Cooperative diversity

In the direct scheme, the destination decodes the data using the signal received from the source node on the first phase where the second phase transmission is omitted so that the relay node is not involved in transmission. The decoding signal received from the source node is written as:

${\displaystyle r_{d,s}=h_{d,s}x_{s}+n_{d,s}\quad }$

While the advantage of the direct scheme is its simplicity in terms of the decoding processing, the received signal power can be severely low if the distance between the source node and the destination node is large. Thus, in the following we consider non-cooperative scheme which exploits signal relaying to improve the signal quality.

In the non-cooperative scheme, the destination decodes the data using the signal received from the relay on the second phase, which results in the signal power boosting gain. The signal received from the relay node which retransmits the signal received from the source node is written as:

${\displaystyle r_{d,r}=h_{d,r}r_{r,s}+n_{d,r}=h_{d,r}h_{r,s}x_{s}+h_{d,r}n_{r,s}+n_{d,r}\quad }$

where ${\displaystyle h_{d,r}}$ is the channel from the relay to the destination nodes and ${\displaystyle n_{r,s}}$ is the noise signal added to ${\displaystyle h_{d,r}}$.

The reliability of decoding can be low since the degree of freedom is not increased by signal relaying. There is no increase in the diversity order since this scheme exploits only the relayed signal and the direct signal from the source node is either not available or is not accounted for. When we can take advantage of such a signal and increase in diversity order results. Thus, in the following we consider the cooperative scheme which decodes the combined signal of both the direct and relayed signals.

For cooperative decoding, the destination node combines two signals received from the source and the relay nodes which results in the diversity advantage. The whole received signal vector at the destination node can be modeled as:

${\displaystyle \mathbf {r} =[r_{d,s}\quad r_{d,r}]^{T}=[h_{d,s}\quad h_{d,r}h_{r,s}]^{T}x_{s}+\left[1\quad {\sqrt {|h_{d,r}|^{2}+1}}\right]^{T}n_{d}=\mathbf {h} x_{s}+\mathbf {q} n_{d}}$

where ${\displaystyle r_{d,s}}$ and ${\displaystyle r_{d,r}}$ are the signals received at the destination node from the source and relay nodes, respectively. As a linear decoding technique, the destination combines elements of the received signal vector as follows:

${\displaystyle y=\mathbf {w} ^{H}\mathbf {r} }$

where ${\displaystyle \mathbf {w} }$ is the linear combining weight which can be obtained to maximize signal-to-noise ratio (SNR) of the combined signals subject to given the complexity level of the weight calculation.

Adaptive scheme selects one of the three modes described above which are the direct, the non-cooperative, and the cooperative schemes relying on the network channel state information and other network parameters.

Using the max-flow min-cut theorem yields the upper bound of full duplex relaying

${\displaystyle C^{+}=\max _{f(X_{1},X_{2})}\min\{I(X_{1};Y_{2},Y_{3}|X_{2}),I(X_{1},X_{2};Y_{3})\}}$

where ${\displaystyle X_{1}}$ and ${\displaystyle X_{2}}$ are transmit information at the source node and the relay node respectively and ${\displaystyle Y_{2}}$ and ${\displaystyle Y_{3}}$ are received information at the relay node and the destination node respectively. Note that the max-flow min-cut theorem states that the maximum amount of flow is equal to the capacity of a minimum cut, i.e., dictated by its bottleneck. The capacity of the broadcast channel from ${\displaystyle X_{1}}$ to ${\displaystyle Y_{2}}$ and ${\displaystyle Y_{3}}$ with given ${\displaystyle X_{2}}$ is

${\displaystyle \max _{f(X_{1},X_{2})}I(X_{1};Y_{2},Y_{3}|X_{2})={\frac {1}{2}}\log(1+(1-\beta )(c_{21}^{2}+c_{31}^{2})P_{1})}$

while the capacity of the multiple access channel from ${\displaystyle X_{1}}$ and ${\displaystyle X_{2}}$ to ${\displaystyle Y_{3}}$ is

${\displaystyle \max _{f(X_{1},X_{2})}I(X_{2},X_{2};Y_{3})={\frac {1}{2}}\log(1+c_{31}^{2}P_{1}+c_{32}^{2}P_{2}+2{\sqrt {\beta c_{31}^{2}c_{32}^{2}P_{1}P_{2}}})}$

where ${\displaystyle \beta }$ is the amount of correlation between ${\displaystyle X_{1}}$ and ${\displaystyle X_{2}}$. Note that ${\displaystyle X_{2}}$ copies some part of ${\displaystyle X_{1}}$ for cooperative relaying capability. Using cooperative relaying capability at the relay node improves the performance of reception at the destination node. Thus, the upper bound is rewritten as

${\displaystyle C^{+}=\max _{0\leq \beta \leq 1}\min \left\{{\frac {1}{2}}\log(1+(1-\beta )(c_{21}^{2}+c_{31}^{2})P_{1}),{\frac {1}{2}}\log(1+c_{31}^{2}P_{1}+c_{32}^{2}P_{2}+2{\sqrt {\beta c_{31}^{2}c_{32}^{2}P_{1}P_{2}}})\right\}}$

Using a relay which decodes and forwards its captured signal yields the achievable rate as follows:

${\displaystyle R_{1}=\max _{f(X_{1},X_{2})}\min\{I(X_{1};Y_{2}|X_{2}),I(X_{1},X_{2};Y_{3})\}}$

where the broadcast channel is reduced to the point-to-point channel because of decoding at the relay node, i.e., ${\displaystyle I(X_{1};Y_{2},Y_{3}|X_{2})}$ is reduced to ${\displaystyle I(X_{1};Y_{2}|X_{2})}$. The capacity of the reduced broadcast channel is

${\displaystyle \max _{f(X_{1},X_{2})}I(X_{1};Y_{2}|X_{2})={\frac {1}{2}}\log(1+(1-\beta )c_{21}^{2}P_{1}).}$

Thus, the achievable rate is rewritten as

${\displaystyle R_{1}=\max _{0\leq \beta \leq 1}\min \left\{{\frac {1}{2}}\log(1+(1-\beta )c_{21}^{2}P_{1}),{\frac {1}{2}}\log(1+c_{31}^{2}P_{1}+c_{32}^{2}P_{2}+2{\sqrt {\beta c_{31}^{2}c_{32}^{2}P_{1}P_{2}}})\right\}}$

The capacity of the TD relay channel is upper-bounded by

${\displaystyle C^{+}=\max _{0\leq \beta \leq 1}\min\{C_{1}^{+}(\beta ),C_{2}^{+}(\beta )\}}$

with

${\displaystyle C_{1}^{+}(\beta )={\frac {\alpha }{2}}\log \left(1+(c_{31}^{2}+c_{21}^{2})P_{1}^{(1)}\right)+{\frac {1-\alpha }{2}}\log \left(1+(1-\beta )c_{31}^{2}P_{1}^{(2)}\right)}$

${\displaystyle C_{2}^{+}(\beta )={\frac {\alpha }{2}}\log \left(1+c_{31}^{2}P_{1}^{(1)}\right)+{\frac {1-\alpha }{2}}\log \left(1+c_{31}^{2}P_{1}^{(2)}+c_{32}^{2}P_{2}+2{\sqrt {\beta C_{31}^{2}P_{1}^{(2)}C_{32}^{2}P_{2}}}\right)}$

• 3GPP long term evolution (LTE) coordinated multipoint transmission/reception (CoMP), making it possible to increase the data rate to and from a mobile situated in the overlap of several base stations.
• 5G
• Mesh network
• Mobile ad hoc network (MANet)
• Wireless mesh network
• Wireless ad hoc network

• Cooperative wireless communications
• Cooperative MIMO
• Diversity schemes
• Dynamic Single Frequency Networks (DSFN)
• Soft handover
• Space–time code
• Multiple-input multiple-output communications (MIMO)
• Multi-user MIMO
• Diversity combining
• Transmit diversity
• Diversity gain