ,1 2)
p y y y y x x x . The signal model on the Gaussian channel is given by,
1 1 2 1
2 1 2 2
1 2 1
2 1 2
Y X X aX Z
Y X bX X Z
Y X cX Z Y X cX Z
= + + +
= + + +
= + +
= + +
(1)
with X the signal transmitted from the source with power P, X ,X1 2 the signal transmitted from the relays with powers PR1 and PR2, respectively, a and b the channel interference coefficients, c the channel coefficient between RS1 and RS2, and Z Z Z Z independent zero mean 1, 2, ,1 2 Gaussian random variables with variances N N N N1, 2, 1, 2, respectively.
4.1 Outer bounds
Applying the “max-flown min cut theorem” [12] we obtain the following outer bounds of the capacity region (CR) of the BRC depicted in Figure 2.
( )
( )
( )
1 2 1 1 2 2 1 2
1 1 2 1
2 1 2 2
; , , , | , , ;
, , ;
R R I X Y Y Y Y X X R I X X X Y
R I X X X Y
+ ≤
≤
≤
(2)
Now assuming that each RS is synchronized only with its associated user and letting,
{ } { }
{ } { }
11 22
* *
11 1 1 1 22 2 2 2
* *
11 2 0 22 1 0
R R
X X X
E X X PP E X X PP
E X X E X X
ϕ ϕ
= +
= =
= =
2
1 1 1 2
1
1
2
2 2 2 1
2
2
1 2
1 2 1 2
2
2
1 1 1 1
R R R
R R R
P P PP a P
R C
N
P P PP b P
R C
N
R R C P
N N N N
ϕ
ϕ
+ + +
≤
+ + +
≤
+ ≤ + + +
(3)
In case of trying to obtain tight outer bounds we should follow [6].
4.2 Achievable Region
Initially, because of the previous work on BRC developed in [13] and [14] for degraded channels, we have considered “superposition coding” for the different signals transmitted to the RSs and users. The main idea is that each user tries to decode his intended signal treating the remaining ones as additive noise. The coding is given by,
(
0,)
1(
0, R1)
2(
0, R2)
X ∼N P X ∼N P X ∼N P (4)
Note that the signal X have to contain the signals devoted for the users 1 and 2 and their associated RSs. We assume that the channel is degraded. Taking into account the different elements in our scenario, the number of interesting cases to study is reduced to the following two:
¾ Option 1: X → RS1 → RS2 → Y1 → Y2
¾ Option 2: X → RS1 → Y1 → RS2 → Y2
However, will see that in general our scenario is not degraded and cannot be expressed using a Markov chain, § 2.2.1.
As a consequence of the non-degraded channel we will consider the Side Information (SI) available in our scenario, because there is only one source. Our scenario combines the broadcast
and the relay channel, which means that at least the source (and maybe the relays) has full knowledge of the signal intended for other users and considered as interference. Therefore it is important to consider how to make good use of this knowledge. For this reason, for a better understanding, first we will study the different cases in which the side information can be considered for the relay channel, taking into account [15]-[17]. This will be done in § 4.2.2.
Afterwards, we will apply these results to our broadcast-relay channel.
4.2.1 EASY CASE FOR BROADCAST-RELAY CHANNEL
In this section we will study easy cases for the broadcast-relay channel. The objective is to obtain some knowledge about the different parameters we have in our scenario and study if it is possible to express it using a Markov chain channel using the common tools to analyze these kinds of channels. However we will see that only some special cases can be considered as degraded channels.
4.2.1.1 No Relays Cooperation and C=0 (No Side Information) and no Interference
In this option we study the case when there is no possible communication between RS (maybe they are placed far away from each other) and additionally the relays only transmit data to its associated users. In that case the RSs work in a scenario without interference. Then we define the same coding structure as “Cover-El-Gamal” [7], that is, coding to two simultaneous relays.
( ) ( ) ( ) ( )
1 2
' '
1 1 1 2
1 1 1 2 2 2
1 2
' '
1 0, 1 1 2 0, 1 2 1 0, 1 2 0, 2
R R
R R
X B B
P P
B X X B X X
P P
X N P X N P X N P X N P
α β α β
α β α β
= +
= + = +
∼ ∼ ∼ ∼
(5)
with Bi the signals devoted for the relay i. The following equations have to be satisfied for the power constraint.
1
1 1
i i
i i i Nrelays
α α β β
+ =
+ = = … (6)
¾ Option 2: X → RS1→ Y1→ RS2→Y2
In this case, user 1 can decode the signal devoted to RS2 and user 2. The achievable region is given by,
( ) ( )
{ }
1 min 1; |1 1 2 2 , , 1; |1 2 2
R ≤ I B Y X X B I X X Y X B (7)
( ) ( )
( ) ( )
2 2 1 2 2 2
2
2 1 1 2 2 1 1 2
; | , , ; ,
min ; | , ; |
I B Y X X I X X Y R I B Y X X I B Y X X
≤
(8)
( ) ( )
( ) ( )
1 1
1 1 1 2 2 1 1 1,1
1
1 1 1 1 1
1 1 2 1 0,1 2,1 1 1 0,1 2,1
1
; |
, ; | R 2 R 2
I B Y X X B C P C SNR
N
P P PP
I X X Y B C C SNR SNR SNR SNR
N
β α β α
α α β
α α β
= =
+ +
= = + +
( )
( )
( )
2 1 1,2
2 1
2 2 1 2
1 1,2
2 1
1 0,2 2,2 1 2 0,2 2,2
1 2 1 2 2
2 2
2 1 1 0,2
2 1
2 1 1 2
1 1
; | 1
2 2 , ;
1
; | degraded
R R
P SNR
I B Y X X C C
SNR
N P
SNR SNR SNR SNR
P P PP
I X X Y C C
N P SNR
I B Y X X C P
N P
β α β α
α α
α α β
α α β
α α
β α α
= + = +
+ + + +
= =
+ +
= +
( )
(
2 1 1 2)
2 1( )
1 1
ness
; | P degradedness
I B Y X X C
N P
β α α
= +
Notice that the signal intended for RS2 can be decoded by user 1 due to the degradedness relation considered in this subsection. For this reason, user 1 is not interfered by RS2.
¾ Option 1: X → RS1→ RS2→ Y1→Y2
In this case the RS1 is the best, RS2 is interfered by the signal devoted to R1, user 1 is interfered by the signal devoted to RS2 (it cannot decode the message) and finally user 2 receives interference of the signals intended to RS1 and user 1. Notice that we assume no interference when both RS are transmitting and receiving (that is the factor c=0).
Therefore, the achievability region is given by
( ) ( )
{ }
1 min 1; |1 1 2 2 , , 1; |1 2 2
R ≤ I B Y X X B I X X Y X B (9)
( ) ( )
{ }
2 min 2; |2 1 2 , , 2; 2
R ≤ I B Y X X I X X Y (10)
( ) ( )
( )
1 1
1 1 1 2 2 1 1 1,1
1
1 0,1 2,1 1 1 0,1 2,1
1 1 1 1 1
1 1 2
1 1 1 0,1
; |
2 2 , ; |
1
R R
I B Y X X B C P C SNR
N
SNR SNR SNR SNR
P P PP
I X X Y B C C
N P SNR
β α β α
α α β
α α β
α α
= =
+ + + +
= =
+ +
These equations changes from the ones presented in the previous subsection just in the “new”
noise received by the user 1 (which comes from the signal intended to RS2 and cannot be decoded by RS1). But, the RS2 cannot help the user 1 to decode the signal intended to user 2, then, despite of presenting a better channel than user 2, suffers interference from the total signal intended to user 2.
The mutual information obtained by the RS2 and the user 2 is the same than in the previous sub- section, because nothing has changed form him in this new scenario.
( )
( )
2 1 1,2
2 1
2 2 1 2
1 1 1,2
2 1 1
1 0,2 2,2 1 2 0,2 2,2
1 2 1 2 2
2 2
2 1 1 0,2
; | 1
2 2 , ;
1
R R
P SNR
I B Y X X C C
SNR
N P
SNR SNR SNR SNR
P P PP
I X X Y C C
N P SNR
β α β α
α β α β
α α β
α α β
α α
= + = +
+ + + +
= =
+ +
However, the following constraints should be satisfied to consider the degradedness of this channel: The RS1 should be able to decode the signal intended to RS2
(
2 1 1 2)
2 1( )
1 1 1
; | P degradedness
I B Y X X C
N P
β α α β
= +
Therefore, this channel cannot be modeled by a MARKOV-CHAIN, that is, it is non-degraded.
Its configuration will be give by,
1
1 2
2
X Y Y Y Y
→ → →
→
4.2.1.2 No Relays Cooperation and C=0 (No Side Information) and Interference
¾ Option 2: X → RS1→ Y1→ RS2→Y2
In this case, the change is in what user 1 can decode. Now it can decode the signal devoted to RS2 and user 2.
( ) ( )
{ }
1 min 1; |1 1 2 2 , , 1; |1 2 2
R ≤ I B Y X X B I X X Y X B (11)
( ) ( )
( ) ( )
2 2 1 2 2 2
2
2 1 1 2 2 1 1 2
; | , , ; ,
min ; | , ; |
I B Y X X I X X Y R I B Y X X I B Y X X
≤
(12)
( ) ( )
( ) ( )
1 1
1 1 1 2 2 1 1 1,1
1
1 1 1 1 1
1 1 2 1 0,1 2,1 1 1 0,1 2,1
1
; |
, ; | R 2 R 2
I B Y X X B C P C SNR
N
P P PP
I X X Y B C C SNR SNR SNR SNR
N
β α β α
α α β
α α β
= =
+ +
= = + +
( )
( )
( )
2 1 1,2
2 2 1 2 2 1
1 1,2
2 1
1 0,2 2,2 1 2 0,2 2,2
1 2 1 2 2
1 2 2
2 1 1 1 0,2 2,1 2
2 1
2 1 1 2
1
; | 1
2 2 , ;
1
; |
R R
R to
P SNR
I B Y X X C C
SNR
N P
SNR SNR SNR SNR
P P PP
I X X Y C C
N P b P SNR SNR
I B Y X X C P N
β α β α
α α
α α β
α α β
α α
β α
= + = +
+ + + +
= =
+ + + +
= +
( )
( ) ( )
1
2 1 1 2 2 1
1 1
degradness
; | degradness
P I B Y X X C P
N P
α β α
α
= +
This case is degraded.
¾ Option 1: X → RS1→ RS2→ Y1→Y2
( ) ( )
{ }
1 min 1; |1 1 2 2 , , 1; |1 2 2
R ≤ I B Y X X B I X X Y X B (13)
( ) ( )
{ }
2 min 2; |2 1 2 , , 2; 2
R ≤ I B Y X X I X X Y (14)
( ) ( )
( )
( )
1 1
1 1 1 2 2 1 1 1,1
1
1 0,1 2,1 1 1 0,1 2,1
1 1 1 1 1
1 1 2 2
1 1 2 1 0,1 2, 2,1
; |
2 2 , ; |
1
R R
R
I B Y X X B C P C SNR
N
SNR SNR SNR SNR
P P PP
I X X Y B C C
N P a P SNR SNR
β α β α
α α β
α α β
α α
= =
+ + + +
= =
+ + + +
( )
( )
( )
( )
2 1 1,2
2 1
2 2 1 2
1 1,2
2 1
1 0,2 2,2 1 2 0,2 2,2
1 2 1 2 2
2 2 2
2 1 1 1 0,2 2, 1,2
2 1 1 2 2 1
1
; | 1
2 2 , ;
1
; |
R R
R
P SNR
I B Y X X C C
N P SNR
SNR SNR SNR SNR
P P PP
I X X Y C C
N P b P SNR SNR
I B Y X X C P N
β α β α
α α
α α β
α α β
α α
β α α
= + = +
+ + + +
= =
+ + + +
= +
( )
1 1
degradedness β P
For this case, user 1 can remove the interference generated by user 2 in case of decoding the signal intended to user 2 (we have assumed only synchronous transmission between each relay and associated users, the signals received from other RSs are considered asynchronous). In that case the MI of user 2 that user 1 can decode is given by,
(
2 1)
1 2 21 1 1
, ; R
R
P a P I X X Y C
N P P
α α
+
= + +
However, this depends on the values of a, N1 and PR1. A similar argument can be considered for the signal intended to user 1 and received by user 2. Therefore, we should consider different values (as in the interference channel (IFC))
This channel is also non-degraded when takes into account the interference. However, for some configuration it is possible to describe it using a Markov-chain model.
4.2.1.3 RESULTS EASY BRC
This section presents some results of the achievable region obtained by the different configuration discussed in previous sections. Firstly, we will show the results obtained by the configuration named “option 2”, which has the following parameters,
( ) ( )
( ) ( )
1 1 2 2
0,1 0,2
1,1 1,2
2,1 2,2
2, 1,2 2, 2,1
2, 1,2 2, 2,1
Option 2
12 0
20 4
12 6
110 110 No Interference
6 6 Interference
X RS Y RS Y
SNR dB SNR dB
SNR dB SNR dB
SNR dB SNR dB
SNR dB SNR dB
SNR dB SNR dB
→ → → →
= =
= =
= =
= − = −
= =
(15)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0
0.5 1 1.5 2 2.5 3
R2 (bps/Hz) R1 (bps/Hz)
Option 2 BC
BRC No Intereference BRC Interference
Figure 3.- AR for the “option 2”, with (7)-(8) and without interference (11)-(12).
Figure 3 presents the AR obtained by “option 2” with and without interference, with the rate of user 1, R1, in the y-axis and the rate of user 2, R2 in the x-axis. It can be seen how the AR curve varies in a different way for each user when the other changes a little bit its rate. Remind that user 1 in this scenario can decode the data intended for user 2, therefore the variation of the AR curve is only due to the variation of the power devoted for user 1. Lets consider the case of no interference between RS, note that when the user 2 does not transmit anything (R2=0, α2=0, α1=1), then user 1 is the relay channel.
( ) ( )
{ }
1 1 1 1,1 1 0,1 2,1 1 1 0,1 2,1
max min , 2
I C SNR C SNR SNR SNR SNR
β β α α α β
= + + (16)
and when user 2 increases it rate there is only changes in the factor, α1<1. In the other hand, for the user 2 when the user 1 does not transmit anything is also working as a relay channel, however when user 1 increases its rate, additionally of decreasing the factor α1 there is an increment of the denominator in (17), producing a fast decrement of the rate obtained by user 2.
2
1 0,2 2,2 1 2 0,2 2,2
2 1 1,2
1 1,2 1 0,2
max min , 2
1 1
SNR SNR SNR SNR
I C SNR C
SNR SNR
β
α α β
β α
α α
+ +
= + +
(17)
In case of considering the interference case, the same reasons apply because user 1 is able to decode the signal received from RS2, an in the case of user 2 the total interference has increased.
For the “option 1” case we have considered the following scenario
( ) ( )
( ) ( )
1 2 1 2
0,1 0,2
1,1 1,2
2,1 2,2
2, 1,2 2, 2,1
2, 1,2 2, 2,1
Option 1
14 0
20 12
12 6
110 110 No Interference
6 6 Interference
X RS RS Y Y
SNR dB SNR dB
SNR dB SNR dB
SNR dB SNR dB
SNR dB SNR dB
SNR dB SNR dB
→ → → →
= =
= =
= =
= − = −
= =
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 0
0.5 1 1.5 2 2.5
R2 (bps/Hz) R1 (bps/Hz)
Option 1 BC
BRC No Intereference BRC Interference
Figure 4.- AR for the “option 1”, (9)-(10)with and without interference (13)-(14).
Figure 4 presents the results of the AR for the “option 1” with and without interference. Notice that for this scenario the configuration is different from the “option 2”, therefore, different results are obtained compared to Figure 3. In this scenario the situation for the second user remains the same as in the previous case. Despite of improving the quality the RS2, it cannot decode the signal intended to user 1 (it is worse than RS1 and it does not have the help of a relay to decode the signal), so the mutual information obtained by the second user follows (17) (but taking into account the current values of the scenario).
For user 1 there is a big difference from the previous case. Notice of the new mutual information obtained by this scenario, (18). The main difference is on the second part of equation (18), considering the signal intended to user 2 as noise. This is because, user 1 does not have the help of any relay to decode the signal intended to user 2, and therefore it has to consider it as additive noise. The new equation produces a large decrement of the R1 when user 2 increases its rate, R2, from 0 to some given value. Notice that there is a decrement of α1 at the numerator and the denominator increases in the second part of (18) whereas for the first part of the equation there is only a low decrement. This performance motivates a decrement of the optimal β1 and at the same time the total mutual information.
( )
( )
1
1 0,1 2,1 1 1 0,1 2,1
1 1 1,1
1 0,1 2, 2,1
max min , 2
1
SNR SNR SNR SNR
I C SNR C
SNR SNR
β
α α β
β α α
+ +
= + +
(18)
4.2.2 RELAY CHANNEL WITH SIDE INFORMATION (SI)
In this section we have studied the cases where the transmitters have knowledge of the interference and adjust their transmissions to be “orthogonal” to the interference, see [15] for the direct channel. This study for the relay channel will help to obtain an achievable region for the non-degraded broadcast channel. Basically this work is based in [15]-[18]. The details of this work and the application to the BRC can be found in Annex A, where there is a copy of the paper submitted to the ISIT 2006.
Firstly, we present the main cases that can be considered for the relay channel. We have identified three:
o The relay and the destination have the same interference
o There is only interference at the destination
o The relay and the destination have different interference (similar to previous case) Notice that for the different options we can consider where the side information is known.
Basically we have considered the case where SI is known at all transmitters and only at the source.
4.2.2.1 Relay and destination with the same interference
In this case we assume that the same interference is affecting the relay and the destination, see Figure 5.
RELAY
N1
SOURCE
I1
I1 I1
N
Figure 5.- Relay channel with the same interference at the relay and the destination
In [18] it has been shown that taking into account the SI when it is available at the source and the relay is the same as the relay channel. Therefore, it is possible to achieve the same rates as the interference is not present using the same coding structure as in the relay channel. But, what happen if the source is the only one who has access to the SI. In that case maybe the coding strategy should be change in order to make good use of the SI.
The signal model is given by,
1 1 1
1 1
Y X I N
Y X X I N
= + +
= + + + (19)
4.2.2.2 Interference at the destination
In this case the interference is only at the destination.
RELAY
N1
N
I2
SOURCE
I2
I2
Figure 6.- Relay with interference at the destination
Assume the case of SI at both transmitters. In that scenario, if we try to use the relay coding, the source can encode its codewords taking into account the interference, and the final result will be the same as the relay channel without interference, [18]. However, when the SI is only available at the source, a new problem arises in the source-relay link because in that link there is not interference. Therefore, if we transmit using the SI, we are decreasing the performance of the source-relay link. In that case, we have to optimize the total performance.
4.2.2.3 Relay and destination with different interference This case is a generalization of the previous one.
RELAY
N1
SOURCE
I1,I2 ?
I1 I2 ? I1
N I2
Figure 7.- Relay channel with different interference at the relay and the destination
4.2.3 NON-DEGRADED BROADCAST RELAY CHANNEL WITH SI
How to apply the previous result to the BRC can be seen in the paper presented in annex A.
5 Preliminary conclusions
This research stage has provided an excellent starting point for an information theoretic study of broadcast relay channels, complementing and extending parts of my Ph.D thesis, cooperative transmission and radio resources management for wireless networks. During this stage I have improved the understanding of the subject.
It has shown than in general the BRC is not degraded and we have studied new achievable regions taking into account the side information available at the different transmitters. BRC always improve the results of the broadcast channels. However, it is difficult to check how far is from capacity region. In the future, we will try to obtain tighter outer bounds of the capacity region and see how far are from our achievable region.
Notice that the BRC is a “new” scenario and there are many possibilities to extend this work:
Rayleigh channel, MIMO, multiplexing-diversity tradeoff, Amplify and forward techniques, ….
During this stage I have “only” worked on the full duplex case, but additionally, this research stage has been an excellent starting point and will not finish at the end of it, because I will continue the joint research with UTD from the UPC.
6 References
[1] A.B.Carleial, “Interference Channels”, IEEE Trans. on Information Theory, vol 24, nº1, January 1978.
[2] A.B.Carleial, “Outer Bounds on the Capacity of Interference Channels”, IEEE Trans. on IT, vol 29, nº4 July 1983
[3] M.H.M. Costa, “On the Gaussian Interference Channel”, IEEE Trans. on IT, vol 31, nº5, September 1985
[4] G.Kramer, “Outer Bounds on the Capacity of Gaussian Interference Channels”, IEEE Trans. on IT, vol 50, nº 3, March 2004
[5] T.S.Han and K.Kobayashi, “A new Achievable Rate Region for the Interference channel”, IEEE Trans. on IT, vol 27, nº1, January 1981.
[6] A.Host-Madsen, “Capacity Bounds for Cooperative Diversity”, Submitted to xxx, available on his web page.
[7] T.M.Cover, A.A.El Gamal, “Capacity Theorems for the Relay Channel”, IEEE Trans. on IT vol 25, nº5, September 1979.
[8] T.M.Cover, “Broadcast Channels”, IEEE Trans. on IT, vol 18,nº 1, January 1972
[9] H.Weingarten, Y.Steinberg, S.Shamai, “The Capacity Region of the Gaussian MIMO Broadcast Channel”, Proc. CISS 2004, Princeton University, March 2004
[10] T.M.Cover, “Comments on Broadcast Channels”, IEEE Trans on IT, vol 44, nº6, October 1998.
[11] S.Vishwanath, N.Jindal, A.Goldsmith, “Duality, Achievable Rates, and Sum-Rate capacity of Gaussian MIMO broadcast channels”, IEEE Trans. on IT, vol 49, n10, October 2003.
[12] T.M.Cover and J.A.Thomas, Elements of Information Theory. New York: Wiley, 1991.
[13] Y.Liang, V.V.Veeravalli, “The Impact of Relaying on the Capacity of Broadcast Channels”, Proc.
of IEEE International Symposium on Information Theory (ISIT), 2004.
[14] A.Reznik, S.R.Kulkarni, S.Verdu, “Broadcast-Relay Channel: Capacity Region Bounds”, Proc. of IEE ISIT 2005.
[15] M.H.M. Costa, “Writing on Dirty Paper”, IEEE Trans. on IT, vol 29, nº3, May 1993.
[16] N.Devroye, P.Mitran, V.Tarokh, “Achievable Rates in Cognitive Radio channels”, Proc. of Conference on Information Sciences and Systems (CISS), March 2005.
[17] N.Devroye, P.Mitran, V.Tarokh, “Achievable Rates in Cognitive Radio channels”, submitted to IEEE Trans of IT, August 2005
[18] Y.H.Kim, A.Sutivong, S.Sigurjonsson, “Multiple User Writing on Dirty Paper”, Proc. of IEEE ISIT 2004
7 Annex A: Paper submitted to ISIT 2006
Channels
Aria Nosratinia
University of Texas at Dallas, Richardson, TX Email: [email protected]
Adrian Agustin and Josep Vidal
Technical University of Catalonia (UPC), Spain Email: {agustin,pepe}@gps.tsc.upc.edu
Abstract— We study the broadcast channel where dedicated relays assist in the broadcast, and furthermore these relays may interfere with each other and the original transmission.
Without loss of generality, it may be assumed that some nodes have superior reception than others. Therefore some nodes have more information than others, and can use dirty paper coding, while others can only transmit by considering the interference as noise. This calls for coding strategies that accommodate unequal knowledge of interference across the nodes. We approach the problem by first analyzing MAC and relay channels with asymmetric information about interference. We then use the strategies developed for asymmetric MAC and relay to calculate achievable rates for the relay-assisted broadcast channel.
I. INTRODUCTION
In the relay channel, a dedicated relay helps a source node’s transmission to a destination [1]. Cover and El Gamal [2]
found capacity regions for the degraded Gaussian relay chan- nel through a number of innovative techniques.
Since many interesting wireless communication scenarios involve a network of multiple users, it is natural to ask how a (group of) relay(s) may be used in such a network. This paper analyzes the scenario with a single source broadcasting to multiple destinations, in the presence of multiple dedicated relays. The relays may interfere with each other, which makes the problem more realistic, but also more challenging.
In the following, we review a few recent results on re- lay channels involving broadcast transmission (several nodes receiving) with a view to their system model. We use the term “cooperation” where the relay nodes are also data sources/sinks, and the term “relay” for the case where a node is purely a relay, and does not source or sink data.
Liang and Veeravalli [3] present two degraded scenarios (in each case the relay has a better channel with respect to the recipient). One scenario consists of a traditional two-user broadcast channel where receiver 1 also acts as a relay for receiver 2. In another scenario, a dedicated relay is helping two receivers. Both these scenarios involve degraded channels.
This work was later extended to a case where both users can transmit relayed information for each other.
Reznik et al. [4] develops capacity region bounds for another type of degraded cooperative system, where the path from the source to the final destination involves multiple hops, and each of the hops involves a combination receiver/relay, i.e., each relay is also a data sink. Therefore, with our terminology,
U1
R1
R2
U2
a
b
Fig. 1. Relay-assisted Gaussian broadcast channel
this constitutes a (degraded) cooperative broadcast system. A serially degraded system as used in [4] can make the problem more tractable, by assuming that the downstream relays do not interfere with upstream relays, while the upstream relays do affect downstream relays (see [4], Figure 3). We did not use a serially degraded model for relay-assisted broadcast (this paper) since the nodes in our graph (Figure 1) do not naturally lend themselves to a linear ordering, and furthermore, we are partially motivated by applications such as wireless communication where interferences in the same frequency band are essentially reciprocal.
In this paper, we consider a broadcast system where dedi- cated relays assist in the communication process. This problem introduces a new topology compared to the works mentioned above, as well as the key difference of multiple dedicated relays and their interference with each other. Figure 1 shows the special case of two receivers and two relays. The channel between the two relays and the two receivers resembles an interference channel, whose capacity region is in general unknown, but capacity bounds have been studied by Sato, Carleial, and others (see e.g. [5]). However, Figure 1 is distinct from the interference channel in an important way: in the standard interference channel, the two transmitters do not share any information, while in the present case, the data of the two relays is provided by a single transmitter, and the “better”
relay in effect can know the transmission of the other relay.
This can be exploited to improve transmission rates.
The general structure of our solution uses arguments similar
Relay 1) and Branch 2 (consisting of User 2 and Relay 2).
We first encode the signals of Branch 1 (which include block Markov coding for the relay), by considering the signals of Branch 2 as noise. Then Branch 2 is encoded, attempting to cancel the interference from Branch 1 using dirty paper coding. This might not be straight forward due to existence of signals from multiple nodes, and we consider the related issues in the sequel. We then consider a second achievable region by interchanging the role of the two branches, i.e., Branch 2 considers all unwanted signals as noise, and Branch 1 uses dirty paper coding. The convex hull of all rate pairs in the two achievable regions yields the final result.
Without loss of generality, one relay will experience a better SNR than the other. Therefore, the branch that is doing dirty paper coding may contain a relay that does not know as much about interference signals as the source node (because it cannot decode the signals sent to the other relay). The source, of course, knows everything, therefore we have a relay channel with asymmetric side information.1 So, as a first step, we calculate achievable rate region for a relay channel with interference, and asymmetric information at source and relay. In Section II we analyze the performance of a MAC channel with asymmetric information, followed by a similar investigation for the relay channel. Subsequently, we use these results to obtain achievable regions for the broadcast relay channel in Section III.
II. ASYMMETRICKNOWLEDGE OFINTERFERENCE
Costa [8] studied the additive white Gaussian noise (AWGN) channel with interference given by,
Yn= Xn+ Sn+ Zn (1)
where Xnis the transmit signal with average power constraint Pn
i=1Xi2 6 nP, Zn the additive noise with distribution N (0, N I) and Snthe interference with distribution N (0, QI), independent of Zn and Xn. It was shown that the channel capacity is unaffected by interference when the interference is noncausally known at the transmitter. Basically, Costa used a modification of the capacity theorem of [9] for the discrete memoryless channels p(y|x, s) with state information Sn noncausally known at the transmitter,
C= max
p(u,x|s)I(U; Y ) − I(U; S) (2) with U an auxiliary random variable. The capacity is found by maximizing over all such U, which leads to a maximizer:
U = X + αS with α∗= P
P+ N (3)
and a capacity that is equivalent to that of AWGN without interference. In [7] the result of Costa is extended to several other channels with known interference, namely the broadcast
1The problem of a relay with side information common to the source and relay was addressed in [7]
S N
User 1
Receiver X1
Fig. 2. Gaussian MAC with interference noncausally known to user 1
channel, MAC, and the relay channel (with interference).
However [7] only considers the case where interference is known at all transmitters and relays. This has a simplifying effect; the achievable region turns out to be the same as the cases without interference, thus no converse needs to be established. Unfortunately, our problem is somewhat different and the results of [7] must be generalized for our purposes.
In the relay-assisted broadcast channel, there are interferences that are not necessarily known to everybody. Thus we must look at channels with asymmetric knowledge of interference.
A. Multiple Access Channel
Although the MAC channel with asymmetric side infor- mation is not directly used in our scheme of relay-assisted broadcast, we mention it here as a stepping stone to the relay channel with asymmetric side information. Figure 2 shows a two-user MAC channel, with interference that is noncausally known for one of the transmitters. The received signal is given by,
Yn = X1n+ X2n+ Sn+ Zn (4) where X1n and X2n are transmit signals with average power constraints Pni=1X`2(i) 6 nP` for ` = 1, 2, Zn the addi- tive noise distributed as N (0, NI) and Sn the interference distributed as N (0, QI), independent of Zn, X1n and X2n.
For a DMC MAC channel with interference noncausally known at the transmitter of User 1, an achievable region is given by the convex hull of the following:
R1 ≤ I(U1; Y |X2) − I(U1; S) (5)
R2 ≤ I(X2; Y |U1) (6)
R1+ R2 ≤ I(U1, X2; Y ) − I(U1; S) (7) U1 is an auxiliary random variable providing an overall distribution p(s)p(u1, x1|s)p(x2|s)p(y|x1, x2, s).
In contrast to [7], in our scenario the interference is only known at one of the transmitters. Thus we consider:
U1= X1+ αS (8)
where α is a parameter to maximize the mutual information, in a manner similar to Costa. Following the same steps as
