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19.5.5 Network with memory In this case we have to slightly modify the associated acyclic network. De nition 1 Given a communication network (G, S) and a positive integer , the associated communication network with memory, denoted as (G [ ] , S), is de ned as follows. The set of nodes in G [ ] includes the node S and all pairs of the type [X, t], where X is a nonsource node in G and t ranges through integers 1 to . Channels in the network (G [ ] , S) belong to one of the three types listed below. For any nonsource nodes X and Y in (G, S); (1) for t , the multiplicity of the channel from S to [X, t] is the same as that of the channel S X in the network (G, S); (2) for t < , the multiplicity of the channel from [X, t] to [Y, t + 1] is the same as that of the channel X Y in the network (G, S); (3) for t < , the multiplicity of channels from [X, t] to [X, t + 1] is equal to t m f G(X ). (4) The communication network (G [ ] , S) is acyclic. (5) Every ow from the source to the node X in the network (G ( ) , S) corresponds to a ow with the same volume from the source to the node [X, t] in the network (G [ ] , S). (6) Every LCM v on the network (G ( ) , S) corresponds to an LCM u on the network (G [ ] , S) such that for all nodes X in G: dim[u([X, ])] = dim[v(X )].
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19.5.6 Construction of a generic LCM on an acyclic network Let the nodes in the acyclic network be sequentially indexed as X 0 = S, X 1 , X 2 , . . . , X n such that every channel is from a smaller indexed node to a larger indexed node. The following procedure constructs an LCM by assigning a vector v(X Y ) to each channel X Y , one channel at a time. { for all channels X Y v(X Y ) = the zero vector; // initialization for ( j = 0; j n; j + +) arrange all outgoing channels X j Y from X j in an arbitrary order; take one outgoing channel from X j at a time { let the channel taken be X j Y ; choose a vector w in the space v(X j ) such that w v(U Z ) : U Z for any collection of at most d 1 channels with
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v(X j ) v(U Z ) : U Z ; v(X j Y ) = w; v(X j+1 ) = the linear span by vectors v(X X j+1 ) on all incoming channels X X j+1 to X j+1 ;
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The essence of the above procedure is to construct the generic LCM iteratively and make sure that in each step the partially constructed LCM is generic. 19.5.7 Time-invariant LCM and heuristic construction In order to handle delays, we can use an element a(z) of F[(z)] to represent the z-transform of a stream of symbols a0 , a1 , a2 , . . . , at , . . . that are sent on a channel, one symbol at a time. The formal variable z is interpreted as a unit-time shift. In particular, the vector assigned to an outgoing channel from a node is z times a linear combination of the vectors assigned to incoming channels to the same node. Hence a TILCM is completely determined by the vectors that it assigns to channels. On the communication network illustrated in Figure 19.18, de ne the TILCM v as v(S X ) = (1 0)T , v(SY ) = (0 1)T , v(X Y ) = (z z 3 )T v(Y W ) = (z 2 z)T and v(W X ) = (z 3 z 2 ) By formal transformation we have for example v(X Y ) = (z z 3 ) = z(1 z 3 )(1 0)T + z(z 3 z 2 )T = z[(1 z 3 )v(S X ) + v(W X )] Thus, v(X Y ) is equal to z times the linear combination of v(S X ) and v(W X ) with coef cients 1 z 3 and 1, respectively. This speci es an encoding process for the channel X Y that does not change with time. It can be seen that the same is true for the encoding process of every other channel in the network. This explains the terminology time-invariant for an LCM. To obtain further insight into the physical process write the information vector as [a(z) b(z)], where a(z) =
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