End-to-End Joint Antenna Selection Strategy and Distributed Compress and Forward Strategy for Relay Channels
© R. Vaze and R. W. Heath Jr. 2009
Received: 15 November 2008
Accepted: 11 May 2009
Published: 15 June 2009
Multihop relay channels use multiple relay stages, each with multiple relay nodes, to facilitate communication between a source and destination. Previously, distributed space-time codes were proposed to maximize the achievable diversity-multiplexing tradeoff; however, they fail to achieve all the points of the optimal diversity-multiplexing tradeoff. In the presence of a low-rate feedback link from the destination to each relay stage and the source, this paper proposes an end-to-end antenna selection (EEAS) strategy as an alternative to distributed space-time codes. The EEAS strategy uses a subset of antennas of each relay stage for transmission of the source signal to the destination with amplifying and forwarding at each relay stage. The subsets are chosen such that they maximize the end-to-end mutual information at the destination. The EEAS strategy achieves the corner points of the optimal diversity-multiplexing tradeoff (corresponding to maximum diversity gain and maximum multiplexing gain) and achieves better diversity gain at intermediate values of multiplexing gain, versus the best-known distributed space-time coding strategies. A distributed compress and forward (CF) strategy is also proposed to achieve all points of the optimal diversity-multiplexing tradeoff for a two-hop relay channel with multiple relay nodes.
Finding optimal transmission strategies for wireless ad-hoc networks in terms of capacity, reliability, diversity-multiplexing (DM) tradeoff , or delay has been a long standing open problem. The multi-hop relay channel is an important building block of wireless ad-hoc networks. In a multi-hop relay channel, the source uses multiple relay nodes to communicate with a single destination. An important first step in finding optimal transmission strategies for the wireless ad-hoc networks is to find optimal transmission strategies for the multi-hop relay channel.
In this paper, we focus on the design of transmission strategies to achieve the optimal DM-tradeoff of the multi-hop relay channel. The DM-tradeoff  characterizes the maximum achievable reliability (diversity gain) for a given rate of increase of transmission rate (multiplexing gain), with increasing signal-to-noise ratio (SNR). The DM-tradeoff curve is characterized by a set of points, where each point is a two-tuple whose first coordinate is the multiplexing gain and the second coordinate is the maximum diversity gain achievable at that multiplexing gain. We consider a multi-hop relay channel, where a source uses relay stages to communicate with its destination, and each relay stage is assumed to have one or more relay nodes. Relay nodes are assumed to be full-duplex. Under these assumptions we find and characterize multi-hop relay strategies that achieve the DM-tradeoff curve (in the two hop case) or come close to the optimum DM-tradeoff curve while outperforming prior work (with more than two hops).
In prior work there have been many different transmit strategies proposed to achieve the optimal DM-tradeoff of the multi-hop relay channel, such as distributed space time block codes (DSTBCs) [2–17], or relay selection [2, 3, 18–23]. The best known DSTBCs [14, 15] achieve the corner points of the optimal DM-tradeoff of the multi-hop relay channel, corresponding to the maximum diversity gain and maximum multiplexing gain, however, fail to achieve the optimal DM-tradeoff for intermediate values of multiplexing gain. Moreover, with DSTBCs [14, 15] the encoding and decoding complexity can be quite large. Antenna selection (AS) or relay selection (RS) strategies have been designed to achieve only the maximum diversity gain point of the optimal DM-tradeoff when a small amount of feedback is available from the destination for a two-hop relay channel in [2, 3, 18–23], and for a multi-hop relay channel in . RS is also used for routing in multi-hop networks [25–27] to leverage path diversity gain. The primary advantages of AS and RS strategies over DSTBCs are that they require a minimal number of active antennas and reduce the encoding and decoding complexity compared to DSTBCs. The only strategy that is known to achieve all points of the optimal DM-tradeoff is the compress and forward (CF) strategy , but that is limited to a -hop relay channel with a single relay node.
In this paper we design an end-to-end antenna selection (EEAS) strategy to maximize the achievable diversity gain for a given multiplexing gain in a multi-hop relay channel. The EEAS strategy chooses a subset of antennas from each relay stage that maximize the mutual information at the destination. The proposed EEAS strategy is an extension of the EEAS strategy proposed in , where only a single antenna of each relay stage was used for transmission. The proposed EEAS strategy is shown to achieve the corner points of the optimal DM-tradeoff corresponding to maximum diversity gain and maximum multiplexing gain. For intermediate values of multiplexing gains, the achievable DM-tradeoff of the EEAS strategy does not meet with an upper bound on the DM-tradeoff, but outperforms the achievable DM-tradeoff of the best known DSTBCs . Other advantages of the proposed EEAS strategy over DSTBCs [14, 15] include lower bit error rates due to less noise accumulation at the destination, reduced decoding complexity, and lesser latency. We assume that the destination has the channel state information (CSI) for all the channels in the receive mode. Using the CSI, the destination performs subset selection, and using a low rate feedback link feedbacks the index of the antennas to be used by the source and each relay stage.
Even though our EEAS strategy performs better than the best known DSTBCs [14, 15], it fails to achieve all points of the optimal DM-tradeoff. To overcome this limitation, we propose a distributed CF strategy to achieve all points of the optimal DM-tradeoff of a -hop relay channel with multiple relay nodes. Previously, the CF strategy of  was shown to achieve all points of the optimal DM-tradeoff of the -hop relay channel with a single relay node in . The result of , however, does not extend for more than one relay node. With our distributed CF strategy, each relay transmits a compressed version of the received signal using Wyner-Ziv coding  without decoding any other relay's message. The destination first decodes the relay signals and then uses the decoded relay messages to decode the source message.
Our distributed strategy is a special case of the distributed CF strategy proposed in , where relays perform partial decoding of other relay messages and then use distributed compression to send their signals to the destination. With partial decoding, the achievable rate expression is quite complicated , and it is hard to compute the SNR exponent of the outage probability. To simplify the achievable rate expression, we consider a special case of the CF strategy  where no relay decodes any other relay's message. Consequently, the derivation for the SNR exponent of the outage probability is simplified, and we show that the special case of CF strategy  is sufficient to achieve the optimal DM-tradeoff for a -hop relay channel with multiple relays.
The rest of the paper is organized as follows. In Section 2, we describe the system model for the multi-hop relay channel and summarize the key assumptions. We review the diversity multiplexing (DM-) tradeoff for multiple antenna channels in Section 3 and obtain an upper bound on the DM-tradeoff of multi-hop relay channel. In Section 4 our EEAS strategy for the multi-hop relay channel is described, and its DM-tradeoff is computed. In Section 5 we describe our distributed CF strategy and show that it can achieve the optimal DM-tradeoff of -hop relay channel with any number of relay nodes. Final conclusions are made in Section 6.
We denote by a matrix, a vector, and the element of . denotes the transpose conjugate of matrix . The maximum and minimum eigenvalue of is denoted by and , respectively. The determinant and trace of matrix is denoted by and . The field of real and complex numbers is denoted by and , respectively. The set of natural numbers is denoted by . The set is denoted by . The set denotes the set , . denotes . The space of matrices with complex entries is denoted by . The Euclidean norm of a vector is denoted by . The , represent the transpose and the transpose conjugate. The cardinality of a set is denoted by . The expectation of function with respect to is denoted by . A circularly symmetric complex Gaussian random variable with zero mean and variance is denoted as . We use the symbol to represent exponential equality, that is, let be a function of , then if , and similarly and denote the exponential less than or equal to and greater than or equal to relation, respectively. To define a variable we use the symbol .
2. System Model
As shown in Figure 1, the channel matrix between the subset of antennas of stage and the subset of antennas of stage is denoted by , , where . Stage represents the source and stage the destination.
In Section 5, we only consider a -hop relay channel and denote the channel matrix between the source and th relay by and between the th relay and destination by . The channel between the source and destination is denoted by and the channel matrix between relay and relay by .
We assume that the CSI is known only at the destination, and none of the relays have any CSI, that is, the destination knows . For Section 5, we assume that the destination knows , and the th relay node knows and . We assume that , and have independent and identically distributed (i.i.d.) entries for all to model the channel as Rayleigh fading with uncorrelated transmit and receive antennas. We assume that all these channels are frequency flat, block fading channels, where the channel coefficients remain constant in a block of time duration and change independently from block to block.
3. Problem Formulation
We consider the design of transmission strategies to achieve the DM-tradeoff of the multi-hop relay channel. In the next subsection we briefly review the DM-tradeoff  for point-to-point channels and obtain an upper bound on the DM-tradeoff of the multi-hop relay channel.
Next, we present an upper bound on the DM-tradeoff of the multi-hop relay channel obtained in .
Lemma 1 (see ).
The upper bound on the DM-tradeoff of multi-hop relay channel is obtained by using the cut-set bound  and allowing all relays in each relay stage to cooperate. Using the cut-set bound it follows that the mutual information between the source and the destination cannot be more than the mutual information between the source and any relay stage or between any two relay stages. Moreover, by noting the fact that mutual information between any two relays stages is upper bounded by the maximum mutual information of a point-to-point MIMO channel with transmit and receive antennas, , then the result follows from (3).
In the next section we propose an EEAS strategy for the multi-hop relay channel and compute its DM-tradeoff. We will show that the achievable DM-tradeoff of the EEAS strategy meets the upper bound at and .
4. Joint End-to-End Multiple Antenna Selection Strategy
In this section we propose a joint end-to-end multiple antenna selection strategy (JEEMAS) for the multi-hop relay channel and compute its DM-tradeoff. In the JEEMAS strategy, a fixed number of antennas are chosen from each relay stage to forward the signal towards the destination using amplify and forward (AF). Before introducing our JEEMAS strategy and analyzing its DM-tradeoff, we need the following definitions and Lemma 2.
Let be a subset of antennas of stage , that is, . Let be the edge joining the set of antennas of stage to the set of antennas of stage , where . Then a path in a multi-hop relay channel is defined as the sequence of edges .
In the next lemma we compute the maximum number of independent paths in a multi-hop relay channel.
Follows directly from [24, Theorem 3] by replacing by .
Now we are ready to describe our JEEMAS strategy for the full-duplex multi-hop relay channel. To transmit the signal from the source to the destination, a single path in a multi-hop relay channel is used for communication. How to choose that path is described in the following. Let the chosen path for the transmission be . Then the signal is transmitted from the subset of antennas of the source and is relayed through subset of antennas of relay stage and decoded by the subset of antennas of the destination. Each antenna on the chosen path uses an AF strategy to forward the signal to the next relay stage, that is, each antenna of stage on the chosen path transmits the received signal after multiplying by , where is chosen to satisfy an average power constraint across antennas of stage .
where the last equality follows from .
Since we assumed that the destination of the multi-hop relay channel has CSI for all the channels in the receive mode, this optimization can be done at the destination, and using a feedback link, the source and each relay stage can be informed about the index of antennas to use for transmission. Next, we evaluate the DM-tradeoff of the JEEMAS strategy by finding the exponent of the outage probability (8).
For the case when , the achievable DM-tradeoff of our hybrid JEEMAS strategy matches with that of the partitioning strategy of . For the case when , however, it is difficult to compare the hybrid JEEMAS strategy with the strategy of  in terms of achievable DM-tradeoff, since an optimization problem has to be solved for the strategy of . For a particular example of the hybrid JEEMAS strategy outperforms the strategy of  as illustrated in Figure 3. Moreover, in  a new partition is required for each , in contrast to our strategy, which has only two modes of operation, one for and the other for .
The following remarks are in order.
Recall that we assumed that , that is, equal number of antennas are selected at each relay stage. The justification of this assumption is as follows. Let us assume that antennas are used from each relay stage. Now assume that all relay stages are using the same number of antennas , except , which is using antennas, , and . Using (26), it can be shown that the achievable DM-tradeoff with , and is a subset of the union of the achievable DM-tradeoff with using (all relay stages using antennas), and (all relay stages using antennas). Thus, it is sufficient to consider same number of antennas from each relay stage. It turns out, however, that different values of provide with different achievable DM-tradeoff's because of the different number of independent paths in the multi-hop relay channel. To optimize over all possible values of we keep as a variable and choose to obtain the best achievable DM-tradeoff.
Remark 3 (CSI Requirement).
With the proposed hybrid JEEMAS strategy, the destination needs to feedback the index of the path with the maximum mutual information to the source and each stage. Recall from the derivation of the achievable DM-tradeoff of the JEEMAS strategy that only paths in a multi-hop relay channel are independent, and control the achievable DM-tradeoff for . Thus, the destination only needs to feedback the index of the best path among independent paths with the maximum mutual information. Consequently the destination only needs to know CSI for paths. For the case when , we need to consider one more path from partition corresponding to and antennas of alternate relay stages. Thus, the CSI overhead is moderate for the proposed EEAS strategy.
Remark 4 (Feedback Overhead).
As explained in Remark 3, to obtain the achievable DM-tradeoff of the hybrid JEEMAS strategy it is sufficient to consider any one set of or independent paths. Let the destination choose a particular set of independent paths. Then each relay node knows on which of the paths of it lies, and depending on the index of the element of from the destination, it knows whether to transmit or remain silent. Thus, only bits of feedback is required from the destination to the source and each stage. Therefore the feedback overhead with the proposed EEAS strategy is quite small and can be realized with a very low-rate feedback link.
In this section we proposed a hybrid JEEMAS strategy that has two modes of operation, one for , where it uses a single antenna of each stage, and the other for , that uses antennas of each stage. The proposed strategy is shown to achieve both the corner points of the optimal DM-tradeoff curve, corresponding to the maximum diversity gain and the maximum multiplexing gain. For intermediate values of multiplexing gain, the diversity gain of our strategy is quite close to that of the upper bound. Even though our strategy does not meet the upper bound, we show that it outperforms the best known DSTBC strategy  with smaller complexity and possess several advantages over DSTBCs as described in . In the next section we propose a distributed CF strategy to achieve the optimal DM-tradeoff of the -hop relay channel.
5. Distributed CF Strategy for 2-hop Relay Channel
Previously in , the CF strategy of  has been shown to achieve the optimal DM-tradeoff of a -hop relay channel with a single relay node ( ) in the presence of direct path between the source and the destination. The result of , however, does not generalize to the case of -hop relay channel with multiple relay nodes. The problem with multiple relay nodes is unsolved, since how multiple relay nodes should cooperate among themselves to help the destination to decode the source message is hard to characterize. A compress and forward (CF) strategy for a -hop relay channel with multiple relay nodes has been proposed in , which involves partial decoding of other relays messages at each relay and transmission of correlated information from different relay nodes to the destination using distributed source coding. The achievable rate expression obtained in , however, is quite complicated and cannot be computed easily in closed form.
The achievable rate expression of the CF strategy  is complicated because each relay node partially decodes all other relay messages. Partial decoding introduces auxillary random variables which are hard to optimize over. To allow analytical tractability, we simplify the strategy of  as follows. In our strategy each relay compresses the received signal from the source using Wyner-Ziv coding similar to , but without any partial decoding of any other relay's message. The compressed message is then transmitted to the destination using the strategy of transmitting correlated messages over a multiple access channel . Our strategy is a special case of CF strategy , since in our case the relays perform no partial decoding. Consequently our strategy leads to a smaller achievable rate compared to . The biggest advantage of our strategy, however, is its easily computable achievable rate expression and its sufficiency in achieving the optimal DM-tradeoff as shown in the sequel. We refer to our strategy as distributed CF from hereon in the paper. Even though the relays do not perform any partial decoding in the distributed CF strategy, in the sequel we show that they still provide the destination with enough information about the source message to achieve the optimal DM-tradeoff. Before describing our distributed CF strategy and showing its optimality in achieving the optimal DM-tradeoff, we present an upper bound on the DM-tradeoff of the -hop relay channel.
Lemma 3 (see ).
Let us assume that all the relay nodes and the destination are colocated and can cooperate perfectly. This assumption can only improve . In this case, the communication model from the source to destination is a point to point MIMO channel with transmit antennas and receive antennas. The DM-tradeoff of this MIMO channel is , and since this point to point MIMO channel is better than our original -hop relay channel, . Next, we assume that the source is co-located with all the relay nodes and can cooperate perfectly for transmission to the destination. This setting is equivalent to a MIMO channel with transmit and receive antenna with DM-tradeoff . Again, this point to point MIMO channel is better than our original -hop relay channel and hence , which completes the proof.
To achieve this upper bound we propose the following distributed CF strategy. Let the rate of transmission from source to destination be . Then the source generates independent and identically distributed according to distribution . Label them . The codebook generation, the relay compression, and transmission remain the same as in , expect that no relay node decodes any other relay's codewords, that is, no partial decoding at any relay node. Relay node generates independent and identically distributed according to distribution and labels them , and for each generates 's, each with probability . Label these and and randomly partition the set into cells .
A Block Markov encoding  together with Wyner-Ziv coding  is used by each relay. Let in block the message sent from the source be , then the source sends . Let the signal received by relay in block be . Then is compressed to using Wyner-Ziv coding  where correlation among is exploited. Then relay determines the cell index in which lies and transmits in block . We consider transmission of blocks of symbols each from the source in which messages will be sent. Each message is chosen from . Thus, as , for fixed , rate is arbitrarily close to . In the first block, the relay has no information about necessary for compression. In this case, however, any good sequence allows each relay to start block Markov encoding . In the last block, the source is silent, and only the relays transmit to destination.
where , are vectors with elements , respectively, is the vector containing , and is the complement of , where . For more detailed error probability analyses we refer the reader to . In the next theorem we compute the outage exponents for (30) and show that they match with the exponents of the upper bound.
CF strategy achieves the DM-tradeoff upper bound (Lemma 3).
To prove the theorem we will compute the achievable DM-tradeoff of the CF strategy (30) and show that it matches with the upper bound.
Next, we compute the values of 's that satisfy the compression rate constraints (32). Note that in (32), we need to satisfy the constraints for each subset . Towards that end, first we consider the subsets of the form and obtain the lower bound on the quantization noise needed to satisfy (32), that is not proportional to for each . It is important to note that should not be proportional to ; otherwise, from (37) it can be concluded that our distributed CF strategy cannot achieve the optimal DM-tradeoff. In the sequel we will point out how to obtain satisfying (32) for all subsets of .
Note that both sides of (44) are functions of ; however, the resulting is not a function of or SNR similar to . Recall that we have only considered the subsets of of the form . For the rest of the subsets also, we can show that the quantization noise required to satisfy (32) is not proportional to . The analysis follows similarly and is deleted for the sake of brevity. Thus, to satisfy (32), we can take the maximum of the required for each subset and use that to analyze the DM-tradeoff. Let the maximum required to satisfy (32) be . Since for each subset is not proportional to , and is also not proportional to .
From here on we follow  to compute the exponent of the .
In this section we proposed a simplified version of the distributed CF strategy of  and showed that it can achieve the optimal DM-tradeoff for the -hop relay channel for any number of relays. In our distributed CF strategy, each relay uses Wyner-Ziv coding to compress the received signal without any partial decoding of other relay messages. After compression, each relay transmits the message to the destination using the strategy for multiple access channel with correlated messages , since the relay compressed messages are correlated with each other. Even though the achievable rate with our strategy is smaller than the one obtained in  (because of no partial decoding at any relay), we show that it is sufficient to achieve the optimal DM-tradeoff. We prove the result by showing that the exponent of the outage probability of our strategy matches with the upper bound on the optimal DM-tradeoff, without requiring the compression noise constraints to be proportional to the .
Generalizing our distributed CF strategy is possible for more than 2-hop relay channel; however, computing the exponents of the outage probability of achievable rate and compression rate constraints is a nontrivial problem.
In this paper we considered the problem of achieving the optimal DM-tradeoff of the multi-hop relay channel. First, we proposed an antenna selection strategy called JEEMAS, where a subset of antennas of each relay stage is chosen for transmission that has the maximum mutual information at the destination. We showed that the JEEMAS strategy can achieve the maximum diversity gain and the maximum multiplexing gain in a multi-hop relay channel. Then we compared the DM-tradeoff performance of the JEEMAS strategy with the best known DSTBC strategy . We observed that the DM-tradeoff of the JEEMAS is better than the DSTBCs , except for the case when the number of antennas at each stage are divisible by the minimum of the antennas across all relay stages, in which case the DM-tradeoffs of JEEMAS and DSTBCs  match.
Next, we proposed a distributed CF strategy for the -hop relay channel with multiple relay nodes and showed that it achieves the optimal DM-tradeoff. Our distributed CF strategy is a special case of the strategy proposed in , where the specializations are done to allow analytical tractability. We showed that if each relay transmits a compressed version of the received signal using Wyner-Ziv coding, it is sufficient to achieve the optimal DM-tradeoff. Our distributed CF strategy can be extended to more than -hop relay channels; however, computing the outage probability exponents is a non-trivial problem.
This work was funded by DARPA through IT-MANET Grant no. W911NF-07-1-0028.
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