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).
Let \({\mathcal {K}}=\{1,2,\ldots,K\}\) denote the set of K helper nodes. Given \(k \in {\mathcal {K}}\), the scheme is at the kth phase, and helper node k acts as the source of a relay network. The remaining helper nodes \({\mathcal {R}}_{k}:= {\mathcal {K}}\setminus \{k\}=\{u_{1}, \ldots, u_{K-1}\}\) are the relays. At the tth channel use of the kth phase, t=1,2,…,2(K−1), node k broadcasts a signal x
k,t
, subject to the power constraint \(\mathbb {E} |{x_{k,t}}|^{2} \leq \text {SNR}\), to all nodes in \({\mathcal {R}}_{k}\) as well as to the repairing node. Due to the half-duplex assumption in Section 2, the nodes in \({\mathcal {R}}_{k}\) can either receive or transmit, but not both at the same time. Therefore, the behavior of each node \(u_{i} \in {{\mathcal {R}}}_{k}\) is set such that it receives the signal from node k when t=2i−1 and transmits to the repairing node when t=2i. More specifically, the signal received by node u
i
at t=2i−1 is given by
$$ r_{u_{i},2i-1} = g_{u_{i}, k} x_{k,2i-1} + z_{u_{i}, k, 2i-1}, $$
(13)
where \(g_{u_{i}, k}\) and \(z_{u_{i}, k, 2i-1}\) are i.i.d. \(\mathbb {C}{\cal {N}}(0,1)\) random variables representing the channel gain from node k to node u
i
and the additive noise, respectively, as defined in (2). Node u
i
then amplifies the signal \(r_{u_{i},2i-1}\) with an amplification factor \(a_{u_{i}, k}\) set such that
$$ {\small{\begin{aligned} \mathbb{E} |{a_{u_{i}, k} r_{u_{i},2i-1}}|^{2} \leq \text{SNR}, \end{aligned}}} $$
(14)
where the expectation is taken with respect to x
k,2i−1 and \(z_{u_{i}, k, 2i-1}\), since \(g_{u_{i}, k}\) is already known to node u
i
. Equivalently, we have
$$ {\small{\begin{aligned} |{a_{u_{i}, k}}|^{2} \leq \frac{\text{SNR}}{1 + \text{SNR} |{g_{u_{i}, k}}|^{2}}. \end{aligned}}} $$
(15)
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.
DMT achieved by Scheme 2
Note firstly that by the symmetry among the phases of Scheme 2, it suffices to analyze the DMT achieved within the first phase, i.e., for k=1, where the helper node 1 acts as the source, and the remaining helper nodes are relays. Thus, for notational convenience, we will henceforth drop the subindex k.
Set N=2(K−1), and let x
t
be a \(\mathbb {C}{\cal {N}}(0,\text {SNR})\) random variable, representing the signal sent by helper node 1 at time instance t for t=1,2,…,N. Then, the signal received by the repairing node at the tth channel use is
$$ {}\underline{y}_{t} =\! \left\{ \begin{array}{ll} \underline{h}_{1} x_{t} + \underline{w}_{t}, & t \; \text{odd,}\\ \underline{h}_{1} x_{t} + a_{i} \underline{h}_{i} \left(g_{i} x_{t-1} + z_{i} \right) + \underline{w}_{t}, & t \; \text{even and}\; i=\frac{t}{2} \,+\,1, \end{array} \right. $$
(19)
where g
i
and z
i
’s are i.i.d. \(\mathbb {C}{\cal {N}}(0,1)\) random variables obtained by re-indexing the corresponding variables in (16) for notational convenience. The amplification factor \(a_{i} \in \mathbb {R}^{+}\), i=2,…,K, is set such that
$$|{a_{i}}|^{2} \leq \frac{\text{SNR}}{1 + \text{SNR} |{g_{i}}|^{2}}. $$
We can equivalently reformulate the received vectors \(\underline {y}_{t}\) in (19) in matrix form, as
$$\begin{array}{@{}rcl@{}} \underline{y} &=& \left[ \begin{array}{cccc} \underline{y}_{1}&\underline{y}_{2}&\cdots&\underline{y}_{N} \end{array} \right]^{\top} \\ &=&\!\!\! \underbrace{\left[\!\!\! \begin{array}{ccccc} \underline{h}_{1} & \underline{0} & \cdots & \underline{0} & \underline{0}\\ a_{2} g_{2} \underline{h}_{2} & \underline{h}_{1} & \cdots & \underline{0} & \underline{0}\\ \vdots & \vdots & \ddots & \vdots & \vdots \\ \underline{0} & \underline{0} &\cdots & \underline{h}_{1} & \underline{0} \\ \underline{0} & \underline{0} &\cdots & a_{K}g_{K} \underline{h}_{K} & \underline{h}_{1} \end{array}\!\!\!\right]}_{H} \underbrace{\left[\!\!\! \begin{array}{c} x_{1}\\ x_{2}\\ \vdots\\ x_{N-1}\\ x_{N} \end{array} \!\!\!\!\right]}_{\underline{x}}\,+\,\!\underbrace{\left[\!\!\! \begin{array}{c} \underline{w}_{1}\\ a_{2} z_{2} \underline{h}_{2} + \underline{w}_{2}\\ \underline{w}_{3}\\ a_{3} z_{3} \underline{h}_{3} + \underline{w}_{4}\\ \vdots\\ \underline{w}_{N-1}\\ a_{K} z_{K} \underline{h}_{K} + \underline{w}_{N} \end{array}\!\! \right]}_{\underline{v}}\!.\\ \end{array} $$
(20)
Given H, the instantaneous mutual information between the transmitted signal \(\underline {x}\) and the received signal \(\underline {y}\) is
$$\begin{array}{@{}rcl@{}} I(\underline{x}; \underline{y} \mid H) &=& \log_{2} \det \left(K_{v} + \text{SNR} H H^{\dag} \right) - \log_{2} \det(K_{v}) \\ &=& \sum_{i=2}^{K} \log_{2} \det \left(I_{2} + \text{SNR} H_{i}^{\dag} K_{i}^{-1} H_{i} \right), \end{array} $$
(21)
where
$$ {\small{\begin{aligned} {}K_{v} \,=\, \mathbb{E} (\underline{v}\, \underline{v}^{\dag}), \ H_{i} \,=\, \left[\! \begin{array}{cc} \underline{h}_{1} & 0\\ a_{i} g_{i} \underline{h}_{i} & \underline{h}_{1} \end{array}\! \right], \text{and}\; K_{i} \,=\, \left[\! \begin{array}{cc} I_{n_{r}} & \\ & I_{n_{r}} + |{a_{i}}|^{2} \underline{h}_{i} \underline{h}_{i}^{\dag} \end{array}\! \right]. \end{aligned}}} $$
(22)
Thus, the outage probability for Scheme 2 is given by
$$ {\begin{aligned} \Pr \left\{ H : \sup_{|{a_{i}}|^{2} \leq \frac{\text{SNR}}{1 + \text{SNR} |{g_{i}}|^{2}} } I \left(\underline{x}; \underline{y} \mid H \right) < 2 K(K-1) r \log_{2} \text{SNR}\right\} \doteq \text{SNR}^{-d_{2}(r)}, \end{aligned}} $$
(23)
where the target information rate 2K(K−1)r log2SNR arises from the facts that
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].
Theorem
2.
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.
Upper and lower bounds on d
2(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 d
2(r). We will comment more on that particular lower bound at the end of this subsection.
To provide bounds on the DMT d
2(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 d
2(r) for Scheme 2.
Theorem
3.
The DMT d
2(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 χ
2 random variable with 2n
r
degrees of freedom, hence it contributes the term n
r
α 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. χ
2 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.
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 d
2(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).
Another upper bound on d
2(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
i
x
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)
Theorem
4.
The DMT d
2(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 χ
2 random variable with 2n
r
degrees of freedom and |g
i
|2 is a χ
2 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.
Remarks on the complexity exponents of Scheme 2
Determining the complexity exponents of the second scheme requires much more effort than determining the DMT. At least two major difficulties must be resolved before any identification of complexity exponents is possible. Notice that the notion of complexity exponents resides in an actual construction of space-time codes for the scheme, and that the complexity exponents can vary from one code to another. Codes with a smaller complexity exponent are more favorable in practice, provided that the codes are optimal in the DMT sense, i.e., achieve the DMT d
2(r). Therefore, we have to at least identify a space-time code for Scheme 2 first. In [39], Yang and Belfiore provided a systematic construction of space-time codes that is approximately universal [48] for NAF-based cooperative relay communications. It is certainly possible to adapt their construction to the transmission using Scheme 2.
The second issue complicating the investigation of complexity exponents arises from the need of an exact characterization of eigenvalues of the matrices \(H_{i}^{\dag } K_{i}^{-1} H_{i}\) for i=2,…,K, appearing in (21). Determining these eigenvalues is particularly difficult. It is in fact the main reason preventing us from obtaining an exact expression for d
2(r) in previous subsections, and we are only able to provide bounds on d
2(r) in this paper.
Nevertheless, it can be seen from (20) that the equivalent channel matrix H is of size (N
n
r
×N) and has linearly independent columns with probability 1. This implies that when applying a sphere decoder to decode the codes—for instance, the code constructed by Yang and Belfiore [39]—transmitted using Scheme 2, the QR decomposition of the matrix H would result in an upper triangular matrix R; hence, there is no ambiguity to be resolved prior to processing the root of the sphere decoding tree. Therefore, the code must be efficiently sphere decodable.