**Cooperative diversity** is a virtual multiple antenna techniques which exploits user diversity by allowing for the destination node to decode the multiple received signal from the relayed and direct nodes in wireless multihop networks. Conventional single hop wireless systems use direct transmission where a receiver node decodes information only based on the received signal from the source node while regarding the relayed signal as interference, whereas the cooperative diversity considers the other signal as contribution. That is, cooperative diversity allows the destination node to decode the information from the combination of two differently received signals from the source and relay nodes. Hence, it can be seen that cooperative diversity is an virtual antenna diversity that uses distributed antennas belonging to different nodes in a wireless network. Note that user cooperation is an alternative definition of cooperative diversity. In user cooperation, we consider an additional fact that each user relays the other user's signal while cooperative diversity can be also achieved by multi-hop relay networking systems.

## Relaying Strategies Edit

Cooperative diversity can be implemented by a few relaying strategies such as the amplify-and-forward, decode-and-forward, and compress-and-forward relaying strategies.### Amplify-and-forward Edit

The amplify-and-forward strategy allows the relay station amplifies the received signal from the source node and forwards to the destination station

### Decode-and-forward Edit

The decode-and-forward strategy allows the relay station decodes the received signal from the source node, re-encodes and forwards to the destination station

### Compress-and-forward Edit

The compress-and-forward strategy allows the relay station compress the received signal from the source node and forwards to the destination without decoding the signal where Wyner-Ziv coding can be used for optimal compression.

## System model Edit

We consider a wireless relay system that consists of source, relay and destination nodes. It is assumed that the channel is in a half-duplex, orthogonal and amplify-and-forward relaying mode. Differently to the conventional direct transmission system, we exploit a time division relaying function where this system can deliver information with two temporal phases.

On the first phase, the source node broadcasts information $ x_{s} $ toward both the destination and the relay nodes. The received signal at the destination and the relay nodes are respectively written as:

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

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

On the second phase, the relay can transmit its received signal to the destination node except the direct transmission mode.

## Signal Decoding Edit

We introduce four schemes to decode the signal at the destination node which are the direct scheme, the non-cooperative scheme, the cooperative scheme and the adaptive scheme. Except the direct scheme, the destination node uses the relayed signal.

### Direct Scheme Edit

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:

- $ 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.

### Non-cooperative Scheme Edit

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:

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

The reliability of the decoding can be low since the number of degree of freedom is not increased by signal relaying. the first order since this scheme exploit only the relayed signal. Thus, in the following we consider the cooperative scheme which decodes the combined signal of both the direct and relayed signals.

### Cooperative Scheme Edit

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:

- $ \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 $ r_{d,s} $ and $ 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:

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

where $ \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 Edit

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.

## Trade-off Edit

It is noteworthy that cooperative diversity can increase the diversity gain at the cost of lossing the wireless resource such as frequency, time and power resources for the relaying phase. Wireless resources are wasted since the relay node uses wireless resources to relay the signal from the source to the destination node. Hence, it is important to remark that there is trade-off between the diversity gain and the waste of the spectrum resource in cooperative diversity.

## Channel Capacity of Cooperative Diversity Edit

In June 2005, A. Høst-Madsen published a paper in-depth analyzing the channel capacity of the cooperative relay network^{[1]}.

We assume that the channel from the source node to the relay node, from the source node to the destination node, and from the relay node to the destination node are $ c_{21} e^{j\varphi_{21}},c_{31} e^{j\varphi_{31}},c_{32} e^{j\varphi_{32}} $ where the source node, the relay node, and the destination node are denoted node 1, node 2, and node 3, subsequently.

### The capacity of cooperative relay channelsEdit

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

- $ 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 $ X_1 $ and $ X_2 $ are transmit information at the source node and the relay node respectively and $ Y_2 $ and $ 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 $ X_1 $ to $ Y_2 $ and $ Y_3 $ with given $ X_2 $ is

- $ \max_{f(X_1,X_2)} I(X_1;Y_2,Y_3|X_2) = \frac{1}{2} \log(1 + (1 - \beta) (c^2_{21} + c^2_{31})P_1 ) $

while the capacity of the multiple access channel from $ X_1 $ and $ X_2 $ to $ Y_3 $ is

- $ \max_{f(X_1,X_2)} I(X_2,X_2;Y_3) = \frac{1}{2} \log(1 + c^2_{31} P_1 + c^2_{32} P_2 + 2 \sqrt{ \beta c^2_{31} c^2_{32} P_1 P_2}) $

where $ \beta $ is the amount of correlation between $ X_1 $ and $ X_2 $. Note that $ X_2 $ copies some part of $ 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

- $ C^+ = \max_{0 \leq \beta \leq 1} \min \left\{ \frac{1}{2} \log(1 + (1 - \beta) (c^2_{21} + c^2_{31}) P_1), \frac{1}{2} \log(1 + c^2_{31} P_1 + c^2_{32} P_2 + 2 \sqrt{ \beta c^2_{31} c^2_{32} P_1 P_2}) \right\} $

### Achievable rate of a decode-and-forward relayEdit

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

- $ 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., $ I(X_1;Y_2,Y_3|X_2) $ is reduced to $ I(X_1;Y_2|X_2) $. The capacity of the reduced broadcast channel is

- $ \max_{f(X_1,X_2)} I(X_1;Y_2|X_2) = \frac{1}{2} \log(1 + (1 - \beta) c^2_{21} P_1 ). $

Thus, the achievable rate is rewritten as

- $ R_1 = \max_{0 \leq \beta \leq 1} \min \left\{ \frac{1}{2} \log(1 + (1 - \beta) c^2_{21} P_1), \frac{1}{2} \log(1 + c^2_{31} P_1 + c^2_{32} P_2 + 2 \sqrt{ \beta c^2_{31} c^2_{32} P_1 P_2}) \right\} $

### Time-Division RelayingEdit

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

- $ C^+ = \max_{0 \leq \beta \leq 1} \min \{ C_1^+(\beta), C_2^+(\beta) \} $

with

- $ 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) $
- $ 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) $

## ReferencesEdit

## See also Edit

### Systems Edit

### Technologies Edit

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

### External references Edit

- A. Sendonaris, E. Erkip, and B. Aazhang, “User Cooperation Diversity Part I and Part II,” IEEE Trans. Commun., vol. 51, no. 11, Nov. 2003, pp. 1927–48.
- J. N. Laneman, D. N. C. Tse, and G. W. Wornell, “Cooperative Diversity in Wireless Networks: Efficient Protocols and Outage Behavior,” IEEE Trans. Inform. Theory, vol. 50, no. 12, pp. 3062-3080, Dec. 2004.
- S. Valentin, et al., Cooperative wireless networking beyond store-and-forward: Perspectives for PHY and MAC design
- Sébastien Simoens, Josep Vidal, Olga Muñoz, COMPRESS-AND-FORWARD COOPERATIVE RELAYING IN MIMO-OFDM SYSTEMS

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