Efficiently sphere-decodable physical layer transmission schemes for wireless storage networks

Consider a wireless DSS with K helper nodes, equipped with n_t transmit antennas each, and a repairing node with n_r receive antennas. Let \(H_{i} \in \mathbb {C}^{n_{r} \times n_{t}}\) be the channel matrix, and \(X_{i} \in \mathbb {C}^{n_{t} \times T}\) the code matrix associated with the ith helper node, where T is the number of channel uses needed for transmitting X_i. The received signal matrix at the repairing node is given by

where \(W \in \mathbb {C}^{n_{r} \times T}\) is a matrix modeling complex additive white Gaussian noise (AWGN). The entries of H_i and W_i are independent and identically distributed (i.i.d.) circularly symmetric complex Gaussian random variables with zero mean and unit variance, a distribution which we henceforth denote as \(\mathbb {C}{\cal {N}}(0,1)\). The code matrices X_i are required to satisfy the average power constraint \(\mathbb {E}||{X_{i}}||^{2} \leq T \cdot \text {SNR}\). It is also assumed throughout the paper that the repairing node has a complete knowledge of channel state information {H_i:i=1,…,K}.

where the entries of G_i,j and Z_i are again modeled as i.i.d. \(\mathbb {C}{\cal {N}}(0,1)\) random variables, and the signal matrices S_j satisfy \(\mathbb {E} ||{S_{j}}||^{2} \leq T' \cdot \text {SNR}\). A complete knowledge of {G_i,j} is assumed to be available at the ith node. Finally, it is assumed throughout the paper that all communication links are half-duplex. A pictorial description of the above channel model is given in Fig. 2.

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2.2 Sphere decoding complexity of state-of-the-art MIMO-MAC codes

A general construction of MIMO-MAC space-time codes was proposed in [26] and was shown to achieve the optimal DMT (5) for any triple (n_t,n_r,K) and multiplexing gain r. More specifically, given n_t and K, the component code \({{\mathcal {C}}}_{i}\) of the ith helper node is taken from an algebraic lattice of lattice rank⁷\(2n_{t}{K_{o}^{2}}\) in [26], and \({{\mathcal {C}}}_{i}\) consists of (n_t×T) matrices with \(|{{\mathcal {C}}_{i}}| \doteq \text {SNR}^{rT}\) and T=n_tK_o, where K_o is the smallest odd integer ≥K.

To estimate the complexity of decoding the overall code \({\mathcal {C}}_{1} \times \cdots \times {\mathcal {C}}_{K}\) using a joint sphere decoder, we follow [35] by using the notion of complexity exponent as a complexity measure.

Definition1 (Complexity exponent [35]).

Given the multiplexing gain r, let \({\mathcal {C}}_{r,k}\) be a lattice code consisting of (n_t×T) codeword matrices with \(|{{\mathcal {C}}_{r,k}}| \doteq \text {SNR}^{rT}\), k=1,…,K. Let \({\mathcal {D}}_{r}\) be a decoder for the overall code \({\mathcal {C}}_{r} = {\mathcal {C}}_{r,1} \times \cdots \times {\mathcal {C}}_{r,K}\), subject to a computational constraint N_max(r), in floating point operations (flops) per T channel uses, in the sense that after N_max(r) flops, the decoder \({\mathcal {D}}_{r}\) must simply terminate, potentially prematurely and before completing the task, thus declaring an error. We then say \({\mathcal {D}}_{r}\) achieves diversity order d(r) with complexity exponent c(r) if \({\mathcal {D}}_{r}\) achieves error probability \(P_{e} \doteq \text {SNR}^{-d(r)}\) using at most \(N_{\max }(r) \doteq \text {SNR}^{c(r)}\) (cf. (3),(4)) flops of computational reserves. □

The above definition means that in order to decode the code \({\mathcal {C}}_{r}\) using a joint sphere decoder, one does not have to decode every received signal matrix, especially when the communication channel is deeply faded. Instead, one can enforce a complexity constraint (also called a halting policy) at the sphere decoder, say at most N_max(r) flops of computational reserves. By choosing N_max(r) large enough such that the probability of any premature termination of the sphere decoder is asymptotically no larger than \(\text {SNR}^{-d^{*}_{n_{t},n_{r},K}(r)}\), the overall error probability at most \(2 \cdot \text {SNR}^{-d^{*}_{n_{t},n_{r},K}(r)}\), thereby achieving the same diversity \(d^{*}_{n_{t},n_{r},K}(r)\).

It was shown in [32, 36] that the complexity exponent for decoding the DMT optimal code [26] is given by

$$ {\small{\begin{aligned} {c}_{n_{t},n_{r},K}(r) &= K_{o} r(Kn_{t}-n_{r}) \cdot \textbf{1}\left(Kn_{t} > n_{r}\right) \\ &\quad + \sup\limits_{\underline{\mu} \in {\mathcal{B}}(r)} K_{o} n_{t}\sum_{i=1}^{v}\left(\frac{r}{n_{t}}-(1-\mu_{i})^{+}\right)^{+}, \end{aligned}}} $$

(6)

where 1(·) is the usual indicator function, v= min{Kn_t,n_r}, (x)⁺:= max{x,0} and

$$ {\begin{aligned} \mathcal{B}(r) \\ &=\left\{ \underline{\mu}=\; [\mu_{1} \ \cdots \mu_{v}]^{\top} \in {\mathbb R}^{v}: \begin{array}{l} \mu_{1} \geq \cdots \geq \mu_{v} \geq 0, \\ \sum_{i=1}^{v} \left(|{Kn_{t}-n_{r}}|+2i-1\right)\mu_{i} \leq d^{*}_{n_{t},n_{r},K}(r) \end{array} \right\}. \end{aligned}} $$

(7)

There is an intuitive explanation for the term K_or(Kn_t−n_r) in (6) when Kn_t>n_r. Recall that the component code \({\mathcal {C}}_{i}\) is taken from a certain subset of an algebraic lattice Λ_i of rank \(2 {n_{t}^{2}}K_{o}\). This means that each codeword matrix X_i of \({\mathcal {C}}_{i}\) is of the form \(X_{i} = \sum _{\ell =1}^{{n_{t}^{2}} K_{o}} x_{i, \ell } C_{i,\ell }\), where \(\{ C_{i,\ell }: \ell =1, \ldots, {n_{t}^{2}} K_{o}\}\) is a basis for Λ_i, and the x_i,ℓ are independent QAM symbols taken from a certain set \({\mathcal {A}} \subset {\mathbb Z}[\, \imath \,]\) of size \(\text {SNR}^{\frac {r}{n_{t}}}\), \(\, \imath \,=\sqrt {-1}\). Thus, we can rewrite (1) as

$$ {\small{\begin{aligned} Y = \sum_{i=1}^{K} \sum_{\ell=1}^{{n_{t}^{2}} K_{o}} H_{i} C_{i,\ell} x_{i,\ell} + W, \end{aligned}}} $$

(8)

or equivalently in a vector form

$$ \underline{y} = H \underline{x} + \underline{w}, $$

(9)

where \(\underline {x}=\left [x_{1,1}, \ldots x_{1,{n_{t}^{2}} K_{o}}, \ldots x_{K,{n_{t}^{2}} K_{o}}\right ]^{\top }\), \(\underline {y}\) is the vectorization of the matrix Y, and H is the corresponding matrix of size \(\left (n_{r} K_{o} n_{t} \times K {n_{t}^{2}} K_{o}\right)\) by (8). When decoding (9) using a sphere decoder, one first performs a QR-decomposition of the matrix H, say H=QR. If Kn_t>n_r, the matrix R is no longer upper triangular; it is a trapezoidal matrix with

$$K {n_{t}^{2}} K_{o}-n_{r} K_{o} n_{t}+1=K_{o} n_{t} \left(K n_{t} - n_{r} \right)+1 $$

nonzero entries in the bottom row. Hence, any sphere decoder for (9) must first resolve – perhaps by brute-force – the \(|{{\mathcal {A}}}|^{K_{o} n_{t} (K n_{t} - n_{r})}=\text {SNR}^{K_{o} r(Kn_{t}-n_{r})}\) ambiguities before processing the root of the sphere decoding tree. The number of ambiguities then forms the first term in (6).

Remark2.

A different definition of complexity exponent has appeared in [41], where Damen et al. studied the number of flops required by a sphere decoder to decode a fixed-rate space-time code at various finite SNR values. In particular, they defined the complexity exponent as the logarithm to base m of the number of flops required by a sphere decoder to complete its task, where m is the length of vector \(\underline {x}\) defined in (9). Below we highlight some of the major differences between Damens’ definition of complexity exponent and the one considered in this paper (cf. Definition 1).

• Damens’ definition focuses on a code with a fixed rate, and Definition 1 concerns more with the theoretical asymptote at high SNR regime when the rate scales linearly with log2SNR.

• Definition 1 considers the possibility of having a halting policy, while Damens’ definition requires the sphere decoder to complete its task at all channel realizations.

Remark3.

In [42] Damen et al. proposed to decode (9) by using GDFE-MMSE preprocessing followed by the sphere decoder when Kn_t>n_r, in hope of making the matrix R upper-triangular and avoiding the need of resolving the ambiguities. However, it can be seen from [32, 36] that at high SNR regime the matrix R – after MMSE-GDFE preprocessing – is ill-conditioned with K_on_t(Kn_t−n_r) number of singular values arbitrarily close to zero. This also explains the appearance of the first term in (6).

On the other hand, when the code has a fixed rate and operates in the low or moderate SNR regime, the MMSE-GDFE approach does offer a certain complexity reduction with a negligible performance loss, as the singular values of R are numerically well-behaved in general. Other approaches for further complexity reductions under such premises are also available in the literature. For instance, Barbero and Thompson [43] proposed a fixed-complexity sphere decoder, where the number of candidates to be searched at the i-th level of sphere decoding tree is at most n_i, thereby yielding a constant complexity \(\prod _{i} n_{i}\). Another way to reduce complexity is through the various orderings of singular values of R. A comprehensive study in this direction can be found for example in [44]. We shall emphasize that the complexity exponents simulated in [43, 44] are both based on Damens’ definition [41] (cf. Remark 2) because of the aforementioned premises.

In Fig. 3, we plot the complexity exponents for the sphere decoding of MAC DMT optimal codes \({{\mathcal {C}}}_{r}\) [26] when n_t=2, K=5, and n_r= 2, 10, 100, respectively. It can be seen that these codes can be efficiently decoded by sphere decoders only when n_r≫Kn_t. Such a requirement is often impossible in practice, particularly in heterogeneous storage networks, where nodes may have only a small number of antennas in use.

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Remark4.

In case of n_r≥Kn_t, it has been shown [45, 46] that the DMT optimal MIMO-MAC lattice codes can be decoded with sub-exponential complexity, i.e., having a complexity exponent asymptotically equal to 0, using the Lenstra-Lenstra-Lovász-based lattice reduction aided regularized lattice decoder. The decoder is a combination of GDFE-MMSE, lattice reduction and sphere decoding, and it has a vanishing gap of performance loss to the exact ML decoding as SNR approaches infinity.

In Section 2, we have seen that there is a fundamental difference between the channel for DSS repair transmission and the classical MIMO-MAC, in the sense that the former includes additional inter-helper communication links. Thus, the MIMO-MAC DMT (5) and the MIMO-MAC codes [26] are no longer optimal in scenarios such as DSS repair transmission. Moreover, due to these additional inter-helper channels, it is expected that the DSS repair transmission can have a higher optimal DMT than (5). This then calls for the design of new transmission schemes with good DMT performance for DSS repairing, which is the first design objective considered in this paper.

The second design objective comes from the observation of high decoding complexity of MIMO-MAC codes [26] in Fig. 3 when Kn_t>n_r. In a DSS, it is often possible that K is large, and n_r is relatively small and fixed. This then calls for the design of new transmission schemes that can yield efficiently sphere-decodable space-time codes avoiding the need to process the ambiguities by brute-force. Potentially, such an aim could be achieved by reducing the number of “active” helper nodes, i.e., reducing the effective value of K in (1), such that the average number of independent QAM symbols received by the repairing node at each channel use be no larger than n_r, as observed from (6).

In the subsequent sections, we will focus on the case of n_t=1 and K≫n_r, and we will provide three transmission schemes, each for a different configuration of the wireless DSS network and for a different design objective. The first scheme is given in Section 4 for the case of two receive antennas and an arbitrary number of helper nodes, each having one transmit antenna. It is based on a simple time sharing among pairs of helpers and is aimed at having a low sphere-decoding complexity at a cost of certain DMT performance-loss due to its neglect of existing inter-helper links. The DMT for this scheme falls between the simple time sharing DMT and optimal MIMO-MAC DMT (5).

Two more elaborate schemes will be presented in Sections 5 and 6, respectively, where we aim to improve the DMT performance at the possible cost⁸ of higher decoding complexity. These schemes take advantage of inter-helper channels and transform the overall DSS network into a series of relay networks, where the conventional half-duplex NAF protocol [37, 38] will be used. In particular, we will see that these schemes can outperform the MIMO-MAC DMT (5) at certain multiplexing gains, simply by exploiting inter-helper communications in the DSS.

We have seen in Section 2.2 that the existing state-of-the-art MIMO-MAC space-time codes [26] could incur an extremely high decoding complexity when the repairing node has only a few number of antennas. Thus, our major aim in this section is to provide a new transmission scheme that can yield space-time codes with reduced decoding complexity. In particular, we would like these potential codes to be efficiently sphere-decodable, by which we mean that the H matrix, when writing the channel input-output relation in a vector form (cf. (9)), has linearly independent columns with probability one.

Besides the desired property of being efficiently sphere-decodable, the complexity of the transmission schemes should also be considered. In other words, if we ignore the existence of inter-helper links (2), then the schemes for DSS repair transmission can be made relatively simple. These are the main objectives of Scheme 1.

With the above, the proposed scheme is the following. For each \(U=\{u_{1}, u_{2}\} \in {\mathcal {U}}\), only helper nodes u₁ and u₂ are allowed to transmit during the active period of U. This implies that the probability of helper node k transmitting equals \(\frac 2K\) for every \(k \in {\mathcal {K}}\). In order to achieve an average multiplexing gain r, each helper node k, when chosen according to U, i.e. k∈U, should actually transmit at a higher multiplexing gain \(\frac {Kr}{2}\). We summarize the above scheme below, and a pictorial description of Scheme 1 is given in Fig. 4.

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The following theorem is a straightforward consequence of [34].

In Fig. 5, we consider the case n_t=1, n_r=2 and K=10, and compare d₁(r) to \(d^{*}_{1,2,10}(r)\), which is the DMT corresponding to all 10 helper nodes transmitting simultaneously. The function d₀(r) is the DMT for the time-division multiple-access (TDMA)-based scheme, by which we mean that each helper node takes turns in an orthogonal manner to transmit information to the repairing node at multiplexing gain Kr. It can be seen that the first proposed scheme outperforms the TDMA scheme in terms of the DMT, and there is a considerable gap between d₁(r) and \(d^{*}_{1,2,10}(r)\). However, the comparison is unfair in the sense that in order to achieve \(d^{*}_{1,2,10}(r)\) the codes in [26] would require exponentially large computational reserves, or equivalently an exponentially long time, for decoding.

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when it is decoded using a sphere decoder with halting policies. In Fig. 6, we compare c₁(r) to the complexity exponent c_1,2,10(r) of the MIMO-MAC code given in [26] for the case n_t=1, n_r=2 and K=10. It can be clearly seen that the proposed scheme can yield a code with with a much lower decoding complexity.

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The aim of Scheme 1 presented in the previous section was to have a small decoding complexity, at a cost of certain DMT performance loss due to the neglect of inter-helper links in DSS repair-communication. In this section as well as the next, we will shift our focus to designing transmission schemes that take into account these inter-helper links and beat the DMT performance \(d^{*}_{n_{t}, n_{r}, K}(r)\).

Consider a DSS repair channel with K helpers, each having n_t=1 transmit antenna, and a repairing node with n_r receive antennas. To make good use of the inter-helper links, we interpret in Scheme 2 some of the links as links of a relay channel. More specifically, in this scheme each of the K helper nodes will take turns acting as the source in a cooperative relay network [37], while the remaining K−1 helper nodes play the role of relays helping the source to send information to the repairing node.

With the above, the proposed scheme is a modification of the NAF protocol [37, 38] for a cooperative relay network with K−1 relays. It consists of K phases, and each phase requires at least 2(K−1) channel uses. Thus, the total number of channel uses required by Scheme 2 is at least 2K(K−1).

Then, at channel use t=2i, node u_i joins node k and sends the amplified signal \(a_{u_{i}, k} r_{u_{i},2i-1}\) to the repairing node.

Since each helper node k is allowed to transmit its own message to the repairing node during the kth phase, its multiplexing gain must be increased to K·r in order to achieve the desired average multiplexing gain r. We now summarize the steps of Scheme 2 below. A pictorial description of Scheme 2 is given in Fig. 7.

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5.2 DMT achieved by Scheme 2 when n_r=1

When n_r=1, it can be seen that the DMT achieved by Scheme 2 is exactly the DMT for the NAF protocol derived by Azarian et al. [38] with K−1 relays and multiplexing gain Kr. Hence, the following result is immediate from [38].

Theorem2.

The DMT achieved by Scheme 2 when n_r=1 is the following

$$ d_{2}(r) \left|{\!~\!}_{n_{r}=1} = (1-Kr)^{+} + (K-1)(1-2Kr)^{+}\right.. $$

(24)

In Fig. 8, we plot the DMT performance achieved by this scheme for the case of K=10 helper nodes. We also include the base-line TDMA scheme for comparison. It can be seen that the proposed scheme has a better DMT performance than \(d^{*}_{n_{t}, n_{r}, K}(r)\) for \(r \leq \frac {1}{2K+1}=\frac {1}{21}\), due to the use of additional inter-helper links.

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5.3 Upper and lower bounds on d₂(r) with general n_r

Analyzing the outage probability (23) turns out to be very challenging in general when the repairing node has multiple antennas, i.e., n_r≥2. Almost all existing works such as [38, 47] consider only the case n_r=1. In [39] Yang and Belfiore investigated the DMT for the MIMO-NAF protocol and provided a lower bound for such DMT. Their result can be modified to yield a lower bound for d₂(r). We will comment more on that particular lower bound at the end of this subsection.

To provide bounds on the DMT d₂(r) for general values of n_r, let U be an (n_r×n_r) unitary matrix such that \( U \, \underline {h}_{1} =\; [\! \begin {array}{cccc} ||{\underline {h}_{1}}||&0&\cdots &0 \end {array} ]^{\top } := \underline {h}. \)

For H_i defined in (22), i=2,…,K, we get

$$ {\text{diag}}(U,U)H_{i} = \left[ \begin{array}{cc} U & \\ & U \end{array} \right] H_{i} = \left[ \begin{array}{cc} \underline{h} & \underline{0}\\ a_{i} g_{i} \underline{\ell}_{i} & \underline{h} \end{array} \right] = S_{i}, $$

(25)

where \(\underline {\ell }_{i} = U \underline {h}_{i}\) has the same probability density function as \(\underline {h}_{i}\), i=2,…,K. Let \( \Sigma _{i} := I_{n_{r}} + |{a_{i}}|^{2} \underline {h}_{i} \underline {h}_{i}^{\dag }\). Clearly, we have the following partial ordering for positive-definite matrices,

$$I_{n_{r}} \prec \Sigma_{i} \prec \left(1 + |{a_{i}}|^{2} ||{\underline{h}_{i}}||^{2} \right) I_{n_{r}} = \left(1 + |{a_{i}}|^{2} ||{\underline{\ell}_{i}}||^{2} \right) I_{n_{r}}, $$

which in turn implies \( \frac {1}{1 + |{a_{i}}|^{2} ||{\underline {\ell }_{i}}||^{2}} I_{n_{r}} \prec \Sigma _{i}^{-1} \prec I_{n_{r}} \). With the above, \(I(\underline {x};\underline {y} \mid H)\) can be upper bounded by

$$\begin{array}{*{20}l} &I(\underline{x}; \underline{y} \mid H)\\ & \leq \sum_{i=2}^{K} \log_{2} \det \left(I_{2} + \text{SNR} H_{i}^{\dag} H_{i} \right) \end{array} $$

(26)

$$\begin{array}{*{20}l} &= \sum_{i=2}^{K} \log_{2} \bigg[ \left(1 + \text{SNR} ||{\underline{h}}||^{2} \right)^{2}+ \text{SNR} |{a_{i} g_{i}}|^{2}||{\underline{\ell}_{i}}||^{2} \\ & \qquad \qquad \quad \; +\text{SNR}^{2} |{a_{i} g_{i}}|^{2}||{\underline{h}}||^{2} \sum_{j=2}^{n_{r}} |{\ell_{i,j}}|^{2} \bigg]. \end{array} $$

(27)

Similarly, set \(c_{i} = \frac {1}{1 + |{a_{i}}|^{2}||{\ell _{i}}||^{2}}\), and \(I(\underline {x}; \underline {y} \mid H)\) is lower bounded by

$$\begin{array}{*{20}l} & I(\underline{x}; \underline{y} \mid H) \\ & \geq \sum_{i=2}^{K} \log_{2} \det \left(I_{2} + \text{SNR} S_{i}^{\dag} \left[ \begin{array}{cc} I_{n_{r}} & \\ & c_{i} I_{n_{r}} \end{array} \right] S_{i} \right) \end{array} $$

(28)

$$\begin{array}{*{20}l} & = \sum_{i=2}^{K} \log_{2} \bigg[ 1 + (1+c_{i}) \text{SNR} ||{\underline{h}}||^{2} + c_{i} |{a_{i} g_{i}}|^{2} ||{\underline{\ell}_{i}}||^{2} \text{SNR} \\ & \quad+ c_{i} \text{SNR}^{2} ||{\underline{h}}||^{4} + {c_{i}^{2}} |{a_{i} g_{i}}|^{2} \text{SNR}^{2} ||{\underline{h}}||^{2} \sum_{j=2}^{n_{r}} |{\ell_{i,j}}|^{2} \bigg]. \end{array} $$

(29)

Equations (27) and (29) then yield the following theorem for bounding the DMT d₂(r) for Scheme 2.

Theorem3.

The DMT d₂(r)of Scheme 2 for a general number n_r≥1 of receive antennas at the repairing node has the following upper bound d_2,U(r) and lower bound d_2,L(r) :

$$ \begin{aligned} d_{2,U}(r) &:= \inf_{g} \ \sup_{b \leq g}\ \inf_{(\alpha, \beta_{1}, \beta_{2}) \in {\mathcal{A}}_{U}(r,b,g)} n_{r} \alpha + (K-1) \beta_{1}& \\ & \qquad +(n_{r}-1) (K-1) \beta_{2} + (K-1) g \end{aligned} $$

(30)

$$ \begin{aligned} d_{2,L}(r) := \inf_{g} \sup_{b \leq g} \inf_{(\alpha, \beta_{1}, \beta_{2}) \in {\mathcal{A}}_{L}(r,b,g)} n_{r} \alpha + (K-1) \beta_{1} \\ + (n_{r}-1) (K-1) \beta_{2} + (K-1) g \end{aligned} $$

(31)

where

$$ {{\begin{aligned} &\mathcal{A}_{U}(r,b,g) \\ &=\! \left\{ \alpha, \beta_{1}, \beta_{2} \in [0,1] : \max \left\{ \begin{array}{l} 2 (1-\alpha),\\ 1+b-g-\min\left\{ \beta_{1}, \beta_{2}\right\},\\ 2+b-g-\beta_{2} \end{array}\right\} \leq 2 K r\right\} \end{aligned}}} $$

(32)

and

$$ {{\begin{aligned} &\mathcal{A}_{L}(r,b,g) \\ &=\! \left\{ \alpha, \beta_{1}, \beta_{2}\in [0,1] : \max \left\{ \begin{array}{l} 1-\alpha,\\ 2-2\alpha-(b-\beta)^{+},\\ 1-\beta+b-g-(b-\beta)^{+},\\ 2-\alpha+b-g-\beta_{2}-2(b-\beta)^{+} \end{array} \right\}\!\leq 2K r \right\}. \end{aligned}}} $$

(33)

Proof.

Note that the random variables g_i’s are i.i.d., hence there is no need to distinguish them in (27) and (29) when deriving the DMT. The same holds also true for a_i, \(\underline {\ell _{i}}\), and its elements ℓ_i,j for i=2,…,K. Thus, we set \(|{a_{i}}|^{2} \doteq \text {SNR}^{b}\), \(|{g_{i}}|^{2} \doteq \text {SNR}^{-g}\), \(||{\underline {h}}||^{2} \doteq \text {SNR}^{-\alpha }\), \(||{\underline {\ell }_{i}}||^{2} \doteq \text {SNR}^{-\beta }\), and \(|{\ell _{i,j}}|^{2} \doteq \text {SNR}^{-\beta _{j}}\) with \(\beta = \min _{j=1,\ldots,n_{r}} \beta _{j}\). Moreover, we note that \(||{\underline {h}}||^{2}\) is a χ² random variable with 2n_r degrees of freedom, hence it contributes the term n_rα to (27). Each \(\underline {\ell }_{i}\) consists of n_r i.i.d. \(\mathbb {C}{\cal {N}}(0,1)\) complex random variables, and there is no need to distinguish ℓ_i,j for i=2,…,K and for j=2,…,n_r as can be seen from (27) and (29). Hence, we can set \(|{\ell _{i,j}}|^{2} \doteq \text {SNR}^{-\beta _{2}}\) for i=2,…,K and for j=2,…,n_r. Similarly, there is no need to distinguish ℓ_i,1 for i=2,…,K, hence we set \(|{\ell _{i,1}}|^{2} \doteq \text {SNR}^{-\beta _{1}}\) for i=2,…,K. Finally, note that |g_i|² and |ℓ_i,j|² are i.i.d. χ² random variables with two degrees of freedom. Plugging the above into (27) and (29) and applying the Laplace principle as in [27] yield the desired upper and lower bounds (30) and (31).

□

In Fig. 9, we plot the DMT bounds d_2,L(r) and d_2,U(r) of Scheme 2 as well as the DMT d^∗_1,2,10(r) with K=10 helper nodes, n_t=1 and n_r=2. While there is a gap between bounds d_2,L(r) and d_2,U(r) when the multiplexing gain r is small, it can be clearly seen that Scheme 2 can offer a better DMT performance than d^∗_1,2,10(r) when r is small. Regarding the sharpness of d_2,L(r) and d_2,U(r), let us focus on the case when r is approaching zero from the right, i.e., when r↓0. Note that there are nine SISO channels from helper node 1 to the remaining helper nodes, and the channel between node 1 and the repairing node is a (1×2) SIMO channel. Therefore, the communication to the repairing node would be in outage if the nine SISO channels and the (1×2) SIMO channel are all in deep fade, thereby yielding a maximal diversity order of 9+2=11. We therefore conclude that the upper bound d_2,U(r) can be further improved.

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As mentioned earlier, Yang and Belfiore ([39], Theorem 2) provided a lower bound on the DMT for MIMO-NAF protocol. Their bound can be modified to become a lower bound for d₂(r) and has the following form

$$ d_{\text{2,L,YB}}(r) = n_{r} \cdot (1-Kr)^{+} + (K-1) \cdot d_{\text{RP}}(2Kr), $$

(34)

where d_RP(r) is the DMT for the Rayleigh product channel \(\underline {h}_{i} \cdot g_{i}\), and an exact expression for d_RP(r) can be found in ([39], Proposition 1).

In Fig. 10, we compare our lower bound d_2,L(r) to the lower bound d_2,L,YB(r) for the case n_t=1, n_r=2 and K=10. It can be clearly seen that, in this case, our bound is shaper than the bound (34).

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5.4 Another upper bound on d₂(r) with general n_r

To obtain another upper bound on the instantaneous mutual information \(I(\underline {x}; \underline {y} \mid H)\), we consider the situation that the repairing node has further knowledge of r_i,t−1=g_ix_t−1+z_i when t=2,4,…,N and \(i=\frac t2 +1\). In this case, define

$$ \underline{y}'_{t} \ = \ \underline{h}_{1} x_{t} + w_{t}, \quad t=1,2,\ldots,N. $$

(35)

Writing \(\underline {y}_{t} = \underline {y}'_{t} + a_{i} \underline {h}_{i} r_{i,t-1}\) for t=2(i−1), it follows that

$$ {}I \left(\underline{x}; \underline{y} \mid H \right) \ \leq \ I \left(\underline{x};\ \underline{y}'_{1}, \ldots, \underline{y}'_{N}, r_{2,1}, r_{3,3}, \ldots, r_{K,N-1} \mid H \right), $$

(36)

and the upper bound has a much simpler expression than \(I \left (\underline {x}; \underline {y} \mid H\right) \). To see this, formulate the received vectors as

$$ \begin{aligned} \underline{y}_{U} := \left[ \begin{array}{c} \underline{y}'_{1}\\ \underline{y}'_{3}\\ \vdots\\ \underline{y}'_{N-1}\\ \underline{y}'_{2}\\ \vdots\\ \underline{y}'_{N}\\ r_{2,1}\\ \vdots\\ r_{K,N-1} \end{array} \right] \ = \ \underbrace{\left[ \begin{array}{ccccccc} \underline{h}_{1} & & & & & & \\ & \_{h}_{1} & & & & & \\ & & \ddots & & &&\\ & && \underline{h}_{1} &&&\\ & &&& \underline{h}_{1} & &\\ & &&&& \ddots & \\ & &&&&& \underline{h}_{1}\\ g_{2} & & & && \\ & \ddots && & &\\ && g_{K} & & &\end{array} \right]}_{H_{U}}\\ \left[\begin{array}{c} x_{1}\\ x_{3}\\ \vdots\\ x_{N-1}\\ x_{2}\\ x_{4}\\ \vdots\\ x_{N} \end{array} \right] + \left[ \begin{array}{c} \underline{w}_{1}\\ \underline{w}_{3}\\ \vdots\\ \underline{w}_{N-1}\\ \underline{w}_{2}\\ \vdots\\ \underline{w}_{N}\\ z_{2}\\ \vdots\\ z_{K} \end{array} \right]; \end{aligned} $$

(37)

then

$$ {\small{\begin{aligned} {}H_{U}^{\dag} H_{U} \ = \ \left[ \begin{array}{cccc} ||{\underline{h}_{1}}||^{2} +|{g_{2}}|^{2} && & \\ & \ddots & &\\ & & ||{\underline{h}_{1}}||^{2} +|{g_{K}}|^{2} & \\ & & & ||{\underline{h}_{1}}||^{2} I_{K-1} \end{array} \right]\,. \end{aligned}}} $$

(38)

This implies that

$$ {\small{\begin{aligned} &I \left(\underline{x}; \underline{y} \mid H \right)\\ & \leq I \left(\underline{x}; \underline{y}_{U} \mid H \right)\\ &= (K-1) \log_{2} (1+ \text{SNR} ||{\underline{h}_{1}}||^{2}) \\ &\quad+ \sum_{i=2}^{K} \log_{2} \left(1 + \text{SNR} \left(||{\underline{h}_{1}}||^{2} +|{g_{i}}|^{2}\right)\right). \end{aligned}}} $$

(39)

Hence, the outage probability for the second scheme is lower bounded by

$$\begin{array}{*{20}l} {\small{\begin{aligned} &\Pr \left\{\!H \!: \sup_{|{a_{i}}|^{2} \leq \frac{\text{SNR}}{1 + \text{SNR} |{g_{i}}|^{2}}} I \left(\underline{x}; \underline{y} \mid H \right)\! < 2 K(K-1) r \log_{2} \text{SNR}\right\}\\ & \geq \Pr \left\{ H: I \left(\underline{x}; \underline{y}_{U} \mid H \right) < 2K(K-1) r \log_{2} \text{SNR} \right\} \end{aligned}}} \end{array} $$

(40)

$$\begin{array}{*{20}l} \,\,\doteq \text{SNR}^{-d_{2,U'}(r)}. \end{array} $$

(41)

Theorem4.

The DMT d₂(r)for Scheme 2 for a general number n_r≥1 of receive antennas at the repairing node is upper bounded by

$$ {{\begin{aligned} {}d_{2,U'}(r) = \left\{ \begin{array}{ll} (n_{r}+K-1)(1-Kr)^{+}, & \text{if \(n_{r} \geq K-1\),}\\ 2n_{r}(1-Kr)^{+} + (K-1-n_{r}) (1-2Kr)^{+},& \text{if \(n_{r} \leq K-1\).} \end{array} \right. \end{aligned}}} $$

(42)

Proof.

Similar to the proof of Theorem 3, it is unnecessary to distinguish the random variables g_i in (39) for i=2,…,K when calculating the DMT. Thus, let \(||{\underline {h}_{1}}||^{2} = \text {SNR}^{-\alpha }\) and |g_i|²=SNR^−β. Note \(||{\underline {h}_{1}}||^{2}\) is a χ² random variable with 2n_r degrees of freedom and |g_i|² is a χ² random variable with 2 degrees of freedom. Plugging the above into (41) and applying the Laplace principle as in [27] gives

$$d_{2,U'} (r) = \inf_{{\mathcal{B}}(r)} n_{r} \alpha + (K-1) \beta, $$

where

$$\begin{array}{@{}rcl@{}} {\mathcal{B}}(r) &=& \left\{ \alpha, \beta \in [0,1] : 1-\alpha + \max\{1-\alpha, 1-\beta\} \ \leq 2K r \right\}\\ &=& \left\{ \alpha, \beta\in [0,1] : 2(1-\alpha) + (\alpha-\beta)^{+} \ \leq \ 2 K r \right\}. \end{array} $$

Solving the above optimization problem gives the desired result. □

In Fig. 11, we plot d_2,L(r), d_2,U(r), and \(\phantom {\dot {i}\!}d_{2,U'}(r)\) for the second proposed scheme with K=10 helper nodes, n_t=1 and n_r=2. It can be seen that \(\phantom {\dot {i}\!}d_{2,L}(r)=d_{2,U'}(r)\) for all values of r, hence we have \(\phantom {\dot {i}\!}d_{2}(r) = d_{2,L}(r)=d_{2,U'}(r)\) in this case.

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In the previous section, we presented a powerful scheme that makes a good use of the inter-helper links to improve the DMT performance of DSS repair transmission. The scheme allows one helper node to transmit information in each phase, and the remaining helper nodes are regarded as relays. Furthermore, we have introduced a novel technique that allows us to upper-bound the DMT for the NAF protocol in a cooperative relay network with multiple antennas at the repairing node. In this section, we will present our third scheme, which can be seen as an enhancement of Schemes 1 and 2 and can provide a further improvement on the DMT performance.

The third proposed scheme concerns the case n_t=1, n_r≥2 and K helper nodes. It allows L helper nodes, L≤ min{n_r,K−1}, to transmit simultaneously and non-cooperatively to the repairing node as well as to the remaining (K−L) helper nodes, which will function as relays⁹ in the network. To achieve an average multiplexing gain r, each of the selected L helper nodes must transmit at a higher multiplexing gain of \(\frac {K}{L} r\). In particular, we could later seek to improve the overall DMT performance by optimizing over the choices of L. Therefore, L can actually be a function of the multiplexing gain r.

Given L, the third scheme consists of \(\binom {K}{L}\) phases, one for each possible L-subset \({{\mathcal {L}}}=\{i_{1}, \ldots, i_{L}\}\) of \({{\mathcal {K}}}\), where \({{\mathcal {K}}}=\{1,2,\ldots,K\}\) is the set of helper nodes. The helper nodes in set \({{\mathcal {L}}}\) transmit simultaneously and non-cooperatively throughout the phase, which has a duration of N=2(K−L) channel uses. The remaining nodes in \({{\mathcal {K}}}\setminus {{\mathcal {L}}}=\{j_{1}, \ldots, j_{K-L}\}\) will function as relays following the NAF protocol. Details of this scheme are given as below, and a pictorial description of this scheme is given in Fig. 12.

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6.1 DMT analysis for Scheme 3

The communication channel deduced from Scheme 3 resembles the multiple-access relay channel (MARC), which was first introduced by Kramer and van Wijngaarden [49]. The DMTs for the two-user and single-relay MARC—in terms of our notation this means n_t=1, n_r=1, K=3 and L=2—using various protocols have been studied in the past. For instance, Azarian et al. [50] investigated the DMT for such MARC using the dynamic-decode-and-forward (DDF) strategy, and Yuksel and Erkip [51] focused on the compress-forward (CF) protocol. Furthermore, a protocol similar to Scheme 3 was proposed in [40] and was termed multiple-access amplify-and-forward (MAF), which is a variation of the NAF protocol. It was found in [40] that the MAF outperforms the DDF in the high multiplexing gain regime and the CF protocol [51] in the low multiplexing gain regime when n_t=1, n_r=1, K=3, and L=2. The MAF thus provides a nice balance between complexity and performance.

Scheme 3 considers a much more complicated scenario than the one in [40], with n_t=1, and general values of n_r, K and L≤ min{n_r,K−1}. To the best of our knowledge, the DMT analysis for the MAF protocol has never been taken to such complexity level. On the other hand, our novel bounding technique employed in the proof of Theorem 4 is extremely powerful and enables us to analyze the DMT for general MARC using the MAF protocol.

To this end, for any subset \({{\mathcal {U}}}=\left \{u_{1}, \ldots, u_{k}\right \} \subseteq {{\mathcal {L}}}\) of the selected helper nodes, let \({{\mathcal {E}}}_{{\mathcal {U}}}\) denote the event that helper nodes u₁,…,u_k are in outage. The probability for \({{\mathcal {E}}}_{{\mathcal {U}}}\) is given by

$${} {\small{\begin{aligned} &\Pr \left\{ {{\mathcal{E}}}_{{\mathcal{U}}} \right\} \\ &=\!\Pr \Big\{\!\left\{\underline{h}_{i}\right\}_{i=1}^{K}, \left\{g_{i,j}\right\}_{i,j=1}^{K} :\! \sup_{a_{j_{s}} \atop s=1,\ldots,K-L} \!I \big(\!\! \left\{ x_{u, t} \!: \!u \in {{\mathcal{U}}} \right\}_{t=1}^{N}; \underline{y}_{1}, \ldots, \underline{y}_{N} \big| \\ &\quad \left\{ x_{u, t} : u \in {{\mathcal{L}}} \setminus {{\mathcal{U}}}\right\}_{t=1}^{N}, \left\{\underline{h}_{i}\right\}_{i=1}^{K}, \left\{g_{i,j}\right\}_{i,j=1}^{K}\! \big) \!<\! \frac{NK}{L} r k \log_{2} \text{SNR}\! \Big\}, \end{aligned}}} $$

(46)

where N=2(K−L). The overall outage probability for the third proposed scheme with given L is

$$ {\small{\begin{aligned} {}P_{{\text{out}},3}(L,r) \ := \ \Pr\left\{ \bigcup_{{{\mathcal{U}}} \subseteq {{\mathcal{L}}}} {{\mathcal{E}}}_{{\mathcal{U}}} \right\} \doteq \ \max_{{{\mathcal{U}}} \subseteq {{\mathcal{L}}}} \Pr \left\{ {{\mathcal{E}}}_{{\mathcal{U}}} \right\} \ \doteq \ \text{SNR}^{-d_{3}(L,r)}. \end{aligned}}} $$

(47)

The technique introduced in Section 5.4 can be applied to yield the following upper bound on d₃(L,r).

Theorem5.

The DMT d₃(L,r) can be upper bounded as

$$ {\small{\begin{aligned} {}d_{3}(L,r) \leq d_{3,U}(L,r) &:= \min_{k=1,\ldots,L} \inf_{{\mathcal{A}}(L,k,r)} \sum_{i=1}^{k} [2i-1\,+\,(n_{r}-k)] \alpha_{i}\\ &\quad+ (K-L) \left[ \sum_{j=1}^{k} \beta_{j} +(n_{r}-k) \beta_{k+1} \right], \end{aligned}}} $$

(48)

where

$${} {{\begin{aligned} &\mathcal{A}(L,k,r) \\ &:= \left\{ \begin{array}{l} \alpha_{1}, \cdots \alpha_{k}, \beta_{1}, \cdots \beta_{k+1}: \\ \ \begin{array}{l} 1 \geq \alpha_{1} \geq \alpha_{2} \geq \cdots \geq \alpha_{k} \geq 0,\\ \alpha_{i} \geq \beta_{i} \geq 0, \quad i=1, 2, \ldots,k, \\ \beta_{k+1} \geq 0 \text{and } \beta_{k+1}=0 \text{if \(n_{r}=k\)},\\ \sum_{i=1}^{k} (1-\alpha_{i}) + \frac12 \max\left\{ \alpha_{1} -\beta_{1},\ldots,\alpha_{k}-\beta_{k}, \beta_{k+1} \right\} < \frac{Krk}{L} \end{array} \end{array}\! \right\}\!. \end{aligned}}} $$

(49)

Proof.

Given any \({\mathcal {U}}=\left \{u_{1}, \ldots, u_{k}\right \} \subseteq {\mathcal {L}}\) of selected helper nodes, we first reformulate the channel input-output relations (45) in matrix form. For the sake of notational convenience, we set \(\phantom {\dot {i}\!}r_{s}=r_{j_{s}, 2s-1}\), \(\phantom {\dot {i}\!}z_{s}=z_{j_{s}, 2s-1}\),

$${}\underline{x}_{t} :=\! \left[ \begin{array}{c} x_{u_{1},t}\\ \vdots\\ x_{u_{k},t} \end{array} \right], \,\,H_{\mathcal{U}} =\! \left[ \begin{array}{ccc} \underline{h}_{u_{1}} & \cdots & \underline{h}_{u_{k}} \end{array} \right], \text{and}\,\,\underline{g}_{s} = \left[ \begin{array}{c} g_{j_{s},u_{1}}\\ \vdots\\ g_{j_{s},u_{k}} \end{array} \right], $$

for \(j_{s} \in {{\mathcal {K}}}\setminus {{\mathcal {L}}}\) and s=1,…,(K−L). Following the same approach as in Section 5.4, we assume the repairing node has further knowledge of r_s for s=1,…,(K−L); therefore, it knows \(\underline {y}'_{t} \ = \ H_{\mathcal {U}} \underline {x}_{t} + \underline {w}_{t}\) for t=1,2,…,N. We then have

$$ {\small{\begin{aligned} {}\underbrace{\left[\! \begin{array}{c} \underline{y}'_{1}\\ \underline{y}'_{3}\\ \vdots\\ \underline{y}'_{N-1}\\ \underline{y}'_{2}\\ \vdots\\ \underline{y}'_{N}\\ r_{1}\\ \vdots\\ r_{K-L} \end{array}\!\right]}_{\underline{y}_{{\mathcal{U}}}} = \underbrace{\left[\!\! \begin{array}{cccccc} H_{{\mathcal{U}}} & & & & & \\ & \ddots & &&&\\ && H_{{\mathcal{U}}} &&&\\ &&& H_{{\mathcal{U}}} & &\\ &&&& \ddots & \\ &&&&& H_{{\mathcal{U}}}\\ \underline{g}_{1}^{\top} & & & &\\ & \ddots & & &\\ && \underline{g}_{{K-L}}^{\top} & &\end{array} \!\!\!\right]}_{H_{\text{eq}}} \underbrace{\left[\!\!\begin{array}{c} \underline{x}_{1}\\ \underline{x}_{3}\\ \vdots\\ \underline{x}_{N-1}\\ \underline{x}_{2}\\ \underline{x}_{4}\\ \vdots\\ \underline{x}_{N} \end{array}\!\! \right]}_{:= \underline{x}} + \left[ \begin{array}{c} \underline{w}_{1}\\ \underline{w}_{3}\\ \vdots\\ \underline{w}_{N-1}\\ \underline{w}_{2}\\ \vdots\\ \underline{w}_{N}\\ z_{1}\\ \vdots\\ z_{{K-L}} \end{array}\!\! \right]. \end{aligned}}} $$

(50)

It follows that

$$ {{\begin{aligned} &I \left(\left. \left\{ x_{u, t} : u \in {{\mathcal{U}}} \right\}_{t=1}^{N} ; \underline{y}_{1}, \ldots, \underline{y}_{N} \right| \left\{ x_{u, t} : u \in {{\mathcal{K}} }\setminus {\mathcal{U}} \right\}_{t=1}^{N},\right.\\ &\qquad\left. \{\underline{h}_{i}\}_{i=1}^{K}, \{g_{i,j}\}_{i,j=1}^{K}\right) \\ &\leq I \left(\underline{x}; \underline{y}_{\mathcal{U}} \mid H_{\text{eq}}\right) \\ &= \frac N2 \log_{2} \det \left(I_{k} + \text{SNR} H_{\mathcal{U}}^{\dag} H_{{\mathcal{U}}} \right)\\ &\quad+ \sum_{s=1}^{\frac N2} \log_{2} \det \left(I_{k} + \text{SNR} H_{{\mathcal{U}}}^{\dag} H_{{\mathcal{U}}} + \text{SNR} \underline{g}_{s}^{*} \underline{g}_{s}^{\top} \right). \end{aligned}}} $$

(51)

Let \(H_{{\mathcal {U}}}^{\dag } H_{{\mathcal {U}}}= E \Lambda E^{\dag }\) be the eigen-decomposition of \(H_{{\mathcal {U}}}^{\dag } H_{{\mathcal {U}}}\), where E is a (k×k) unitary matrix, Λ=diag(λ₁,⋯,λ_k), and 0<λ₁≤⋯≤λ_k are the nonzero ordered eigenvalues of \(H_{\mathcal {U}}^{\dag } H_{\mathcal {U}}\), since \(\text {rank}(H_{{\mathcal {U}}}^{\dag } H_{{\mathcal {U}}})=k\) with probability one. The instantaneous mutual information \(I \left (\underline {x}; \underline {y}_{{\mathcal {U}}} \mid H_{\text {eq}} \right) \) can be further simplified to

$$ {\small{\begin{aligned} &I \left(\underline{x}; \underline{y}_{{\mathcal{U}}} \mid H_{\text{eq}} \right)\\ &= N \sum_{s=1}^{k} \log_{2} (1 + \text{SNR} \lambda_{s}) \\ &\quad+ \sum_{s=1}^{\frac N2} \log_{2} \left(I_{k} + \left(I_{k} + \text{SNR} \Lambda \right)^{-1} \text{SNR} E^{\dag} \underline{g}_{s}^{*} \underline{g}_{s}^{\top} E \right)\\ &= N \sum_{s=1}^{k} \log_{2} (1 + \text{SNR} \lambda_{s})\\ & \quad+\! \sum_{s=1}^{\frac N2} \log_{2}\! \left(\!\!1 \,+\, \sum_{j=1}^{k} \frac{\text{SNR}}{1 + \text{SNR} \, \lambda_{j}} |{v_{s,j}}|^{2} \,+\, \sum_{j=k+1}^{n_{r}} \text{SNR} |{v_{s,j}}|^{2} \right)\!, \end{aligned}}} $$

(52)

where we have set \(\underline {v}_{s}=E^{\dag } \underline {g}_{s}^{*}\), which is a length- n_r random vector with i.i.d. \(\mathbb {C}{\cal {N}}(0,1)\) entries. It follows that

$$ {\small{\begin{aligned} {}\Pr\left\{ {{\mathcal{E}}}_{{\mathcal{U}}}\right\} \!\geq\! \Pr\left\{ I \left(\underline{x}; \underline{y}_{{\mathcal{U}}} \mid H_{\text{eq}} \!\right) \!<\! \frac{NK}{L} rk \log_{2} \text{SNR} \right\} \!\doteq \text{SNR}^{-d_{\mathcal{U}}(r)}. \end{aligned}}} $$

(53)

We set

• \(\lambda _{s} = \text {SNR}^{-\alpha _{s}}\) for \(s=1, \ldots, \frac N2\), with each α_s contributing the term (2s−1+(n_r−k))α_s to the overall diversity order.

• \(|{v_{s,j}}|^{2} = \text {SNR}^{-\beta _{s,j}} = \text {SNR}^{-\beta _{j}}\) for \(s=1, \ldots, \frac N2\) and j=1,…,k, since there is no need to distinguish v_s,j in these cases when applying the Laplace principle to (53). Each β_j, j=1,…,k, contributes the term \(\frac N2 \beta _{j}\) to the overall diversity order.

• \(|{v_{s,j}}|^{2} = \text {SNR}^{-\beta _{s,j}} = \text {SNR}^{-\beta _{k+1}}\) for \(s=1, \ldots, \frac N2\) and j=k+1,…,n_r, for the same reason. The factor β_k+1 contributes the term \(\frac N2 (n_{r}-k) \beta _{k+1}\) to the overall diversity order.

It follows from the above that

$$ {\small{\begin{aligned} {}d_{{\mathcal{U}}}(r)& := \inf_{{{\mathcal{A}}}'(L,k,r)} \sum_{i=1}^{k} [2i-1+(n_{r}-k)] \alpha_{i} + (K-L) \sum_{j=1}^{k} \beta_{j} \\ &\quad+ (K-L)(n_{r}-k) \beta_{k+1}\,, \end{aligned}}} $$

(54)

where

$${} {{\begin{aligned} &\mathcal{A}'(L,k,r) :=\\ & \left\{ \begin{array}{l} \alpha_{1}, \ldots, \alpha_{k}, \beta_{1}, \ldots, \beta_{k+1} \in {\mathbb R}:\\ \qquad \begin{array}{l} \alpha_{1} \geq \alpha_{2} \geq \cdots \geq \alpha_{k} \geq 0, \ \beta_{1}, \beta_{2}, \ldots, \beta_{k+1} \geq 0,\\ \beta_{k+1}=0 \text{if} n_{r}=k,\\ \sum_{i=1}^{k} (1-\alpha_{i})^{+} + \frac12 \max\big\{ (1-\beta_{j}- (1-\alpha_{j})^{+})^{+}, \ldots,\\ \qquad\qquad(1-\beta_{k}- (1-\alpha_{k})^{+})^{+}, \beta_{k+1} \big\} < \frac{Krk}{L} \end{array} \end{array} \right\} \end{aligned}}} $$

(55)

Finally, the upper bound d_3,U(L,r) is obtained after minimizing \(d_{{\mathcal {U}}}(r)\) for all possible subsets \({{\mathcal {U}}} \subseteq {{\mathcal {L}}}\) (or equivalently for all k=1,…,L) and after simplifying the constraints in (55). □

By optimizing over all possible L=1,2,…, min{n_r,K−1} for d_3,U(r), we obtain an upper bound on the DMT performance for the third scheme.

Corollary6.

The DMT performance for Scheme 3 is upper bounded by

$$ d_{3,U}(r) := \max_{L=1, \ldots, \min\{K-1,n_{r}\}} d_{3,U}(L,r). $$

(56)

In Fig. 13, we illustrate the overall picture for the case of n_t=1, n_r=2, and K=10. While we do not yet have a lower bound or a tight DMT result, we believe that Scheme 3 is indeed likely to be superior¹⁰ to all other schemes presented in this paper, namely the TDMA scheme, Schemes 1 and 2.

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In particular, we note that the DMT upper bound for Scheme 3 achieves the maximal possible multiplexing gain of \(\frac {2}{10}=0.2\), which is the same as the TDMA scheme and the MIMO-MAC. Such possibility for the optimality of Scheme 3 turns out to be generally true, at least from the viewpoint of the upper bound (51). To see this, note that by (51), we have

$$\begin{array}{@{}rcl@{}} &&\mathbb{E} \left[ \frac{I \left(\underline{x}; \underline{y}_{\mathcal{L}} \vert H_{\text{eq}} \right)}{N \log_{2} \text{SNR}}\right]\\ &&\;= \frac{\mathbb{E} \frac 12 \log_{2} \det \left(I_{L} + \text{SNR} H_{\mathcal{L}}^{\dag} H_{\mathcal{L}} \right)+ \frac12 \mathbb{E} \log_{2} \det \left(I_{L} + \text{SNR} H_{\mathcal{L}}^{\dag} H_{\mathcal{L}} + \text{SNR} \underline{g}_{1}^{*} \underline{g}_{1}^{\top} \right)}{\log_{2} \text{SNR}} \\ &&\;= \frac12 \min\{L, n_{r}\} + \frac12 \min\{L, n_{r}+1\} + o(1), \end{array} $$

(57)

as SNR→∞, where the last equality follows from the asymptotic analysis of the degrees-of-freedom (DoF) for the MIMO channel [27, 52] and from the fact that \(H_{{\mathcal {L}}}\) is a channel matrix of size (n_r×L), and \(H_{{\mathcal {L}}}^{\dag } H_{{\mathcal {L}}} + \underline {g}_{1}^{*} \underline {g}_{1}^{\top } = FF^{\dag }\) with \(F=\;[H_{{\mathcal {L}}}^{\dag }\ \underline {g}_{1}^{*}]\) is a matrix of size (L×(n_r+1)). Eq. 57 shows that the channel capacity resulting from Scheme 3 equals L· log2SNR+O(log2SNR) in high SNR regime for L≤ min{n_r,K−1}, and such an amount of capacity is shared by the L selected helper nodes. In other words, each selected helper node gets 1· log2SNR+O(log2SNR) bits per channel use as the maximal achievable transmission rate. Note that in Scheme 3 the selected helper node must transmit at a higher multiplexing gain \(\frac {K}{L} r\) such that the average multiplexing gain equals r. This then implies

$$ \frac{K}{L} r \log_{2} \text{SNR} \leq 1 \cdot \log_{2} \text{SNR} + o(\log_{2}\text{SNR}), $$

(58)

i.e., \(r \leq \frac LK\). Now, with L=n_r<K we see that Scheme 3 achieves the maximal possible multiplexing gain of \(\frac {n_{r}}{K}\) for each helper node, same as MIMO-MAC [28], where the maximal possible multiplexing gain is given by \(\frac {\min \{Kn_{t}, n_{r}\}}{K} = \frac {n_{r}}{K}\).

¹ Device-to-device (D2D) communication networks provide one such example, see e.g. [53, 54].

² Preliminary results related to this work were reported in the Global Wireless Summit 2014 GWS’14 [18] (invited abstract which is considered a preprint), 21st International Symposium on Mathematical Theory of Networks and Systems MTNS’14 [19] (short invited abstract, Scheme 1), and 2014 International Symposium on Information Theory and Its Applications (ISITA) [20] (Schemes 1–3, now combined to Scheme 2). We point out that the numbering of the schemes has been changed so that the schemes previously called 2 and 3 [20] have been combined to Scheme 2, and the new scheme is hence now called Scheme 3 and has not appeared anywhere before. This paper extends the results by additional proofs for the bounds related to Scheme 2, and with a completely new scheme, Scheme 3, that improves upon the other schemes.

³ By efficiently sphere-decodable space-time code we mean that the code can be sphere-decoded without the need of performing an exhaustive search for part of the symbols before starting processing the root of a sphere-decoding tree. See discussions in Section 2.2.

⁴ By half-duplex we mean each node can choose to either transmit or receive, but not both at the same time.

⁵ Such a comparison might not seem fair to some readers as (5) assumes no inter-helper links. However, the DMT (5) is the best DMT result that can be found in the related literature.

⁶ This latter condition might seem unrealistic in certain (logical) distributed storage codes. However, it would be extremely difficult to determine the mutual information between the helpers and the repairing node if one takes into account the shared information among helper nodes.

⁷ A lattice is a discrete abelian subgroup of a real or complex vector space, and its rank is given by its rank as a module over \({\mathbb Z}\). By an algebraic lattice we refer to one constructed from a number field extension or a division algebra, see e.g. [29].

⁸ It is unfortunate that measuring the exact complexity exponents for these schemes is extremely complicated, and we are unable to complete the task in this paper. Nevertheless, it can still be seen that these schemes can yield efficiently sphere-decodable space-time codes without the need of resolving ambiguities when processing the sphere-decoding tree.

⁹ Here we have implicitly assumed K−L≥1 such that at least one helper node will function as a relay.

¹⁰ Cf. the corresponding upper bound \(\phantom {\dot {i}\!}d_{2,U'}(r)=d_{2}(r)\) for Scheme 3 that turned out to be tight.