Coordinate Interleaved Distributed SpaceTime Coding for TwoAntennaRelays Networks
Abstract
Distributed space time coding for wireless relay networks when the source, the destination and the relays have multiple antennas have been studied by Jing and Hassibi. In this setup, the transmit and the receive signals at different antennas of the same relay are processed and designed independently, even though the antennas are colocated. In this paper, a wireless relay network with single antenna at the source and the destination and two antennas at each of the relays is considered. A new class of distributed space time block codes called Coordinate Interleaved Distributed SpaceTime Codes (CIDSTC) are introduced where, in the first phase, the source transmits a length complex vector to all the relays and in the second phase, at each relay, the inphase and quadrature component vectors of the received complex vectors at the two antennas are interleaved and processed before forwarding them to the destination. Compared to the scheme proposed by JingHassibi, for , while providing the same asymptotic diversity order of , CIDSTC scheme is shown to provide asymptotic coding gain with the cost of negligible increase in the processing complexity at the relays. However, for moderate and large values of , CIDSTC scheme is shown to provide more diversity than that of the scheme proposed by JingHassibi. CIDSTCs are shown to be fully diverse provided the information symbols take value from an appropriate multidimensional signal set.
Cooperative communication, distributed spacetime coding, coordinate interleaving, coding gain.
I Introduction and Preliminaries
Cooperative diversity is proved to be an efficient means of achieving spatial diversity in wireless networks without the need of multiple antennas at the individual nodes. In comparison with single user colocated multiple antenna transmission, cooperative communication is based on the relay channel model where a set of distributed antennas belonging to multiple users in the network cooperate to encode the signal transmitted from the source and forward it to the destination so that the required diversity order is achieved [2, 3, 4, 5].
In [6], the idea of spacetime coding devised for point to point colocated multiple antenna systems is applied for a wireless relay network with single antenna nodes and PEP (Pairwise error probability) of such a scheme was derived. It is shown that in a relay network with a single source, a single destination with single antenna relays, distributed space time coding (DSTC) achieves the diversity of a colocated multiple antenna system with transmit antennas and one receive antenna, asymptotically.
Subsequently, in [7], the idea of [6] is extended to relay networks where the source, the destination and the relays have multiple antennas. But, cooperation between the multiple antennas of each relay is not used, i.e., the colocatedness of the antennas is not exploited. Hence, a total of relays each with a single antenna is assumed in the network instead of a total of antennas in a smaller number of relays. With this set up, for a network with antennas at the source, antennas at the destination and a total of antennas at relays, for large values of , the PEP of the network, varies with as
if  
if 
In particular, the PEP of the scheme in [7] for large when specialized to with 2 antennas at relays is upperbounded by,
(1) 
where is the minimum singular value of where S and are the two distinct codewords of a distributed space time block code and is the total power per channel use used by all the relays for transmitting an information vector.
Following the work of [7], constructions of distributed space time block codes for networks with multiple antenna nodes are presented in [8], [9].
We refer cooperative diversity schemes in which multiple antennas of a relay do not cooperate i.e when the transmitted vector from every antenna is function of only the received vector in that antenna, or when every relay has only one antenna as Regular Distributed SpaceTime Coding (RDSTC).
The key idea in the proposed scheme is the notion of vector coordinate interleaving defined below:
Definition 1
Given two complex vectors we define a Coordinate Interleaved Vector Pair of denoted as to be the pair of complex vectors where given by
or equivalently,
(2) 
(3) 
The notion of coordinate interleaving of two complex variables has been used in [10] to obtain singlesymbol decodable STBCs with higher rate than the well known complex orthogonal designs. Definition 1 is an extension of the above technique to two complex vectors. The idea of vector coordinate interleaving has been used in [11] in order to obtain better diversity results in fast fading MIMO channels.
In this paper, we show that multiple antennas at the relays can be exploited to improve the performance of the network. Towards this end, a single antenna source and a single antenna destination with two antennas at each of the relays is considered. Also, the two phase protocol as in [7] is assumed where the first phase consists of transmission of a length complex vector from the source to all the relays (not to the destination) and the second phase consists of transmission of a length complex vector from each of the antennas of the relays to the destination, as shown in Fig.1. The modification in the protocol we introduce is that the two received vectors at the two antennas of a relay during the first phase is coordinate interleaved as defined in Definition 1. Then, multiplying the coordinate interleaved vector with the predecided antenna specific unitary matrices, each antenna produces a length vector that is transmitted to the destination in the second phase. The collection of all such vectors, as columns of a matrix constitutes a codeword matrix and collection of all such codeword matrices is referred as coordinate interleaved distributed space time code (CIDSTC). The contributions of this paper may be summarized as follows in more specific terms:

For , an upper bound on the PEP of our scheme with fully diverse CIDSTC, at large values of the total power is derived.

For the PEP of the RDSTC scheme in [7] with fully diverse DSTBC is upper bounded by the expression given in (1). Comparing this bound, with ours, for equal number of antennas, a term appears in the numerator of the PEP expression of our scheme instead of the term . This improvement in the PEP comes just by vector coordinate interleaving at every relay the complexity of which is negligible.

It is shown that CIDSTC scheme provides asymptotic coding gain compared to the corresponding RDSTC scheme.

CIDSTC in variables is shown not to provide full diversity if the variables take values from any 2dimensional signal set.

Multidimensional signal sets are shown to provide full diversity for CIDSTCs whose choice depends on the design in use.

The number of channel uses needed in the proposed scheme is at least where as only is needed in an RDSTC scheme. With for both the schemes, through simulation, it is shown that CIDSTC gives improved BER (Bit Error Rate) performance over that of RDSTC scheme.
Notations: Through out the paper, boldface letters and capital boldface letters are used to represent vectors and matrices respectively. For a complex matrix X, the matrices , , , , , and denote the conjugate, transpose, conjugate transpose, determinant, Frobenious norm, real part and imaginary part of X respectively. and denotes the identity matrix and the zero matrix respectively. Absolute value of a complex number , is denoted by and is used to denote the expectation of the random variable A circularly symmetric complex Gaussian random vector x with mean and covariance matrix is denoted by . The set of all integers and complex numbers are denoted by and respectively and j is used to denote Through out the paper refers to .
The remaining content of the paper is organized as follows: In Section II, the signal model and a formal definition of CIDSTC is given along with an illustrative example.
The pairwise error probability (PEP) expression for a CIDSTC is obtained in Section III using which it is shown that (i) CIDSTC scheme gives asymptotic diversity gain equal to the total number of antennas in the relays and (ii) offers asymptotic coding gain compared to the corresponding RDSTCs. Constructions of CIDSTCs along with conditions on the full diversity of CIDSTCs are provided in Section IV. In Section V, simulation results are presented to illustrate the superiority of CIDSTC schemes. Concluding remarks and possible directions for further work constitute Section VI.
Ii signal model
The channel from the source node to the th antenna of the th relay is denoted as and the channel from the th antenna of the th relay to the destination node is represented by for and as shown in Fig.1. The following assumptions are made in our system model:

All the nodes are subjected to half duplex constraint.

Fading coefficients are i.i.d with coherence time interval, .

All the nodes are synchronized at the symbol level.

Destination knows all the fading coefficients .
In the first phase the source transmits a length complex vector from the codebook = consisting of information vectors such that = 1 for all , so that is the average transmit power used at the source every channel use. When the information vector s is transmitted, the received vector at the th antenna of the th relay is given by
where is the additive noise vector at the th antenna of the th relay. In the second phase, all the relay nodes are scheduled to transmit length vectors to the destination simultaneously. In general, the transmitted signals from the different antennas of the same relay can be designed as a function of the received signals at both the antennas of the relay. We use one such technique which is very simple; every relay manufactures a CIVP using the received vectors and as given in (2) and (3), i.e., It is straight forward. to verify that Each relay is equipped with a pair of fixed unitary matrices and , one for each antenna and process the above CIVP as follows: The and the antennas of the th relay are scheduled to transmit
(4) 
respectively. The average power transmitted by each antenna of a relay per channel use is . The vector received at the destination is given by
(5) 
where is the additive noise at the destination. Using (4) in (5), y can be written as
where

The additive noise, n in the above equivalent MIMO channel is given by,
with . Since , we have

The equivalent channel h is given by
(6) where

The matrix,
is the equivalent codeword matrix. Henceforth, by codeword matrix will be meant only this equivalent matrix even though the transmitted vectors from the antennas constitute a matrix.
The collection of codeword matrices shown below when s runs over the codebook ,
(7) 
will be called the Coordinate Interleaved Distributed SpaceTime code (CIDSTC).
Proposition 1
The random variables for all and are independent and also
The proof is straight forward.
Example 1
Consider and Let the relay specific unitary matrices and be
The equivalent channel is
and to be explicit, with where are complex variables which may take values from a signal set like QAM, PSK etc.
Iii pairwise error probability
Since the relay specific matrices are unitary, w and are independent Gaussian random variables and since are known at the receiver, n is a Gaussian random vector with
= and = .
Assume that is a codeword in the CIDSTC given in (7). When both and are known, is also a Gaussian random vector with
and
The maximum likelihood (ML) decoding is given by
(8) 
Iiia Chernoff bound on the PEP.
Lemma 1
Assume where is a CIDSTC. With the ML decoding as in (8), the probability of decoding to when S is transmitted given that are known at the destination has the following Chernoff bound [6]:
(9) 
where
and
where
We refer to a CIDSTC as fully diverse if U is a full rank matrix for every codeword pair.
Lemma 2
If U is of full rank and the minimum singular value of U is denoted by , then the PEP in (9) averaged over satisfies
(10) 
where
See Appendix A.
The power allocation problem of our model is the same as the one considered in [6] with single antenna relays. As introduced in the result of Lemma 1, where has the gamma distribution with mean and variance being . For very large values of , we can make the approximation and hence . Since for every channel use, the power used at the source and every antenna of a relay are and respectively, total power is . Therefore,
Thus, achieves the above equality when and . Since we have used the approximation , the above power allocation is valid only for large values of as in [6]. With this optimum power allocation, when we have ,
IiiB Derivation of Diversity order for Large R
The upperbound on the PEP in (10) needs to be averaged over ’s and ’s to obtain the diversity order of the CIDSTC scheme. A simple approximate derivation of the diversity order considering large number of relays in the network is presented. When is large, with high probability and .
Theorem 1
Assume and the CIDSTC is fully diverse. For large total transmit power , the probability of decoding to when S is transmitted is upper bounded as
(11) 
See Appendix B. The term in the right hand side of (11) can be written as . Hence, the diversity of the wireless relay network with CIDSTC is
whereas the diversity of the scheme in [7] is
Asymptotically, both the expressions
can be taken to be equal, and hence the diversity gain is approximately in both the schemes. However, for moderate values of the second term is larger than the first one and this difference depends on So, our scheme performs better than the one in [7] by an amount that depends on
The PEP of the scheme in [7] for large when specialized to with 2 antennas at relays is upperbounded by (1).
Using (1) and (11), the fractional change in PEP of CIDSTC with respect the one in [7] can be written as
(12) 
For a specified PEP, the following scenarios may occur: The total power, required by the CIDSTC may be smaller than that of the RDSTC or viceverse. In the former case, since we have already shown that the PEP of CIDSTC drops at a faster rate than RDSTC, the value of required to achieve a PEP below the specified PEP will be lesser for CIDSTC compared to RDSTC.
In the event of the latter case, from (12) we see that, depending on the value of the corresponding value of for CIDSTC for the specified PEP may be more or less than that of the value for RDSTC. However, at large , dominates the above ratio and hence the expression in (12) increases with increase in .
Upperbounds on the PEP in (1) and (11) are plotted for , , using different values of and in Figure 2, Figure 3 and Figure 4 over several values of the total power, . Figures 2  4 provide useful information on the PEP behavior of RDSTC and CIDSTC for different values of and . In particular, these figures provide information on the power levels beyond which CIDSTC starts out performing RDSTCs and the power levels below which RDSTCs outperforms CIDSTCs. It is to be noted that the power level at which the crossover in the performance between the two schemes takes place depends on the values of and . It is also interesting to observe that irrespective of the values of and , there exists a sufficiently large total power such that for , CIDSTC outperforms RDSTC. However, the plot shows that for all practical purposes, asymptotic coding gain provided by CIDSTC is meaningful only for the case when .
In Figures 2, 3 and 4, the upperbounds on the PEP in (1) and (11) are compared at lower values of also. Since the upperbounds on the PEP is derived assuming a large value of , the above plots may not provide actual behavior of our scheme at lower values of , which corresponds to PEP in the range of to . Plots in the above figures show that RDSTC outperforms CIDSTC at lower values of for the cases when and , but we caution the reader once again to note that these plots may not provide the correct information since the derived bound is no longer valid at lower power values.
IiiC Receiver complexity of CIDSTC
From the results of Theorem 1, a necessary condition on the full diversity of CIDSTC is . This implies that the number of complex variables transmitted from the source to relays in the first phase is at least twice the total number of antennas at all the relays. Therefore, CIDSTC is a design in at least 4 complex variables where as a RDSTC for the same setup is a design in at least 2 variables. With this, the ML decoder for CIDSTC has to decode at least 4 complex variables every codeword use where as the ML decoder of RDSTC has to decode at least 2 symbols every codeword use. Thus CIDSTC increases the ML decoding complexity at the receiver even though the additional complexity in performing coordinate interleaving of symbols at the relays is very marginal.
Iv on the full diversity of CIDSTC
In this section, we provide conditions on the signal set such that the CIDSTC in variables is fully diverse. First, we show that CIDSTC in variables is not fully diverse if the variables take values from any 2dimensional signal set. Towards that end, let be a fully diverse RDSTC for relays and given by,
where and take values from some 2dimensional signal set. Using the above RDSTCs, we can construct a CIDSTC for relays each having two antennas by assigning the relay matrices of RDSTC to every antenna of our setup. Therefore, CIDSTC is of the form,
where . By permuting the columns, can be written as,
(13) 
where,
From (13), every codeword S of is of the form where and . From Section III, a CIDSTC, is said to be fully diverse if is full rank, for every such that .
The difference matrix of two codewords is given by,
(14) 
where and such that . Also, and such that . Since and are fully diverse, and are full rank.
(15) 
(16) 
where .
The following proposition shows that is not full rank even if and are full rank.
Proposition 2
If variables ’s take value from a 2dimensional signal set, then CIDSTC is not fully diverse.
Suppose complex variables ’s take value from any 2dimensional signal set, then can possibly take values such that . Since , some of the rows of are linearly dependent. Also, identical rows of will also be linearly dependent. Therefore, and together make the corresponding rows of linearly dependent.
Example 2
For the CIDSTC in Example 1, if , then is given by,
It can be observed that the first and the third row of are linearly dependent and hence CIDSTC in Example 1 is not fully diverse.
From the results of the Proposition 2, full diversity of CIDSTC can be obtained by making the real variables for take values from an appropriate multidimensional signal set. In particular, the signal set needs to be chosen such that such that is full rank for every pair of codewords. The determinant of will be a polynomial in variables for . Therefore, a signal set has to be chosen to make determinant of nonzero for every pair of codewords. A particular choice of the signal set depends on the design in use. However, it is to be noted that, more than one multi dimensional signal set can provide full diversity for a given design.
In the rest of this section, we provide a multidimensional signal set, for the CIDSTC, in Example 1 such that, when the variables take values from , the CIDSTC is fully diverse. Towards that direction, real and imaginary components of det for any pair of codewords is given in (17) and (IV) respectively.
(17) 
(18) 
Full diversity for the code in Example 1 can be obtained by using a signal set which is carved out of a rotated lattice such that either the real or the imaginary components of det is non zero for any pair of codewords [16]. In general, the variable ’s of the vector can take values from say, a M  PAM set where M is any natural number. The dimensional real vector z is rotated using the generator, G of a rotated lattice to generate a lattice point as using which a complex vector is transmitted to all the relays. The signal set is identified using computer search as . As an example, ’s is allowed to take values from and the generator of the lattice, G is found to be in (19).
(19) 
The matrix G in (19) is obtained using computer search. Through simulations, it has been verified that if the vector takes value from the above signal set , then the determinant of is non zero for any pair of codewords of and hence is fully diverse. In general, for CIDSTCs of any dimension, appropriate signal sets needs to be designed so as to make the code fully diverse.
V simulations
In this section, we provide simulation results for the performance comparison of CIDSTC and RDSTC for a wireless network with two relay nodes (Figure 5) and a single relay node (Figure 6). Optimal power allocation strategy discussed in Subsection IIIA has been used in our simulation setup though the strategy is not optimal for smaller values of R. Even though the power allocation used is not optimal, CIDSTCs are found to perform better than their corresponding RDSTCs. We have used the Bit Error Rate (BER) which corresponds to errors in decoding every bit as error events of interest. For the network with 2 relays, since we need , for CIDSTC, we use the channel coherence time of channel use for both the schemes.
The real and imaginary parts of information symbols are chosen equiprobably from a 2 PAM signal set and are appropriately scaled to maintain the unit norm condition. Simulations are carried out using the linear designs in variables , as given in (20) and (21) for RDSTC and CIDSTC respectively. It can be verified that design in (21) is of the required form given in (7).
The design in (20) is four group decodable, i.e., the variables can be partitioned into four groups and the ML decoding can be carried out for each group of variables independently of the variables in the groups and the variables of each groups need to be jointly decoded [14]. The corresponding four groups of real variables are, . We use sphere decoding algorithm for ML decoding [15]. Though the design in (20) is four group decodable, the design in (21) is not four group ML decodable. Full diversity is obtained by making every group of real variables choose values from a rotated lattice constellation [16] whose generator given by,
(20) 
(21) 
BER comparison of the two schemes using the above designs is shown in Figure 5. The plot shows that CIDSTC in (21) performs better than the RDSTC in (20) by close to 1.5 to 2 db.
Simulation results comparing the BER performance of CIDSTC in Example 1 with its corresponding RDSTC is shown in Figure 6. Full diversity is obtained by choosing a rotated lattice constellation (Section IV) whose generator is given by (19). The plot shows the superiority of the design in Example 1 over its RDSTC counterpart by 1 db. Simulation results comparing the BER performance of CIDSTC in Example 1 with RDSTC from random coding is also shown in Figure 6 which shows the superiority of CIDSTC by 1.75  2 db at larger values of P.
Vi Discussion
The technique of coordinate interleaved distributed spacetime coding at the relays was introduced for wireless relay networks having relays each having two antennas. For , we have shown that CIDSTC provides coding gain compared to the scheme when transmit and receive signals at different antennas of the same relay are processed independently. This improvement is at the cost of only a marginal additional complexity in processing at the relays. Condition on the full diversity of CIDSTCs is also presented. Some of the possible directions for future work is to extended the above technique to relay networks where the source and the destination nodes have multiple antennas. Also, if the relays have more than two antennas then a general linear processing need to be employed in the place of CIVP and new performance bounds need to be derived.
Appendix A Proof of Lemma 2
The channel h as given in (6) can be written as the product Gk of G and k where
and
Since U is Hermitian and positive definite, we can write , where D is diagonal matrix containing the eigen values of in (9). Since, U is of full rank, the right hand side of the resulting following PEP expression
can be upperbounded by replacing D by where is the minimum singular value of Then, we have,
where
Since the set of random variables are independent (from Proposition 1) and distributed as we have,
and hence