Ai,n (t) pm t i,n (t) , m = 0, . . . , M 1 in .NET

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Ai,n (t) pm t i,n (t) , m = 0, . . . , M 1
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where n i (t) is the ith receiver AWGN with variance N0 , i,n (t) is the delay of the link between the ith and the nth node, including the asynchronism of the beginning of transmission for each node n, and Ai,n (t) is the product of a complex fading coef cient i,n (t), the transmit power Pt and the channel average gain, e.g. (1 + di,n ) i,n (log normal fading), where di,n is the distance, and i,n the decay constant between the ith and nth nodes. The following assumptions are used: (a1) Ai,n (t) and i,n (t) are constant over multiple symbol durations Ts ; the nodes are quasi-stationary for a time much greater than Ts . (a2) The delays are i,1 < i,2 i,N , where the minimum delay i,1 corresponds to the leader. To avoid ISI, the upper bound for the effective symbol rate is Rs = 1/Ts 1/ , where is the maximum delay spread of si,m (t) for all i. The
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(t i )2 Si,m (t) dt
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t Si,m (t) dt Si,m (t) dt
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and thus, = maxi i . Echoes that come from farther away are strongly attenuated (by d ); therefore, the echoes received at node i are nonnegligible only for those coming from nodes within a certain distance d, which essentially depends on the transmit power and path loss. Hence, Rs can be increased by lowering the transmit power, capitalizing on spatial bandwidth reuse. In the reach back problem, however, the delay spread is supi [ i,N i,1 ] because the receiver is roughly at the same distance from all nodes. (a3) Ts is xed for all nodes to c1 , where c1 is a constant taken to satisfy the ISI constraint. With (a3), we guarantee that no ambiguity will occur at the nodes in timing their responses. The transmission activity of the node is solely dependent on the signal that the node receives. Based on the evolution of si,m (t), we can distinguish two phases: (1) the earlier receive phase, when the upstream waves of signals approach the node, and (2) the period after the ring instant, which we call the rest phase, where the node hears the echoes of the downstream wave of signals fading away (for the regenerative case, the ring instant occurs shortly after the time when the node has accumulated enough energy to detect the signal). The switching between the two modes can be viewed as a very elementary form of time-division duplex (TDD). (a4) The leader (and also the nodes in the regenerative case) transmits pulses with complex envelope pm (t) having limited double-sided bandwidth W and, approximately, duration T p . By sampling at the Nyquist rate, N p = T p W is the approximate length of the sequence { pm (k/W )} of samples. Multipath propagation can be simply included in the model by increasing the number of terms in the summation in si,m (t); therefore, it does not require special attention. In fact, when we neglect the propagation of errors and noise that occurs in the case of regenerative and nonregenerative repeaters, respectively, the OLA itself is equivalent to a multipath channel, created by a set of active scatterers. In the regenerative case, the ideal OLA response is
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(19.55)
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The nonregenerative OLA scattering model is more complex due to the feedback effect, which implies that not one but several signal contributions are scattered by each source.
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The received OLA response is
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For every possible path in the network, there is a contribution to the summation in Equation (19.55a) that has an amplitude equal to the product of all the path link gains traveled so far and a delay equal to the sum of all the path delays. Theoretically, the number of re ections N because the signals and their ampli ed versions keep cycling in the network and adding up. If properly controlled, the contributions will keep adding up and then opportunistically serve the purpose of enhancing the signal. Hence, the key for the nonregenerative design is to control the noise that accompanies the useful signal. In both regenerative and nonregenerative cases, the received signal can be rewritten as the following convolution: ri (t) = gi (t)* pm (t) + n i (t) (19.56)
where gi (t) is the network impulse response, which is analogous to that of a multipath channel. Based on Equation (19.46), the idea is to let the nodes operate as regenerative and nonregenerative repeaters and avoid any complex coordination procedure to forward their signals at the network layer and share the bandwidth at the MAC layer. In addition, no channel state information is used. The information ow is carried forward using receivers that are capable of tracking the unknown network response gi (t) or, directly, the signature waveforms si,m (t) = gi (t)* pm (t). We should expect that the OLA behaves as a frequencyselective channel. Nodes mobility causes changes of the response gi (t) over time. If most of the network is stationary and N is large, the inertia of the system will be such that mobile nodes will cause small changes in gi (t). Since the transmission channel is bandlimited with passband bandwidth W , the signature waveform pm (t) will have to be bandlimited and, therefore, uniquely expressible through its samples taken at the Nyquist rate 1/Tc , where Tc = 1/W . In general, pm (t) corresponds to a nite number of samples N p and is approximately time limited with duration T p N p /W . Introducing the vectors pm , gi and ri such that {pm }k = pm (kTc ), k = 0, . . . N p 1 {gi }k = sinc( W )gi (kTc + li TC )d , {ri }k = ri (kTc + li Tc )k = 0, . . . , Ni + N p 2 {ni }k = n i (kTc + li Tc ), k = 0, . . . , Ni + N p 2 where Ni is the number of samples needed to represent gi (t), we have