Advanced wireless networks 4g technologies phần 10 doc

JWBK083-19 JWBK083-Glisic March 7, 2006 20:8 Char Count= 0 NETWORK CODING 771 1.1 1.15 1.2 1.25 1.3 1.35 1.4 1.45 1.5 20 30 40 50 60 70 80 Number of users End-to-end delay (s) × 10 −3 Figure 19.16 End-to-end delay vs number of nodes in the network. (b) U Z T W Y X U Z T S W Y X (a) U Z T S W Y U Z T S W Y Figure 19.17 Illustration of network coding. As in Chapter 7, we define a communication network as a pair (G, S), where G is a finite directed multigraph and S (source) is the unique node in G without any incoming edges. A directed edge in G is called a channel in the communication network (G, S). A channel in graph G represents a noiseless communication link on which one unit of information (e.g. a bit) can be transmitted per unit time. The multiplicity of the channels from a node X to another node Y represents the capacity of direct transmission from X to Y. We assume that, every single channel has unit capacity. At the source S, information is generated and multicast to other nodes on the network in the multihop fashion where every node can pass on any of its received data to other nodes. At each nonsource node which serves as a sink, the complete information generated at S is recovered. Now the question is how fast each sink node can receive the complete information. As an example, consider the multicast of two data bits, b 1 and b 2 , from the source S in the communication network depicted by Figure 19.17(a) as both nodes Y and Z. One solution is to let the channels ST, TY, TW and WZ carry the bit b 1 and channels SU, UZ, UW and WY carry the bit b 2 . Note that, in this scheme, an intermediate node sends out a data bit only if it receives the same bit from another node. For example, the node T receives the bit b 1 and sends a copy on each of the two channels TY and TW. Similarly, the node U receives the bit b 2 and sends a copy into each of the two channels UW and UZ.We assume that there is no processing delay at the intermediate nodes. JWBK083-19 JWBK083-Glisic March 7, 2006 20:8 Char Count= 0 772 NETWORK INFORMATION THEORY Unlike a conserved physical commodity, information can be replicated or coded. The notion of network coding refers to coding at the intermediate nodes when information is multicast in a network. Let us now illustrate network coding by considering the communi- cation network depicted in Figure 19.17(b). Again, we want to multicast two bits b 1 and b 2 from the source S to both the nodes Y and Z. A solution is to let the channels ST, TW and TY carry the bit b 1 , channels SU, UW and UZ carry the bit b 2 , and channels WX, XY and XZ carry the exclusive-OR b 1 ⊕ b 2 . Then, the node Y receives b 1 and b 1 ⊕ b 2 , from which the bit b 2 = b 1 ⊕ ( b 1 ⊕ b 2 ) can be decoded. Similarly, the node Z can decode the bit b 1 from b 2 and b 1 ⊕ b 2 as b 1 = b 2 ⊕ ( b 1 ⊕ b 2 ) . The coding/decoding scheme is assumed to have been agreed upon beforehand. In order to discuss this issue in more detail, in this section we first introduce the notion of a linear-code multicast (LCM). Then we show that, with a ‘generic’ LCM, every node can simultaneously receive information from the source at rate equal to its max-flow bound. After that, we describe the physical implementation of an LCM, first when the network is acyclic and then when the network is cyclic followed by a presentation of a greedy algorithm for constructing a generic LCM for an acyclic network. The same algorithm can be applied to a cyclic network by expanding the network into an acyclic network. This results in a ‘time-varying’ LCM, which, however, requires high complexity in implementation. After that, we introduce the time-invariant LCM (TILCM). Definition 1 Over a communication network a flow from the source to a nonsource node T is a collection of channels, to be called the busy channels in the flow, such that: (1) the subnetwork defined by the busy channels is acyclic, i.e. the busy channels do not form directed cycles; (2) for any node other than S and T, the number of incoming busy channels equals the number of outgoing busy channels; (3) the number of outgoing busy channels from S equals the number of incoming busy channels to T. The number of outgoing busy channels from S will be called the volume of the flow. The node T is called the sink of the flow. All the channels on the communication network that are not busy channels of the flow are called the idle channels with respect to the flow. Definition 2 For every nonsource node T on a network (G, S), the maximum volume of a flow from the source to T is denoted max flow G (T ), or simply mf(T) when there is no ambiguity. Definition 3 A cut on a communication network (G, S) between the source and a nonsource node T means a collection C of nodes which includes S but not T. A channel XY is said to be in the cut C if X ∈ C and Y  ∈ C. The number of channels in a cut is called the value of the cut. 19.5.1 Max-flow min-cut theorem (mfmcT) For every nonsource node T, the minimum value of a cut between the source and a node T is equal to mf (T). Let d be the maximum of mf (T) over all T. In the sequel, the symbol  will denote a fixed d-dimensional vector space over a sufficiently large base field. The JWBK083-19 JWBK083-Glisic March 7, 2006 20:8 Char Count= 0 NETWORK CODING 773 information unit is taken as a symbol in the base field. In other words, one symbol in the base field can be transmitted on a channel every unit time. Definition 4 An LCM v on a communication network (G, S) is an assignment of a vector space v(X) to every node X and a vector v(XY) to every channel XY such that (1) v(S) = ; (2) v(XY) ∈ v(X) for every channel XY; and (3) for any collection ℘ of nonsource nodes in the network {v(T ):T ∈ ℘} = {v(XY):X ∈ ℘Y ∈ ℘}. The notation · is for linear span. Condition (3) says that the vector spaces v(T) on all nodes T inside ℘ together have the same linear span as the vectors v(XY) on all channels XY to nodes in ℘ from outside ℘. LCM v data transmission: The information to be transmitted from S is encoded as a d-dimensional row vector, referred to as an information vector. Under the transmission mechanism prescribed by the LCM v, the data flowing on a channel XY is the matrix product of the information (row) vector with the (column) vector v(XY). In this way, the vector v(XY) acts as the kernel in the linear encoder for the channel XY. As a direct consequence of the definition of an LCM, the vector assigned to an outgoing channel from a node X is a linear combination of the vectors assigned to the incoming channels to X. Consequently, the data sent on an outgoing channel from a node X is a linear combination of the data sent on the incoming channels to X. Under this mechanism, the amount of information reaching a node T is given by the dimension of the vector space v(T) when the LCM v is used. Coding in Figure 19.17(b) is achieved with the LCM v specified by v(ST) = v(TW) = v(TY) = ( 10 ) T v(SU) = v(UW) = v(UZ) = ( 01 ) T (19.58) and v(WX) = v(XY) = v(XZ) = ( 11 ) T where ( ) T stands for transposed vector. The data sent on a channel is the matrix product of the row vector (b 1 b 2 ) with the column vector assigned to that channel by v. For instance, the data sent on the channel WX is (b 1 , b 2 )v(WX) = (b 1 , b 2 ) ( 11 ) T = b 1 + b 2 Note that, in the special case when the base field of  is GF(2), the vector b 1 + b 2 reduces to the exclusive-OR b 1 ⊕ b 2 in an earlier example. Proposition P1 For every LCM v on a network, for all nodes T dim[v(T)] ≤ mf (T ). Toproveitfixa nonsource node T and any cut C between the source and T v(T ) ⊂v(Z):Z ∈ C= v(YZ):Y ∈ C and Z ∈ C. Hence, dim[v(T)] ≤ dim(v(YZ):Y ∈ C and Z ∈ C), which is at most equal to the value of the cut. In particular, dim[v(T )] is upper-bounded JWBK083-19 JWBK083-Glisic March 7, 2006 20:8 Char Count= 0 774 NETWORK INFORMATION THEORY by the minimum value of a cut between S and T, which by the max-flow min-cut theorem is equal to mf(T). This means that mf (T) is an upper bound on the amount of information received by T when an LCM v is used. 19.5.2 Achieving the max-flow bound through a generic LCM In this section, we derive a sufficient condition for an LCM v to achieve the max-flow bound on dim[v(T )] in Proposition 1. Definition An LCM v on a communication network is said to be generic if the following condition holds for any collection of channels X 1 Y 1 , X 2 Y 2 , ,X m Y m for 1 ≤ m ≤ d :(∗) v(X k ) ⊂  {v(X j Y j ): j = k}  for 1 ≤ k ≤ m if and only if the vectors v(X 1 Y 1 ),v(X 2 Y 2 ), ,v(X m Y m ) are linearly independent. If v(X 1 Y 1 ),v(X 2 Y 2 ), ,v(X m Y m ) are linearly in- dependent, then v(X k ) ⊂  {v(X j Y j ): j = k}  since v(X k Y k ) ∈ v(X k ). A generic LCM re- quires that the converse is also true. In this sense, a generic LCM assigns vectors which are as linearly independent as possible to the channels. With respect to the communication network in Figure 19.17(b), the LCM v defined by Equation (19.58) is a generic LCM . However, the LCM u defined by u(ST) = u(TW) = u(TY) = ( 10 ) T u(SU) = u(UW) = u(UZ) = ( 01 ) T (19.59) and u(WX) = u(XY) = u(XZ) = ( 10 ) T is not generic. This is seen by considering the set of channels {ST, WX} where u(S) = u(W) =  1 0  ,  0 1  Then u(S) ⊂u(WX) and u(W) ⊂u(ST),butu(ST) and u(WX) are not linearly inde- pendent. Therefore, u is not generic. Therefore, in a generic LCM v any collection of channels XY 1 , XY 2 , ,XY m from a node X with m ≤ dim[v(X)] must be assigned linearly independent vectors by v. Theorem T1 If v is a generic LCM on a communication network, then for all nodes T, dim [v(T )] = mf(T ). To prove it, consider a node T not equal to S. Let f be the common value of mf(T ) and the minimum value of a cut between S and T . The inequality dim[v(T)] ≤ f follows from Proposition 1. So, we only have to show that dim[v(T )] ≥ f. To do so, let dim(C) = dim(v(X, Y ):X ∈ C and Y  ∈C)forany cut C between S and T . Wewill show that dim[v(T)] ≥ f by contradiction. Assume dim[v(T )] < f and let A be the collection of cuts U between S and T such that dim(U) < f. Since dim[v(T )] < f implies V \{T }∈A, where v is the set of all the nodes in G, A is nonempty. JWBK083-19 JWBK083-Glisic March 7, 2006 20:8 Char Count= 0 NETWORK CODING 775 By the assumption that v is a generic LCM, the number of edges out of S is at least d, and dim({S}) = d ≥ f . Therefore, {S} ∈ A. Then there must exist a min- imal member U ∈ A in the sense that for any Z ∈ U\{S} = φ,U\{Z} ∈ A. Clearly, U ={S} because {S} ∈ A. Let K be the set of channels in cut U and B be the set of boundary nodes of U, i.e. Z ∈ B if and only if Z ∈ U and there is a channel (Z, Y) such that Y ∈ U. Then for all W ∈ B, v(W) ⊂  v(X, Y ):(X, Y) ∈ K  which can be seen as follows. The set of channels in cut U\{W} but not in K is given by {(X, W):X ∈ U\{W}}. Since v is an LCM v(X, W):X ∈ U\{W} ⊂ v(W). If v(W) ⊂ v(X, Y ):(X, Y) ∈ K , then v(X  , Y  ):X  ∈ U\{W}, Y  ∈ U\{W} the subspace spanned by the channels in cut U\{W}, is contained by  v(X, Y ):(X, Y) ∈ K  . This im- plies that dim(U\{W}) ≤ dim(U) < f isacontradiction.Therefore,forall W ∈ B,v(W) ⊂  v(X, Y ):(X, Y) ∈ K  . For all (W, Y ) ∈ K, since  v(X, Z):(X, Z) ∈ K \{(W, Y )}  ⊂  v(X, Y ):(X, Y) ∈ K  ,v(W)  ⊂  v(X, Y ):(X, Y) ∈ K  implies that v(W) ⊂  v(X, Z): (X, Z) ∈ K \{(W, Y)}  . Then, by the definition of a generic LCM {v(XY):(X, Y) ∈ K }is a collection of vectors such that dim(U) = min(|K|, d). Finally, by the max-flow min-cut theorem, |K|≥ f , and since d ≥ f, dim(U) ≥ f . This is a contradiction to the assumption that U ∈ A. The theorem is proved. An LCM for which dim[v(T)] = mf(T) for all T provides a way for broadcasting a message generated at the source S for which every nonsource node T receives the message at rate equal to mf(T). This is illustrated by the next example, which is based upon the assumption that the base field of  is an infinite field or a sufficiently large finite field. In this example, we employ a technique which is justified by the following arguments. Lemma 1 Let X, Y and Z be nodes such that mf(X) = i, mf(Y) = j, and mf(Z) = k, where i ≤ j and i > k. By removing any edge UX in the graph, mf(X) and mf(Y) are reduced by at most 1, and mf(Z) remains unchanged. To prove it we note that, by removing an edge UX, the value of a cut C between the source S and node X (respectively, node Y) is reduced by 1 if edge UXis in C, otherwise, the value of C is unchanged. By the mfmcT, we see that mf(X) and mf(Y) are reduced by at most 1 when edge UX is removed from the graph. Now consider the value of a cut C between the source S and node Z.IfC contains node X, then edge UX is not in C, and, therefore, the value of C remains unchanged upon the removal of edge UX.IfC does not contain node X, then C is a cut between the source S and node X.BythemfmcT, the value of C is at least i. Then, upon the removal of edge UX, the value of C is lower-bounded by i −1 ≥ k. Hence, by the mfmcT, mf(Z) remains to be k upon the removal of edge UX. Example E1 Consider a communication network for which mf(T ) = 4, 3 or 1 for nodes T in the network. The source S is to broadcast 12 symbols a 1 , ,a 12 taken from a sufficiently large base field F. (Note that 12 is the least common multiple of 4, 3 and 1.) Define the set T i ={T : mf(T ) = i}, for i = 4, 3, 1. For simplicity, we use the second as the time unit. We now describe how a 1 , ,a 12 can be broadcast to the nodes in T 4 ,T 3 , T 1 ,in3,4and12s, respectively, assuming the existence of an LCM on the network for d = 4, 3, 1. JWBK083-19 JWBK083-Glisic March 7, 2006 20:8 Char Count= 0 776 NETWORK INFORMATION THEORY (1) Let v 1 be an LCM on the network with d = 4. Let α 1 = (a 1 a 2 a 3 a 4 ),α 2 = (a 5 a 6 a 7 a 8 ) and α 3 = (a 9 a 10 a 11 a 12 ). In the first second, transmit α 1 as the in- formation vector using v 1 , in the second second, transmit α 2 , and in the third second, transmit α 3 . After 3 s, after neglecting delay in transmissions and computations all the nodes in T 4 can recover α 1 ,α 2 and α 3 . (2) Let r be a vector in F 4 such that {r} intersects trivially with v 1 (T ) for all T in T 3 , i.e.{r,v 1 (T )} = F 4 for all T in T 3 . Such a vector r can be found when F is sufficiently large because there are a finite number of nodes in T 3 . Define b i = α i r for i = 1, 2, 3. Now remove incoming edges of nodes in T 4 , if necessary, so that mf(T ) becomes 3 if T is in T 4 , otherwise, mf(T ) remains unchanged. This is based on Lemma 1). Let v 2 be an LCM on the resultingnetwork with d = 3. Let β = (b 1 b 2 b 3 ) and transmit β as the information vector using v 2 in the fourth second. Then all the nodes in T 3 can recover β and hence α 1 ,α 2 and α 3 . (3) Lets 1 and s 2 be two vectors in F 3 such that  {s 1 , s 2 }  intersects with v 2 (T ) trivially for all T in T 1 , i.e.{s 1, s 2 ,v 2 (T )} = F 3 for all T in T 1 . Define γ i = βs i for i = 1, 2. Now remove incoming edges of nodes in T 4 and T 3 , if necessary, so that mf(T) becomes 1 if T is in T 4 or T 3 , otherwise, mf(T) remains unchanged. Again, this is based on Lemma 1). Now let v 3 be an LCM on the resulting network with d = 1. In the fifth and the sixth seconds, transmit γ 1 and γ 2 as the information vectors using v 3 . Then all the nodes in T 1 can recover β. (4) Let t 1 and t 2 be two vectors in F 4 such that  {t 1 , t 2 }  intersects with  {r,v 1 (T )}  trivially for all T in T 1 , i.e.  {t 1 , t 2 , r,v 1 (T )}  = F 4 for all T in T 1 . Define δ 1 = α 1 t 1 and δ 2 = α 1 t 2 . In theseventhand eighth seconds, transmit δ 1 and δ 2 as the information vectors using v 3 . Since all the nodes in T 1 already know b 1 , upon receiving δ 1 and δ 2 ,α 1 can then be recovered. (5) Define δ 3 = α 2 t 1 and δ 4 = α 2 t 2 . In the ninth and tenth seconds, transmit δ 3 and δ 4 as the information vectors using v 3 . Then α 2 can be recovered by all the nodes in T 1 . (6) Define δ 5 = α 3 t 1 and δ 6 = α 0 t 2 . In the eleventh and twelveth seconds, transmit δ 5 and δ 6 as the information vectors using v 3 . Then α 3 can be recovered by all the nodes in T 1 . So, in the ith second for i = 1, 2, 3, via the generic LCM v 1 , each node in T 4 receives all four dimensions of α i , each node in T 3 receives three dimensions of α i , and each node in T 1 receives one dimension of α i . In the fourth second, via the generic LCM v 2 , each node in T 3 receives the vector β, which provides the three missing dimensions of α 1 ,α 2 and α 3 (one dimension for each) during the first3sofmulticast by v 1 . At the same time, each node in T 1 receives one dimension of β. Now, in order to recover β, each node in T 1 needs to receive the two missing dimensions of β during the fourth second. This is achieved by the generic LCM v 3 in the fifth and sixth seconds. So far, each node in T 1 has received one dimension of α i for i = 1, 2, 3 via v 1 during the first 3 s, and one dimension of α i for i = 1, 2, 3 from β via v 2 and v 3 during the fourth to sixth seconds. Thus, it remains to provide the six missing dimensions of α 1 ,α 2 and α 3 (two dimensions for each) to each node in T 1 , and this is achieved in the seventh to the twelfth seconds via the generic LCM v 3 . The previous scheme can be generalized to arbitrary sets of max-flow values. JWBK083-19 JWBK083-Glisic March 7, 2006 20:8 Char Count= 0 NETWORK CODING 777 19.5.3 The transmission scheme associated with an LCM Let v be an LCM on a communication network (G, S), where the vectors v(SX) assigned to outgoing channels SX linearly span a d-dimensional space. As before, the vector v(XY ) assigned to a channel XY is identified with a d-dimensional column vector over the base field of  by means of the choice of a basis. On the other hand, the total information to be transmitted from the source to the rest of the network is represented by a d-dimensional row vector, called the information vector. Under the transmission scheme prescribed by the LCM v, the data flowing over a channel XY is the matrix product of the information vector with the column vector v(XY). We now consider the physical realization of this transmission scheme associated with an LCM. A communication network (G, S) is said to be acyclic if the directed multigraph G does not contain a directed cycle. The nodes on an acyclic communication network can be sequentially indexed such that every channel is from a smaller indexed node to a larger indexed node. On anacyclic network, a straightforward realization of the abovetransmission scheme is as follows. Take one node at a time according to the sequential indexing. For each node, ‘wait’ until data is received from every incoming channel before performing the linear encoding. Then send the appropriate data on each outgoing channel. This physical realization of an LCM over an acyclic network, however, does not apply to a network that contains a directed cycle. This is illustrated by the following example. Example 2 Let p, q, and r be vectors in , where p and q are linearly independent. Define v(SX) = p,v(SY) = q and v(WX) = v(XY) = v(YW) = r. This specifies an LCM v on the network illustrated in Figure 19.18 if the vector r is a linear combination of p and q. Otherwise, the function v gives an example in which the law of information flow is observed for every single node but not observed for every set of nodes. Specifically, the law of information flow is observed for each of the nodes X, Y and W, but not for the set of nodes {X, Y, W}. Now, assume that p = (1 0) T , q = (0 1) T and r = (1 1) T . Then, v is an LCM. Write the information vector as (b 1 b 2 ), where b 1 and b 2 belong to the base field of . According to the transmission scheme associated with the LCM, all three channels on the directed cycle transmit the same data b 1 + b 2 . This leads to the logical problem of how any of these cyclic channels acquires the data b 1 + b 2 in the first place. In order to discuss the transmission scheme associated with an LCM over a network containing a directed cycle, we need to introduce the parameter of time into the scheme. Instead of transmitting a single data symbol (i.e. an element of the base field of ) through Y WW XU S Figure 19.18 An LCM on a cyclic network. JWBK083-19 JWBK083-Glisic March 7, 2006 20:8 Char Count= 0 778 NETWORK INFORMATION THEORY each channel, we shall transmit a time-parameterized stream of symbols. In other words, the channel will be time-slotted. As a consequence, the operation of coding at a node will be time-slotted as well. 19.5.4 Memoryless communication network Given a communication network (G, S) and a positive integer τ, the associated memoryless communication network denoted as (G (τ ) , S) is defined as follows. The set of nodes in G (τ ) includes the node S and all the pairs of the type [X, t], where X is a nonsource node in G and t ranges through integers 1 – τ . The channels in the network (G (τ ) , S) belong to one of the three types listed below. For any nonsource nodes X and Y in (G, S): (1) for t ≤ τ , the multiplicity of the channel from S to [X, t] is the same as that of the channel SX in the network (G, S); (2) for t <τ, the multiplicity of the channel from [X, t]to[Y, t + 1] is the same as that of the channel XY in the network (G, S); (3) for t <τ, the multiplicity of the channel from [X, t]to[X,τ] is equal to max flow G (X) = mfG(X). Lemma 2 The memoryless communication network (G (τ ) , S) is acyclic. Lemma 3 There exists a fixed number ε, independent of τ, such that for all nonsource nodes X in (G, S), the maximum volume of a flow from S to the node [X,τ]in(G (τ ) , S) is at least τ − ε times mfG(X). For proof see Li et al. [47]. Transmission of data symbols over the network (G (τ ) , S) may be interpreted as ‘memo- ryless’ transmission of data streams over the network (G, S) as follows: (1) A symbol sent from S to [X, t]in(G (τ ) , S) corresponds to the symbol sent on the channel SX in (G, S) during the time slot t. (2) A symbol sent from [X, t]to[Y, t +1] in (G (τ ) , S) corresponds to the symbol sent on the channel XY in (G, S) during the time slot t +1. This symbol is a linear combination of symbols received by X during the time slot t and is unrelated to symbols received earlier by X. (3) The channels from [X, t]to[X, t] for t <τ signify the accumulation of received information by the node X in (G, S) over time. Since this is an acyclic network, the LCM on the network (G (τ ) , S) can be physically realized in the way mentioned above. The physical realization can then be interpreted as a memoryless transmission of data streams over the original network (G, S). JWBK083-19 JWBK083-Glisic March 7, 2006 20:8 Char Count= 0 NETWORK CODING 779 19.5.5 Network with memory In this case we have to slightly modify the associated acyclic network. Definition 1 Given a communication network (G, S) and a positive integer τ, the associated communi- cation network with memory, denoted as (G [ τ ] , S), is defined as follows. The set of nodes in G [τ ] 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 τ . Channels in the network (G [τ ] , S) belong to one of the three types listed below. For any nonsource nodes X and Y in (G, S); (1) for t ≤ τ , the multiplicity of the channel from S to [X, t] is the same as that of the channel SX in the network (G, S); (2) for t <τ, the multiplicity of the channel from [X, t]to[Y, t +1] is the same as that of the channel XY in the network (G, S); (3) for t <τ, the multiplicity of channels from [X, t]to[X, t +1] is equal to t × mfG(X). (4) The communication network (G [ τ ] , S) is acyclic. (5) Every flow from the source to the node X in the network (G (τ ) , S) corresponds to a flow with the same volume from the source to the node [X, t] in the network (G [ τ ] , S). (6) Every LCM v on the network (G (τ ) , S) corresponds to an LCM u on the network (G [ τ ] , S) such that for all nodes X in G: dim[u([X,τ])] = dim[v(X)]. 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(XY) to each channel XY, one channel at a time. { for all channels XY v(XY) = the zero vector; // initialization for ( j = 0; j ≤ n; j ++) { arrange all outgoing channels X j YfromX 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(UZ):UZ ∈ ξ  for any collection ξ of at most d −1 channels with JWBK083-19 JWBK083-Glisic March 7, 2006 20:8 Char Count= 0 780 NETWORK INFORMATION THEORY v(X j ) ⊂  v(UZ):UZ ∈ ξ  ; v(X j Y) = w; } v(X j+1 ) = the linear span by vectors v(XX j+1 ) on all incoming channels X X j+1 to X j+1 ; } } 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)ofF[(z)] to represent the z-transform of a stream of symbols a 0 , a 1 , a 2 , ,a t , 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, define the TILCM v as v(SX) = (1 0) T ,v(SY) = (0 1) T ,v(XY) = (zz 3 ) T v(YW) = (z 2 z) T and v(WX) = (z 3 z 2 ) By formal transformation we have for example v(XY) = (zz 3 ) = z(1 − z 3 )(1 0) T + z(z 3 z 2 ) T = z[(1 − z 3 )v(SX) + v(WX)] Thus, v(XY) is equal to z times thelinearcombination of v(SX)andv(WX)withcoefficients 1 − z 3 and 1, respectively. This specifies an encoding process for the channel XY 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) =  j≥0 a j z j and b(z) =  j≥0 b j z j belong to F[(z)]. The product of the information (row) vector with the (column) vector assigned to that channel represents the data stream transmitted over a channel [a(z) b(z)] · v(SX) = [a(z) b(z)] ·(1 0) T = a(z) → (a 0 , a 1 , a 2 , a 3 , a 4 , a 5 , ,a t , )[a(z) b(z)] ·v(SY) = b(z) → (b 0 , b 1 , b 2 , b 3 , b 4 , b 5 , ,b t , )[a(z) b(z)] ·v(XY) = za(z) + z 3 b(z) → (0, a 0 , a 1 , a 2 + b 0 , a 3 + b 1 , a 4 + b 2 , , a t−1 + b t−3 , )[a(z) b(z)] ·v(YW) = z 2 a(z) + zb(z) → (0, b 0 , a 0 + b 1 , a 1 + b 2 , a 2 + b 3 , a 3 + b 4 , , [...]... systems, Wireless Personal Commun., vol 16, 2001, pp 15–67 [30] L.-C Wang, K Chawla and L.J Greenstein, Performance studies of narrow-beam trisector cellular systems, Int J Wireless Inform Networks, vol 5, 1998, pp 89 102 [31] S Glisic et al Effective capacity of advanced wireless cellular networks (invited paper), in PIMRC2005, Berlin, 11–14 September 2005 [32] P.Gupta and P.R Kumar, The capacity of wireless. .. 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However, the LCM u defined by u(ST) = u(TW) = u(TY) = ( 10 ) T u(SU) = u(UW) = u(UZ) = ( 01 ) T (19.59) and u(WX) = u(XY) = u(XZ) = ( 10 ) T is not generic. This is seen by considering the set. a ‘good’ TILCM, then JWBK083-19 JWBK083-Glisic March 7, 2006 20:8 Char Count= 0 CAPACITY OF WIRELESS NETWORKS USING MIMO TECHNOLOGY 783 the heuristic algorithm calls for adjustments on the coefficients. TILCM can be further expanded to cover the block {Z} and then the block {T }. 19.6 CAPACITY OF WIRELESS NETWORKS USING MIMO TECHNOLOGY In this section an information theoretic network objective function