Transmit Power and Snr Analysis

Given the system model and cooperative transmit protocol described in section II we consider the problem of efficiently allocating transmit power in order to achieve a pair of fixed SNR targets, denoted as SNRi and SNRj at the source and the destination. In the absence of cooperation, the orthogonality of the sources makes the solution to this problem straightforward.

The SNR target will be satisfied if

 

Denoting the transmit power source i in frequency slot f as Pi[f], the transmit power in each frequency slot are taken from section I

The total transmit power for source i is given as

Pi=Pi[f₁]+Pi[f₂f₁] and the total transmit power for source j is given as

Pj=Pj[f₁f₂]+Pj[1 f ₁], the total transmit power over all sources is given as Ptotal=Pi+Pj

From the power expressions of eq (9) and eq (10), we can define a ―cooperation ratio‖

parameter for each source in the system. The i^th source‘s cooperation ratio is defined as the ratio of the power of the i^th source‘s cooperative retransmission to the power of the original transmission of source j (i≠j). Using the results from eq (9) and eq (10) we can write the cooperation ratios for the i^th source as

Җi=

 

The next step in our analysis is to derive expressions for the SNR of TR nodes at the destination under the assumption that the destination optimally combines the observations { rcij, řcij } to maximize SNR.

Assuming that all of the noise terms in the observations are mutually independent as well as independent of the data, it can be shown that maximal ratio combining at the destination maximizes the SNR of rcij and řcij. The maximal ratio combining coefficients can be written as

i 2 1

And the resulting SNRs with maximal ratio combining at the destination can be expressed as SNRi=

Substituting the cooperation ratios and the transmit powers from eqs (9) and (10) into the above eqs (12) and (13) we can rewrite the SNR of each source as

SNRi =

We note that specification of the SNR targets {SNRi,SNRj} as well as the cooperation ratios { Җi, Җj } fully determine the minimum transmit powers and bandwidth for all TR nodes and maximum cooperation between them.

4.3. Canonical Examples

This section presents the circular network with cooperative protocols that motivate our algorithms and demonstrate the potential of cooperative network coding. In this case we consider a wireless network with n nodes. We assume that every node is a source that wants to broadcast information to all other nodes. Also, each node broadcasts at a fixed range and therefore, each transmission consumes the same

amount of energy. Let Tnc denote the total number of transmission required to broadcast one information unit to all nodes when we use network coding. Similarly, let Tw denote the required number of transmissions when we do not use network coding. We are interested in calculating ^nc

w

T T . Figure 6: A circular configuration

Consider n nodes placed at equal distances around a circle as depicted as circular network in Fig.6. Assume that each node can successfully broadcast information to its two neighbours. For example, node b1 can broadcast to nodes a1 and a2.

Lemma 1: For the circular network it holds that 1. Without network coding Tw  n  2 2. With network coding Tnc 1

2 n

 .

Proof: Since a node can successfuly broadcast to its two nearest neighbours, each broadcast transmission can transfer at most one unit of information to two recivers. We have n1 receivers to cover and thus the best energy efficiency we may hope for is ¹

2 n

per information unit. When forwardng information bit we may consider a single source broadcasting to n1 receivers. The first transmission reaches two receivers. Each additional transmission can contribute one unit of information to one receiver.For the case of forwarding, it is easy to see that a simple flooding algorithm achieves the bound in Lemma 1. For network coding consider the following scheme.

Assume that n is an even number. Partition the n nodes in two sets A ={ a1,...

2

an } and B ={ b1,...

2

bn } of size 2

n each, such that every node in A has as nearest neighbors two nodes in B. For example, Fig. 1 depicts a circular configuration with n=8 nodes. It is sufficient to show that we can broadcast one information unit from each node in set A to all nodes in sets A and B using Tnc

2

 n transmissions.

We can then repeat this procedure symetrically to broadcast the information from the nodes in B. Let { x1,...

2

xn } denote the information units associated with the nodes in A. Consider the following

transmission scheme that operates in 4

n steps. Each step has two phases, where first nodes in A transmit and nodes in B receive and then nodes in B transmit and nodes in A receive.

Algorithm Network Coding cooperative protocols (NC) Step 1:

Phase 1: The nodes in set A transmit their information to their nearest neighbors such that each node bi receives xi and xi+1 as shown in Fig.1.

Phase 2: The nodes in set B simply add the information symbols they receive and broadcast it.

For example, node ai receives xi+1+xi from node bi and xi-1+xi from node bt-1. Thus node ai has the information units from sources ai-1 and ai+1.

Step k neighbouring node, k neighbouring node1:

Phase 1: Each node in A transmits the sum of the two information units it received in phase 2, step k1.

Phase 2: Each node in B transmits the sum of the two Information units it received in phase 1, step k.

Lemma 2: There exist schemes that achieve the lower bounds in lemma 1. Thus