Advanced wireless networks 4g technologies phần 10 doc

At the sourceS , 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 othernodes.. Defini

NETWORK CODING 771

1.1 1.15 1.2 1.25 1.3 1.35 1.4 1.45 1.5

Y X

T

Y X

Figure 19.17 Illustration of network coding

As in Chapter 7, we define acommunication 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 inG is called a channel in the communication network (G , S) A channel in

graphG 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 nodeX to

another nodeY represents the capacity of direct transmission from X to Y We assume that,

every single channel has unit capacity

At the sourceS , 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 othernodes At each nonsource node which serves as a sink, the complete information generated

information As an example, consider the multicast of two data bits,b1 andb2, from thesourceS 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 b1 and channels

SU, UZ, UW and WY carry the bit b2 Note that, in this scheme, an intermediate node sendsout a data bit only if it receives the same bit from another node For example, the nodeT

receives the bitb1and sends a copy on each of the two channelsTY and TW Similarly, the

assume that there is no processing delay at the intermediate nodes

Unlike a conserved physical commodity, information can be replicated or coded Thenotion of network coding refers to coding at the intermediate nodes when information ismulticast 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 bitsb1andb2

from the sourceS to both the nodes Y and Z A solution is to let the channels ST, TW and

TY carry the bit b1, channelsSU, UW and UZ carry the bit b2, and channelsWX, XY and XZ

carry the exclusive-ORb1⊕ b2 Then, the nodeY receives b1andb1⊕ b2, from which thebitb2= b1⊕ (b1⊕ b2) can be decoded Similarly, the nodeZ can decode the bit b1fromb2

andb1⊕ b2asb1= b2⊕ (b1⊕ b2) The coding/decoding scheme is assumed to have beenagreed upon beforehand In order to discuss this issue in more detail, in this section we firstintroduce the notion of alinear-code multicast (LCM) Then we show that, with a ‘generic’

LCM, every node can simultaneously receive information from the source at rate equal toits max-flow bound After that, we describe the physical implementation of an LCM, firstwhen 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 samealgorithm can be applied to a cyclic network by expanding the network into an acyclicnetwork 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 aflow from the source to a nonsource node T is a collection

of channels, to be called thebusy 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) forany node other thanS 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 toT The number of outgoing busy channels from S will

be called thevolume 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 theidle

channels with respect to the flow

Definition 2

For every nonsource nodeT on a network (G , S), the maximum volume of a flow from the

source toT is denoted max f lo w G(T ), or simply mf(T) when there is no ambiguity.

Definition 3

means a collectionC of nodes which includes S but not T A channel XY is said to be in the

cutC 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 nodeT , 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

NETWORK CODING 773

information unit is taken as a symbol in the base field In other words, one symbol in thebase field can be transmitted on a channel every unit time

Definition 4

An LCMv 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 vectorsv(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 LCMv specified by

Note that, in the special case when the base field of is GF(2), the vector b1+ b2reduces

to the exclusive-ORb1⊕ b2in an earlier example

Proposition P1

For every LCMv on a network, for all nodes T dim[v(T )] ≤ m f (T ) To prove it fix a

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

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 thatmf (T) is an upper bound on the amount of information received by T when

an LCMv is used.

19.5.2 Achieving the max-flow bound through a generic LCM

In this section, we derive a sufficient condition for an LCMv to achieve the max-flow bound

on dim[v(T )] in Proposition 1.

Definition

condition holds for any collection of channelsX1Y1, X2Y2, , X mYmfor 1≤ m ≤ d : (∗) v(X k)⊂5{v(X j Y j) : j = k}6for 1≤ k ≤ m if and only if the vectors v(X1Y1), v(X2Y2), , v(X mYm) are linearly independent If v(X1Y1), v(X2Y2), , v(X mYm) are linearly in-

dependent, thenv(X k)⊂5{v(X jY j) : j = k}6sincev(X kYk)∈ v(X k) A generic LCM quires that the converse is also true In this sense, a generic LCM assigns vectors which are

re-as linearly independent re-as possible to the channels

With respect to the communication network in Figure 19.17(b), the LCMv defined by

Equation (19.58) is a genericLCM However, the LCM u defined by

and

is not generic This is seen by considering the set of channels{ST, W X} where

u(S) = u(W) =

7

10



,



01

8

Thenu(S) ⊂ u(WX) and u(W) ⊂ u(ST), but u(ST) and u(WX) are not linearly

inde-pendent Therefore,u is not generic Therefore, in a generic LCM v any collection of

channelsXY1, XY2, , XY mfrom a nodeX with m ≤ dim[v(X)] must be assigned linearly

independent vectors byv.

Theorem T1

Ifv is a generic LCM on a communication network, then for all nodes T, dim [v(T )] =

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) for any cut C between S and T We will show

that dim[v(T )] ≥ f by contradiction Assume dim[v(T )] < f and let A be the collection of

cutsU between S and T such that dim(U ) < f Since dim[v(T )] < f implies V \{T } ∈ A,

wherev is the set of all the nodes in G, A is nonempty.

NETWORK CODING 775

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 cutU \{W}, is contained by v(X, Y ) : (X, Y ) ∈ K  This

im-plies that dim(U\{W}) ≤ dim(U) < f is a contradiction Therefore, for all 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

sinced ≥ 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 sourceS for which every nonsource node T receives the message

at rate equal tomf(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

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 edgeUX, the value of a cut C between the

sourceS and node X (respectively, node Y ) is reduced by 1 if edge U X is in C, otherwise,

the value ofC is unchanged By the mfmcT, we see that mf(X) and mf(Y) are reduced by

at most 1 when edgeUX is removed from the graph Now consider the value of a cut C

between the sourceS and node Z If C contains node X , then edge UX is not in C, and,

therefore, the value ofC remains unchanged upon the removal of edge UX If C does not

contain nodeX , then C is a cut between the source S and node X By the mfmcT, the value

ofC 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 whichmf(T ) = 4, 3 or 1 for nodes T in the network.

The sourceS is to broadcast 12 symbols a1, , a12taken from a sufficiently large base

field F (Note that 12 is the least common multiple of 4, 3 and 1.) Define the set Ti = {T :

m f (T ) = i}, for i = 4, 3, 1 For simplicity, we use the second as the time unit We now

describe howa1, , a12 can be broadcast to the nodes in T4 ,T3, T1, in 3, 4 and 12s,

respectively, assuming the existence of an LCM on the network ford = 4, 3, 1.

(1) Let v1 be an LCM on the network with d = 4 Let α1= (a 1 a2 a3 a4) , α2=

(a5a6 a7 a8) and α3= (a 9 a10 a11 a12) In the first second, transmit α1 as the formation vector usingv1, in the second second, transmitα2, and in the third second,transmitα3 After 3s, after neglecting delay in transmissions and computations all

in-the nodes in T4can recoverα 1 , α 2andα 3.

(2) Letr be a vector in F4 such that{r} intersects trivially with v1(T ) for all T in

T3 , i.e.{r, v1(T )} = F4 for allT in T3 Such a vector r can be found when F is

sufficiently large because there are a finite number of nodes in T3 Definebi = α ir

fori = 1, 2, 3 Now remove incoming edges of nodes in T 4, if necessary, so that

mf(T ) becomes 3 if T is in T4, otherwise,mf(T ) remains unchanged This is based on

Lemma 1) Letv2be anLCM on the resulting network with d = 3 Let β = (b 1 b2 b3)and transmitβ as the information vector using v2in the fourth second Then all the

nodes in T3can recoverβ and hence α1, α2and α3.

(3) Lets1ands2be two vectors inF3such that{s1, s2} intersects with v2(T ) trivially forallT in T1 , i.e.{s1, s2, v2(T )} = F3for allT in T1 Defineγ i = βs i fori = 1, 2.

Now remove incoming edges of nodes in T4 and T3, if necessary, so thatmf(T )

becomes 1 ifT is in T4or T3, otherwise,mf(T ) remains unchanged Again, this is

based on Lemma 1) Now letv3be anLCM on the resulting network with d= 1 In

the fifth and the sixth seconds, transmitγ1andγ2as the information vectors using

v3 Then all the nodes in T1can recoverβ.

(4) Lett1 andt2 be two vectors in F4 such that{t1, t2} intersects with {r, v1(T )}

trivially for allT in T1 , i.e {t1, t2, r, v1(T )} = F4for allT in T1 Define δ1= α1t1

andδ2= α1t2 In the seventh and eighth seconds, transmitδ1andδ2as the informationvectors usingv3 Since all the nodes in T1already knowb1, upon receivingδ1and

δ2, α1can then be recovered

(5) Defineδ 3 = α 2t1andδ 4 = α 2t2 In the ninth and tenth seconds, transmitδ 3andδ 4asthe information vectors usingv3 Thenα 2 can be recovered by all the nodes in T1

(6) Defineδ 5 = α 3t1andδ 6 = α 0t2 In the eleventh and twelveth seconds, transmit δ 5

andδ 6as the information vectors usingv 3 Then α 3can be recovered by all the nodes

in T1.

So, in theith second for i = 1, 2, 3, via the generic LCM v1, each node in T4receivesall four dimensions ofα i, each node in T3receives three dimensions ofα i, and each node

in T1 receives one dimension ofα i In the fourth second, via the generic LCM v2, each

node in T3receives the vectorβ, which provides the three missing dimensions of α1, α2

andα3(one dimension for each) during the first 3 s of multicast byv1 At the same time,

each node in T1receives one dimension ofβ Now, in order to recover β, each node in T1

needs to receive the two missing dimensions ofβ during the fourth second This is achieved

by the generic LCMv3in the fifth and sixth seconds So far, each node in T1has receivedone dimension ofα i fori = 1, 2, 3 via v1 during the first 3 s, and one dimension ofα i

fori = 1, 2, 3 from β via v2andv3during the fourth to sixth seconds Thus, it remains

to provide the six missing dimensions ofα1, α2andα3(two dimensions for each) to each

node in T1, and this is achieved in the seventh to the twelfth seconds via the generic LCM

v3 The previous scheme can be generalized to arbitrary sets of max-flow values

NETWORK CODING 777

19.5.3 The transmission scheme associated with an LCM

Letv be an LCM on a communication network (G, S), where the vectors v(SX) assigned

to outgoing channelsSX linearly span a d-dimensional space As before, the vector v(XY )

assigned to a channelXY 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 ad-dimensional

row vector, called the information vector Under the transmission scheme prescribed bythe LCMv, the data flowing over a channel XY is the matrix product of the information

vector with the column vectorv(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 largerindexed node On an acyclic network, a straightforward realization of the above transmissionscheme is as follows Take one node at a time according to the sequential indexing Foreach node, ‘wait’ until data is received from every incoming channel before performing thelinear encoding Then send the appropriate data on each outgoing channel This physicalrealization of an LCM over an acyclic network, however, does not apply to a network thatcontains a directed cycle This is illustrated by the following example

Example 2

Letp , q, and r be vectors in , where p and q are linearly independent Define v(SX) =

p , v(SY ) = q and v(W X) = v(XY ) = v(Y W) = r.

This specifies anLCM 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 ofnodes Specifically, the law of information flow is observed for each of the nodesX , 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

Y WW XU

S

Figure 19.18 An LCM on a cyclic network

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 nodeS and all the pairs of the type [X , t], where X is a nonsource node in G

andt 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 nodesX and Y in (G , S):

(1) fort ≤ τ, the multiplicity of the channel from S to [X, t] is the same as that of the

channelS X in the network (G , S);

(2) fort < τ, the multiplicity of the channel from [X, t] to [Y, t + 1] is the same as

that of the channelX Y in the network (G , S);

(3) for t < τ, the multiplicity of the channel from [X, t] to [X, τ] is equal to

Lemma 2

The memoryless communication network (G(τ) , S) is acyclic.

Lemma 3

There exists a fixed numberε, 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 ryless’ transmission of data streams over the network (G , S) as follows:

‘memo-(1) A symbol sent fromS to [X , t] in (G(τ) , S) corresponds to the symbol sent on the

channelS X 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 X Y 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 nodeX 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

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 networkwith memory, denoted as (G[τ] , S), is defined as follows The set of nodes

inG[τ]includes the nodeS 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 nodesX and Y in (G , S);

(1) fort ≤ τ, the multiplicity of the channel from S to [X, t] is the same as that of the

channelS X in the network (G , S);

(2) fort < τ, the multiplicity of the channel from [X, t] to [Y, t + 1] is the same as that

of the channelX Y in the network (G , S);

(3) for t < τ, the multiplicity of channels from [X, t] to [X, t + 1] is equal to t ×

m f G(X )

(4) The communication network (G[τ] , S) is acyclic.

(5) Every flow from the source to the nodeX 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) EveryLCM 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 X0= S, X1, X2, , X n

such that every channel is from a smaller indexed node to a larger indexed node Thefollowing procedure constructs an LCM by assigning a vectorv(XY ) to each channel XY ,

one channel at a time

{

for all channels X Y

v(XY ) = the zero vector; // initialization

ξ of at most d − 1 channels with

v(X j)⊂ v(U Z) : U Z ∈ ξ;

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; }

}

The essence of the above procedure is to construct the generic LCM iteratively and makesure 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 elementa(z) of F[(z)] to represent the z-transform

of a stream of symbolsa0, a1, a2, , a t , that are sent on a channel, one symbol at

a time The formal variablez is interpreted as a unit-time shift In particular, the vector

assigned to an outgoing channel from a node isz 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 inFigure 19.18, define the TILCMv as

Thus,v(XY ) is equal to z times the linear combination of v(SX) and v(W X) with coefficients

1− z3and 1, respectively This specifies an encoding process for the channelX Y that does

not change with time It can be seen that the same is true for the encoding process of everyother 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

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) → (a0, a1, a2, a3, a4, a5, , a t , )[a(z) b(z)] · v(SY )

= b(z) → (b0, b1, b2, b3, b4, b5, , b t , )[a(z) b(z)] · v(XY )

= za(z) + z3b(z) → (0, a0, a1, a2+ b0, a3+ b1, a4+ b2, ,

at−1+ b t−3, )[a(z) b(z)] · v(Y W)

= z2a(z) + zb(z) → (0, b0, a0+ b1, a1+ b2, a2+ b3, a3+ b4, ,

NETWORK CODING 781

at−2+ b t−1, )[a(z) b(z)] · v(W X)

= z3a(z) + z2b(z) → (0, 0, b0, a0+ b1, a1+ b2, a2+ b3, ,

at−3+ b t−2, )

Adopt the convention thatat = b t = 0 for all t < 0 Then the data symbol flowing over the

channelX Y , for example, at the time slot t is at−1+ b t−3for allt ≥ 0 If infinite loops are

allowed, then the previous definition of TILCMv is modified as follows:

v(SX) = (1 0)T, v(SY ) = (0 1)T

v(XY ) = (1 − z3)−1(z z3)T, v(Y W) = (1 − z3)−1(z2 z)T

and

v(W X) = (1 − z3)−1(z3 z2)The data stream transmitted over the channelX Y , for instance, is represented by

That is, the data symbolat−1+ a t−4+ a t−7+ · · · + b t−3+ b t−6+ b t−9+ · · · is sent on the

channel X Y at the time slot t This TILCM v, besides being time invariant in nature, is a

‘memoryless’ one because the following linear equations allows an encoding mechanismthat requires no memory:

There are potentially various ways to define ageneric TILCM and, to establish desirable

dimensions of the module assigned to every node In this section we present a heuristicconstruction procedure based on graph-theoreticalblock decomposition of the network For

the sake of computational efficiency, the procedure will first remove ‘redundant’ channelsfrom the network before identifying the ‘blocks’ so that the ‘blocks’ are smaller A channel

in a communication network is said to beirredundant if it is on a simple path starting at

the source otherwise, it is said to beredundant Moreover, a communication network is

said to beirredundant if it contains no redundant channels In the network illustrated in

Figure 19.19, the channelsZ X , T X and T Z are redundant.

The deletion of a redundant channel from a network results in a subnetwork with the sameset of irredundant channels Consequently, the irredundant channels in a network define an

Wss X

Z

T

Figure 19.19 Redundant channels in a network

irredundant subnetwork It can be also shown that, ifv is an LCM (respectively, a TILCM)

on a network, then v also defines an LCM (respectively, a TILCM) on the subnetwork

that results from the deletion of any redundant channel In addition we say that two nodesare equivalent if there exists a directed path leading from one node to the other and viceversa An equivalence class under this relationship is called ablock in the graph The source

node by itself always forms a block When every block ‘contracts’ into a single node, theresulting graph is acyclic In other words, the blocks can be sequentially indexed so thatevery interblock channel is from a smaller indexed block to a larger indexed block.For the construction of a ‘good’ TILCM, smaller sizes of blocks tend to facilitate thecomputation The extreme favorable case of the block decomposition of a network is whenthe network is acyclic, which implies that every block consists of a single node The oppositeextreme is when all nonsource nodes form a single block exemplified by the network illus-trated in Figure 19.19 The removal of redundant channels sometimes serves for the purpose

of breaking up a block into pieces For the network illustrated in Figure 19.19, the removal

of the three redundant channels breaks the block{T, W, X, Y, Z} into the three blocks {T }, {W, X, Y } and {Z}.

In the construction of a ‘good’ LCM on anacyclic network, as before, the procedure

takes one node at a time according to the acyclic ordering of nodes and assigns vectors tooutgoing channels from the taken node For a general network, we can start with the trivialTILCMv on the network consisting of just the source and then expand it to a ‘good’ TILCM

v that covers one more block at a time.

The sequential choices of blocks are according to the acyclic order in the block position of the network Thus, the expansion of the ‘good’ TILCMv at each step involves

decom-onlyincoming channels to nodes in the new block A heuristic algorithm for assigning

vec-torsv(XY ) to such channels XY is for v(XY ) to be z times an arbitrary convenient linear

combination of vectors assigned to incoming channels toX In this way, a system of linear

equations of the form ofAx = b is set up, where A is a square matrix with the dimension

equal to the total number of channels in the network andx is the unknown column vector

whose entries arev(XY ) for all channels XY The elements of A and b are polynomials in z.

In particular, the elements ofA are either ±1, 0, or a polynomial in z containing the factor z.

Therefore, the determinant ofA is a formal power series with the constant term (the zeroth

power ofz) being ±1, and, hence is invertible in F(z) According to Cramer’s rule, a unique

solution exists This is consistent with the physical intuition because the whole network

is completely determined once the encoding process for each channel is specified If thisunique solution does not happen to satisfy the requirement for being a ‘good’ TILCM, then

CAPACITY OF WIRELESS NETWORKS USING MIMO TECHNOLOGY 783

the heuristic algorithm calls for adjustments on the coefficients of the linear equations onthe trial-and-error basis

After a ‘good’ TILCM is constructed on the subnetwork formed by irredundant channels

in a given network, we may simply assign the zero vectors to all redundant channels

Example

After the removal of redundant channels, the network depicted by Figure 19.19 consists

of four blocks in the order of{S}, {W, X, Y }, {Z} and {T } The subnetwork consisting

of the first two blocks is the same as the network in Figure 19.18 When we expand thetrivial TILCM on the network consisting of just the source to cover the block{W, X, Y },

a heuristic trial would be

v(SX) = (1 0) T v(SY ) = (0 1) T

together with the following linear equations:

v(XY ) = zv(SX) + zv(W X) v(Y W) = zv(SY ) + zv(XY )

and

v(W X) = zv(Y W).

The result is the memoryless TILCMv in the preceding example This 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 is formulated, whichtakes full advantage of multiple input multiple output (MIMO) channels The demand forefficient data communications has fueled a tremendous amount of research into maximizingthe performance of wireless networks Much of that work has gone into enhancing the design

of the receiver, however considerable gains can be achieved by optimizing the transmitter

A key tool for optimizing transmitter performance is the use of channel reciprocity ploitation of channel reciprocity makes the design of optimal transmit weights far simpler Ithas been recognized that network objective functions that can be characterized as functions

Ex-of the output signal-to-interference noise ratio (SINR) are well suited to the exploitation

of reciprocity, since it permits us to relate the uplink and downlink objective functions[48–50] It is desirable, however, to relate the network objective function to informationtheory directly, rather thanad-hoc SINR formulations, due to the promise of obtaining

optimal channel capacity This section demonstrates a link between the information oretic, Gaussian interference channel [51] and a practical network objective function thatcan be optimized in a simple fashion and can exploit channel reciprocity The formulation

the-of the objective function permits a water filling [52] optimal power control solution, thatcan exploit multipath modes, MIMO channels and multipoint networks

Consider the network of transceivers suggested by Figure 19.20 Each transceiver islabeled by a node number The nodes are divided into two groups The group 1 nodestransmit in a given time-slot, while the group 2 nodes receive In alternate time-slots, group

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 Thisassumption allows the channel between any transmit antenna and any receive antenna to bemodeled as a single complex gain and is consistent with an orthogonal frequency divisionmultiplexing (OFDM) modulation scheme All transceivers are assumed to have eithermultiple antennas or polarization diversity, for both transmit and receive processing Thelink connecting nodes may experience multipath reflections For this analysis we assumethat the receiver synchronizes to a given transmitter and removes any propagation delayoffsets 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 2transceivers When node2is transmitting during the uplink, we model the channel between

the two nodes as a complex matrix H12(1, 2) of dimensionM1(1)× M2(2), whereM1(1)

is the size of the antenna array at node1, andM2(2) is the size of the array at node2.Polarization diversity is treated like additional antennae In the next TDD time slot, duringthe downlink, node1 will transmit and the channel from 1 to2 is described by thecomplex M2(1)× M1(2) matrix H21(2, 1) For every node pair (1, 2) that forms acommunications link in the network, we assign a MIMO channel link number, indexed byk

orm, 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 Wealso define the mapping from the MIMO link number to the associated group 1 node,1(k)and the associated group 2 node2(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 possiblymore than one transmission mode The set of all transmission modes over the entire network

is indexed byq 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 linknumbers will map to its MIMO channel link, via the mappingk(q) Because our network will

CAPACITY OF WIRELESS NETWORKS USING MIMO TECHNOLOGY 785

exploit channel reciprocity, we assume that every uplink symbol stream indexed byq has

an associated downlink symbol stream assigned to the same index Theqth uplink symbolstream is spread by an M1(q)× 1 complex vector g1(q), where M1(q)≡ M1{1[k(q)]}

Similarlyg2(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(qMc(k))

(19.61)and

G1(k)≡g1(q1), g1(q2), , g1(qMc(k))

(19.62)wherek(qi)= k and there are M c(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) (19.63)

x2(n; k)= i2(n; k)+ H21(k, k)G1(k)d1(n; k) (19.64)

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 trix for downlink transmission,n is a time/frequency index, that represents an independent

ma-reuse of the channel, either due to adjacent frequency channels (e.g adjacent OFDM

chan-nels) or due to multiple independent time samples, i1(n; k) is the interference vector seen

at node1(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)=

m =k

Hr t(k , m)G t(m)dt(n; m)+ ε r(n, k) (19.65)

whereε 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

thatt = 2 if r = 1, otherwise t = 1 when r = 2.

su-Considering therefore the channel model described by Equations (19.63) and (19.65), wecan write the mutual information between the source data vector and the received data vector

as [52, 53]:

I [xr(k); dt(k)]= log2I + R−1

i r i r(k)Rst s t(k)= C r t(k; Gt) (19.66)

where, Gt ↔ {g t(q)} represents all transmit weights stacked into a single parameter vector

and where we neglect then dependency for the random vectors xr(n; k) and dt(n; k) Theinterference and the signal covariance matrices are defined as:

and where Rε r ε r(k) is the covariance of the background noise vectorε r(n, k) The covariance

of the source statistics is assumed governed by the transmit weights Gt(k), therefore the

covariance of dt(k) is assumed to be the identity matrix

Let us now consider the introduction of complex linear beam-forming weights at thereceiver For each low level linkq, we assign a receive weight vector wr(q), r ∈ {1, 2} for

use at receiver r [k(q)] The weights, in theory, are used to copy the transmitted information

for the uplink case, highlighting the firstMc(k) low-level links, all assumed to be associatedwith the same MIMO-channel link The MIMO channel link indexk and sample index n are

omitted for simplicity The downlink case is the same after the 1 and 2 indices are swapped.Owing to the additional processing [52], the mutual information either remains the same

or is reduced by the application of the linear receive weights,

CAPACITY OF WIRELESS NETWORKS USING MIMO TECHNOLOGY 787

Figure 19.21 Uplink transceiver channel model

In addition to inequality, Equation (19.75), it can be also shown [54, 55] that, if the sourcesymbolsdt(q) are mutually independent then,

where the collection of all receive weights is stacked into the parameter vector Wr ↔

{wr(q)} Equations (19.75) and (19.76) demonstrate that

The decoupled capacity is implicitly a function of thetransmit weights gt(q) and the receive

Gaussian interference channel and hence the upper bounds in Equation (19.78)

First we write ˆdt(n; q) as,

Now the mutual information on the right-hand side of Equation (19.55a) can be written as,

wherek = k(q) In the following we will also use the fact that scrambling the transmit

weights by an orthonormal scrambling matrix does not change the values of the mutualinformations for the Gaussian interference channel In other words, if

where ˜ G ↔ {˜gt(q)} is the parameter vector of all stacked, scrambled transmit weights, ˜gt(q)

drawn from the columns of ˜ G t(k) The proof is based on the fact that the mutual information

is completely determined by the statistics, R i r i r(k) and Rs t s t(k) From Equations (19.67)

and (19.47), these depend only on the outer products Gt(m)GH

t (m), which are invariant

with respect to orthonormal scrambling because ˜ Gt(m) ˜GHt (m)= Gt(m)GH

t(m).Therefore

replacing Gt(m) with ˜Gt(m) does not change the mutual information Based on this we

have the following relation For any set of network wide transmit weights G, there exists a set of receive weights ˆ w r(q) and transmit weights ˜gt(q) such that for all k,

In References [54, 55] a technique called locally enabled global optimization (LEGO)

is designed to fully exploit the reciprocity theorem This technique transforms the timization over the transmit powers to one over the set of achievable output SINRs

op-CAPACITY OF WIRELESS NETWORKS USING MIMO TECHNOLOGY 789

in Equation (19.85) The LEGO algorithm can be efficiently implemented using cal information, and requires only an estimate of the post beamforming interferencepower, yr(q)≡ wH

lo-r(q)Ri r i r wr(q), and an estimate of the post-beamforming channel gain,

(3) For each linkp, estimate the associated post-beamforming interference power y1(p)

and relay this information back to the base station

(4) Update the base receiver weights for every link during uplink transmission:

γ (p)= α[ ˆγ (p) − γ (p)] + γ (p), for some 0 < α ≤ 1(α is initially set to 1).

(7) Update the downlink transmit powersπ2(q):

illus-modes [i.e eachQ(m) contains 16 links] The performance of LEGO is compared with a

standard power management algorithm, that seeks to transmit a constant power for eachlink from each base station, and a single antenna network, that uses frequency divisionmultiplexing to isolate each RU into a separate channel Background radiation is assumed

to be thermal white noise at room temperature, with an added 10 dB noise figure As can beseen from the figure, the performance improvement of the LEGO algorithm is significant

km

15 20 25 30 35 40 45 50 55 60 0

1 2 3 4 5 6 7 8 9 10

Smallest capacity

vs largest transmit power

Single antenna

dBm

LEGO Standard power management

(a)

(b)

Figure 19.22 (a) Cell network geometry; (b) LEGO performance: worst case capacity vs

worst transmit power (Reproduced by permission of IEEE [55].)

19.7 CAPACITY OF SENSOR NETWORKS WITH MANY-TO-ONE TRANSMISSIONS

In sensor networks the many-to-one throughput capacity is theper source data throughput,

when all or many of the sources are transmitting to a single fixed receiver or sink [58–63].Earlier in this chapter we have shown that the achievable per node throughput in a wirelessnetwork isθW /(n log n), whereW is the transmission capacity and n is the total number

of nodes in the network

CAPACITY OF SENSOR NETWORKS WITH MANY-TO-ONE TRANSMISSIONS 791

The result was based on the assumption that communications are one-to-one, and thatsources and destinations are randomly chosen It does not apply to scenarios where there arecommunication hot spots in the network Since many-to-one communication causes the sink

to become a point of traffic concentration, the throughput achievable per source node inthis case is reduced In this section we are only interested in the case where every sourcegets an equal (on average) amount of original data (not including relayed data) across to thesink This is because otherwise throughput can be maximized by having only the sensorsclosest to the destination transmit Equal share of throughput from every sensor is desiredfor applications like imaging where each sensor represents a certain region of the wholefield and data from each part are equally important When distributed data compression isused, this is again approximately the case However, when conditional coding is used thismay no longer be true, since the amount of processed data can vary from source to source

In order to achieve the above goal we use the following assumptions about the networkarchitecture

19.7.1 Network architecture

(1) The network is deployed in a field of circular shape There aren nodes, sources (we

will use nodes, sources and sensors interchangeably in subsequent discussions) ployed in a network A sink/destination is located at the center of the network/circle.Each node is not only a source of data, but also a relay for some other sources toreach the sink

de-(2) A network where the nodes are randomly placed following a uniform distributionwill be referred to as a randomly deployed network or a random network In such anetwork we have no direct control over the exact location of the nodes A networkwhere we can determine the exact locations of the nodes will be referred to a as anarbitrary network

(3) Two network organizational architectures are considered: (a) a flat architecture wherenodes communicate with the sink via possibly multi-hop routes by using peer nodes

as relays; and (b) hierarchical architecture, where clusters are formed so that sourceswithin a cluster send their data (via a single hop or multihop depending on thesize of the cluster) to a designated node known as the clusterhead The cluster-head can potentially perform data aggregation and processing and then forwarddata to the sink In this study, we will assume that the clusterheads serve as simplerelays and no data aggregation is performed We will also assume that the com-munication between nodes and clusterheads and communication between cluster-heads and the sink are on separate frequency channels so that the two layers do notinterfere

(4) Throughout the section we will assume that the sources transmit following a schedulethat consists of time slots

(5) To simplify the resulting expressions, we assume the field has an area of 1 Nodesshare a common wireless channel using omnidirectional antennas We assume nodesuse a fixed transmission power and achieve a fixed transmission range We adopt thecommonly used interference model, as earlier in this chapter LetXi andX be two

sources with distancedi , jbetween them Then the transmission fromXi toX jwill

be successful if and only if

di , j ≤ r and d k , j (19.76)

for any source Xk that is simultaneously transmitting In the followingr will be

referred to as the transmission range There are two interference concepts here Anode may interfere with another node that is transmitting if it is within distance2r

within 2r

r around the first transmitting node and a circle of radius r

transmitting node If the intended receiver is located within the overlapping areathe transmission will fail because of interference Therefore the two nodes need

to be at least 2rnode that is receiving if it is within distance r

following both will be generally referred to as interference range The distinctionwill be clear from the context Also note that this interference model, Equation(19.76), essentially implies that no nodes can receive more than one transmission

at a time We will also assume that no node can transmit and receive at the sametime

(6) The network scenario is depicted in Figure 19.23 The sink is placed at the center ofthis field It receives all data generated by sources in the network

(7) In the followingW refers to the transmission capacity of the channel in a flat network.

In a hierarchical networkW refers to the transmission capacity of the channel used

within clusters.Wrefers to the transmission capacity of the channel used from theheads to the sink The capacity is derived as a function of the transmission range,assuming the transmission range can provide connectivity

Sink Sources Single-hop Multihop

Figure 19.23 Many-to-one network scenario

CAPACITY OF SENSOR NETWORKS WITH MANY-TO-ONE TRANSMISSIONS 793

19.7.2 Capacity results

In this section we summarize capacity results for thenetwork defined above For formal

proofs of the results the reader is referred to References [58–63]

19.7.2.1 Capacity in a flat network

(1) The maximum per node throughput in the network is upper bounded byW /n.

(2) λ = W/n can be achieved when every source can directly reach the sink.

(3) λ = W/n is not achievable if not every source can directly reach the destination and

(4) λ = W/n may be achieved in an arbitrary network when not every source can directly

reach the destination andWhen the sink cannot directly receive from every source in the network, and assuming thatthe channel allocation does not take into account difference in traffic load, thenλ = W/n

is not achievable with high probability regardless of the value of

use the upper bound on throughput by deriving the maximum number of simultaneoustransmissions

Denote by Ar the area of a circle of radius r , i.e Ar = πr2 Let random variableVr

denote the number of nodes within an area of sizeArand assume a total area of 1 We thenhave [58–63]:

(5) In a randomly deployed network withn nodes,

(6) If a network has randomly deployed sources and the transmission ranger is such that

not all sources can directly reach the sink, then with high probability the throughputupper boundλ = W/n is not achievable.

(7) A randomly deployed network using multihop transmission for many-to-one munication can achieve throughput

com-λ ≥ Wπr2−√ε /n

with high probability, when no knowledge of the traffic load is assumed andε is as

given in (1)

In the following we will use a concept ofvirtual sources As an example consider a simple

network consisting of three sources and a sink, shown in Figure 19.24(a) The distancebetween adjacent nodes isr Regardless of the value of

interferes with all other sources in this network Therefore only one source can transmit

at a time The number of interfering neighbors for any of the sources is two, which is thehighest degree of the graph that represents the interference relationship in this network.Thus a schedule of length 3 allows all sources to transmit once during the schedule The

Figure 19.24 (a) Chain network (b) virtual sources.

load on the source closest to the sink, source 3, is 3λ, since it carries the traffic of all three

sources The achievable throughput is then calculated as 3λ = W/3, thus λ = W/9.

The way the schedule was calculated previously assigned the same share of the resources(time) to all the sources Since we used the source with the highest need of resource (the onecarrying the most traffic) to calculate the amount of resource needed, every other source iswasting resource In our example we are giving every source the possibility of making threetransmissions Source 3 does indeed need all three transmissions, but source 2 only needstwo and source 1 only needs one, hence a total of six transmissions Now consider a similarnetwork, only this time we have three sources that can reach the sink, shown in Figure19.24(b) We create a schedule where each one of the sources gets to transmit once andonce only However this time not all sources generate data Using labels shown in Figure19.24(b), source 1 generates a packet and transmits it to source 2a Source 2a relays thepacket to source 3a, which then relays it to the sink Then source 2b generates a packetand transmits it to source 3b, which relays it to the sink Finally source 3c generates andtransmits a packet to the destination We can view each raw of sources in this network as anequivalent of a single source in the previous example, i.e 2a and 2b combined are equivalent

to 2 in Figure 19.24(a); 3a, 3b and 3c combined are equivalent to 3, in terms of interferenceand traffic load We will define sources 2a, 2b and 3a, 3c as virtual sources in the sense that

they each represent one actual source in the network but they are co-located in one physicalsource Adopting this concept, in this network the highest number of interfering neighbors

is five (with a total of six virtual sources all in one interference area) and therefore thereexists a schedule of length 6 that enables every virtual source to transmit once Since thetraffic load is the same for all virtual sources, the resources will be shared equally and nosource will be wasting its share In this case we getλ = W/6 Note that this is the largest λ

that could be obtained for the example in Figure 19.24(a) This concept allows us to define

a ‘traffic load-aware’ schedule in the following way

(1) For each source node, create one virtual source for every source node whose trafficgoes through this node, including itself

(2) Counting all the virtual sources we can determine the number of interfering neighbors(virtual sources)k The new maximum degree of the interference graph is then k− 1

CAPACITY OF SENSOR NETWORKS WITH MANY-TO-ONE TRANSMISSIONS 795

(3) A schedule of lengths ≤ k exists which is equally shared among virtual sources.

(4) The achievable throughput per node is simply the share obtained by any virtualsource in the network, i.e.λ = W/s ≥ W/k.

The concept of virtual source is used in References [58–63] to prove the followingtheorems:

(8) A randomly deployed network using multi-hop transmission for many-to-one munication can achieveλ ≥ W/ h=

com-knowledge of the traffic load is assumed and

creates a cluster containing the sources closest to it Within each cluster the communication

is either via a single hop or via multihop, while the communication from clusterheads to thesink is assumed to be done via a single hop on a different channel We assume that clusterheads cannot transmit and receive simultaneously In order to avoid boundary problems,

we will assume there is at least a distance of 2(2r

will also assume that each cluster covers an area of same size, as though not necessarilythe same shape Following these two assumptions and using result (5), we have with highprobability that the number of nodes in each cluster is within (α nn) of n /H, where α nis suchthat limn→∞α n /n = ε Therefore the clusters essentially form a Voronoi tessellation of the

field, where every cluster (or Voronoi cell) contains a circle of radius 2r

sources located near the boundary between two clusters will not have a higher number ofinterfering neighbors (in terms of virtual sources), due to low traffic load, than the onescloser to the clusterheads Thus previous results are directly applicable and we do not have

to be concerned with the boundary

The question of interest is whether there exists an appropriate number of clustersH that

would allow the network to achieveλ = W/n with high probability using clustering, when

clusterheads have the same transmission capacityW as the sources That W /n remains to

be the upper bound is again obvious considering the fact that the sink cannot receive frommore than one node (at rateW ), and that there are n sources in the network In References

[58–63] the following results are proven:

(10) In a network using clustering, where cluster heads have the same transmissioncapacityW as the sources, there exists an appropriate number of clusters H and

an appropriate range of transmissionr that would allow the network to achieve

λ = W/n with high probability as n → ∞ The range of transmission r must

(11) In a network using clustering, where cluster heads have transmission capacity

W, there exists an appropriate number of clustersH and an appropriate range of

transmissionr , as n → ∞ , that allows the network to achieve λ = W/n with

high probability.W/n is also the upper bound on throughput in this scenario The

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Energy-efficient Wireless Networks

20.1 ENERGY COST FUNCTION

In Chapter 5 we discussed the impact of MAC layer protocols on energy efficiency, includingTCP controlled retransmissions In this chapter, we extend this analysis to the network layerand focus on routing algorithms We discuss how the error rate associated with a link affectsthe overall probability of reliable delivery, and consequently the energy associated with thereliable transmission of a single packet For any particular linki, j between a transmitting

node i and a receiving node j , let T i , j denote the transmission power and pi , j represent

the packet error probability Assuming that all packets are of a constant size, the energy

involved in a packet transmission, Ei , j , is simply a fixed multiple of Ti , j

Any signal transmitted is affected by two different factors: attenuation due to the medium,

and interference with ambient noise at the receiver The attenuation is proportional to D K,

where D is the distance between the receiver and the transmitter The bit error rate associated

with a particular link is essentially a function of the ratio of the received signal power to the

ambient noise In the constant-power scenario, Ti , j is independent of the characteristics ofthe linki, j and is a constant In this case, a receiver located further away from a transmitter

will suffer greater signal attenuation (proportional to D K) and will, accordingly, be subject

to a larger bit-error rate In the variable-power scenario, a transmitter node adjusts T i , j to

ensure that the strength of the (attenuated) signal received by the receiver is independent of

D and is above a certain threshold level Th The minimum transmission power associated with a link of distance D in the variable-power scenario is Tm = Th × γ × D K, whereγ is

a constant and K is the coefficient of channel attenuation (K ≥ 2) Since Th is typically a

technology-specific constant, we can see that the minimum transmission energy over such

a link varies as Em(D) ∝ D K

Advanced Wireless Networks: 4G Technologies Savo G Glisic

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If links are considered error-free, then minimum hop paths are the most energy-efficientfor the fixed-power case Similarly, in the absence of transmission errors, paths with a largenumber of small hops are typically more energy efficient in the variable power case However

in the presence of link errors, none of the above choices may give optimal energy efficientpaths We now analyze the consequences of this behavior for the variable-power scenarioand end-to-end (EER) and hop-by-hop (HHR) packet retransmission techniques The anal-ysis for the fixed-power scenario is simpler, and is a special case of the variable-powerscenario Energy consumption for additional signal processing in the transmitter/receiver(modulation/demodulation) will be neglected Modification of the models to include theselosses is straightforward

In the EER case, a transmission error on any link leads to an end-to-end retransmission

over the path Given the variable-power formulation of E m, it is easy to see why breaking up a

link of distance D into two shorter links of distance D1and D2 such that D1+ D2= D

always reduces the total Em To elaborate on this, let us consider communication between

a sender (S) and a receiver (R) separated by a distance D Let N represent the total number

of hops between S and R, so that N − 1 represents the number of forwarding nodes i:

i = {2, , N}, with node i referring to the (i − 1)th intermediate hop in the forwarding

path Node 1 refers to S and node N + 1 refers to R In this case, the total energy spent

in simply transmitting a packet once (without considering whether or not the packet was

reliably received) from the sender to the receiver over the N− 1 forwarding nodes is:

where Di , j refers to the distance between nodes i and j and α is a proportionality constant.

To understand the tradeoffs associated with the choice of N− 1, we compute the lowest

possible value of Etfor any given layout of N− 1 Using very simple symmetry arguments,

it is easy to see that the minimum transmission energy case occurs when each of the hops

are of equal length D /N In that case, Etis given by

We now consider how the choice of N affects the probability of transmission errors and the

consequent need for retransmissions Clearly, increasing the number of intermediate hopsincreases the likelihood of transmission errors over the entire path Assuming that each of

the N links has an independent packet error rate of plink, the probability of a transmission

error over the entire path, denoted by p, is given by p = 1 − (1 − plink)N.The number of transmissions (including retransmissions) necessary to ensure the suc-

cessful transfer of a packet between S and D is then a geometrically distributed random variable X , such that Pr{X = k} = p k−1× (1 − p), ∀k The mean number of individual

packet transmissions for the successful transfer of a single packet is thus 1/(1 − p) Since each such transmission uses total energy Etgiven above, the total expected energy required

in the reliable transmission of a single packet is given by:

MINIMUM ENERGY ROUTING 803

0.5 1.0 1.5 2.0 2.5 3.0

Figure 20.1 Total energy costs ( K= 2, EER).

By treating N as a continuous variable and differentiating, it follows that the optimal value

of the number of hops, Noptis given by:

Nopt= −(K − 1)/ log(1 − plink) (20.3)The existence of the optimum value is demonstrated in Figure 20.1

In the case of the HHR model, the number of transmissions on each link is independent

of the other links and is geometrically distributed The total energy cost for the HHR case with N intermediate nodes, with each hop being of distance D /N and having a link packet error rate of plink, is

In this case, it is easy to see that the total energy required always decreases with increasing

N One should be aware that in a practical system at some point when N is sufficiently

large, the signal processing energy will become comparable with the energy spent fortransmissions

20.2 MINIMUM ENERGY ROUTING

Energy-aware routing protocols typically compute the shortest-cost path, where the costassociated with each link is some function of the transmission (and/or reception) energyassociated with the corresponding nodes To adapt such minimum cost route determinationalgorithms (such as Dijkstra’s or the Bellman–Ford algorithm) for energy-efficient reliablerouting, the link cost must now be a function of not just the associated transmission energy,but the link error rates as well A link is assumed to exist between node pair{i, j} as long

as node j lies within the transmission range of node i This transmission range is uniquely

defined for the constant-power case For the variable-power case, this range is really the

maximum permissible range corresponding to the maximum transmission power of a sender Let Ei , j be the energy associated with the transmission of a packet over link li , j , and pi , jbe

the link packet error probability associated with that link In the fixed-power scenario, Ei , j

is independent of the link characteristics; in the variable-power scenario, Ei , j is a function

of the distance between nodes i and j Now, the routing algorithm’s job is to compute the

shortest path from a source to the destination that minimizes the sum of the energy costsover each constituent link

Choosing path P for communication between S and D implies that the total energy cost

Choosing a minimum-cost path from node 1 to node N+ 1 is thus equivalent to choosing

the path P that minimizes Equation (20.5) It is thus easy to see that the corresponding link cost for link Li , j , denoted Ci , j , is given by Ci , j = E i , j /(1 − p i , j) Ad-hoc routing protocols,

discussed in Chapter 13, such as AODV, DSR and TORA, can use this link cost to pute the appropriate energy-efficient routes Some of the existing energy-efficient routingtechniques, e.g PARO, can also be easily adapted to use this new link cost formulation tocompute minimum-energy routes Thus, in such a modified version of the PARO algorithm,

com-an intermediate node C would offer to interject itself between two nodes A com-and B if the sum of the link cost CA ,C + C C ,B was less than the ‘direct’ link cost CA ,B

In end-to-end retransmissions, the total energy cost along a path contains a multiplicative

term involving the packet error probabilities of the individual constituent links In fact,assuming that transmission errors on a link do not stop downstream nodes from relayingthe packet, the total energy cost can be now expressed as:

in-Ci , j = Ei , j

where L = 2, 3, , and is chosen to be identical for all links Clearly, if the exact path

length is known and all nodes on the path have identical link error rates and transmission

costs, L should be chosen equal to that path length However, in accordance with current

routing schemes, we require that a link should associate only a single link cost with itself,irrespective of the lengths of specific routing paths that pass through it Therefore, we

need to fix the value of L independent of the different paths that cross a given link If better knowledge of the network paths is available, then L should be chosen to be the average path length of this network Higher values of L impose progressively stiffer penalties on links

with non-zero error probabilities Given this formulation of the link cost, the minimum-cost

MAXIMIZING NETWORK LIFETIME 805

path computation effectively computes the path with the minimum “approximate” energycost given by:

As before, protocols like AODV, DSR, TORA and PARO can use this new link cost function

to make their routing decisions

20.3 MAXIMIZING NETWORK LIFETIME

We now discuss how we can include the retransmission-aware formulation of the link cost

in an algorithm, that attempts to increase the operational lifetime of multihop wireless

networks Unlike previous protocols, maximum reliable packet carrying capacity (MRPC)

considers both the node characteristics (residual battery energy at the transmitting node) andthe link characteristics (link distance and link error rates), while evaluating the suitability

of alternative paths Given the current battery power levels at the different nodes, MRPC

selects a route that has the maximum reliable packet carrying capacity among all possible

paths, assuming no other cross-traffic passes through the nodes on that path

To formalize the algorithm, let us assume that the residual battery power at a certain

instance of time at node i is Bi As before, let the transmission energy required by node i

to transmit a packet over linki, j to node j be E i , j Let the source and destination nodesfor a specific session (route) be S and D respectively If the route-selection algorithm then selects a path P from S to D that includes the link i, j, then the maximum number of

packets that node i can forward over this link is clearly Bi /E i , j Accordingly, we can define

a node-link metric, Mi , jfor the linki, j as:

P = min

The MRPC algorithm then selects the candidate route Pcthat maximizes the ‘lifetime’ of

communication between S and D Formally, the chosen route is such that:

Pc= arg max{ P |P ∈ all possible routes} (20.10)Given the cost and lifetime formulations for MRPC it is then easy to use a modified version

of Dijkstra’s minimum cost algorithm for decentralized route computation

To apply Dijkstra’s algorithm for determining the minimum-cost path, the distance metricfrom any node to the given destination should be defined as the value of Pover the optimal

path from that node to D Now consider a node A that sees advertisements from its neighbors,

{X, Y, Z, }, with corresponding distance metrics X, Y, Z , for a given destination

D Node A can then compute the best path to D (using its optimal neighbor) by using the

following simple algorithm:

(1) For each of the neighboring nodes ( j ∈ {X, Y, Z, }), compute the link cost M A , j

using Equation (20.8)

(2) For each of the neighboring nodes ( j ∈ {X, Y, Z, }) compute the potential new

value of potusing pot( A , j) = min{M A , j , j }.

(3) Select as the next-hop neighbor towards D the node which results in the maximum

value of pot, i.e choose node k such that k= arg maxj ε{X,Y,Z}{ pot( A , j)} and

assign A = pot( A , k).

Using this recursive formulation allows all nodes in the ad hoc network to iteratively build their optimal route towards a specific destination D The distance-vector formulation

presented here can easily be incorporated in protocols, such as AODV, DSR and TORA,

that are specifically designed for ad hoc mobile environments.

The basic MRPC formulation for power-aware routing does not need to specify the value

of the transmission energy cost associated with a specific link Note that Equation (20.8)

is expressed as a function of a generic link cost Mi , j Accordingly, by specifying different forms of Mi , jit is possible to tailor the MRPC mechanism for specific technologies and/orscenarios

For the fixed-power scenario, the energy involved in a single packet transmission attempt,

Ei , j, is a constant for alli, j and is independent of the distance between neighboring

nodes i and j For the variable-power scenario, Ei , j will typically be∝ D K

i , j , where Di , j

is the distance between nodes i and j A routing algorithm for reliable packet transfer

should include the link’s packet error probability in formulating the transmission energycost By ignoring the packet error probability, the link cost concentrates (wrongly) only on

the energy spent in transmitting a single packet The correct metric is the effective packet

transmission energy for reliable transmission, which includes the energy spent in one or morere-transmissions that might be necessary in the face of link errors A transmission energy

metric of the form Ci , j = E i , j /(1 − p i , j)L was suggested, where pi , j is the link’s packeterror probability, L ≥ 1 For hop-by-hop re-transmissions L should be chosen to be 1 In

the absence of hop-by-hop re-transmissions (i.e re-transmissions are only performed

end-to-end), the transmission cost is well approximated by L ∈ [3, 5] Power-aware routing has

been studied in a number of papers [1–42]

The conditional MRPC (CMRPC) algorithm is the MRPC equivalent of the conditional min–max minimum battery cost routing (CMMBCR) algorithm presented in Toh et al [2].

The CMMBCR algorithm is based on the observation that using residual battery energy as

the sole metric throughout the lifetime of the ad hoc network can actually lower the overall

lifetime, since it never attempts to minimize the total energy consumption Accordingly,the CMMBCR algorithm uses regular minimum-energy routing as long as there is even onecandidate path, where the remaining battery power level in all the constituent nodes liesabove a specified thresholdγ When no such path exists, CMMBCR switches to MMBCR,

i.e it picks the path with the maximum residual capacity on the ‘critical node’

The CMRPC algorithm differs from CMMBCR in that the cost-functions at all timesinclude the link-specific parameters (e.g error rates) as defined earlier in this chapter Thealgorithm can thus be specified as follows Let be the set of all possible paths between

MAXIMIZING NETWORK LIFETIME 807

the source S and destination D and let  represent the set of paths such that: for any route

Q ∈ , Q ≥ γ In other words  represents the set of paths whose most critical nodes

have a lifetime greater than a specified threshold The routing scheme thus consists of thefollowing actions:

(1) If = Ø (there are one or more paths with > γ , the algorithm selects a path

¯

Q= arg max

Q ∈ { Q |Q ∈ }

The thresholdγ is a parameter of the CMRPC algorithm A lower value of γ implies a

smaller protection margin for nodes nearing battery power exhaustion Accordingly, theperformance of the CMRPC algorithm will be a function ofγ

The performance example is based on network topology shown in Figure 20.2 The

corner nodes and the mid-points of each side of the rectangular grid were chosen as trafficsources and destinations; the bold lines in the figure show the session end-points [1].Each (source, destination) pair had two simultaneous sessions activated in the oppositedirection, giving rise to a total of 16 different sessions For the results reported here, eachsession consisted of a UDP traffic generated by a CBR source whose inter-packet gap wasdistributed uniformly between 0.1 and 0.2 s The error rate on each link was independentlydistributed uniformly between (0.05, pmax) Varying values of pmaxwere used Routes wererecomputed at 2 s intervals in these simulations to capture the effect of changes in theresidual packet capacity on the link metrics

Whenever nodes died (when its battery power gets completely drained) during the course

of a simulation, the simulation code would check whether the graph became partitioned Thesimulations were run until each of the 16 sessions failed to find any route from their source

to the corresponding destination To avoid the termination of a simulation due to batterypower exhaustion at source or destination nodes, all source and sink nodes were configured

AFigure 20.2 Simulation scenario

60 80 100 120 140 0

5 10 15 20 25

30

min-hop min-energy mmbcr mrpc cmmbcr cmrpc

To study the performance of the various algorithms, experiments were performed where

the maximum transmission radius, R, of each node was varied Figure 20.2 shows the set

of neighboring nodes for a corner node when the transmission radius is set to 1.5 Theexpiration sequence, as well as the node expiry times were noted, for each simulation

The expiration sequence (sorted in ascending order of the expiration times) provides a

useful indicator of how each algorithm affects the lifetime of the individual nodes, and the

entire network In addition to the expiration sequence, the total packet throughput was also

calculated by counting the total number of packets successfully received at the destination

nodes, and the energy costs per packet by dividing the total energy expenditure by the total

packet throughput Except for the expiration sequences, all other metrics were obtained

by averaging over multiple runs The results are shown in Figures 20.3–20.5 From theseresults one can see that CMRPC/MRPC outperforms other options

20.4 ENERGY-EFFICIENT MAC IN SENSOR NETWORKS

Among the requirements for MACs in wireless sensor networks, energy efficiency is ically the primary goal In these systems, idle listening is identified as a major source ofenergy wastage Measurements show that idle listening consumes nearly the same power asreceiving Since in sensor network applications traffic load is very light most of the time, it

typ-is often desirable to turn off the radio when a node does not participate in any data delivery.Some schemes put (scheduled) idle nodes in power-saving mode (SMAC) and switch nodes

to full active mode when a communication event happens Although a low duty cycle MAC

is energy-efficient, it has three side-effects

(1) It increases the packet delivery latency At a source node, a sampling reading mayoccur during the sleep period and has to be queued until the active period Anintermediate node may have to wait until the receiver wakes up before it can forward

hop energymmbcr mrpc cmmbcr cmrpc6000

min-0 1 2 3 4 5 6

Figure 20.4 (a) Total packet throughput; (b) average transmission energy per received

packet (UDP sources), R = 1.5 (Reproduced by permission of IEEE [1].)

0 1 2 3 4 5 6

min-Algorithms

hop energymmbcr mrpc cmmbcr cmrpc

min-Algorithms

Figure 20.5 CMRPC: (a) total packet throughput; (b) average transmission energy per

received packet vs the protection threshold (Reproduced by permission ofIEEE [1].)

809

a packet received from its previous hop This is called sleep latency in SMAC, and it

increases proportionally with hop length by a slope of schedule length (active periodplus sleep period)

(2) A fixed duty cycle does not adapt to the varying traffic rate in sensor network Afixed duty cycle for the highest traffic load results in significant energy wastagewhen traffic is low while a duty cycle for low traffic load results in low message datadelivery and long queuing delay Therefore it is desirable to adapt the duty cycleunder variant traffic load

(3) A fixed synchronous duty cycle may increase the possibility of collision If boring nodes turn to active state at the same time, all may contend for the channel,making a collision very likely There are several possibilities to reduce sleep delayand adjust duty cycle to the traffic load Those mechanisms are either implicit, inwhich nodes remain active on overhearing an ongoing transmission or explicit, inwhich there are direct duty cycle adjusting messages In adaptive listening, a node thatoverhears its neighbor’s transmission wakes up for a short period of time at the end

neigh-of the transmission, so that if it is the next hop neigh-of its neighbor, it can receive themessage without waiting for its scheduled active time A node also can keep listen-ing and potentially transmitting as long as it is in an active period An active periodends when no activation event has occurred for a certain time The activation timeevents include reception of any data, the sensing of communication on the radio, theend-of-transmission of a node’s own data packet or acknowledgement, etc

If the number of buffered packets for an intended receiver exceeds a threshold L, the sender

can signal the receiver to remain on for the next slot A node requested to stay awake sends

an acknowledgement to the sender, indicating its willingness to remain awake in the nextslot The sender can then send a packet to the receiver in the following slot The request isrenewed on a slot-by-slot basis

However, in previous mechanisms (whether explicit or implicit), not all nodes beyondone hop away from the receiver can overhear the data communication, and therefore packet

forwarding will stop after a few hops This data forwarding interruption problem causes

sleep latency for packet delivery

DMAC employs a staggered active/sleep schedule to solve this problem and enable continuous data forwarding on the multihop path In DMAC, data prediction is used to

enable active slot request when multiple children of a node have packets to send in a same

sending slot, while the more to send packet is used when nodes on the same level of the

data gathering tree with different parents compete for channel access

20.4.1 Staggered wakeup schedule

For a sensor network application with multiple sources and one sink, the data delivery paths

from sources to sink are in a tree structure, a data gathering tree Flows in the data gathering

tree are unidirectional from sensor nodes to sink There is only one destination, the sink Allnodes except the sink will forward any packets they receive to the next hop The key insight

in designing a MAC for such a tree is that it is feasible to stagger the wake-up scheme so

that packets flow continuously from sensor nodes to the sink DMAC is designed to deliver data along the data gathering tree, aiming at both energy efficiency and low latency.

paper), inPIMRC2005, Berlin, 11–14 September 2005.

[32] P.Gupta and P.R Kumar, The capacity of wireless networks, ... for wirelessnetworks,Proc IEEE Wireless Commun Networking Conf., 2000, p 294.

[39] J Laneman, G.Wornell and D Tse, An efficient protocol for realizing tive diversity in wireless networks, ... such

a link varies as Em(D) ∝ D K

Advanced Wireless Networks: 4G Technologies< /small> Savo G Glisic

C