New aspects arise when generalizing two-way relay network with one relay to two-way relay network with multiple relays. To study the essential features of the two-way multiple-relay network, we focus on the case of two relays in our work. The problem of how two terminals, equipped with multiple antennas, exchange messages with the help of two relays is studied. Five transmission strategies, namely, amplify-forward (AF), hybrid decode amplify forward (HLC), hybrid decode amplify forward (HMC), decode forward (DF), and partial decode forward (PDF), are proposed. Their designs are based on a variety of techniques including network coding, multiplexed coding, multi-input multi-output transmission, and multiple access with common information. Their performance is compared with the cut-set outer bound. It is shown that there is no dominating strategy and the best strategy depends on the channel conditions. However, by studying their multiplexing gains at high signal-tonoise (SNR) ratio, it is shown that the AF scheme dominates the others in high SNR regime.
1. Introduction
Relay channel, which considers the communication between a source node and a destination with the help of a relay node, was introduced by van der Meulen in [1]. Based on this channel model, Cover and El Gamal developed coding strategies known as decode-forward (DF) and compress-forward (CF) in [2]. These techniques now become standard building blocks for cooperative and relaying networks, which have been extensively studied in the literature (e.g., [3, 4]).
For many applications, communication is inherently two-way. A typical example is the telephone service. In fact, the study of two-way channel is not new and can be traced back to Shannon's work in 1961 [5]. However, the model of two-way relay channel, though natural, did not attract much attention. Recently, probably due to the advent of network coding [6] in the last decade, there is a growing interest in this model. The application of DF and CF to two-way relay channel was considered in [7]. The half-duplex case was studied in [8, 9]. The results in [10] showed that feedback is beneficial only in a two-way transmission. Network coding for the two-way relay channel was studied in [11, 12]. Physical layer network coding based on lattices is considered recently [13], and shown to be within 0.5 bit from the capacity in some special cases [14].
Anzeige
All the aforementioned works are for one relaying node. It is easy to envisage that in real systems, more than one relay can be used. Schein in [15] started the investigation of the network with one source-destination pair and two parallel relays in between. This model was further studied in [16] under the assumption of half-duplex relay operations. For one-way multiple-relay networks in general, cooperative strategies were proposed and studied in [17]. We remark that a notable feature that does not exist in the single-relay case is that the multiple relays can act as a virtual antenna array so that beamforming gain can be reaped at the receiver. In this paper, we follow this line of research and consider two-way communications. Two-relays are assumed, for this simple model already captures the essential features of the more general multiple-relay case. We are interested in knowing how different techniques can be used to construct transmission strategies for the two-way two-relay network and how they perform under different channel conditions. In particular, we apply the idea of network coding to both the physical layer and the network layer. Besides, channel coding techniques for multiple access channel (MAC) and multi-input multi-output (MIMO) channel are also employed. Several transmission strategies are thus constructed and their achievable rate regions are derived.
We remark that the channel model that we consider in this paper is also called the restricted two-way two-relay channel [7]. This means that the signal from a source node depends only on the message to be transmitted, but not on the received signal at the source. Besides, our results are obtained under the half-duplex assumption, which is realistic for practical systems. Each node is assumed to transmit one half of the time and receive during the other half of the time. The performance of our proposed strategies can be further improved if the ratio of transmission time and receiving time is optimized. We do not consider this more general case, since it complicates the analysis but provides no new insights.
This paper is organized as follows. Our network model is described in Section 2. Some basic coding techniques are reviewed in Section 3. Based on these coding techniques, several transmission strategies are devised in Section 4. Their performance at high signal-to-noise ratio regime is analyzed in Section 5. The rate regions of these strategies are compared under some typical channel realizations in Section 6. The conclusion is drawn in Section 7.
2. Channel Model and Notations
The two-way two-relay (TWTR) network consists of four nodes: two terminals and , and two parallel relays 1 and 2 (see Figure 1). Terminals and want to exchange messages with the help of the two relays. We assume there is no direct link between the two terminals and between the two-relays. Furthermore, all of the nodes are half-duplex. The total communication time, , are divided into two stages, each of which consists of time slots. In the first stage, the terminals send signals and the relays receive. In the second stage, the relays send signals and the terminals receive. The solid arrows in Figure 1 correspond to stage 1 and the dashed arrows correspond to stage 2.
Figure 1
Model of two-way two-relay network. The labels of the arrows indicate the corresponding link gains.
Figure 2
Decoded messages at the two-relays in the DF strategy.
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Suppose that terminals and are equipped with antennas, whereas each of relays 1 and 2 has only one antenna. For and , we use to denote the transmit signal from node , and to denote independently and identically distributed (i.i.d.) Gaussian noise with distribution . The channel is assumed static and the channel gain from node to is denoted by an -dimensional column vector . We assume channel reciprocity holds so that = . In the first stage, the outputs of the network at time , are given by
(1)
(2)
In the second stage, for , the outputs at the terminal nodes are
(3)
(4)
where , is the transmit symbol of relay , for is a Gaussian random vector with each component i.i.d according to . We assume that the link gains , , , and are time-invariant and known to all nodes. We have the following power constraints in each stage:
(5)
for , and
(6)
for , where , , , and denote the power constraints on terminals and and relays 1 and 2, respectively.
Let and be the data rates of terminal and , respectively. In a period consisting of channel symbols ( symbols for each phase), terminal wants to send one of the symbols to terminal , and terminal wants to send one of the symbols to terminal . A code for the TWTR network consists of two message sets and , two encoding functions
(7)
two relay functions
(8)
and two decoding functions
(9)
For , terminal transmits the codeword in stage one, where is the message to be transmitted. For , relay applies the function to its received signal and transmits the resulting signal in the second stage. Let the received signals at terminals and be and , respectively. In this paper, we will use a superscript "" to indicate a sequence of length . So and are sequences of length , with each component equal to a vector in . After the second stage, terminal decodes the message from the other source node by . We note that the decoding function uses the message from source terminal as input as well. We say that a decoding error occurs if or . The average probability of error is
(10)
A rate pair is said to be achievable if there exists a sequence of codes, satisfying the power constraints in (5) and (6), with as .
Although the terminals are equipped with antennas, the transmitted signals from the terminals are essentially 2 dimensional. To see this, we observe that the first term in the right hand side of (1), namely, , is a projection of in the direction of . Any signal component of orthogonal to will not be picked up by relay 1. Likewise, from (2), we see that any signal component of orthogonal to will not be sensed by relay 2. There is no loss of generality, if we assume that the signals transmitted from the terminals take the following form:
(11)
for , where is an matrix, and the two components in represent the projections of on and . We consider the 2-dimensional vector as the input to the channel at node . The power constraint in (5) can be written as
(12)
for .
Notations
We will treat random vectors and as input signals at terminal and , respectively, and let and denote their corresponding covariance matrices. For and , let
(13)
be the signal to noise ratio of the signal received at relay from terminal . Shannon's capacity formula is denoted by . Also, for matrices, we let , where denote the identity matrix. The reason for the factor of 0.25 before the log function, instead of a factor of 0.5 in the original capacity formula, is due to the fact that the total transmission time is divided into two stages of equal length. All logarithms in this paper are in base 2. The set of non-negative real numbers is denoted by . Gaussian distribution with mean zero and covariance matrix is denoted by .
3. Review of Coding Techniques and Capacity Regions from Information Theory
The proposed transmission strategies are based on a host of existing coding techniques and capacity results. A review of them is given in this section.
3.1. Physical-Layer Network Coding
In wireless channel, the channel is inherently additive; the received signal is a linear combination of the transmitted signals. This fact is exploited for the two-way relay channel in [18‐21]. Consider the following single-antenna two-way network with two sources and one relay in between. There is no direct link between the two sources, and the exchange of data is done via the relay node in the middle. Let be the transmitted signal from source , for . The transmission is divided into two phases. In the first phase, the relay receives
(14)
where is an additive noise. For simplicity, it is assumed that both link gains from the sources to the relay are equal to one. In the second phase, the relay amplifies the received signal , and transmits a scaled version of , where is a scalar chosen so that the power requirement is met. Since source 1 knows , the component within the received signal at source 1 can be treated as known interference, and hence be subtracted. Similarly, source 2 can subtract from the received signal. Decoding is then based on the signal after interference subtraction.
3.2. Multiplexed Coding
Multiplexed coding [22] is a useful coding technique for multi-user scenarios in which some user knows the message of another user a priori. Consider the two-way relay channel as in the previous paragraph. Node 1 wants to send message to node 2 via the relay node, and node 2 wants to send message to node 1 via the relay node. For , let be the number of bits used to represent message . The transmission of the nodes is divided into two phases. In the first phase, the two source nodes transmit. Suppose that the relay node is able to decode and . For the encoder at the relay, we generate a array of codewords. Each codeword is independently drawn according to the Gaussian distribution such that the total power of each codeword is less than or equal to . In the second phase, the relay node sends the codeword in the -entry in this array. Suppose that the received signal at source node is corrupted by additive white Gaussian noise with variance , for . At source 1, since is known, the decoder knows that one of the codewords in the row corresponding to had been transmitted. Out of these codewords, it then declares the one based on the maximal likelihood criterion. By the channel coding theorem for the point-to-point Gaussian channel, source 1 can decode reliably at a rate of . Likewise, by considering the columns in the array of codewords, source 2 can decode at a rate of .
Multiplexed coding can be implemented using concepts from network coding. We assume, without loss of generality, that . We identify the possible messages from source node with the vectors in the -dimensional vector space over the finite field of size 2, , and identify the messages from source node with a subspace of of dimension , say . We generate Gaussian codewords independently, one for each vector in . To send messages and in the second phase, the relay node transmits the codeword corresponding to , where the addition is performed using arithmetics in . The output of the decoder at node 1 is a vector in . We subtract from it the vector in corresponding to . If there is no decoding error, this gives the codeword corresponding to , and the value of is recovered.
Now let us consider node 2. Since is known a priori, node 2 is certain that the signal transmitted from the relay is associated with one of the vectors in the affine space . The message can be estimated by comparing the likelihood function of the codewords associated with . It can be seen that the maximal data rate is the same as in the array approach mentioned in the previous paragraph, but the size of the codebook at the relay reduces from to .
3.3. Capacity Region for MIMO Channel
Consider a MIMO channel with transmit antennas and receive antennas, with the link gain matrix denoted by a real matrix . The channel output equals
(15)
where is the -dimensional channel input and is an -dimensional zero-mean colored Gaussian noise vector with covariance matrix . Without loss of information, we whiten the noise by pre-multiplying both sides of (15) by . The transformed channel output is thus
(16)
The covariance matrix of the noise vector is now the identity matrix. By the capacity formula for MIMO channel with white Gaussian noise [23], the capacity for the MIMO channel in (15) is given by
(17)
where denotes the covariance matrix of . Using the identity
(18)
which holds for any matrix and matrix , we rewrite (17) as
(19)
3.4. Capacity Region for Multiple-Access Channel (MAC)
The channel output of the two-user single-antenna Gaussian multiple-access channel is given by
(20)
where is the signal from user , for , and is an additive white Gaussian noise with variance . Each of the two users wants to send some bits to the common receiver. Suppose that the power of user is limited to , for . The rate pair , where is the data rate of user , is achievable in the above 2-user MAC if and only if it belongs to
(21)
(22)
(23)
(24)
We refer the reader to [24] for more details on the optimal coding scheme for MAC.
4. Channel-Network Coding Strategies
We develop five transmission schemes for TWTR network. In the first scheme (AF), the received signals at both relay nodes are amplified and forwarded back to terminals and . In the second and third scheme (HLC, HMC), one of the relays employs the amplify forward strategy, while the other decodes the messages from terminals and . In the fourth scheme (DF), both relays decode the messages from terminals and . In the last strategy (PDF), another mixture of decode-forward and amplify-forward strategy is described.
4.1. Amplify Forward (AF)
In this strategy, relay node () buffers the signal received in the first stage, and amplifies it by a factor of . The amplified signal
(25)
is then transmitted in the second stage. At the end of the second stage, each terminal, who has the information of itself, subtracts the corresponding term and obtains the desired message from the residual signal.
By putting (25) into (3), we can write the received signal at terminal as
(26)
Here, we have replaced and by their 2-dimensional representations and . Since terminal knows its own input as well as the link gains and amplifying factors, the signal component containing as a factor can be subtracted from . The residual signal is
(27)
The message from terminal can then be decoded using a decoding algorithm for point-to-point MIMO channel. The received signal at terminal is treated similarly.
Theorem 1.
A rate pair is achievable by the AF strategy if
(28)
where
(29)
, and and are covariance matrices, such that the following power constraints:
(30)
(31)
are satisfied.
Proof.
The residual signal (27) at terminal can be written as plus a noise vector with covariance matrix . The residual signal at terminal equals plus a noise vector with covariance matrix . Therefore, after self-signal subtraction, the resultant channels can be considered MIMO channels with two transmit antennas and receive antennas. From (19), we obtain the rate constraints in (28). The inequalities in (30) are the power constraints for terminals and , and those in (31) are the power constraints for relays 1 and 2.
4.2. Hybrid Decode-Amplify Forward with Linear Combination (HLC)
In this strategy, relay 1 decodes the messages from terminals and , and meanwhile, relay 2 employs the amplify-forward strategy. In order to obtain beamforming gain, after decoding the two messages, relay 1 reconstructs the codewords corresponding to the decoded messages and sends a linear combination of them in the second stage.
In the first stage, relay 1 and terminals and form a multiple-access channel with relay 1 as the destination node. We use the optimal encoding scheme for MAC at terminals and , and the optimal decoding scheme at relay 1. In the second stage, relay 1 decodes and reconstructs and , and then transmits a linear combination
(32)
for some and . Relay 2 amplifies by a scalar factor and transmits .
At terminal , after subtracting the signal component that involves (t), we get
(33)
At terminal , the residual signal after subtraction is
(34)
The decoding is done by using decoding method for MIMO channel.
Theorem 2.
A rate pair is achievable by the HLC strategy if
(35)
(36)
(37)
where
(38)
, , , and and are covariance matrices such that the following power constraints:
(39)
(40)
(41)
are satisfied.
In (35), the product of a real number and a set is defined as .
Proof.
From the rate constraints for MAC channel in (22)–(24), we have the rate constraints for relay 1 in (35). We multiply by a factor of one half because the first phase only occupies half of the total transmission time.
The conditions in (36) and (37) are derived from the capacity formula for MIMO channel with colored noise in (19). The inequalities in (39) are the power constraints for sources and . The inequalities in (40) and (41) are the power constraints for relays 1 and 2, respectively.
The parameters , , , and can be obtained by running an optimization algorithm. For example, we can aim at maximizing a weighted sum . The values of , , and which maximize the weighted sum are chosen.
4.3. Hybrid Decode-Amplify Forward with Multiplexed Coding (HMC)
As in the previous strategy, relay 1 decodes and forwards the messages from and , and relay 2 amplifies and transmits the received signal. However, in this strategy, relay 1 re-encodes the messages into a new codeword to be sent out in the second stage. Terminals and decode the desired messages based on multiplexed coding.
Theorem 3.
A rate pair is achievable by the HMC strategy if and satisfy
(42)
(43)
(44)
where
(45)
(46)
(47)
, are covariance matrices satisfying (39), and satisfies (41).
Proof.
The proof is by random coding argument and we will sketch the proof below. More details can be found in [25].
Our objective is to show that any rate pair that satisfies the condition in the theorem is achievable. For , terminal randomly generates a Gaussian codebook with codewords with length , satisfying the power constraint in (5). Label the codewords by , for . For relay 1, we generate a array of Gaussian codewords of length and power . The codeword in row and column is denoted by , and satisfies the power constraint in (6).
After the first stage, relay 1 is required to decode both messages from terminals and . This can be accomplished with arbitrarily small probability of error if the rate constraints for MAC in (22) to (24) are satisfied. This corresponds to the rate constraint in (42). Let the estimated messages from and be and .
In the second stage, relay 1 transmits . Relay 2 amplifies its received signal and transmits . From (41), the amplified signal has average power no more than .
After subtracting the term , which is known to terminal , the residual signal at terminal is
(48)
Note that terminal A knows its message , and with probability arbitrarily close to one if (42) is satisfied. The idea of multiplexed coding can then be used. In (48), the covariance matrix of the signal in square bracket is given by in (46), and the covariance of the noise term is given by . Applying the capacity expression, we obtain the rate constraint in (44). In a similar manner, we obtain (43).
4.4. Decode Forward (DF)
In the DF strategy, terminal node , () splits the message into two parts: the common part and the private part . The two common messages are transmitted via both relay nodes. The private message is decoded by relay 1 only, and can be interpreted as going through the path from terminal to relay 1 to terminal . Symmetrically, the private part of message is decoded by relay 2 only, and can be interpreted as going through the path from terminal to relay 2 to terminal . After the first stage, relay 1 decodes the common messages of both terminals and the private message of terminal . Relay 2 decodes the common messages of both terminals and the private message of terminal . The encoding and decoding schemes in the first stage is similar to those developed by Han and Kobayashi for the interference channel (IC) in [26]. Since both relays have access to the common messages, the channel in the second stage can be considered a multiple access channel with common information. Furthermore, since terminals and have information of themselves, we can further improve the rate region by the idea of multiplexed coding.
We have the following characterization of the rate region for the DF strategy:
Theorem 4.
For , let and be the rates of the private and common messages, respectively, from terminal . Let denote for , and let , , , and denote covariance matrices, and
(49)
for , and . For . A rate pair is achievable if we can decompose and such that
(50)
(51)
(52)
(53)
(54)
(55)
(56)
(57)
for some nonnegative and .
Details of the DF coding scheme and the proof of Theorem 4 are given in the Appendix.
4.5. Partial Decode Forward (PDF)
In the PDF strategy, both relays decode the message of terminal . Each relay then subtracts the reconstructed signal of terminal from the received signal. Call the resulting signal the residual signal. The message of terminal is re-encoded into a new codeword, and linearly combined with the residual signal. This linear combination is then transmitted in the second stage. Since both relays know the message of terminal , the two-relays can jointly re-encode the message of terminal using some encoding scheme for a MIMO channel with two transmit antennas and receive antennas.
Theorem 5.
A rate pair is achievable by the PDF strategy if it satisfies
(58)
(59)
(60)
where
(61)
and and , are covariance matrices such that the following power constraints hold
(62)
(63)
for . (Here, denotes the th diagonal entry in .)
Proof.
The two-relays treat the signal originated from terminal as noise, and decode the message of terminal . The rate requirement in (58) guarantees that the message of terminal can be decoded with arbitrarily small probability of error at both relays. Let the decoded message of terminal be denoted by .
For , the reconstructed signal is then subtracted from . The residual signal at relay is .
At the relays, we employ two Gaussian codebooks for the re-encoding of the message from terminal . For each message , we generate two correlated codewords and , with mean zero and each pair of symbols at any distributed according to a covariance matrix . At relay , the decoded message is re-encoded into , which is a codeword with power . In the second stage, relay transmits
(64)
for some amplifying factor . The inequality in (63) ensures that the power constraint is satisfied at the relays.
At the end of stage 2, terminal subtracts the signal component that involves and from its received signal and obtains
(65)
From the capacity formula for MIMO channel (19), terminal can recover the message from terminal reliably if (60) is satisfied.
For the decoding in terminal , we subtract all terms involving , and get
(66)
This is equivalent to a MIMO channel with link gain matrix and colored noise. Recall that is the covariance matrix of the encoded signal. By the capacity formula of MIMO channel (19), we obtain the rate constraint in (59).
Remark 1.
We note that the matrices , , and , for , are invertible. Indeed, by checking that is strictly positive for all non-zero , we see that the matrix is positive definite, and hence invertible.
5. Performance in High SNR Regime
In this section, we compare the performance of the five strategies described in the previous section in the high Signal-to-Noise Ratio (SNR) regime.
For fixed powers and link gains, let denote the sum rate as a function of the noise variance . We use the multiplexing gain (also called degree of freedom) [27], defined by
(67)
as the performance measure at high SNR. At high SNR, that is, when is very small, we can approximate the sum rate by if the multiplexing gain is equal to .
Consider the multiplexing gain of the AF scheme. When the sum rate is maximized subject to the rate constraints (28) in Theorem 1, the equalities in (28) hold. We can assume without loss of generality that
(68)
(69)
We first suppose that the covariance matrices and , and the amplifying constants and , are fixed. Note that if the power constraint in (31) holds, then it continues to hold if becomes smaller. Therefore, when , the power constraints in (30) and (31) are satisfied.
Each of the expressions in (68) and (69) can be written in the form
(70)
where is a matrix that equals
(71)
(72)
By singular value decomposition [28, Chapter ], we can factor as , where and are unitary matrices, and is a diagonal matrix with non-negative diagonal entries . The number of positive diagonal entries in is precisely the rank of . We can rewrite (70) as
(73)
Suppose that is equal to . The determinant
(74)
in (73) can be expanded as a polynomial in , with the degree equal to the rank of . Therefore, the limit
(75)
depends only on the rank of the matrix , and equals 0, 0.5, or 1, if the rank of is 0, 1, or 2, respectively. The problem of determining the multiplexing gain now reduces to determining the rank of the matrices in (71) and (72).
Recall that the rank function satisfies the following properties [28, page 13]: (i) if and are square invertible matrices, then for all matrix , whenever the matrix multiplications are well-defined; (ii) for all matrices , we have . Consider the matrix in (72). After replacing by its definition, we can express the matrix in (72) as
(76)
where denotes the diagonal matrix . We assume that and have full rank. This assumption holds with probability one if the link gains are generated from a continuous probability distribution function such as Rayleigh. Also, we assume that , , and are of full rank. This assumption does not incur any loss of generality, because they are design parameters that we can choose. We can perturb them infinitesimally, and the resulting matrices will be of rank two, but the value on the right hand side of (69) deviates negligibly. By property (i), and the fact that , , and are invertible matrices, the rank of the matrix in (76) is equal to the rank of . Then we get
(77)
Similarly, we can show that the rank of the matrix in (71) is equal to two.
For fixed invertible covariance matrices and , and positive real numbers and ,
(78)
Since the above argument holds for all invertible and , and positive and , we conclude that the multiplexing gain of the AF strategy is equal to 2.
For HLC and HMC, relay 1 is required to decode the messages of the terminals, and in both schemes the sum rate is subject to the sum rate constraint in the MAC channel in the first phase. The multiplexing gains of both the HLC and HMC strategies are limited by
(79)
Similarly, the multiplexing gain of DF is also limited by the decoding of messages at the relays. The rate constraints (50) and (51) imply that it is no more than 0.5.
The multiplexing gain of the PDF scheme is somewhere in between the multiplexing gains of AF and DF. The transmission from terminal to terminal can be considered AF, while the transmission from terminal to terminal in the other direction is limited by the message decoding after stage 1. From (58), we get
(80)
and from (60), we have
(81)
provided that the has full rank. Therefore, its maximal multiplexing gain is 1.5.
We summarize the performance of the five schemes at high SNR in Table 1. We can see that the AF strategy has the highest multiplexing gain. It is well known that the maximal multiplexing gain of the Gaussian MIMO channel with two transmit antennas and two received antennas is equal to two [23]. We see that at high SNR, the AF strategy behaves like a transmission scheme achieving full multiplexing gain in the MIMO channel with two transmit antennas and two received antennas.
Table 1
Multiplexing gains of the transmission schemes in the high SNR regime.
Scheme
AF
HMC, HLC, DF
PDF
Multiplexing gain
2
0.5
1.5
6. Numerical Examples
We compare the information rates achievable by the proposed strategies in Section 4 with the cut-set outer bound in [29]. Since the derivation is straightforward, we state the outer bound without proof. For , and , let
(82)
Theorem 6 (Outer bound).
A rate pair is achievable in the TWTR network only if it satisfies
(83)
for some real number between 0 and 1, and covariance matrices and such that holds for .
We select several typical channel realizations and show the corresponding achievable rate regions in Figure 3 to Figure 8. To simplify the calculation, we consider the single antenna case where . The power constraint is set to and the noise variance is set to .
Figure 3
The achievable rate regions when all link gains are large.
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In Figure 3, we plot the rate regions when all link gains are large (the link gain is 10 for all links). As mentioned in the previous section, the AF strategy has the largest multiplexing gain in the high SNR regime. We can see in Figure 3 that the AF strategy achieves the largest sum rate.
In Figures 4 and 5, we consider the case where relay 1 has larger link gains than relay 2. In Figure 4, the link gains and are the same. In this case, HMC dominates all other strategies. In Figure 5, the two link gains, and , are not equal. In this case, HLC dominates HMC. HLC performs better in this asymmetric case because of its ability to adjust power between signals and utilize the beamforming gain.
Figure 4
The achievable rate regions when one relay has large link gains (symmetric case).
Figure 5
The achievable rate regions when one relay has large link gains (symmetric case).
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×
When both relays are close to one of the terminals, PDF has the best performance, as can be seen in Figure 6. The reason is that both relays are able to decode reliably the message from the closer terminal, and then they cooperatively forward the message to the other terminal using MIMO techniques.
Figure 6
The achievable rate regions when both relays are close to terminal.
×
Figures 7 and 8 presents two scenarios in which DF dominates all other transmission strategies. We remark that DF is quite flexible in that it has many tunable parameters. The case where both and are relatively large is shown in Figure 7. Another case where and are larger than and is shown in Figure 8. In both cases, DF is much better than other strategies.
Figure 7
The achievable rate regions and the outer bound.
Figure 8
The achievable rate regions and the outer bound.
×
×
We can further summarize the numerical results in Table 2. It is not supposed to be a precise description on the relative merits of the schemes. Instead, it provides a rough guideline for easy selection of a suitable scheme. In the table, "G" refers to "the channel condition is good" and "B" refers to "the channel condition is bad." We say that a channel is good if its link gain is two to three times, or more, than the link gain of a bad channel.When all the link gains are large, we should use AF. In the case when one pair of the opposite links of the network is good,whereas the other pair is weak, DF provides larger throughput. If one of the relays is good but the other relay is bad, HMC or HLC should be used. PDF scheme is the best one in the scenario where one of the sources has large link gains but the other does not.
Table 2
Performance guideline for the two-way two-relay network in the medium SNR regime.
Scheme
G
G
G
G
AF
G
B
B
G
DF
G
G
B
B
HMC, HLC
G
B
G
B
PDF
7. Conclusion
We have devised several transmission strategies for the TWTR network, each of which is derived from a mix-and-match of several basic building blocks, namely, amplify-forward strategy, decode-forward strategy, and physical-layer network coding, and so forth. We can see from the numerical examples that there is no single transmission strategy that can dominate all other strategies under all channel realizations. In other words, transmission strategy should be tailor-made for a given environment. In this paper, we have investigated the pros and cons of different building blocks and demonstrated how they can be used to construct transmission strategies for the TWTR network. We believe that the idea can be applied to other relay networks as well.
While in this paper we only consider the case where there are only two-relays, the ideas of our proposed schemes can be applied to the case with more than two-relays. In particular, AF and PDF can be directly implemented without any change. As for DF, HMC, and HLC, the design may be more complicated, since we have to determine which relay to decode which source's message. On the other hand, the idea behind remains the same.
In our work, we have assumed that the channels are static. When link gains are time varying, our result reveals that a static strategy can only be suboptimal. To fully exploit the available capacity of the network, adaptive strategies that can switch between several modes are needed. How to determine a good strategy based on channel state information is an open problem. It is especially difficult if the switching is based on local information only, and we leave it for future work.
Appendix
Proof of Theorem 4
The following information-theoretic argument shows that any rate pair satisfying the conditions in Theorem 4 is achievable.
Codebook Generation
For , the common message of terminal is drawn uniformly in and the private message from . For , we generate independent sequences of length . In each sequence, the components are vectors drawn independently with distribution . Label the generated sequences by for . Generate independent sequences of length , with each component drawn independently with distribution . Label the generated sequences by for . Set
(A1)
By (56) and (57), with very high probability the power constraints on node and node are satisfied.
There is a common codebook for relay 1 and relay 2. We generate an array of codewords with rows and columns. The codewords have length and each component is drawn independently from . Label the codewords by , for and .
For relay 1, we generate codewords, indexed by , , , and denoted by
(A2)
Each of them is drawn independently with each component generated from . Let be the linear combination
(A3)
Since is strictly less than 1, satisfies the power constraint of node 1 with very high probability.
For relay 2, we generate codewords, labeled by
(A4)
for , , . The components of each codeword are generated independently from . Let be
(A5)
The codeword satisfies the power constraint of node 2 by the hypothesis that .
Encoding
For source node , to send the message , it sends to the relays.
In the second stage, relay 1 and relay 2 transmit and . The messages indicated by is the estimated version of the original message.
Decoding
For , the channel output at relay is
(A6)
The receiver at relay 1 treats the signal component as noise, and tries to decode , and . It reduces to a MAC with two users, but three independent messages; two messages from node and one message from node . In order to decode these three messages reliably, we need the requirement in (50). Likewise, we have the requirement in (51) for correct decoding at node 2.
Relay 2 treats the signal component as noise, and tries to decode , and . This can be done with arbitrarily small error if the condition in (51) holds.
In the second stage, terminal receives
(A7)
Assuming that and , the channel is equivalent to a two-user MAC with common information, in which both users send , and one of the users sends the private message . The decoding is done by typicality as in [30, chapter 8], with the additional functionality of multiplexed coding. The decoder at terminal A searches for and such that , , and are jointly typical. From the capacity region of MAC with common information [30, page 102], we obtain the following rate requirements
(A8)
where is the mutual information function. This gives the conditions in (54) and (55).
Similarly, we have the conditions in (52) and (53) for successful decoding in terminal . This completes the proof of Theorem 4.
Acknowledgment
This work is supported by a grant from the City University of Hong Kong (Project no. SRG 7002386).
Open Access
This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (
https://creativecommons.org/licenses/by/2.0
), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
and
, and two parallel relays 1 and 2 (see Figure 1). Terminals
and
want to exchange messages with the help of the two relays. We assume there is no direct link between the two terminals and between the two-relays. Furthermore, all of the nodes are half-duplex. The total communication time,
, are divided into two stages, each of which consists of
time slots. In the first stage, the terminals send signals and the relays receive. In the second stage, the relays send signals and the terminals receive. The solid arrows in Figure 1 correspond to stage 1 and the dashed arrows correspond to stage 2.
and
are equipped with
antennas, whereas each of relays 1 and 2 has only one antenna. For
and
, we use
to denote the transmit signal from node
, and
to denote independently and identically distributed (i.i.d.) Gaussian noise with distribution
. The channel is assumed static and the channel gain from node
to
is denoted by an
-dimensional column vector
. We assume channel reciprocity holds so that
=
. In the first stage, the outputs of the network at time
, are given by
, the outputs at the terminal nodes are
,
is the transmit symbol of relay
,
for
is a Gaussian random vector with each component i.i.d according to
. We assume that the link gains
,
,
, and
are time-invariant and known to all nodes. We have the following power constraints in each stage:
, and
, where
,
,
, and
denote the power constraints on terminals
and
and relays 1 and 2, respectively.
and
be the data rates of terminal
and
, respectively. In a period consisting of
channel symbols (
symbols for each phase), terminal
wants to send one of the
symbols to terminal
, and terminal
wants to send one of the
symbols to terminal
. A
code for the TWTR network consists of two message sets
and
, two encoding functions
, terminal
transmits the codeword
in stage one, where
is the message to be transmitted. For
, relay
applies the function
to its received signal and transmits the resulting signal in the second stage. Let the received signals at terminals
and
be
and
, respectively. In this paper, we will use a superscript "
" to indicate a sequence of length
. So
and
are sequences of length
, with each component equal to a vector in
. After the second stage, terminal
decodes the message from the other source node by
. We note that the decoding function
uses the message from source terminal
as input as well. We say that a decoding error occurs if
or
. The average probability of error is
is said to be achievable if there exists a sequence of
codes, satisfying the power constraints in (5) and (6), with
as
.
antennas, the transmitted signals from the terminals are essentially 2 dimensional. To see this, we observe that the first term in the right hand side of (1), namely,
, is a projection of
in the direction of
. Any signal component of
orthogonal to
will not be picked up by relay 1. Likewise, from (2), we see that any signal component of
orthogonal to
will not be sensed by relay 2. There is no loss of generality, if we assume that the signals transmitted from the terminals take the following form:
, where
is an
matrix, and the two components in
represent the projections of
on
and
. We consider the 2-dimensional vector
as the input to the channel at node
. The power constraint in (5) can be written as
.
random vectors
and
as input signals at terminal
and
, respectively, and let
and
denote their corresponding
covariance matrices. For
and
, let
from terminal
. Shannon's capacity formula is denoted by
. Also, for
matrices, we let
, where
denote the
identity matrix. The reason for the factor of 0.25 before the log function, instead of a factor of 0.5 in the original capacity formula, is due to the fact that the total transmission time is divided into two stages of equal length. All logarithms in this paper are in base 2. The set of non-negative real numbers is denoted by
. Gaussian distribution with mean zero and covariance matrix
is denoted by
.
be the transmitted signal from source
, for
. The transmission is divided into two phases. In the first phase, the relay receives
is an additive noise. For simplicity, it is assumed that both link gains from the sources to the relay are equal to one. In the second phase, the relay amplifies the received signal
, and transmits a scaled version
of
, where
is a scalar chosen so that the power requirement is met. Since source 1 knows
, the component
within the received signal at source 1 can be treated as known interference, and hence be subtracted. Similarly, source 2 can subtract
from the received signal. Decoding is then based on the signal after interference subtraction.
to node 2 via the relay node, and node 2 wants to send message
to node 1 via the relay node. For
, let
be the number of bits used to represent message
. The transmission of the nodes is divided into two phases. In the first phase, the two source nodes transmit. Suppose that the relay node is able to decode
and
. For the encoder at the relay, we generate a
array of codewords. Each codeword is independently drawn according to the Gaussian distribution such that the total power of each codeword is less than or equal to
. In the second phase, the relay node sends the codeword in the
-entry in this array. Suppose that the received signal at source node
is corrupted by additive white Gaussian noise with variance
, for
. At source 1, since
is known, the decoder knows that one of the
codewords in the row corresponding to
had been transmitted. Out of these
codewords, it then declares the one based on the maximal likelihood criterion. By the channel coding theorem for the point-to-point Gaussian channel, source 1 can decode reliably at a rate of
. Likewise, by considering the columns in the array of codewords, source 2 can decode at a rate of
.
. We identify the
possible messages from source node
with the vectors in the
-dimensional vector space over the finite field of size 2,
, and identify the
messages from source node
with a subspace of
of dimension
, say
. We generate
Gaussian codewords independently, one for each vector in
. To send messages
and
in the second phase, the relay node transmits the codeword corresponding to
, where the addition is performed using arithmetics in
. The output of the decoder at node 1 is a vector in
. We subtract from it the vector in
corresponding to
. If there is no decoding error, this gives the codeword corresponding to
, and the value of
is recovered.
is known a priori, node 2 is certain that the signal transmitted from the relay is associated with one of the vectors in the affine space
. The message
can be estimated by comparing the likelihood function of the
codewords associated with
. It can be seen that the maximal data rate is the same as in the array approach mentioned in the previous paragraph, but the size of the codebook at the relay reduces from
to
.
transmit antennas and
receive antennas, with the link gain matrix denoted by a real
matrix
. The channel output equals
is the
-dimensional channel input and
is an
-dimensional zero-mean colored Gaussian noise vector with covariance matrix
. Without loss of information, we whiten the noise by pre-multiplying both sides of (15) by
. The transformed channel output is thus
is now the
identity matrix. By the capacity formula for MIMO channel with white Gaussian noise [23], the capacity for the MIMO channel in (15) is given by
denotes the
covariance matrix of
. Using the identity
matrix
and
matrix
, we rewrite (17) as
is the signal from user
, for
, and
is an additive white Gaussian noise with variance
. Each of the two users wants to send some bits to the common receiver. Suppose that the power of user
is limited to
, for
. The rate pair
, where
is the data rate of user
, is achievable in the above 2-user MAC if and only if it belongs to
and
. In the second and third scheme (HLC, HMC), one of the relays employs the amplify forward strategy, while the other decodes the messages from terminals
and
. In the fourth scheme (DF), both relays decode the messages from terminals
and
. In the last strategy (PDF), another mixture of decode-forward and amplify-forward strategy is described.
(
) buffers the signal received in the first stage, and amplifies it by a factor of
. The amplified signal
as
and
by their 2-dimensional representations
and
. Since terminal
knows its own input
as well as the link gains and amplifying factors, the signal component containing
as a factor can be subtracted from
. The residual signal is
can then be decoded using a decoding algorithm for point-to-point MIMO channel. The received signal at terminal
is treated similarly.
is achievable by the AF strategy if
,
and
and
are
covariance matrices, such that the following power constraints:
can be written as
plus a noise vector with covariance matrix
. The residual signal at terminal
equals
plus a noise vector with covariance matrix
. Therefore, after self-signal subtraction, the resultant channels can be considered MIMO channels with two transmit antennas and
receive antennas. From (19), we obtain the rate constraints in (28). The inequalities in (30) are the power constraints for terminals
and
, and those in (31) are the power constraints for relays 1 and 2.
and
, and meanwhile, relay 2 employs the amplify-forward strategy. In order to obtain beamforming gain, after decoding the two messages, relay 1 reconstructs the codewords corresponding to the decoded messages and sends a linear combination of them in the second stage.
and
form a multiple-access channel with relay 1 as the destination node. We use the optimal encoding scheme for MAC at terminals
and
, and the optimal decoding scheme at relay 1. In the second stage, relay 1 decodes and reconstructs
and
, and then transmits a linear combination
and
. Relay 2 amplifies
by a scalar factor
and transmits
.
, after subtracting the signal component that involves
(t), we get
, the residual signal after subtraction is
is achievable by the HLC strategy if
,
,
, and
and
are
covariance matrices such that the following power constraints:
and a set
is defined as
.
and
. The inequalities in (40) and (41) are the power constraints for relays 1 and 2, respectively.
,
,
, and
can be obtained by running an optimization algorithm. For example, we can aim at maximizing a weighted sum
. The values of
,
,
and
which maximize the weighted sum
are chosen.
and
, and relay 2 amplifies and transmits the received signal. However, in this strategy, relay 1 re-encodes the messages into a new codeword to be sent out in the second stage. Terminals
and
decode the desired messages based on multiplexed coding.
is achievable by the HMC strategy if
and
satisfy
,
are
covariance matrices satisfying (39), and
satisfies (41).
that satisfies the condition in the theorem is achievable. For
, terminal
randomly generates a Gaussian codebook with
codewords with length
, satisfying the power constraint in (5). Label the codewords by
, for
. For relay 1, we generate a
array of Gaussian codewords of length
and power
. The codeword in row
and column
is denoted by
, and satisfies the power constraint in (6).
and
. This can be accomplished with arbitrarily small probability of error if the rate constraints for MAC in (22) to (24) are satisfied. This corresponds to the rate constraint in (42). Let the estimated messages from
and
be
and
.
. Relay 2 amplifies its received signal and transmits
. From (41), the amplified signal has average power no more than
.
, which is known to terminal
, the residual signal at terminal
is
, and
with probability arbitrarily close to one if (42) is satisfied. The idea of multiplexed coding can then be used. In (48), the covariance matrix of the signal in square bracket is given by
in (46), and the covariance of the noise term is given by
. Applying the capacity expression, we obtain the rate constraint in (44). In a similar manner, we obtain (43).
, (
) splits the message
into two parts: the common part
and the private part
. The two common messages are transmitted via both relay nodes. The private message
is decoded by relay 1 only, and can be interpreted as going through the path from terminal
to relay 1 to terminal
. Symmetrically, the private part of message
is decoded by relay 2 only, and can be interpreted as going through the path from terminal
to relay 2 to terminal
. After the first stage, relay 1 decodes the common messages of both terminals and the private message of terminal
. Relay 2 decodes the common messages of both terminals and the private message of terminal
. The encoding and decoding schemes in the first stage is similar to those developed by Han and Kobayashi for the interference channel (IC) in [26]. Since both relays have access to the common messages, the channel in the second stage can be considered a multiple access channel with common information. Furthermore, since terminals
and
have information of themselves, we can further improve the rate region by the idea of multiplexed coding.
, let
and
be the rates of the private and common messages, respectively, from terminal
. Let
denote
for
, and let
,
,
, and
denote
covariance matrices, and
,
and
. For
. A rate pair
is achievable if we can decompose
and
such that
and
.
. Each relay then subtracts the reconstructed signal of terminal
from the received signal. Call the resulting signal the residual signal. The message of terminal
is re-encoded into a new codeword, and linearly combined with the residual signal. This linear combination is then transmitted in the second stage. Since both relays know the message of terminal
, the two-relays can jointly re-encode the message of terminal
using some encoding scheme for a MIMO channel with two transmit antennas and
receive antennas.
is achievable by the PDF strategy if it satisfies
and
,
are
covariance matrices such that the following power constraints hold
. (Here,
denotes the
th diagonal entry in
.)
as noise, and decode the message of terminal
. The rate requirement in (58) guarantees that the message of terminal
can be decoded with arbitrarily small probability of error at both relays. Let the decoded message of terminal
be denoted by
.
, the reconstructed signal
is then subtracted from
. The residual signal at relay
is
.
. For each message
, we generate two correlated codewords
and
, with mean zero and each pair of symbols at any
distributed according to a
covariance matrix
. At relay
, the decoded message
is re-encoded into
, which is a codeword with power
. In the second stage, relay
transmits
. The inequality in (63) ensures that the power constraint is satisfied at the relays.
subtracts the signal component that involves
and
from its received signal and obtains
can recover the message from terminal
reliably if (60) is satisfied.
, we subtract all terms involving
, and get
and colored noise. Recall that
is the covariance matrix of the encoded signal. By the capacity formula of MIMO channel (19), we obtain the rate constraint in (59).
,
,
and
, for
, are invertible. Indeed, by checking that
is strictly positive for all non-zero
, we see that the matrix is positive definite, and hence invertible.
denote the sum rate
as a function of the noise variance
. We use the multiplexing gain (also called degree of freedom) [27], defined by
is very small, we can approximate the sum rate by
if the multiplexing gain is equal to
.
is maximized subject to the rate constraints (28) in Theorem 1, the equalities in (28) hold. We can assume without loss of generality that
and
, and the amplifying constants
and
, are fixed. Note that if the power constraint in (31) holds, then it continues to hold if
becomes smaller. Therefore, when
, the power constraints in (30) and (31) are satisfied.
is a
matrix that equals
], we can factor
as
, where
and
are
unitary matrices, and
is a diagonal matrix with non-negative diagonal entries
. The number of positive diagonal entries in
is precisely the rank of
. We can rewrite (70) as
is equal to
. The determinant
, with the degree equal to the rank of
. Therefore, the limit
, and equals 0, 0.5, or 1, if the rank of
is 0, 1, or 2, respectively. The problem of determining the multiplexing gain now reduces to determining the rank of the matrices in (71) and (72).
and
are square invertible matrices, then
for all matrix
, whenever the matrix multiplications are well-defined; (ii) for all
matrices
, we have
. Consider the matrix in (72). After replacing
by its definition, we can express the matrix in (72) as
denotes the diagonal matrix
. We assume that
and
have full rank. This assumption holds with probability one if the link gains are generated from a continuous probability distribution function such as Rayleigh. Also, we assume that
,
, and
are of full rank. This assumption does not incur any loss of generality, because they are design parameters that we can choose. We can perturb them infinitesimally, and the resulting matrices will be of rank two, but the value on the right hand side of (69) deviates negligibly. By property (i), and the fact that
,
, and
are invertible
matrices, the rank of the matrix in (76) is equal to the rank of
. Then we get
and
, and positive real numbers
and
,
and
, and positive
and
, we conclude that the multiplexing gain of the AF strategy is equal to 2.
to terminal
can be considered AF, while the transmission from terminal
to terminal
in the other direction is limited by the message decoding after stage 1. From (58), we get
has full rank. Therefore, its maximal multiplexing gain is 1.5.
, and
, let
is achievable in the TWTR network only if it satisfies
between 0 and 1, and
covariance matrices
and
such that
holds for
.
. The power constraint is set to
and the noise variance is set to
.
and
are the same. In this case, HMC dominates all other strategies. In Figure 5, the two link gains,
and
, are not equal. In this case, HLC dominates HMC. HLC performs better in this asymmetric case because of its ability to adjust power between signals and utilize the beamforming gain.
.
and
are relatively large is shown in Figure 7. Another case where
and
are larger than
and
is shown in Figure 8. In both cases, DF is much better than other strategies.
satisfying the conditions in Theorem 4 is achievable.
, the common message of terminal
is drawn uniformly in
and the private message from
. For
, we generate
independent sequences of length
. In each sequence, the components are
vectors drawn independently with distribution
. Label the generated sequences by
for
. Generate
independent sequences of length
, with each component drawn independently with distribution
. Label the generated sequences by
for
. Set
and node
are satisfied.
rows and
columns. The codewords have length
and each component is drawn independently from
. Label the codewords by
, for
and
.
codewords, indexed by
,
,
, and denoted by
. Let
be the linear combination
is strictly less than 1,
satisfies the power constraint of node 1 with very high probability.
codewords, labeled by
,
,
. The components of each codeword are generated independently from
. Let
be
satisfies the power constraint of node 2 by the hypothesis that
.
, to send the message
, it sends
to the relays.
and
. The messages indicated by
is the estimated version of the original message.
, the channel output at relay
is
as noise, and tries to decode
,
and
. It reduces to a MAC with two users, but three independent messages; two messages from node
and one message from node
. In order to decode these three messages reliably, we need the requirement in (50). Likewise, we have the requirement in (51) for correct decoding at node 2.
as noise, and tries to decode
,
and
. This can be done with arbitrarily small error if the condition in (51) holds.
receives
and
, the channel is equivalent to a two-user MAC with common information, in which both users send
, and one of the users sends the private message
. The decoding is done by typicality as in [30, chapter 8], with the additional functionality of multiplexed coding. The decoder at terminal A searches for
and
such that
,
,
and
are jointly typical. From the capacity region of MAC with common information [30, page 102], we obtain the following rate requirements
is the mutual information function. This gives the conditions in (54) and (55).
. This completes the proof of Theorem 4.