NETWORK INFORMATION THEORY in .NET

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Figure 19.20 Network layout. 2 nodes transmit, while group 1 nodes receive, according to a time division duplex (TDD) transmission scheme. The channel is assumed to be channelized into several narrow band frequency channels, whose bandwidth is small enough so that all multipath is assumed to add coherently. This assumption allows the channel between any transmit antenna and any receive antenna to be modeled as a single complex gain and is consistent with an orthogonal frequency division multiplexing (OFDM) modulation scheme. All transceivers are assumed to have either multiple antennas or polarization diversity, for both transmit and receive processing. The link connecting nodes may experience multipath re ections. For this analysis we assume that the receiver synchronizes to a given transmitter and removes any propagation delay offsets. The channel, for a given narrow band, between the transmitter and receiver arrays is therefore modeled as a complex matrix multiply. Let 1 be an index into the group 1 transceivers, and 2 an index into the group 2 transceivers. When node 2 is transmitting during the uplink, we model the channel between the two nodes as a complex matrix H12 ( 1 , 2 ) of dimension M1 ( 1 ) M2 ( 2 ), where M1 ( 1 ) is the size of the antenna array at node 1 , and M2 ( 2 ) is the size of the array at node 2 . Polarization diversity is treated like additional antennae. In the next TDD time slot, during the downlink, node 1 will transmit and the channel from 1 to 2 is described by the complex M2 ( 1 ) M1 ( 2 ) matrix H21 ( 2 , 1 ). For every node pair ( 1 , 2 ) that forms a communications link in the network, we assign a MIMO channel link number, indexed by k or m, in order to label all such connections established by the network. Obviously not every node pair is necessarily assigned a link number, only those that actually communicate. We also de ne the mapping from the MIMO link number to the associated group 1 node, 1 (k) and the associated group 2 node 2 (k) by the association of k with the link [ 1 (k), 2 (k)]. Because each channel is MIMO, a given node will transmit multiple symbols over possibly more than one transmission mode. The set of all transmission modes over the entire network is indexed by q and p. This index represents all the symbol streams that are transmitted from one node to another and therefore represents a lower level link number. The low level link numbers will map to its MIMO channel link, via the mapping k(q). Because our network will
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CAPACITY OF WIRELESS NETWORKS USING MIMO TECHNOLOGY
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exploit channel reciprocity, we assume that every uplink symbol stream indexed by q has an associated downlink symbol stream assigned to the same index. The q th uplink symbol stream is spread by an M1 (q) 1 complex vector g1 (q), where M1 (q) M1 { 1 [k(q)]}. Similarly g2 (q) is the associated transmit vector for the downlink. For each node, we group the transmit vectors into a single matrix, G2 (k) g2 (q1 ), g2 (q2 ), . . . , g2 (q Mc (k) ) and G1 (k) g1 (q1 ), g1 (q2 ), . . . , g1 (q Mc (k) ) (19.62) (19.61)
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where k(qi ) = k and there are Mc (k) transmission modes associated with MIMO link k. With these conventions the signal model can be written as: x1 (n; k) = i1 (n; k) + H12 (k, k)G2 (k)d2 (n; k) x2 (n; k) = i2 (n; k) + H21 (k, k)G1 (k)d1 (n; k) (19.63) (19.64)
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where x1 (n; k) is the received complex data vector at sample n, and node 1 (k), H12 (k, m) H12 [ 1 (k), 2 (m)] is the M1 [ 1 (k)] M2 [ 2 (m)] complex MIMO channel matrix for downlink transmission, n is a time/frequency index, that represents an independent reuse of the channel, either due to adjacent frequency channels (e.g. adjacent OFDM channels) or due to multiple independent time samples, i1 (n; k) is the interference vector seen at node 1 (k) due to the other transmitting nodes as well as due to background radiation, and d2 (n; k) is the Mc (k) 1 downlink information symbol vector, transmitted for sample n. The analogous model for the uplink case is shown in (4). The interference vector can be written as ir (n; k) =
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Hr t (k, m)G t (m)dt (n; m) + r (n, k)
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(19.65)
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where r (n, k), r = {1, 2}, is the background radiation noise vector seen by the receiving node r (k) and t = {2, 1} is the transmission timeslot indicator. The convention is adopted that t = 2 if r = 1, otherwise t = 1 when r = 2. 19.6.1 Capacity metrics In this section we will be interested in the asymptotic maximum achievable throughput under certain network constraints: (1) We will assume that signal energy from all the nodes in the network, other than the one from the link being processed, is treated as interference. This is due to dif culty of synchronizing a node to multiple interference sources and reduces the complexity of the receiver. (2) The interference ir (n; k) is modeled as complex circular Gaussian noise. The superposition of many distant interferers will appear Gaussian by the central limit theorem. Considering therefore the channel model described by Equations (19.63) and (19.65), we can write the mutual information between the source data vector and the received data vector
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