Cascaded orthogonal space–time block codes for wireless multihop relay networks
 Rahul Vaze^{1}Email author and
 Robert W HeathJr.^{2}
https://doi.org/10.1186/168714992013113
© Vaze and Heath; licensee Springer. 2013
Received: 20 November 2012
Accepted: 18 March 2013
Published: 25 April 2013
Abstract
Distributed space–time block coding is a diversity technique to mitigate the effects of fading in multihop wireless networks, where multiple relay stages are used by a source to communicate with its destination. This article proposes a new distributed space–time block code called the cascaded orthogonal space–time block code (COSTBC) for the case where the source and destination are equipped with multiple antennas and each relay stage has one or more multiple antenna relays. Each relay stage is assumed to have receive channel state information (CSI) for all the channels from the source and all relays from previous stages to itself, while the destination is assumed to have receive CSI for all the channels. To construct the COSTBC, multiple orthogonal space–time block codes (OSTBCs) are used in cascade by the source and each relay stages. In the COSTBC, each relay stage separates the constellation symbols of the OSTBC sent by the preceding relay stage using its CSI, and then transmits another OSTBC to the next relay stage. COSTBCs are shown to achieve the maximum diversity gain in a multihop wireless network with linear decoding complexity thanks to the connection to OSTBCs. Several explicit constructions of COSTBCs are also provided, and their performance is simulated in different relay configurations.
Keywords
1 Introduction
Distributed space–time block coding (DSTBC) is a technique to improve reliability in relayassisted communication, where one or more relays help the source to communicate with its destination. Relayassisted communication is likely to occur in large wireless networks, such as adhoc or sensor network, where the destination is possibly out of the source’s communication range. Relayassisted communication is also used in a cellular wireless networks to improve the performance of cell edge users, and has been incorporated in modern wireless standards such as IEEE 802.16j, and 3GPP LTE Advanced.
In DSTBCs, relay antennas are used together with the source antennas in a distributed manner to transmit a space–time block code (STBC) [1] to the destination. By introducing redundancy in space and time, DSTBCs increase the reliability of the communication by increasing the diversity gain, defined as the negative of the exponent of the signaltonoise ratio (SNR) in the pairwise error probability expression at high SNR [1].
In prior work, maximum diversity gain achieving DSTBC constructions have been proposed for the twohop network [2–21], and for the multihop network [22–24]. Even though these DSTBC constructions [2–24] achieve the maximum diversity gain, the decoding complexity of most of them, except [14–21], is very high, thereby limiting their use in practical deployment. Construction of DSTBCs with low decoding complexity is practically important as highlighted by the fact that the Alamouti code is the most practically used code not only because it achieves the maximum diversity gain, but also because it requires minimum decoding complexity. Moreover, the DSTBC constructions with low decoding complexity [14–21] are limited to twohop network with single antenna equipped source, destination, and the relay nodes.
In this article, we design maximum diversity gain achieving DSTBCs with lowdecoding complexity for a multihop wireless network where the source, the destination, and the relay nodes are equipped with multiple antennas. In the proposed DSTBC, called the cascaded orthogonal space–time block code (COSTBC), an orthogonal spacetime code (OSTBC) [25] is used by the source, and subsequently by each relay stage to communicate with its adjacent relay stage. OSTBCs are considered because of their single symbol decodable property [25, 26], i.e., with the maximum likelihood decoding each constellation symbol of the OSTBC can be decoded independently of other constellation symbols. We assume that each relay has receive channel state information (CSI) for all the channels from the source to itself, while the destination is assumed to have receive CSI for all the channels. With COSTBCs, in the first time slot, the source transmits an OSTBC to the first relay stage. Using the orthogonality property of the OSTBC and the available CSI, each relay of the first relay stage separates the different OSTBC constellation symbols from the received signal, and transmits a codeword vector in the next time slot, such that the matrix obtained by stacking all the codeword vectors transmitted by the different relays of the first relay stage is an OSTBC. These operations are repeated by subsequent relay stages. With COSTBCs, no signal is decoded at any of the relays, therefore COSTBC construction with single antenna relays is equivalent to COSTBC construction with multiple antenna relays. Thus, without loss of generality, in this article, we only consider COSTBC construction for single antenna relays. We note that for the code construction each relay is required to have receive CSI for all the channels from the source and all relays from previous stages to itself, while the destination is assumed to have receive CSI for all the channels.
1.1 Our contributions

We show that COSTBCs achieve the maximum diversity gain in a multihop wireless network when each symbol of the code is decoded independently (non maximumlikelihood decoding), resulting in linear decoding complexity similar to single symbol decodable codes.

We prove that for a twohop network and when the destination has a single antenna, by adding channel coefficientdependent noise terms to the received signals, COSTBCs have the single symbol decodable property for any number of source and relay antennas. Thus, by paying a penalty in terms of coding gain because of extra noise, COSTBCs provide significant decoding complexity gain.
A part of this article has been presented at [27, 28]. Due to space limitation, the studies [27, 28] contain only the results of this article without any proofs. In this article, detailed proofs of the results, together with explicit code construction, and some simulation results are described.
1.2 Comparison with prior work
Previous constructions of maximum diversity gain achieving DSTBCs with low decoding complexity (single symbol decodable) [14–21] are limited to a twohop network with single antenna nodes. COSTBCs, in comparison, achieve the maximum diversity gain with linear decoding complexity (similar to [14–21]) in a multihop network with multiple antenna equipped nodes, even though they do not have the single symbol decodable property. For the multihop network, the focus of [23, 24] is on the construction of DSTBCs that can achieve the optimal diversity multiplexing tradeoff [29]. In comparison to the strategies of [23, 24], COSTBCs only achieve the maximum diversity gain and fall short of achieving the maximum multiplexing gain because of the use of OSTBCs. The decoding complexity of COSTBC, however, is significantly less (linear) than the strategies of [23, 24] and makes COSTBCs amenable for practical implementation in comparison to [23, 24], where STBCs with high decoding complexity are used. Thus, COSTBCs are well suited for relayassisted communication where relays are used to improve the cell coverage, by improving reliability of the users at the cell edge, while requiring low decoding complexity.
Notation: Let A denote a matrix, a a vector and a_{ i } the i th element of a. diag(A) represents a vector consisting of diagonal entries of A. The determinant and trace of matrix A are denoted by det(A) and tr(A). The vector consisting of the diagonal entries of A is denoted by diag(A). The field of real and complex numbers are denoted by $\mathbb{R}$ and $\mathbb{C}$, respectively. The space of M × N matrices with complex entries is denoted by ${\mathbb{C}}^{M\times N}$. The Euclidean norm of a vector a is denoted by a. An m × m identity matrix is denoted by I_{ m }, and 0_{ m } is as an all zero m × m matrix. The superscripts ^{ T },^{∗},^{ ‡ } represent the transpose, transpose conjugate, and element wise conjugate. The expectation of function f(x) with respect to x is denoted by $\mathbb{E}\left\{f\right(x\left)\right\}$. A circularly symmetric complex Gaussian random variable x with zero mean and variance σ^{2} is denoted as $x\sim \mathcal{C}N(0,{\sigma}^{2})$. We use the symbol =. to represent exponential equality, i.e., let f(x) be a function of x, then f(x) =. x^{ a } if $\underset{x\to \infty}{lim}\frac{log\left(f\right(x\left)\right)}{logx}=a$ and similarly ≤. and ≥. denote the exponential less than or equal to and greater than or equal to relation, respectively. We use the symbol := to define a variable.
2 System model
As shown in Figure 1, the channel between the source and the i th relay of the first stage of relays is denoted by ${\mathbf{h}}_{i}=\phantom{\rule{0.3em}{0ex}}{[{h}_{1i}\phantom{\rule{1em}{0ex}}{h}_{2i}\phantom{\rule{1em}{0ex}}\dots \phantom{\rule{1em}{0ex}}{h}_{{M}_{0}i}]}^{T},\phantom{\rule{1em}{0ex}}i=1,2,\dots ,{M}_{1}$, between the j th relay of relay stage s and the k th relay of relay stage s + 1 by ${f}_{\mathit{\text{jk}}}^{s},\phantom{\rule{1em}{0ex}}s=0,1,\dots ,N2,\text{\xe6}=1,2,\dots ,{M}_{s},\phantom{\rule{1em}{0ex}}k=1,2,\dots ,{M}_{s+1}$ and the channel between the relay stage N − 1 and the ℓ th antenna of the destination by ${\mathbf{g}}_{\ell}={[{g}_{1\ell}\phantom{\rule{1em}{0ex}}{g}_{2\ell}\phantom{\rule{1em}{0ex}}\dots \phantom{\rule{1em}{0ex}}{g}_{{M}_{N1}\ell}]}^{T},\phantom{\rule{1em}{0ex}}\ell =1,2,\dots ,{M}_{N}$. We assume that ${\mathbf{h}}_{i}\in {\mathbb{C}}^{{M}_{0}\times 1}$, ${f}_{\mathit{\text{jk}}}^{s}\in {\mathbb{C}}^{1\times 1}$, ${\mathbf{g}}_{l}\in {\mathbb{C}}^{{M}_{N1}\times 1}$ are independent and identically distributed (i.i.d.) $\mathcal{C}N(0,1)$ entries for all i, j, k, ℓ, s. We assume that the m^{ th } relay of n^{ th } stage knows ${\mathbf{h}}_{i},{f}_{\mathit{\text{jk}}}^{s},\phantom{\rule{1em}{0ex}}\forall \phantom{\rule{1em}{0ex}}i,\phantom{\rule{1em}{0ex}}j,\phantom{\rule{1em}{0ex}}k,s=1,2,\dots ,n2,\phantom{\rule{1em}{0ex}}{f}_{\mathit{\text{jm}}}^{n1}\phantom{\rule{1em}{0ex}}\forall j$, and the destination knows ${\mathbf{h}}_{i},{f}_{\mathit{\text{jk}}}^{s},{\mathbf{g}}_{l},\phantom{\rule{1em}{0ex}}\forall \phantom{\rule{1em}{0ex}}i,j,k,l,s$. We further assume that all these channels are frequency flat and block fading, where the channel coefficients remain constant in a block of time duration T_{ c } and change independently from blocktoblock. We assume that the T_{ c } is at least max{M_{0}, M_{1}, …, M_{N − 1}}.
Remark 1.
Duplexing: Duplexing is an important consideration in multihop relay networks. For example, if relay stage n is receiving the signal from relay stage N−1 and relay stage n+1 is transmitting to relay stage n+2 simultaneously, then relay stage n will receive back flow of signals from relay stage n+1 that it has already transmitted. Since each relay stage uses an amplify and forward strategy, most of the power at relay stage n will then be used to retransmit signals that have been transmitted before. A related paper [24] claims that back flow can be allowed with successive relay stages transmitting simultaneously without decreasing the diversity gain. That is true, however, in the limit of extremely large transmit power, and not applicable for any realistic transmit power level.
This problem is unique to greater than twohop relay network and is not well understood. To avoid this situation, a rate penalty of onethird is unavoidable for both fullduplex and halfduplex relay operation, where every third relay stage is switched on alternatively one at a time. For example, in first time slot communication happens between relay stages $01,34,67,\dots \phantom{\rule{0.3em}{0ex}}$, while in the second time slot communication happens between relay stages $12,45,78,\dots \phantom{\rule{0.3em}{0ex}}$, in third time communication happens between relay stages $23,56,89,\dots \phantom{\rule{0.3em}{0ex}}$, and so on, with periodic repetitions.
2.1 Problem formulation
Definition 1
(STBC) [30] A rate L/T T × N_{ t }design D is a T × N_{ t } matrix with entries that are complex linear combinations of L complex variables s_{1}, s_{2}, …, s_{ L } and their complex conjugates. A rate L / T T × N_{ t } STBC S is a set of T × N_{ t } matrices that are obtained by allowing the L variables s_{1}, s_{2}, …, s_{ L } of the rate L / T T × N_{ t } design D to take values from a finite subset ${\mathbb{C}}^{f}$ of the complex field $\mathbb{C}$. The cardinality of S$={\mathbb{C}}^{f}{}^{L}$, where $\left{\mathbb{C}}^{f}\right$ is the cardinality of $\mathcal{C}$. We refer to s_{1}, s_{2}, …, s_{ L } as the constituent symbols of the STBC.
Definition 2
Definition 3
P_{ e }(E) is the pairwise error probability using DSTBC Φ, and E is the sum of the transmit power used by each node in the network.
The problem we consider in this article is to design DSTBCs that achieve the maximum diversity gain in an Nhop network. To identify the limits on the maximum possible diversity gain in an Nhop network, an upper bound on the diversity gain achievable with any DSTBC is presented next.
Theorem 1
The diversity gain d_{ Φ } of DSTBC Φ for an Nhop network is upper bounded by min{M_{ n }M_{n + 1}}, n = 0, 1, …, N − 1.
Proof
See Proposition 2.1 of [23]. □
Theorem 1 implies that the maximum diversity gain achievable in an Nhop network is equal to the minimum of the maximum diversity gain achievable between any two relay stages, when all the relays in each relay stage are allowed to collaborate. In the next section, we propose COSTBCs that are shown to achieve this upper bound on the diversity gain.
3 COSTBC
In this section, we introduce the COSTBC design for an Nhop network. Before introducing COSTBCs, we need the following definitions.
Definition 4
With T ≥ N_{ t }, a rate L/T T × N_{ t } STBC S is called fullrank or fully diverse or is said to achieve the maximum diversity gain if the difference of any two matrices M_{1}, M_{2} ∈ S is fullrank, i.e., minM_{1 }^{≠}M_{2}, M_{1}, M_{2 }^{∈} _{ S } rank(M_{1} − M_{2}) = N_{ t }.
Definition 5
A rateL/K K × K STBC S is called an OSTBC if the design D from which it is derived is orthogonal, i.e.,DD^{∗} = (s_{1}^{2} + ⋯ + s_{ L }^{2})I_{ K }.
Definition 6
Let S be a rateL / K K × K STBC. Then, if the maximum likelihood (ML) decoding of S is such that each of the constituent symbols s_{ i }, i = 1, …, L of S can be decoded independently of s_{ j } ∀ i ≠ j i, j = 1, …, L, then S is called a single symbol decodable STBC.
Remark 2
OSTBCs are single symbol decodable STBCs [25].
With these definitions we are now ready to describe COSTBCs for an Nhop network.
where ${\widehat{\mathbf{n}}}_{k}^{1}$ is an L × 1 vector with entries that are uncorrelated and $\mathcal{C}N(0,1)$ distributed.
∀ k = 1, 2 …, M_{1}, l = 1, 2, … L, where ${\mathbf{A}}_{k}^{1}\left(l\right)$ and ${\mathbf{B}}_{k}^{1}\left(l\right)$ denote the l th column of A_{ k } and B_{ k }, respectively, and ${\mathbf{S}}_{1}:=\phantom{\rule{0.3em}{0ex}}[{\mathbf{A}}_{1}^{1}\mathbf{s}+{\mathbf{B}}_{1}^{1}{\mathbf{s}}^{\u2020}\dots {\mathbf{A}}_{{M}_{1}}^{1}\mathbf{s}+{\mathbf{B}}_{{M}_{1}}^{1}{\mathbf{s}}^{\u2020}]$ is an OSTBC.
is of the form (1), where S_{1} is also an OSTBC similar to S_{0}. Thus, repeating the operations illustrated in (2), (3), and (4), and using matrices ${\mathbf{A}}_{k}^{n}$, ${\mathbf{B}}_{k}^{n},\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}k=1,\dots ,{M}_{n},n=2,\dots ,N1$ satisfying (5), an OSTBC is transmitted from each relay stage to construct the COSTBC.
where $\mathbf{R}:=\mathbb{E}\left\{\mathbf{w}{\mathbf{w}}^{\ast}\right\}$ is the noise covariance matrix. Note that if R is a scaled identity matrix, then the ML decision rule (8) is equal to $\sum _{j=1}^{L}\underset{j}{\text{min}}f\left({s}_{j}\right)$, where f(s_{ j }) is a function of s_{ j } that does not depend on s_{ k }, k ≠ j, since ${\mathbf{S}}_{N1}^{\ast}{\mathbf{S}}_{N1}$ is a scaled identity matrix. Thus, the COSTBCs are single symbol decodable if R is a scaled identity matrix.
even though this is not ML decoding. We consider this decoding rule to ensure linear decoding complexity and show that COSTBCs achieve maximum diversity gain with this rule.
Theorem 2
COSTBCs achieve the diversity gain upper bound(Theorem 1) in an Nhop network with decoding rule (10).
Proof
See Appendix 1. □
The basic idea behind the proof of Theorem 2 is to exploit the orthogonality of OSTBCs transmitted by each relay stage.
For the special case of N = 2 and M_{2} = 1, we next show that the COSTBCs can be made single symbol decodable by degrading the received signal by adding some channel coefficientdependent noise terms as discussed before.
Theorem 3
COSTBCs are single symbol decodable STBCs after adding some channel coefficientdependent noise terms to the received signals for N = 2 and M_{2} = 1.
Proof
See Appendix Appendix 2. □
Remark 3
CSI: We note that for decoding of COSTBCs, global CSI is required at the destination. The requirement of destination having global CSI regarding all the underlying channels has been made in several recent related papers, including [23, 24]. Actually, this is a common assumption made by all papers that consider amplifyandforward protocol. Since mostly, only twohop communication is considered, the CSI requirement is somewhat limited compared to the case of multihop communication, the topic of this paper and [22–24]. Acquiring such CSI in practice is a challenge, however, using techniques like Grassmannian codebooks, CSI about all channels can be acquired by the destination through the relay nodes by dedicating the start of time slots for training purposes. In particular, relay in stage 1 can get the CSI between source and itself by using pilots and channel estimation. Thereafter, by using Grassmannian codebooks it can forward the CSI it has gathered to the next relay stage in addition to sending pilots for the relays in the next stage to gather CSI between relay stages. Repeating this procedure all nodes can get the required CSI.
Another assumption about CSI we make for our code construction to work is that CSI is available at each relay node for channels preceding itself which is not required for other related works [22–24]. Since the CSI required at the destination for any amplifyandforward protocol has to be transmitted through to the destination through the relays, CSI can safely be assumed to be available at each relay node as well. Thus, this is also not a limiting assumption.
Discussion: In this section, we constructed COSTBCs by cascading OSTBCs at each relay stage. OSTBCs are cascaded at each relay stage by first separating each constellation symbol of the OSTBC transmitted from the preceding relay stage, and then transmitting another OSTBC to the next relay stage. The proposed OSTBC cascading strategy is novel, and different than other approaches that use Alamouti code or OSTBC in a distributed manner [12, 31].
We showed that the single symbol decodable property of OSTBCs is lost by cascading OSTBCs to construct COSTBCs. Using the orthogonality property of the OSTBCs, however, we showed that the maximum diversity gain can be achieved by COSTBCs even when each source transmitted symbol is decoded independently. Therefore, COSTBCs have decoding complexity that is linear in the number of symbols transmitted by source in one codeword, which is quite critical for practical implementations. Since independent symbol decoding is not ML, COSTBCs entail an unavoidable coding gain loss, however, we show that at least in terms of diversity gain there is no loss compared to ML decoding. We also showed that the COSTBCs are single symbol decodable for a twohop wireless network N=2 when the destination has only a single antenna M_{2}=1, by adding some channel coefficientdependent noise terms to the received signal.
4 Explicit code constructions
In this section, we explicitly construct COSTBCs that achieve the maximum diversity gain in an Nhop network. The ingredient OSTBCs can be borrowed from [25, 32, 33], similar to [34]. We present examples of COSTBCs for N = 2, M_{0} = M_{1} = 2 using the Alamouti code [26], N = 2, M_{0} = M_{1} = 4 using the rate 3 / 4 4 antenna OSTBC [25] and N = 2, M_{0} = M_{1} = 4 using the rate 3/4 4 antenna OSTBC and the Alamouti code.
Example 1
for m = 1, 2. We define ${\stackrel{~}{h}}_{m}:={h}_{1m}{}^{2}+{h}_{2m}{}^{2}$, ${\eta}_{1m}:=({n}_{1m}{h}_{1m}^{\ast}+{n}_{2m}^{\ast}{h}_{2m})$, and ${\eta}_{2m}:=({n}_{1m}{h}_{2m}^{\ast}{n}_{2m}^{\ast}{h}_{1m})$.
for m = 1, 2.
the STBC S_{1} formed by the two relays is equal to ${\mathbf{S}}_{\mathit{\text{ala}}}^{T}$ which is also an OSTBC as required. Note that A_{ i }, B_{ i }i = 1, 2 satisfy the requirements of (5). We call this the cascaded Alamouti code.
Example 2
It is easy to verify that $\mathit{\text{tr}}\left({\mathbf{A}}_{i}^{\ast}{\mathbf{A}}_{i}+{\mathbf{B}}_{i}^{\ast}{\mathbf{B}}_{i}\right)=3$ and ${\mathbf{A}}_{i}^{\ast}{\mathbf{B}}_{i}={\mathbf{B}}_{i}^{\ast}{\mathbf{A}}_{i},i=1,2,3,4$ as required. Then the STBC S_{1} = S_{0} using these A_{ i }, B_{ i }i = 1, 2, 3, 4, which is a rate 3/4 OSTBC as described above.
In both the previous examples, we constructed a COSTBC for the N = 2hop case by repeatedly using the same OSTBC at both the source and the relay stage. Using a similar procedure, it can be seen that when M_{ i } = M_{ j } ∀ i, j = 0, 1, …, N − 1, i ≠ j we can construct COSTBCs by using particular OSTBC for M_{0} antennas at the source and each relay stage, e.g., if $\mathcal{O}$ is an OSTBC for M_{0} antennas, then by using ${\mathbf{S}}_{n}=\mathcal{O},\phantom{\rule{1em}{0ex}}n=0,1,\dots ,N1$ we obtain a maximum diversity gain achieving COSTBC. OSTBC constructions for different number of antennas can be found in [25]. In the next example, we construct COSTBC for M_{0} = 4 and M_{1} = 2 by cascading the rate 3/4 4 antenna OSTBC with the Alamouti code.
Example 3
Let N = 2, M_{0} = 4, and M_{1} = 2. We choose S_{0} to be the rate 3/4 4 antenna OSTBC (12) and S_{1} to be the Alamouti code. The COSTBC is constructed as follows.
Let S_{0} given by (12) be the transmitted rate 3/4 4 antenna OSTBC from the source. Then the received signal at relay node m, m = 1, 2 is ${\mathbf{r}}^{m}\phantom{\rule{0.3em}{0ex}}=\phantom{\rule{0.3em}{0ex}}\sqrt{{E}_{0}}{\mathbf{S}}_{0}{\left[\begin{array}{cccc}{h}_{1m}& {h}_{2m}& {h}_{3m}& {h}_{4m}\end{array}\right]}^{T}\phantom{\rule{0.3em}{0ex}}+{\left[\begin{array}{cccc}{n}_{1}^{m}& {n}_{2}^{m}& {n}_{3}^{m}& {n}_{4}^{m}\end{array}\right]}^{T}$. Using CSI the received signal r^{ m } can be transformed into ${\widehat{\mathbf{r}}}^{m}$, where ${\widehat{\mathbf{r}}}^{m}:={\left[\begin{array}{ccc}{\widehat{r}}_{1}^{m}& {\widehat{r}}_{2}^{m}& {\widehat{r}}_{3}^{m}\end{array}\right]}^{T}=\sqrt{{E}_{0}}{\left[\begin{array}{ccc}{\u0125}_{m}{s}_{1}& {\u0125}_{m}{s}_{2}& {\u0125}_{m}{s}_{3}\end{array}\right]}^{T}+{\left[\begin{array}{ccc}{\widehat{n}}_{1}^{m}& {\widehat{n}}_{2}^{m}& {\widehat{n}}_{3}^{m}\end{array}\right]}^{T}$ and ${\u0125}_{m}=\sqrt{\sum _{i=1}^{{M}_{0}}{h}_{\mathit{\text{im}}}{}^{2}}$. Then in the next time slot, the relay m, m = 1, 2 transmits ${\mathbf{A}}_{m}{\left[\begin{array}{cc}{\widehat{r}}_{1}^{m}& {\widehat{r}}_{2}^{m}\end{array}\right]}^{T}+{\mathbf{B}}_{m}{\left[\begin{array}{cc}{\widehat{r}}_{1}^{m}& {\widehat{r}}_{2}^{m}\end{array}\right]}^{T\u2020}$ where A_{ m }, B_{ m } are given in (11).
These operations are repeated at the source and each relay stage in subsequent time slots. In the next time slot, signal s_{3} received in the previous time slot and s_{1} received in the current time slot is transmitted from relay stage 1 to the destination. Clearly, the relay stage transmits an Alamouti code which is an OSTBC and hence leads to a COSTBC construction for M_{0} = 4, M_{1} = 2.
Using a similar technique as illustrated in this example, COSTBCs can be constructed for different number of source antenna and relay node configurations by suitably adapting different OSTBCs.
5 Simulation results
In this section, we provide simulation results to illustrate the bit error rates (BERs) of COSTBCs for 2 and 3hop networks. In all the simulation plots, E denotes the total power used by all nodes in the network, i.e., ${E}_{0}+\sum _{n=1}^{N1}{M}_{n}{E}_{n}=E$ and the additive noise at each relay and the destination is complex Gaussian with zero mean and unit variance. By equal power allocation between the source and each relay stage we mean ${E}_{0}={M}_{n}{E}_{n}=\frac{E}{N},\phantom{\rule{1em}{0ex}}\forall n=1,\dots ,N1$.
From all the simulation plots, it is clear that COSTBCs require large transmit power to obtain reasonable BERs with multihop wireless networks. This is a common phenomenon across all the maximum diversity gain achieving DSTBCs for multihop wireless networks that use AF [3, 5, 9]. With AF, the noise received at each relay gets forwarded towards the destination and limits the received SNR at the destination, however, without using AF it is difficult to achieve the maximum diversity gain in a multihop wireless network.
6 Conclusion
In this article, we designed COSTBCs that achieve the maximum diversity gain in a multihop wireless network with low decoding complexity. We then gave an explicit construction of COSTBCs for various numbers of source, destination, and relay antennas. The only restriction that COSTBCs impose is that the source and all the relay stages have to use an OSTBC. It is well known that high rate OSTBCs do not exist; therefore, the COSTBCs have rate limitations. For future work it will be interesting to see whether the OSTBC requirement can be relaxed without sacrificing the maximum diversity gain and minimum decoding complexity of the COSTBCs.
Appendix 1
We prove Theorem 2 using induction. First we show that COSTBCs achieve the maximum diversity gain for N = 2, and then extend the result for a khop network, where k is any arbitrary natural number.
The outage probability P_{out}(R) is defined as P_{out}(R) := P(I(s;r) ≤ R), where S is the input and R is the output of the channel and I(s; r) is the mutual information between S and R[36]. Let d_{out}(r) be the SNR exponent of P_{out} with rate of transmission R scaling as r logSNR, i.e., $log{P}_{\text{out}}(rlog\mathsf{\text{SNR}})\stackrel{.}{=}{\mathsf{\text{SNR}}}^{{d}_{\text{out}}\left(r\right)}$. Then, if ${P}_{e}\left(\mathsf{\text{SNR}}\right)\stackrel{.}{=}{\mathsf{\text{SNR}}}^{d\left(r\right)}$, then from [29], and the compound channel argument [24], ${P}_{\text{out}}(rlog\mathsf{\text{SNR}})\stackrel{.}{=}{P}_{e}\left(\mathsf{\text{SNR}}\right),\phantom{\rule{1em}{0ex}}d\left(r\right)\stackrel{.}{=}{d}_{\text{out}}\left(r\right)$. Therefore, to compute d(r), it is sufficient to compute d_{out}(r). In the following, we compute d_{out}(r) for the COSTBC with a 2hop network.
Let ${z}_{l}:=\sum _{j=1}^{{M}_{2}}{z}_{\ell}^{j}$. Even though ${z}_{\ell}^{j}$’s are not independent, any linear combination of ${z}_{l}^{j}$’s is Gaussian, therefore z_{ ℓ } is $\mathcal{C}N(0,{\sigma}^{2})$ distributed for some σ^{2}. Note that σ^{2} depends on the channel coefficients, however, as shown in Theorem 2.3 of [24], z_{ ℓ } is white in the scale of interest and without loss of generality z_{ ℓ } can be modeled as $\mathcal{C}N(0,1)$, i.e., independent of channel coefficients in the outage analysis.
Thus, ${P}_{\text{out}}(rlog\mathsf{\text{SNR}})\stackrel{.}{=}{\mathsf{\text{SNR}}}^{\text{min}\{{M}_{0}{M}_{1},\phantom{\rule{1em}{0ex}}{M}_{1}{M}_{2}\}(1r)},r\le 1$, and we have that d_{out}(r) = min{M_{0}M_{1}, M_{1}M_{2}}(1 − r), r ≤ 1. Thus, the maximum diversity gain of the COSTBCs is d_{out}(0) = min{M_{0}M_{1}, M_{1}M_{2}} which equals the diversity gain upper bound (Theorem 1). Thus, we have shown that COSTBCs achieve the maximum diversity gain in a 2hop network. Next, using induction, we prove the result for any khop network.
where θ_{k − 1} is the normalization constant so as to ensure the average power constraint of E_{k − 1} is satisfied at the relay stage k − 1, s_{ ℓ } is the ℓ th, ℓ = 1, 2, …, L symbol transmitted from the source, c_{ i } is the channel gain experienced by s_{ ℓ } at the i th antenna of the destination, and z_{ l } is the AWGN with variance ${\sigma}_{k}^{2}$.
for each ℓ = 1, …, L, where n_{ ℓ i } = n_{ ℓ } / M_{ k }. Recall from induction hypothesis that the diversity gain of COSTBCs with channel c_{ i }, ∀ i (15) is α := min{min{M_{ n }M_{n + 1}}, M_{k − 1}}, n = 0, 1, …, k − 2, by restricting the destination of the khop network to have only single antenna, and with channel $\sum _{i=1}^{{M}_{k}}{c}_{i}$ is min{M_{ n }M_{n + 1}}, n = 0, 1, …, k − 1, respectively. Thus, if the diversity gain of COSTBCs with channel q_{ i } (17) is min{min{M_{ n }M_{n + 1}}, M_{k − 1}, M_{k + 1}} n = 0, 1, …, k − 2, then, since $\sum _{j=1}^{{M}_{k+1}}{g}_{\mathit{\text{ij}}}{}^{2}$ are independent ∀ i, it follows that the diversity gain of COSTBCs with channel $\sum _{i=1}^{{M}_{k}}{q}_{i}$ is min{M_{ n }M_{n + 1}}, n = 0, 1, …, k. Next, we show that the diversity gain of COSTBCs with channel q_{ i } is min{min{M_{ n }M_{n + 1}}, M_{k − 1}, M_{k + 1}}, n = 0, 1, …, k − 2.
Using the definition of diversity gain, it follows that the diversity gain of COSTBCs with channel q_{ i } is equal to min{α, M_{k + 1}}, which implies that the diversity gain of COSTBCs with the received signal model (16) is min{α M_{ k }, M_{ k }M_{k + 1}}. Note that the upper bound on the diversity gain (Theorem 1) is also min{α M_{ k }, M_{ k }M_{k + 1}}, and hence we conclude that the COSTBCs achieve the maximum diversity gain in an Nhop network.
Appendix 2
In this section, we prove that COSTBCs have the single symbol decodable property for N=2 and M_{2} = 1.
where ${\mathbf{S}}_{1}=[{\mathbf{A}}_{1}^{1}\mathbf{s}+{\mathbf{B}}_{1}^{1}{\mathbf{s}}^{\u2020}\phantom{\rule{1em}{0ex}}{\mathbf{A}}_{2}^{1}\mathbf{s}+{\mathbf{B}}_{2}^{1}{\mathbf{s}}^{\u2020}\phantom{\rule{1em}{0ex}}\dots \phantom{\rule{1em}{0ex}}{\mathbf{A}}_{{M}_{1}}\mathbf{s}+{\mathbf{B}}_{{M}_{1}}{\mathbf{s}}^{\u2020}]$.
Let $\mathbf{R}=\mathbb{E}\left\{\mathbf{w}{\mathbf{w}}^{\ast}\right\}$ be the noise covariance matrix.
where ${\mathbf{D}}_{j}=({\mathbf{A}}_{j}^{1}{\mathbf{A}}_{j}^{1\ast}+{\mathbf{B}}_{j}^{1}{\mathbf{B}}_{j}^{1\ast})$ is a diagonal matrix whose diagonal entries are either zero or one since A_{ j } and B_{ j } are constituents of an OSTBC, and the number of ones in D_{ j } is equal to k, if A_{ j } and B_{ j } are constituents of a rate k/n OSTBC. Note that the locations of nonzero entries of D_{ j } can be different for different j’s, and hence R is not necessarily a scaled identity matrix. Therefore, because of the nondiagonal structure of R^{ − 1}, the ML decoding metric (8) cannot be split in several terms, where each term is a function of only one of the constituent symbols of S_{1}. Therefore in general, COSTBCs are not single symbol decodable. The problem can, however, be fixed easily by adding an additional channel coefficientdependent noise vector to the received signal at the receiver as follows.
which is a scaled identity matrix. Hence, using y_{ a } (instead of y) to decode S_{1}, the ML decoding metric (8) splits in L different terms, where each term is a function of only one of the constituent symbols of S_{0}, and the COSTBC is single symbol decodable for N = 2, and M_{2} = 1.
Declarations
Acknowledgements
This material is based upon work supported by the Army Research Laboratory Labs W911NF1010420 and by the National Science Foundation NSFCCF1218338.
Authors’ Affiliations
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