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.

Definition 1.

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 .

Definition 2.

Two paths and are called independent if .

In the next lemma we compute the maximum number of independent paths in a multi-hop relay channel.

Lemma 2.

The maximum number of independent paths in a multi-hop relay channel is

Proof.

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 .

Therefore with AF by each antenna subset on the chosen path, the received signal at the subset of antennas of the destination at time of a multi-hop relay channel is

where and are functions of channel coefficients , ensures that the power constraint at each stage is met, is a function of 's, is the complex Gaussian noise with zero mean and unit variance added at stage , and . Since the destination has the CSI, accumulated noise is white and Gaussian distributed. From hereon in this paper we assume that the accumulated noise at the destination for all the multi-hop relay channels is white Gaussian distributed without explicitly mentioning it. Let be the covariance matrix of , then by multiplying to the received signal we have

where is a matrix with entries. Note that is a function of channel coefficients .

We propose to use successive decoding at the destination with the JEEMAS strategy, similar to [24]. With successive decoding, the destination tries to decode only at time assuming that all the symbols have been decoded correctly. Assuming that at time all the symbols have been decoded correctly, the received signal (7) can be written as

since the channel coefficients are known at the destination. Let the probability of error in decoding from (8) be , then the probability of error in decoding from (7) with successive decoding is

where the last equality follows from [24].

From (8) it is clear that is the same for any , since the channel coefficients do not change for time instants. Therefore without loss of generality we compute an upper bound on to upper bound . Next, we describe our JEEMAS strategy and compute an upper bound on of the JEEMAS strategy to evaluate its DM-tradeoff. Let . Let , then the mutual information of path is

Then the JEEMAS strategy chooses the path that maximizes the mutual information at the destination, that is, it chooses path , if

Thus defining , the mutual information of the chosen path is

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).

From [1] we know that , where is the outage probability of (8). Therefore it is sufficient to compute an upper bound on the outage probability of (8) to upper bound . With the proposed EEAS strategy, the outage probability of (8) can be written as

From [14, 15] can be dropped from the DM-tradeoff analysis without changing the outage exponent, since [14], that is, the maximum or the minimum eigenvalue of does not scale with . Thus,

We first compute the DM-tradeoff of the JEEMAS strategy for the case when there exists such that , and then for the general case.

If , then by Lemma 2, the total number of independent paths in a multi-hop relay channel is . Thus,

since from (14) for any .

From [14]

where

where , and . Thus, , and the DM-tradeoff of the JEEMAS strategy is given by

For the general case when , let , for some and . Then partition the multi-hop relay channel into two parts, the first partition containing antennas of each stage, such that the chosen set of antennas by the JEEMAS strategy , and the second partition containing the rest antennas of each stage. By reordering the index of antennas, without loss of generality, let contain antennas to of each relay stage, and let contain antennas to of stage . Recall that the JEEMAS strategy chooses those antennas of each stage that have the maximum mutual information at the destination. Thus,

where , and is the channel matrix between to antennas of stage and to antennas of stage . Note that the channel coefficients in are not independent of the channel coefficients in , and therefore we cannot write as the product of

To circumvent this problem, let , where is the channel matrix between the last antennas of stage and the last antennas of stage of partition , and is the channel matrix between the last antennas of stage and the last antennas of stage of partition . Basically we pick and antennas alternatively, note that use of more antennas increases the mutual information of the channel, and consequently reduces the outage probability. Since uses a subset of antennas of , therefore from (19),

Since the channel coefficients in are independent of the channel coefficients of ,

Therefore,

since the number of independent paths in partition is .

From [14], where

, where and is the nondecreasing ordered version of , . Thus,

Therefore, using (16), the DM-tradeoff of the JEEMAS strategy is

.

Recall that in the JEEMAS strategy the design parameter is , the number of antennas to use from each stage. To obtain the best lower bound on the DM-tradeoff of JEEMAS strategy one needs to find out the optimal value of . From (26), it follows that using a single antenna , maximum diversity gain point can be achieved. Similarly, choosing , the maximum multiplexing gain point can also be achieved. For intermediate values of , however, it is not apriori clear what value of maximizes the diversity gain. After tedious computations it turns out that choosing provides with the best achievable DM-tradeoff for . Thus, we propose a hybrid JEEMAS strategy, where for use , and for use . Our approach is similar to [15], where for each an optimal partition of the multi-hop relay channel is found by solving an optimization problem. We compare the achievable DM-tradeoff of our hybrid JEEMAS strategy and the strategy of [15] for and in Figures 2 and 3.

For the case when , the achievable DM-tradeoff of our hybrid JEEMAS strategy matches with that of the partitioning strategy of [15]. For the case when , however, it is difficult to compare the hybrid JEEMAS strategy with the strategy of [15] in terms of achievable DM-tradeoff, since an optimization problem has to be solved for the strategy of [15]. For a particular example of the hybrid JEEMAS strategy outperforms the strategy of [15] as illustrated in Figure 3. Moreover, in [15] 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.

Remark 1.

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 2.

Using the DM-tradeoff analysis of the JEEMAS strategy, we can obtain the DM-tradeoff of an antenna selection strategy for the point-to-point MIMO channel by considering a multi-hop relay channel with , transmit, and receive antennas such that (). Surprisingly we could not find this result in literature and provide it here for completeness sake. Let , and the transmitter uses antennas out of antennas that have maximum mutual information at the destination, then the DM-tradeoff is given by

. The proof follows directly from (26).

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.

Discussion

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 [15] with smaller complexity and possess several advantages over DSTBCs as described in [24]. In the next section we propose a distributed CF strategy to achieve the optimal DM-tradeoff of the -hop relay channel.