Abstract
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.
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).
-hop relay channel with a single relay node.
-hop relay channel with multiple relay nodes. Previously, the CF strategy of [
-hop relay channel with a single relay node in [
-hop relay channel with multiple relays.
-hop relay channel with any number of relay nodes. Final conclusions are made in Section 6.
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 
.
antennas wants to communicate with a destination terminal with 
antennas via 
stages of relays as shown in Figure 
th relay stage has 
relays and, the 
th relay of 
th stage has 
antennas 
. The total number of antennas in the 
th relay stage is 
. In Section 5 we consider a 
-hop relay channel with 
relay nodes, where the 
relay has 
antennas and 
. We assume that the relays do not generate their own data, and each relay stage has an average power constraint of 
. We assume that the relay nodes are synchronized at the frame level. To keep the relay functionality and relaying strategy simple we do not allow relay nodes to cooperate among themselves. For Section 4 we assume that there is no direct path between the source and the destination, but we relax this assumption in Section 5 for the 
-hop relay channel. The absence of the direct path is a reasonable assumption for the case when relay stages are used for coverage improvement, and the signal strength on the direct path is very weak. We also assume that relay stages are chosen in such a way that all the relay nodes of any two adjacent relay stages are connected to each other, and there is no direct path between relay stage 
and 
. This assumption is reasonable for the case when successive relay stages appear in increasing order of distance from the source toward the destination, and any two relay nodes are chosen to lie in adjacent relay stages if they have sufficiently good SNR between them. In any practical setting there will be interference received at any relay node of stage 
because of the signals transmitted from relay nodes of relay stage 
and 
. Due to relatively large distances between nonadjacent relay stages, however, this interference is quite small and we account for that in the additive noise term. The system model is similar to the fully connected layered network with intralayer links [
stages.
of antennas of stage 
and the subset 
of antennas of stage 
is denoted by 
, 
, where 
. Stage 
represents the source and stage 
the destination.
-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 
.
. 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.
be a family of codes, one for each 
. The multiplexing gain of 
is 
if the data rate 
of 
scales is 
with respect to 
, that is,
is defined as the rate of fall of probability of error 
of 
with respect to 
is called the diversity gain at rate 
, and the curve joining 
for different values of 
characterizes the DM-tradeoff. The DM-tradeoff for a point-to-point multi antenna channel with 
transmit and 
antennas has been computed in [
and then computing the exponent 
, where
, for 
.
is upper bounded by the piecewise linear function connecting the points 
where
.
transmit and 
receive antennas, 
, then the result follows from (3).
and 
.
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.
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 
.
and 
are called independent if 
.
by 
.
. 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 
.
subset of antennas of the destination at time 
of a multi-hop relay channel is
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
is a matrix with 
entries. Note that 
is a function of channel coefficients 
.
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
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
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
, if
, the mutual information of the chosen path is
, 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
can be dropped from the DM-tradeoff analysis without changing the outage exponent, since 
[
does not scale with 
. Thus,
such that 
, and then for the general case.
, then by Lemma 2, the total number of independent paths in a multi-hop relay channel is 
. Thus,
for any 
.
, and 
. Thus, 
, and the DM-tradeoff of the JEEMAS strategy is given by
, 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,
, 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
, 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),
are independent of the channel coefficients of 
,
is 
.
where
, where 
and 
is the nondecreasing ordered version of 
, 
. Thus,
.
, 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 [
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 [
and 
in Figures 
, the achievable DM-tradeoff of our hybrid JEEMAS strategy matches with that of the partitioning strategy of [
, however, it is difficult to compare the hybrid JEEMAS strategy with the strategy of [
the hybrid JEEMAS strategy outperforms the strategy of [
, in contrast to our strategy, which has only two modes of operation, one for 
and the other for 
.
, 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.
, 
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).
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.
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.
, 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 [
-hop relay channel.
-hop relay channel with multiple relay nodes in the presence of a direct path between the source and the destination. For this 
-hop relay channel we propose a distributed compress and forward (CF) strategy to achieve the optimal DM-tradeoff. The signal model for this section is as follows. We consider a 
-hop relay channel with 
relay nodes, where the 
relay has 
antennas, and 
. The source and destination are assumed to have 
and 
antennas, respectively. We assume that the source and each relay have an average power constraint of 
. Different transmit power constraints do not change the DM-tradeoff. Let the signal transmitted from the source be 
, and from the relay node 
let it be 
, respectively. Then,
is the received signal at the destination, and 
is the signal received at relay 
.
-hop relay channel with a single relay node (
) in the presence of direct path between the source and the destination. The result 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 [
-hop relay channel.
.
. 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.
. 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 [
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 
.
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 [
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 
[
necessary for compression. In this case, however, any good sequence allows each relay to start block Markov encoding [
, the codeword sent by source in block 
is decoded. At the end of block 
, the destination first decodes 
for each 
by looking for a jointly typical 
and 
. If 
, 
can be decoding reliably. Next, given that 
's have been decoded correctly for each 
, the destination tries to find a set 
of 
such that 
is jointly typical. The destination declares that 
were the correctly sent codewords if 
. After decoding 
and 
the destination decodes 
if 
is jointly typical. With this distributed CF strategy,
, 
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 [
, where 
is 
vector with covariance matrix 
. Also, we choose 
and 
to be complex Gaussian with covariance matrices 
, and 
, and independent of each other, respectively. Next, we compute the various mutual information expressions to derive the achievable DM-tradeoff of the CF strategy. By the definition of the mutual information,
is defined as
. From (28),
'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 
.
, from (32), for each relay 
, we need to satisfy
is defined as
is defined as
; however, the resulting 
is not a function of 
or SNR similar to [
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 
.
, it is sufficient to find the negative of the exponent of the 
of outage probability at the destination, where outage probability 
is defined as
. Then choose 
such that
's that satisfy (47), since 
is not proportional to 
.
by constant does not change the DM-tradeoff.
.
; therefore, using [
and 
of 
and 
. Notice that, however, 
is the mutual information between the source and the destination by choosing the covariance matrix to be 
and allowing all the relays and the destination to cooperate perfectly. From [
as the covariance matrix does not change the optimal DM-tradeoff; therefore, 
. Similar argument holds for 
, by noting that 
is the mutual information between the source and the destination if all the relays and the source were co-located and could cooperate perfectly, while using covariance matrix 
, where
. Thus, the achievable DM-tradeoff with CF strategy meets the upper bound (Lemma 3).
-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 [
.
-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 [
-hop relay channels; however, computing the outage probability exponents is a non-trivial problem.