Table Of ContentA General Method for the Design of Tree Networks
Under Communication Constraints
Alla Tarighati, and Joakim Jalde´n
ACCESS Linnaeus Centre, Department of Signal Processing,
KTH Royal Institute of Technology, Stockholm, Sweden
Email: {allat, jalden}@kth.se
5 Abstract—We consider a distributed detection system with optimization gives a necessary, but in general not sufficient,
1 communicationconstraints,whereseveral nodesarearranged in condition for an optimal strategy [4].
0 anarbitrarytreetopology,undertheassumptionofconditionally
2 independentobservations. We proposea cyclic design procedure Whilederivingdecisionfunctionforonenodeintheperson-
using the minimum expected error probability as a design by-person methodology for the design of nodes in a general
n
criterion while adopting a person-by-person methodology. We
treenetwork(includingparallelandtandemnetworks)allother
a
design each node jointly together with the fusion center, while
J nodes and the FC are assumed to have already been designed
other nodes are kept fixed, and show that the design of each
2 nodeusingtheperson-by-personmethodologyisanalogoustothe and remain fixed. Focusing on person-by-personoptimality, a
2 designofanetworkwithtwonodes,anetworkwhichwereferto typical result is that if observations at the nodes are indepen-
as the restricted model. We further show how the parameters in dentconditionedontruehypothesis,likelihoodratioquantizers
] therestrictedmodelforthedesignof anodeinthetreenetwork
T are person-by-personoptimal, while the optimal thresholdsin
canbefoundinacomputationallyefficientmanner.Theproposed
I numerical methodology can be applied for the design of nodes thequantizersaregivenbythesolutionofsystemsofnonlinear
.
s arrangedinarbitrarytreetopologieswitharbitrarychannelrates equations,withasmanyvariablesasthenumberofthresholds.
c for the linksbetween nodes and for a general M-ary hypothesis This however makes the computation of optimal thresholds
[
testing problem. intractable, even for a moderate size network [4], [6].
2 IndexTerms—Decentralizeddetection,Bayesiancriterion,tree
v topology, person-by-person optimization. Themaincontributionofthisworkistoshowthat–contrary
5 to previous claims – it is possible to under the person-by-
0 person methodology design a distributed detection network
I. INTRODUCTION
8 arranged in an arbitrary tree topology with a reasonable
0
We considera distributed,ordecentralized,hypothesistest- computational burden. In order to do that we modify the
0
ingprobleminageneraltreenetworkconfiguredasadirected person-by-personmethodology for the design of nodes in the
.
1
graph, where observations are made at spatially separated tree topology by letting the FC update its decision function
0
nodes. The root of the graph is the fusion center (or FC), and in every iteration together with the nodes. In other words,
5
1 informationfromnodespropagatetowardtheFC. Ifthenodes we adopt a person-by-person methodology in which at every
: are able to communicate all their data to the FC, there is no iteration each node is designed jointly together with the FC.
v
fundamental difference from the centralized case, where the We furtherassumethattheFCusesthemaximuma-posteriori
i
X classical solution is to use threshold tests on the likelihood (MAP) rule to make the final decision, which is motivated by
r ratios of the received data at the FC. However if there are theoptimalityoftheMAPrulewhentheperformancecriterion
a
communication constraints on the links between the nodes, is global error probability, or error probability at the FC.
the nodes need to carry out some processing and give a
In order to obtain a tractable solution during the design
summarized, or quantized, version of their data as output.
of a node, let say m (together with the FC), all other
The problem of optimal decentralized hypothesis testing 0
nodes are modeled using a Markov chain and it will be
has gained noticeable interest over the last 30 years, see
shown that the design of m is analogous to the design of
for instance [1]–[5] and references therein. A common goal 0
a special case of a network with only two nodes (which is
in these references is to find a strategy which optimizes a
here called the restricted model). Then we will show how the
performancemeasure, like minimizingthe errorprobabilityat
parameters for this restricted model can be found recursively
theFC.However,itisdifficulttoderivetheoptimalprocessing
in a computationally efficient way from the original network.
strategies at the nodes in distributed networks, even for small
size networks. Therefore most of the works on this topic Thispaperisorganizedasfollows.InSectionIIwe present
focusonperson-by-personoptimizationasa practicalwayfor our model in detail and formulate the problem. In Section III
the design of decentralizednetworks.Using person-by-person we introduce the restricted model and describe how it can be
optimization, it is guaranteed that the overall performance at obtainedfromtheoriginaltreenetwork.We presentnumerical
the FC is improved (or, at least not worsened) with every examples to illustrate the benefit of proposed approach in
iteration of the algorithm. Unfortunately person-by-person Section IV and Section V concludes the paper.
m3 m6 equal to the cardinality of its observation space. Let C be
l
the set of all leaves and C be the set of all relays in the
m5 r
m2 network. In a tree network, each leaf mi ∈ Cl using its own
m4 observation xi ∈ Xi makes a decision ui ∈ Mi and sends
it through a rate-constrained channel (e ) to its immediate
i,j
m1 successor m (which is a relay or FC). Each relay m ∈ C ,
j i r
using input messages from all of its immediate predecessors
FC
I , makes a decision u ∈ M and sends it through a rate-
i i i
uN constrained channel to its immediate successor. Eventually
m7 m10 the fusion center makes the final decision uN from the set
{0,1,...,M −1} in an M-ary hypothesis testing problem.
In this paper, we restrict our attention to discrete observation
m8 m9
spaces X where m ∈ C . However we wish to stress that
i i l
Fig.1. Anexample network(observations arenotshown). any continuous observation space can be approximated by
a discrete observation space, by representing the continuous
space by a set of intervals indexed by x from the discrete
i
space [7], [8].
II. PRELIMINARIES
The channel between node m and its successor is consid-
i
We describe a tree network by a directed and acyclic
ered to be an error-freebut rate-constrainedchannelwith rate
graph where the fusion center is the root of the network and
R bits. The output of node m is then from a discrete set
i i
itnhfeorromoatt.ioWnefldoewnsotferotmheetvreeerynneotwdoeroknbya uTni,qu(eVp,aEth),towwhaerrde Mi with cardinality kMik=2Ri. Without loss of generality
we assume that the output of node m is from the discrete
i
V ={m1,m2,...,mN}isthesetofN nodesandE ={ei,j} set Mi = {1,...,2Ri}. In this setup each node is a scalar
isthesetofdirectededgesfromnodem tonodem .Without
i j quantizer which maps its inputs to an outputmessage using a
lossofgeneralityweassumethatnodem isthefusioncenter
N decision function γ . A leaf m maps its observation x to an
i l l
(labelled FC in Fig. 1).
output message u using the function γ :X →M , i.e.,
l l l l
We say that node m is the predecessor of node m if
i j
there is a directed path from node mi to node mj, and say γl(xl)=ul,
that node m is a successor of node m . Accordingly we
j i whereasarelaym mapsitsinputvectorcontainingmessages
r
say that node m is an immediate predecessor of node m if
i j from its immediate predecessors {u : m ∈I } to an output
i i r
e ∈ E (equivalently node m is the immediate successor
i,j j message u using the function γ :M →M , i.e.,
of node m ). This is exemplified in Fig. 1 where nodes r r Ir r
i
m and m are predecessors of node m , while m in an γ ({u :m ∈I })=u .
3 2 1 2 r i i r r
immediate predecessor of node m . The set of all immediate
1 M is defined as the product of alphabet of immediate
predecessors to node m is denoted by I , and the set of all Ir
i i predecessors to node m . For example, relay m in Fig. 1
immediate predecessorsto the fusion center is denotedby I . r 7
f has three immediate predecessors, I = {m ,m ,m },
Forinstance,I ={m ,m ,m } inFig. 1. We also defineS 7 8 9 10
f 1 4 7 i with decision spaces M , M and M , respectively, and
asasetconsistingofnodem andallitssuccessors,excluding 8 9 10
i M = M × M × M . We will use the terminology
the FC. In other words, S is the set of all nodes the input I7 8 9 10
i “message” and “index” interchangeably to denote the output
messages of node m pass through to reach the FC, e.g.,
i of a node in the network.
S = {m ,m ,m } in the example of Fig. 1. We further
6 6 5 4
define the last successor ls(m ) of node m as the last node We assume that the observations at the leaves, conditioned
i i
that the input of node m passes through before it reaches on the hypotheses, are independent. Then acyclicity of the
i
the FC, i.e., ls(m ) = m , if m ∈ S and m ∈ I . We let network implies that the inputs to each relay, and also to the
i l l i l f
T ,(V ,E )definethesub-treeofthenetworkwithnodem FC, areconditionallyindependent.We furtherassumethatthe
i i i i
as its root, where V and E are the set of nodes and directed conditional probability masses of the observations are known
i i
edges in sub-tree Ti. and denoted Pj(xl) , P(xl|Hj), j = 0,1,...,M −1, for
We assume that there are two types of nodes in the tree an M-ary hypothesis testing problem.
network: leaves and relays. A leaf is a node which makes Inthispaper,theobjectiveistoarriveatasimplemethodfor
observation and a relay is a node that only receives messages the design of the nodes decision functions, γ ,...,γ , in the
1 N
from its immediate predecessors. In Fig. 1 node m and m treenetwork,insuchawaythattheglobalerrorprobabilityat
3 2
exemplify a leaf and a relay, respectively. Without loss of theFCisminimized.Inordertoderivethedecisionfunctionat
generality nodes which both make an observation and receive anode,weusetheperson-by-personmethodologyandassume
messages from their immediate predecessors are considered that all other nodes have already been designed and remain
to be relays, since every observation can equivalently be fixed. However, in contrast to previous works, we treat the
considered as the output of a leaf with output cardinality FC in a different way than the other nodes: the FC decision
a knowntransitionprobabilityP(w|z,H) thatdependsonthe
Phenomenon
P(y|H) P(v|H) present hypothesis H. P(y|H) and P(v|H) are probabilistic
H
mapping from the observation space to the discrete sets M
y
and M , respectively.
v
y v
Weshownextthatundertheperson-by-personmethodology,
k P(w|z,H) FC the design of a node, let say m , in an arbitrary tree network
0
z w
isanalogoustothedesignofnodekintherestrictedmodelfor
aparticularinstanceoftheparametersoftherestrictedmodel.
Fig.2. Restricted modelforthedesignofnodesintreetopology. To see this, let
x if m ∈C
y , 0 0 l (3)
(cid:26) {ui :mi ∈I0} if m0 ∈Cr,
functionγ isalwaysupdatedtogetherwiththenodedecision
N
function γl currently being optimized. be the completeinputof nodem0 in the originalnetworkand
If the decision functions of all the leaves and all the relays let
are fixed, the optimal fusion center will use the maximum a- v ,{ui :mi ∈If, mi 6=ls(m0)}, (4)
posteriori(MAP)ruleinordertomaketheglobaldecisionu
N be the completeinputof the FC from its immediatepredeces-
in favor of one of the hypotheses. Defining u as the vector
f sorsI ,excludingthenodethathasm asitspredecessor(the
f 0
containing the messages from the immediate predecessors of
immediate predecessor of FC whose the path from m goes
the FC, i.e., u , {u : m ∈ I }, the FC decides on the 0
f i i f throughit to reach the FC), and assume that the fusion center
hypothesis H if [9]
mˆ intherestrictedmodelusestheMAPrule(1).Theconditional
π P(u |H )=max π P(u |H ) , (1) PMFs of the inputs to node k and FC in the restricted model
mˆ f mˆ j f j
j are,dueto the independencyofobservationsandacyclicityof
(cid:8) (cid:9)
whereπ ,P(H )isthea-priorprobabilityofhypothesisH , the network, then given by
j j j
and where j ∈ {0,1,...,M −1} for the M-ary hypothesis P (x ) if m ∈C
j 0 0 l
testing problem. The expected minimum error probability in
P (y)= (5)
estimating H given an input message vector uf from the set j Pj(ui) if m0 ∈Cr,
MIf ,mYi∈IfMi is [10] and ui:mYi∈I0
P (v)= P (u ). (6)
j j i
P =1− max π P(u |H ) . (2)
E j j f j ui:mYi∈If
ufX∈MIf (cid:8) (cid:9) mi6=ls(m0)
Knowingtheconditionalprobabilitiesoftheinputmessagesto The transition probability P(w|z,Hj) is then simply the
the FC from its immediate predecessors, the error probability transition probability from u0 to uK, where uK is the output
PE at the fusion center can be uniquely computed. message of the last successor of m0, i.e., mK ,ls(m0). The
Inthenextsectionwewillshowthatthedesignofanodein key point is that under the person-by-personmethodologyfor
thenetworkisanalogoustothedesignofanode(labeledbyk) the design of node m0 together with the FC, all other nodes
intherestrictedmodelasshowninFig.2,wheretheFCinboth in the network remain fixed. This implies that Pj(v), Pj(y)
networksusetheMAPrule(1)asthefusiondecisionfunction. and P(w|z,Hj) remain fixed and together with the structure
We will further show how the conditional probabilities in the of the restricted model, they capture all the important aspects
restrictedmodelcanberecursivelycomputedfromtheoriginal of the joint design problem of m0 and the FC. In the rest
tree network. of this section we will show how the transition probabilities
P(w|z,H ) in the restricted model can be found from the
j
III. RESTRICTED MODEL original network and describe a recursive method for the
Consider the distributed network with two nodes, k and computation of the conditional probability masses Pj(y) and
FC, illustrated in Fig. 2. The fusion center FC using its input P (v) in the restricted model from the original tree network.
j
messages w and v makes a decision according to the MAP
rule (1). Let w and v be from the discrete sets M and First, consider an arbitrary node in a tree network, say
w
M , respectively. Conditioned on hypothesis H , the input m , and assume that this node has |I | = L +1 immediate
v j i i i
messages y and v are independent with known conditional predecessors, containing node m . With a slight abuse
i−1
probability masses P (y) and P (v), respectively. Node k of notation we define the set of immediate predecessors of
j j
maps its input y from a discrete set M to an output z node m as I , {m˜ ,...,m˜ ,m }, as exemplified in
y i i 1 Li i−1
from a discrete set M according to a decision function Fig. 3. We also define u˜ ∈ M˜ as the output message
z l l
γ : M → M , i.e., γ (y) = z. The index z then passes of node m˜ , l = 1,...,L . Conditioned on H , each index
k y z k l i j
througha discrete channelwhich maps it to the index w with u ∈ M at the input of node m is mapped to output
i−1 i−1 i
m
i−1 messageu ofnodem fromthediscrete setM , whichis
K K K
thenusedasaninputto theFC. TheMarkovpropertyimplies
m˜Li ui−1 that the transition probability from u0 to uK is given by
u˜
. Li Pj(uK|u0)= ... Pj(uK,uK−1,...,u1|u0)
.
. Xu1 uXK−1
u˜ K
m˜ 1
1 u = ... P (u |u ,...,u )
i j i i−1 0 (8)
Xu1 uXK−1iY=1
K
Fig.3. Nodemi (shadedcircle)anditsLi+1immediatepredecessors,Ii. = ... P (u |u ).
j i i−1
Xu1 uXK−1iY=1
index ui ∈Mi according to function Equivalently, in matrix form if we define Pj0→K(m,n) ,
P (u =m|u =n), then (8) implies
j K 0
γ (u˜ ,...,u˜ ,u )=u ,
i 1 Li i−1 i P0→K =PeK−1,K ×...×Pe1,2 ×Pe0,1. (9)
with probability P (u |u ) , P(u |u ,H ) which is j j j j
j i i−1 i i−1 j
equal to As there is only one directed path from node m0 to mK,
we omitted the corresponding edge labels in P0→K. Thus,
P (u |u )=P (γ (u˜ ,...,u˜ ,u )|u ) j
j i i−1 j i 1 Li i−1 i−1 using(9) wecanreplaceallnodesbetweenm0 andtheFCby
= P (u˜ ,...,u˜ ) a single hypothesis dependent transition probability given by
j 1 Li
(u˜1,...,u˜Li)X∈γi−1(ui−1,ui) (7) rPejs0t→riKct,edwhmeonddeels(iwgnhiinchgmis0e.qDuiuvrailnegntthtoedtheesidgensoigfnnoodfenokdienmthe
0
= P (u˜ )...P (u˜ ),
j 1 j Li inactualnetwork)everychanneltransitionprobabilityPj(w =
(u˜1,...,u˜Li)X∈γi−1(ui−1,ui) m|z =n) is replaced by the corresponding (m,n)th entry of
P0→K.
j
where Pj(u˜l) , P(u˜l|Hj) is the conditional PMF of u˜l, In formingthe restricted modelfor the design of m0 in the
and where γ−1(u ,u ) is the set of all input messages originalnetwork,inadditiontochanneltransitionprobabilities
i i−1 i
(u˜ ,...,u˜ ) that satisfy γ (u˜ ,...,u˜ ,u ) = u . It P(w|z,H), the transition probabilities P(y|H) and P(v|H)
1 Li i 1 Li i−1 i
should be mentioned that, conditioned on the hypothesis, the should be also determined. The input y to the node k is the
input messages to node mi are independent. Now we can complete input messages to node m0 in the original tree. If
state the first important result from (7) as following: consider node m0 is a leaf then it only makes observationand y =x0.
node mi and the set of its immediate predecessors, Ii = However, if node m0 is a relay, then y is a vector containing
{m˜ ,...,m˜ ,m }. Node m hasa Markovianbehaviorin input messages from its immediate predecessors according to
1 Li i−1 i
the sense that, conditioned on the hypothesis and the input (3) and P(y|H) is defined as (5). In the following we will
message ui−1, its output message ui depends only on the show how Pj(ui) in a tree network [corresponding to Pj(y)
inputs from the immediate predecessors {m˜ ,...,m˜ } and intherestrictedmodel]canbefoundinarecursivemanner.To
1 Li
not the sequence of messages preceding mi−1. The transition this end, consider again node m0 in the original tree network
probabilities for this Markov chain is found using (7). The and its immediate predecessors mi ∈ I0. Suppose that node
transition probability matrix from input index ui−1 to output mi ∈I0receivesmessagesfromitsLiimmediatepredecessors
index ui, conditioned on hypothesis Hj, is denoted Peji−1,i Ii , {mˆ1,...,mˆLi} and maps its input vector (denoted by
where the superscript ei−1,i indicates the specific edge that (uˆ1,...,uˆLi))toanoutputmessageuiaccordingtoadecision
corresponds to the desired input ui−1. Peji−1,i is with size function γi :Mˆ1×...×MˆLi →Mi, i.e.,
kM k×kM k, and its (m,n)th entry is defined as [11]
i i−1 γ (uˆ ,...,uˆ )=u .
i 1 Li i
Pei−1,i(m,n),P (u =m|u =n).
j j i i−1 Then the probability masses P (u ) at the output of node m
j i i
Consequently,eachrelaynodem canberepresentedbyM|I | are given by
i i
transition probabilitymatricesPek,i, where m ∈I , and j =
j k i Pj(ui)=Pj(γi(uˆ1,...,uˆLi))
0,...,M −1.
= P (uˆ ,...,uˆ )
Consider again node m0 and set S0 ={m0,m1,...,mK}. j 1 Li
Thereisexactlyonedirectedpathfromm0 totheFC.Assume (uˆ1,...,uˆLXi)∈γi−1(ui) (10)
ltthh=aatt1nn,oo.dd.e.e,mKml1−+1i1s.istThhteheeiFmiCmmmiesdetidhaiteaetiemsusmcucecedcsiesasotserorsoufocfcneonsdosoedremomf0nlaofndoder =(uˆ1,...,uˆLXi)∈γi−1(Puij)(uˆ1)...Pj(uˆLi),
m .Afterpassingthroughm tom ,theoutputmessageu whereγ−1(u )isthesetofallinputvectors(uˆ ,...,uˆ )that
K 1 K 0 i i 1 Li
of node m from the discrete set M is mapped to an output satisfy γ (uˆ ,...,uˆ ) = u . The last equation is the result
0 0 i 1 Li i
of the fact that the inputs to each node in the tree network, l1 l2 l3 l4
conditioned on the hypothesis, are independent.
Equation (10) shows how the probability masses of the
output of node ui can be found based on the probability r1 r2
masses of its inputs and its decision function γ . Consider
i
sub-tree T in the network with node m as its root. In the
0 0 FC
person-by-person methodology used for the design of node
m we assume that all other nodes (except the FC) are kept
0 uN
fixed, including all the predecessors of node m in its sub-
0
tree T . Starting from the immediate predecessors of m and
0 0 Fig.4. 2-symmetric2-uniform treenetwork.
going backward in the sub-tree, the probability masses of the
output of each node can consequently be found based on
the probability masses of its input and its decision function
methodswhichuseperson-by-personmethodology)leadsonly
[cf.(10)].Eventually,foraleafm inT theprobabilitymasses
l 0
to locally optimum solutions, which also depends on the
at the output are given by
initialization of the nodes. However, in the next section we
Pj(ul)= Pj(xl). (11) show by a numerical example that good performance can
xl∈γXl−1(ul) nevertheless be obtained.
Thus, the required PMFs at m0 (represented by Pj(y) in the IV. EXAMPLES
restricted model) can be found by going forward from the
In what follows we present some results from the applica-
leavesinT towardnodem .Usingthesameapproach,P (v)
0 0 j
tion of the proposed method in the design of a tree network.
in the restricted model can be found in a recursive manner.
We will consider a 2-symmetric 2-uniform tree network, as
The minimum error probability at the FC in the restricted
defined in [13] given in Fig. 4. The primary reason for
model (Fig. 2) is a function of the parameters γ , P(y|H),
k
choosing such a simple network is to be able to assess the
P(v,|H) and P(w|z,H), i.e.,
performanceoftheproposedmethodthroughcomparisonwith
P =F γ ,P(y|H),P(v,|H),P(w|z,H) . (12) previousresultsfortreenetworkandforspecificchannelrates.
E,min k
(cid:0) (cid:1) Weassumetheleavesl ,...,l makeobservationsx ,...,x ,
Once the parameters in the restricted model are found, the 1 4 1 4
respectively, and the relays r ,r summarize the messages
error probability at the FC, when using the MAP criterion, is 1 2
received from their corresponding immediate predecessors
given by
and the FC makes the final decision u in favor of one
N
P =1− max π P (v)P (w) , (13) hypothesis. We consider the case of binary hypothesis testing
E j j j
Xv Xw j (cid:8) (cid:9) M = 2, where real valued observations are, conditioned
on the hypothesis, independent and identically distributed.
where P (w),P(w|H ) is
j j The observation model at each leaf, where each observation
P (w)= P (z)P (w|z) consists of an antipodal signal ±a in unit-variance additive
j j j
z∈XMz (14) white Gaussian noise ni,i=1,...,4, is given by
= P (y)P (w|z), H :x =−a+n
j j 0 i i
z∈XMz y∈γXk−1(z) H1 :xi =+a+ni.
and where γ−1(z) is the set of all input messages (vectors) The per channel signal-to-noise ratio (SNR) is then defined
k
y that satisfiy γ (y) = z. Equations (13) and (14) show how as E = |a|2. We further assume equally likely hypotheses
k
the error probability at the FC is affected by the parameters (π =π =0.5). Channels between the nodes are considered
0 1
in the restricted channel (especially γ ). error-free but rate-constrained where the rate of the leaf-to-
k
Thegoalofthispaperishowevernottoshowhowγ canbe relaylinksareequaltoR bits,andtherateoftherelay-to-FC
k l
designed(togetherwiththeFC)intherestrictedmodel.Rather, linksare equalto R bits. This impliesthatthe leaves’ output
r
the goal is just to show how the optimization problem for massagesare fromthe set {1,...,2Rl} andthe relays’ output
γ can be formulatedcompactly. However,in [12] a clear-cut messages are from the set {1,...,2Rr}. The FC using the
k
guidelineforthedesignofγ withareasonablecomputational MAPrule(1)makesfinaldecisionu fromtheset{H ,H }.
k N 0 1
burdenisproposed.Itisin[12]shownthatthedesignofnode In our simulations we initialized the relays with random
k in the restricted model can also be done in a person-by- functions, while for R = 1 we initialized the leaves in all
l
person manner in terms of the input set; an output index z is methodswiththeoptimallocaldecisionfunctions.ForR >1
l
assigned to a specific input y, while the assigned indices to weuniformlyquantizedthetwodecisionregionsoftheR =1
l
other inputs are fixed. initialization.
In closing, we emphasize that the proposed method for A performance comparison of the designed tree networks
the design of nodes in the general tree topology (like other for different rate pairs (R ,R ) and for different per channel
l r
−0.5 −0.5
−1−.51|bCrSb |bCrSb |bbCrS |bCrSb |bbCrS |bCrSb |bCrSb |bC −−1.51**||bb **||bb **||bb **||bb **||bb **|| || |
bb
rSb |bC ** | |
PE −2 rSb |bC PE −2 bb ** | |
0 0 bb
og1−2.5 rSb |bC og1−2.5 | Tree R=1 ** | |
l l * Tree R=2 bb
−3 bC Tay et al.’s method (1,1) rSb −3 b Tree R=3 ** |
| Proposed method (1,1) | Parallel R=1 bb
Proposed method (2,1) * Parallel R=2 *
rS
−3.5 b Proposed method (1,2) −3.5 b Parallel R=3 *
Optimum linear detector Optimum linear detector bb
−4 −4
−5 −4 −3 −2 −1 0 1 2 3 4 5 −5 −4 −3 −2 −1 0 1 2 3 4 5
SNR (dB) SNR (dB)
Fig.5. Errorprobabilityperformanceofthedesigned2-symmetric2-uniform Fig. 6. Comparison of error probability performance of the designed tree
tree network for different rate pairs (Rl,Rr), and for different per channel networkwithratepairs (R,R)andparallel networkwithrateR.
SNRs.
situations where the optimal decision functions are obvious.
SNRs is illustrated in Fig. 5. The results of the proposed For example, consider a 2-symmetric 2-uniform tree network
method are compared to the optimum un-constrained linear with rate pair (1,2), where the leaves send one-bit messages
detector (which is optimum for this problem) applied to the to the relays and the relays send two-bit messages to the FC.
set of all inputs and results due to Tay et al.’s method which In this case, an optimal relay would simply put the one-bit
leads to the optimal error exponent for an r-symmetric tree received messages from its predecessors together and send
[13] for rate pair (1,1). In that case, the relays use an AND the resulting two-bit message to the FC, which means the
strategy and the leaves have the same threshold γ on their performance of the (optimal) (1,2) tree network is the same
observations.Usinganexhaustivesearch,wefoundthebestγ as the performance the (optimal) parallel network with one-
whichminimizestheerrorprobabilityattheFC,giventhatthe bit channel rates. This is consistent with Fig. 5 and Fig. 6,
FC uses the MAP rule. The simulation results in Fig. 5 show where using the proposed design method yields the same
that for rate pair (1,1) the proposed method gives the same performanceintermsoferrorprobabilityforbothcases,which
result as the asymptotically optimum solution, and increasing indicates that the proposed method is working as expected.
the rate of the links gives better performance. Also, note that
the performanceof designed tree networks for rate pair (1,2) V. CONCLUSION
coincides with that for rate pair (2,1) for equally probable Inthispaper,we haveconsideredthedistributedhypothesis
hypotheses. It should however be mentioned that it is not a testing problem in a general tree network where the nodes
general result and for other a-prior probability assignments, make observationswhich are, conditionedon the true hypoth-
the resulting curves do not show the same performance. esis, independent. We have shown that the design of nodes
Theproposedmethodforthedesignofgeneraltreenetwork under the person-by-person methodology is analogous to the
can also be used for the design of parallel networks. It is a design of a two-node network, the restricted model, in which
well known statement that the performance of any optimum thedecisionfunctioncanbedesignedefficiently(cf.[12]).We
tree network is dominatedby the performanceof an optimum also have shown how the parameters of the restricted model
parallel network, for an equal number of observations [6]. for the design of a node in the general tree can be formed in
Fig. 6 shows the error probability performance of designed a recursive and computationally efficient manner.
treeandparallelnetworks,wheretherateofallthelinksinthe
treenetworkandtherateofallthelinksintheparallelnetwork ACKNOWLEDGMENT
areequaltoR.AsisillustratedinFig.6,forthesamechannel Thiswork has been supportedin partby the ACCESS seed
rates, the parallel network outperformsthe correspondingtree project DeWiNe.
network. As the rate of the links increases, the performance
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