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Modeling and Mi iga ing E o s in Belie P opaga ion o Dis ibu ed De ec ion
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Abdi, Younes; Ris aniemi, Tapani
Abdi, Y., & Ris aniemi, T. (2021). Modeling and Mi iga ing E o s in Belie P opaga ion o
Dis ibu ed De ec ion. IEEE T ansac ions on Communica ions, 69(5), 3286-3297.
h ps://doi.o g/10.1109/TCOMM.2021.3056679
2021
3286 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 69, NO. 5, MAY 2021
Modeling and Mi iga ing E o s in Belie
P opaga ion o Dis ibu ed De ec ion
Younes Abdi ,Membe , IEEE, and Tapani Ris aniemi ,Senio Membe , IEEE
Abs ac — We s udy he beha io o he belie -p opaga ion
(BP) algo i hm a ec ed by e oneous da a exchange in a wi eless
senso ne wo k (WSN). The WSN conduc s a dis ibu ed mul i-
dimensional hypo hesis es o e bina y andom a iables. The
join s a is ical beha io o he senso obse a ions is modeled
by a Ma ko andom ield whose pa ame e s a e used o build
he BP messages exchanged be ween he sensing nodes. Th ough
linea iza ion o he BP message-upda e ule, we analyze he
beha io o he esul ing e oneous decision a iables and de i e
closed- o m ela ionships ha desc ibe he impac o s ochas ic
e o s on he pe o mance o he BP algo i hm. We hen de elop
a decen alized dis ibu ed op imiza ion amewo k o enhance
he sys em pe o mance by mi iga ing he impac o e o s ia
a dis ibu ed linea da a- usion scheme. Finally, we compa e
he esul s o he p oposed analysis wi h he exis ing wo ks
and isualize, ia compu e simula ions, he pe o mance gain
ob ained by he p oposed op imiza ion.
Index Te ms— Dis ibu ed sys ems, coope a i e communica-
ions, likelihood- a io es , communica ion e o s, compu a ion
e o s, blind signal p ocessing, message-passing algo i hms, lin-
ea da a- usion, ac o g aphs.
I. INTRODUCTION
DESIGN o s a is ical in e ence sys ems o en in ol es
analysis and modeling o he collec i e beha io
o a g oup o andom a iables and hei in e ac ions.
Consequen ly, ac o g aphs, which a e commonly used o
cap u e he in e dependencies be ween co ela ed andom a i-
ables, p o ide a powe ul amewo k o de eloping e ec i e
low-complexi y in e ence algo i hms in a ious ields such
as wi eless communica ions, image p ocessing, combina o ial
op imiza ion, and machine lea ning, see e.g., [1]–[3]. Belie
p opaga ion (BP) [4] is a well-known s a is ical in e ence algo-
i hm ha wo ks based on pa allel message-passing be ween
he nodes in a ac o g aph. BP is some imes e e ed o as
he sum-p oduc algo i hm.
When wo king wi h he BP algo i hm, we should bea
in mind ha digi al compu a ion and digi al communica ion
a e bo h e o -p one p ocesses in gene al. The messages
exchanged be ween he nodes in a wi eless ne wo k can always
be ad e sely a ec ed by e o s caused by un eliable ha dwa e
Manusc ip ecei ed Ma ch 12, 2020; e ised Sep embe 6, 2020 and
Janua y 25, 2021; accep ed Janua y 25, 2021. Da e o publica ion Feb ua y 3,
2021; da e o cu en e sion May 18, 2021. The associa e edi o coo dina ing
he e iew o his a icle and app o ing i o publica ion was A. Cohen.
(Co esponding au ho : Younes Abdi.)
Younes Abdi is wi h he Facul y o In o ma ion Technology, Uni e si y o
Jy äskylä, 40014 Jy äskylä, Finland (e-mail: younes.[email p o ec ed]).
Tapani Ris aniemi, deceased, was wi h he Facul y o In o ma ion
Technology, Uni e si y o Jy äskylä, 40014 Jy äskylä, Finland (e-mail:
apani. is aniemi@jyu. i).
Colo e sions o one o mo e igu es in his a icle a e a ailable a
h ps://doi.o g/10.1109/TCOMM.2021.3056679.
Digi al Objec Iden i ie 10.1109/TCOMM.2021.3056679
componen s, quan iza ion p ocesses, app oxima e ep esen a-
ions, wi eless channel impai men s, e c. E en hough he
BP algo i hm has been ex ensi ely s udied in he li e a-
u e, we ha e a he limi ed knowledge abou how s ochas ic
e o s in messages a ec he belie s ob ained and how hese
e oneous belie s in luence he esul o s a is ical in e ence
schemes implemen ed by he BP algo i hm. This e i o y is
di icul o explo e mainly due o he nonlinea i ies in he BP
message-passing i e a ion.
In [5], we ha e de eloped a sys ema ic amewo k o
analyzing he beha io o BP and op imizing i s pe o mance
in a dis ibu ed de ec ion scena io. In pa icula , we ha e
shown ha he decision a iables buil by he BP algo i hm
a e, app oxima ely, linea combina ions o he local likeli-
hoods in he ne wo k. Consequen ly, we ha e de i ed in [5]
closed- o m ela ionships o he sys em pe o mance me ics
and o mula ed a dis ibu ed op imiza ion scheme o achie e
a nea -op imal de ec ion pe o mance. Mo eo e , we ha e
discussed he ela ionship be ween he BP and he max-p oduc
algo i hms in [6] whe e we ex end he p oposed ame-
wo k in [5] o op imize he pe o mance o he max-p oduc
algo i hm in a dis ibu ed de ec ion scena io. In his pape ,
we u he ex end ha amewo k o gain insigh in o he
impac o compu a ion and communica ion e o s, in a BP
i e a ion, on he esul ing decision a iables and o e ec i ely
mi iga e ha impac . Examples o BP being used in dis ibu ed
de ec ion can be ound in [7]–[10].
Accumula ion o message e o s and hei ad e se e ec
on he pe o mance o BP is analyzed in [11] whe e he
message e o s a e modeled as unco ela ed andom a iables
o ind p obabilis ic gua an ees on he magni ude o e o s
a ec ing he belie s. The wo k in [11] is inspi ed by obse ing
he beha io and s abili y o digi al il e s, in he p esence
o quan iza ion e ec s, which can be analyzed eliably by
assuming unco ela ed beha io in he co esponding andom
e o s [12]. Such a modeling app oach is in line wi h he
on Neumann model o noisy ci cui s [13], which conside s
ansien aul s in logic ga es and wi es as message and
node compu a ion noise ha is bo h spa ially and empo ally
independen [14].
The beha io o BP implemen ed on noisy ha dwa e is
in es iga ed in [15] whe e i is obse ed ha unde he
so-called con ac ing mapping condi ion [16], he dis ance
be ween successi e messages in a noise- ee BP dec eases
by he numbe o i e a ions. Consequen ly, in he p esence
o ha dwa e (o compu a ion) noise, he aul y messages ha
iola e his end can be de ec ed and disca ded (censo ed)
om he BP i e a ions. Such an app oach is e med censo ing
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ABDI AND RISTANIEMI: MODELING AND MITIGATING ERRORS IN BELIEF PROPAGATION FOR DISTRIBUTED DETECTION 3287
BP in [15] and is shown o pe o m well when he ha dwa e
noise dis ibu ion has a la ge mass a ze o and non-negligible
masses a some poin s su icien ly away om ze o. As an
al e na i e app oach, he so-called a e aging BP (ABP) is
also p oposed in [15]. In his me hod, as he name implies,
an a e age o he messages up o he las i e a ion is sa ed and
hen used, ins ead o he ac ual messages, o build he belie s.
This me hod is p oposed and i s con e gence is es ablished
o gene al ze o-mean compu a ion noise dis ibu ions. Again,
he on Neumann model is used in [15] o analyze he beha io
o message e o s.
In his pape , we use he ac ha he BP algo i hm and he
linea da a- usion scheme a e elegan ly ela ed o each o he in
he con ex o dis ibu ed de ec ion. Fo una ely, he e al eady
exis s a ich collec ion o scien i ic wo ks in he li e a u e ha
in es iga e low-complexi y de ec o s uc u es based on linea
usion in a ious design scena ios [17]–[21]. In many o hese
wo ks, he da a-exchange p ocess wi hin he senso ne wo k is
assumed ad e sely a ec ed by non-ideali ies in he unde lying
communica ion links. Hence, dealing wi h e oneous da a is
a amilia challenge aced when designing wi eless senso
ne wo ks (WSN). We use his knowledge o cope wi h he
impac o message e o s on dis ibu ed de ec ion sys ems
ealized by BP.
I is a common p ac ice o e e o he channels o e
which he da a exchange be ween he sensing nodes is con-
duc ed as epo ing channels o dis inguish hem om he
channels o e which he a ge signal is de ec ed, which
a e e e ed o as lis ening channels. The impo ance o he
p esen wo k can be highligh ed by no ing ha dis ibu ed
de ec ion sys ems can be highly sensi i e o he epo ing
e o s. This phenomenon is illus a ed in [22, Sec. IV-C]
by a simple example ha shows ha he o e all de ec ion
pe o mance canno go beyond he limi s dic a ed by he
epo ing channel condi ions i espec i e o he signal- o-
noise- a io (SNR) le els o he a ge signal expe ienced a
he lis ening channels. This e ec is u he s udied and
quan i ied o se e al ha d- and so -decision usion schemes
in [23]–[25] whe e i is shown ha he epo ing chan-
nel e o s can ha e a signi ican impac on he de ec ion
pe o mance.
In he exis ing li e a u e, he epo ing channels ha e
been commonly conside ed nonideal o accoun o ealis ic
da a-exchange p ocesses be ween he ne wo k nodes [17],
[20], [21], [26]–[30]. A majo applica ion scena io o dis-
ibu ed de ec ion sys ems is spec um sensing in cogni i e
adio ne wo ks (CRN) whe e communica ion be ween he
sensing nodes is ypically conduc ed wi hou ha ing access o
dedica ed spec um bands. This means ha he da a exchange
be ween hose nodes could ace s ingen cons ain s in e ms
o ansmi powe and bandwid h. Consequen ly, many wo ks
on dis ibu ed de ec ion in CRNs ocus on epo ing links
wi h bandwid h o powe cons ains, see e.g., [27]. These
cons ain s a e ypically aken in o accoun , in he sys em
modeling and op imiza ion, by non-ideal epo ing links ha
in oduce non-ze o bi e o p obabili ies (BEP) in digi al [21]
and unco ela ed noise in analog epo ing schemes [20]. Since
he link noise le el and BEP a e mono onically ela ed o each
o he [21, Eq. (12)], simila app oaches can be used in bo h
analog and digi al cases o mi iga e he impac o epo ing
e o s, see [17], [21].
In his pape , we a e ocused on a BP-based decen alized
dis ibu ed de ec ion scheme. We iew message e o s in he
BP i e a ion as epo ing e o s and app oxima e he messages
by a linea exp ession o s udy he impac o e oneous
da a-exchange on he BP algo i hm and o cla i y how i
a ec s he pe o mance o he esul ing dis ibu ed de ec ion.
We de i e app oxima e exp essions ha measu e he s eng h
o he cumula i e e o s ha a ec he BP-based decision
a iables. These exp essions a e in he o m o mean-squa ed
e o (MSE) le els. We compa e he MSE le els ob ained
wi h he one in [11] o gain insigh in o he beha io o
BP and o see how compu a ion and communica ion e o s
p opaga e h oughou he unde lying ac o g aph. Ou analysis
closely p edic s he ex en o he de ia ion o he e oneous
decision a iables, ob ained by an e oneous BP i e a ion, om
hei ac ual alues. This is a signi ican imp o emen o e
Ihle ’s bound in [11]. Mo eo e , based on he p oposed linea
app oxima ion, we show ha ABP is e ec i e in alle ia ing
message e o s and alls sho o mi iga ing he impac o
e oneous local likelihood a ios (LLRs) on he esul ing
decision a iables.
We also show, unde p ac ical assump ions, ha he decision
a iables buil by an e oneous BP a e dis u bed by a sum
o independen e o componen s whose collec i e impac can
be modeled, app oxima ely, by Gaussian andom a iables.
Consequen ly, we es ablish he p obabili y dis ibu ion o he
esul ing e oneous decision a iables, de i e he pe o mance
me ics o he BP-based dis ibu ed de ec ion in closed o m,
and p opose a wo-s age op imal linea usion scheme o
cope wi h he impac o e o s on he sys em pe o mance.
We hen de elop a blind adap a ion algo i hm o ealize he
p oposed wo-s age op imiza ion when he s a is ics desc ibing
he adio en i onmen a e no a ailable ap io i. The p oposed
blind adap a ion modi ies he pa ame e s o he BP and he
decision h eshold a each node, in acco dance wi h he e o
s a is ics and channel condi ions, o mi iga e he impac o
e o s and o enhance he de ec ion pe o mance. To summa-
ize, we ex end he wo ks in [5] and [6] by he ollowing
con ibu ions:
•We analyze he beha io o he BP algo i hm in he
p esence o message (and likelihood) e o s and de i e
i s pe o mance me ics in closed o m in a dis ibu ed
de ec ion subjec o hose e o s.
•We build a dis ibu ed op imiza ion amewo k o he
sys em ha akes in o accoun and e ec i ely mi iga es
he impac o e oneous da a exchange in BP.
Mo eo e ,
•We ex end he wo k in [11] by p oposing a igh e
e o bound ha mo e accu a ely desc ibes he impac
o message e o s on he decision a iables buil by he
BP algo i hm.
•We ex end he wo k in [15] by analyzing he beha io o
ABP. Ou wo k sheds ligh on ABP’s e ec i eness and
sho comings.
He e is an o e iew o he pape o ganiza ion: In Sec. II,
we b ie ly explain he use o linea usion and BP in dis ibu ed
3288 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 69, NO. 5, MAY 2021
de ec ion and p o ide he ela ed o mula ions. In Sec. III,
we discuss e o s in BP and model hei impac on he
decision a iables ob ained. In Sec. IV, we iew BP as a
dis ibu ed linea usion and o mula e he p oposed op imiza-
ion amewo k. In Sec. V, we conduc compu e simula ions
o e i y ou analysis and o illus a e how e ec i ely he
p oposed me hod mi iga es he impac o e o s in a WSN wi h
aul y de ices. Finally, we p o ide ou concluding ema ks in
Sec. VI.
II. LINEAR FUSION AND BELIEF PROPAGATION
FOR DISTRIBUTED DETECTION
We conside Nbina y andom a iables, ep esen ed by
x=[x1,...,x
N]T, whose s a us a e es ima ed based on N
obse a ions deno ed Y=[y1,...,yN]made by a ne wo k
o Nsensing nodes. Each node, say node i, which in ends
o es ima e he s a us o xi, collec s Kobse a ion samples,
deno ed by yi=[yi(1),...,y
i(K)]T, and exchanges in o -
ma ion wi h o he nodes in he ne wo k o ealize oge he
a mul idimensional hypo hesis es as ˆ
x=max
xp(x|Y)=
maxxp(Y|x)p(x). This es can be conduc ed wi h low
implemen a ion complexi y in wo al e na i e ways ha a e
explained in he ollowing.
A. Linea Da a-Fusion
Linea usion has been ex ensi ely used in he con ex o
spec um sensing whe e he aim is o de ec he p esence
o absence o a a ge signal by e alua ing noisy obse a-
ions made h oughou a WSN. Fo b e i y, we explain he
uni- a ia e case whe e we ha e a single bina y a iable x∈
{0,1}. The op imal app oach o such a de ec ion is known
o be he so-called likelihood- a io es (LRT) [31], which is
conduc ed by e alua ing he LLR, i.e., by ˆx=1{λLRT −τ}
whe e
λLRT ln p(Y|x=1)
p(Y|x=0)=
N

i=1
γi(1)
whe e
γiln p(yi|x=1)
p(yi|x=0)=sT
iyi−1
2si2(2)
whe e γiis e e ed o as he local LLR a node i.By
1{·} we ep esen he indica o unc ion ha e u ns one
i i s a gumen is posi i e and e u ns ze o o he wise. τis
a de ec ion h eshold selec ed ia a a ge alse-ala m a e.
Eq. (2) indica es ha , he LRT is a ma ched- il e ing p ocess,
which equi es he a ge signal si o be known ap io ia
he sensing nodes. Mo eo e , o Gaussian obse a ions, γi
in (2) ollows a Gaussian dis ibu ion. In p ac ice, he local
sensing p ocess is ealized by ene gy de ec ion, due o i s ease
o implemen a ion and because i s s uc u e does no equi e
he a ge signal o be known. Ene gy de ec ion is ealized by
γi1
Kyi2and he senso ou comes a e combined linea ly
o build a global es s a is ic [17]–[21], i.e.,
λLF 
N

i=1
wiγi=wTγ(3)
whe e w[w1,...,w
N]Tand γ[γ1,...,γ
N]T. Then,
λLF is compa ed agains τ o conduc he hypo hesis es ,
i.e., ˆx=1{λLF −τ}.wcan be se o maximize he
de ec ion p obabili y. Acco ding o he cen al limi heo em
(CLT) [32], when he numbe o signal samples Kis la ge
enough [17]–[21], he ou come o ene gy de ec ion ollows
a Gaussian dis ibu ion and we can model he es summa y
λLF, gi en he s a us o x, as a Gaussian andom a iable. Con-
sequen ly, he de ec o pe o mance can be op imized by he
well-known Neyman-Pea son app oach [31]. This op imiza ion
is o mula ed as
w∗=a gmin
w
Q−1(α)wTΣ0w−wTδ
wTΣ1w(4)
whe e δμ1−μ0while μbE[γ|x=b]and Σb
co (γ|x=b)and Q−1(·)deno es he in e se o he Q-
unc ion. αdeno es he a ge alse-ala m p obabili y a which
he de ec ion p obabili y is maximized. This non-con ex p ob-
lem is sol ed in [17], [19], [20]. F om hese wo ks, we know
ha he pe o mance o linea usion is close o he LRT
pe o mance. Al e na i ely, we can maximize he so-called
de lec ion coe icien o he de ec o . This app oach, which
has a low compu a ional complexi y and leads o a good
pe o mance le el, is ealized by
w∗=a gmax
wΔ2(w),s. ., w=1 (5)
whe e
Δ2(w)(E[λLF|x=1]−E[λLF|x=0])
2
Va [λLF|x=0] =wTδ2
wTΣ0w(6)
Consequen ly, by using he Rayleigh-Ri z inequali y [17],
w∗is ob ained in closed o m as w∗=Σ−1
0δ/
Σ−1
0δ
.
Ex ension o he linea de ec ion s uc u e in (3) o N a iables
is discussed in [18] in he con ex o mul iband spec um
sensing.
B. Belie P opaga ion
We model he senso ne wo k s uc u e conce ned by an
MRF de ined on an undi ec ed g aph G=(V,E).In his
model, he se o e ices Vco esponds o he se o ne wo k
nodes while each edge (i, j)∈E ep esen s a possible
connec ion be ween nodes iand j. Each node, say node i,
is associa ed wi h a andom a iable xiand he edge (i, j)
models a possible co ela ion be ween xiand xj. This model
i s well in o he commonly-used ad-hoc ne wo k con igu a-
ions in which majo ne wo k unc ionali ies a e conduc ed
h ough pai wise i.e., one-hop, links be ween he nodes loca ed
close o each o he . This design me hod is based on he
common assump ion ha nodes loca ed close enough o each
o he o one-hop communica ion, expe ience some le els o
co ela ion be ween hei senso ou comes.
By using he MRF, we w i e p(x|Y)as a p oduc o
uni a ia e and bi a ia e unc ions, i.e.,
p(x|Y)∝
n∈V
φn(xn)
(i,j)∈E
ψij (xi,x
j)(7)
No e ha ∝in (7) e e s o a no maliza ion ha ensu es
xp(x|Y)=1and includes bu is no limi ed o 1/p(Y).
When including he bi a ia e e ms in he p oduc , each edge
in he ac o g aph is included in he p oduc only once. This
is ealized by doing he mul iplica ion on i<jwhile i∈N
j.
ABDI AND RISTANIEMI: MODELING AND MITIGATING ERRORS IN BELIEF PROPAGATION FOR DISTRIBUTED DETECTION 3289
We use Nj o deno e he se o neighbo s o node jin he
g aph, i.e., Nj{k:(k,j)∈E}. By using (7), we o mula e
he message ecei ed a node j om node kas
μ(l)
k→j(xj)∝
xk
φk(xk)ψkj (xk,x
j)
n∈Nj
k
μ(l−1)
n→k(xk)(8)
whe e by Nj
kNk {j}we deno e all nodes connec ed o
node kexcep o node j. We deno e by b(l)
j(xj) he belie ,
abou he s a us o xj, o med a node j, which is ob ained ia
mul iplying he po en ial a node jby he messages ecei ed
om all i s neighbo s, i.e.,
b(l)
j(xj)∝φj(xj)
k∈Nj
μ(l)
k→j(xj)(9)
The belie s a e used as es ima es o he desi ed ma ginal
dis ibu ions, i.e., b(l)
j(xj)≈p(xj|Y). By adop ing he
commonly-used exponen ial model [4] o ep esen he ap io i
p obabili y measu e de ined on x,weha e
p(x)∝exp ⎛
⎝
n∈V
θnxn+
(i,j)∈E
Jij xixj⎞
⎠(10)
Fo no a ional con enience, we use bipola bina y a iables,
i.e., xj∈{−1,+1}in ou o mula ions o BP. Fo a gi en x,
we assume he local obse a ions o be mu ually independen .
Consequen ly, as explained in [5, Sec. I-B], we ha e
p(x|Y)∝
n∈V
p(yn|xn)eθnxn
(i,j)∈E
eJij xixj(11)
Hence, by using (11), he BP messages a e buil as
μ(l)
k→j(xj)∝
xk
p(yk|xk)eθkxkeJkj xkxj
n∈Nj
k
μ(l−1)
n→k(xk)
(12)
and he belie s a i e a ion la e exp essed as
b(l)
j(xj)∝p(yj|xj)eθjxj
k∈Nj
μ(l)
k→j(xj)(13)
In he log domain, (12) and (13) con e , espec i ely, as cla -
i ied in Appendix A, o
m(l)
k→j=S⎛
⎝Jkj,γ
k+
n∈Nj
k
m(l−1)
n→k⎞
⎠(14)
λ(l)
j=γj+
k∈Nj
m(l)
k→j(15)
whe e
λ(l)
jln b(l)
j(xj=+1)
b(l)
j(xj=−1) (16)
m(l)
k→jln μ(l)
k→j(xj=+1)
μ(l)
k→j(xj=−1) (17)
deno e, espec i ely, he es ima ed likelihood a io a node j
and he message sen o node j om node kwhile S(a, b)
ln 1+ea+b
ea+eband γkln p(yk|xk=+1)
p(yk|xk=−1) =sT
kyk−1
2sk2.In his
model, yk=1
2(xk+1)sk+nkdeno es he signal ecei ed
a node k. Hence, xk=−1indica es ha he a ge signal
skis absen lea ing he he spec um ee whe e node k
ope a es. I xk=+1, hen he co esponding spec um band
is occupied. Jkj ’s a e calcula ed as in Eq. (16) in [5] by
p ocessing a window o Tsensing ou comes. No e ha θk
in (14) is me ged in o γkwi hou ha ing any impac on he
es o he analysis.
A e l∗i e a ions, λ(l∗)
jis compa ed, as a decision a iable,
agains a de ec ion h eshold τja node j o decide he s a us
o xj, i.e., ˆxj=1{λ(l∗)
j−τj}. By a linea app oxima ion
o (14), we ha e [5]
m(l)
k→j≈cjk ⎛
⎝γk+
n∈Nj
k
m(l−1)
n→k⎞
⎠(18)
whe e cjk (e2Jkj −1)
(1+eJkj )2. This app oxima ion is ob ained by he
i s -o de Taylo se ies expansion, i.e., S(a, b)≈Sb(a, 0)b
whe e Sb(a, b)=∂S(a, b)/∂b. By using (18) we see ha
liml→∞ λ(l)
j≈λjwhe e
λjγj+
k∈Nj
cjkγk+
k∈Nj
n∈Nj
k
cjkcknγn
+
k∈Nj
n∈Nj
k
m∈Nk
n
cjkckncnmγm+... (19)
The e o e, his app oxima ion e eals ha , gi en enough ime,
all he local likelihood a ios obse ed in he ne wo k a e
almos linea ly combined a node j o calcula e i s decision
a iable λj. We ha e shown in [5] ha , he con e gence
o his linea message-passing algo i hm is gua an eed when
|cj,k|<1
maxn|Nn|−1,∀(j, k)∈E. The linea combina ion
in (19) can be exp essed as λj=N
i=1 ajiγi,whichis
compac ly s a ed in ma ix o m as
λ=Aγ (20)
whe e λ[λ1,...,λ
N]Tand A[a1,...,aN]Twhile
aj[aj1,...,a
jN]T. He e we de i e he ela ionship
be ween Aand cjk’s in (19) as
A≈I+
∞

n=1
Cn−D∞

n=1
Cn(21)
whe e C[cjk]N×Nand D(X)deno es a diagonal ma ix
whose main diagonal is equal o ha o X. The p oo is
p o ided in Appendix B.
I is now clea ha o ha e con e gence in he
message-passing i e a ion (18), he spec al adius o Chas o
be less han one. This c i e ion may be used o impose bounds
on cjk’s o gua an ee he con e gence o he algo i hm. Al e -
na i ely, he con e gence can be gua an eed, wi hou dealing
wi h he complexi ies o inding he spec al adius, by using
he con ac ing mapping condi ion as we ha e discussed in [5].
We use (21) in he ollowing sec ion o de i e an es ima ion
o he e o s eng h a ec ing he decision a iables buil by
an e oneous BP.
III. ERRORS IN BELIEF PROPAGATION
Eq. (14) shows ha a each BP i e a ion each node c ea es
i s messages in e ms o i s local LLR alue as well as he
messages ecei ed om he neighbo ing nodes a he p e ious
i e a ion. In ou sys em model, we assume ha he local LLRs
and he BP messages a e e oneous. As in [11] and [15],

3290 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 69, NO. 5, MAY 2021
we use he on Neumann app oach o modeling he join
s a is ical beha io o e o s.
A. E o Model and Analysis
Since he messages a e mul iplied oge he o build he
belie s, we o mula e hem as mul iplica i e pe u ba ions
a ec ing ue (i.e., e o - ee) message alues, i.e.,
˜μ(l)
k→j(xj)=μ(l)
k→j(xj)ε(l)
k→j(xj)(22)
whe e ˜μ(l)
k→j(xj)deno es he e oneous message sen o node
j om node ka i e a ion lwhile ε(l)
k→j(xj)deno es he
co esponding e o , which is conside ed in his pape as a
s ochas ic p ocess.
Eq. (22) di e s om he model used in [11] in he sense
ha he e o model in ha wo k measu es he di e ence
be ween he messages a i e a ion lwi h hei coun e pa s
a he ixed poin o he message-passing i e a ion. In o he
wo ds, he e o model in [11] measu es he de ia ion o he
messages a each i e a ion om hei inal alue eached by
BP a e con e gence. The s ochas ic e o we discuss he e is
b ie ly s udied in [11] unde he no ion o addi ional e o .
By exp essing he messages in he he log domain, we ha e
˜m(l)
k→jln ˜μ(l)
k→j(xj=+1)
˜μ(l)
k→j(xj=−1) =m(l)
k→j+ν(l)
k→j(23)
whe e
ν(l)
k→jln ε(l)
k→j(xj=+1)
ε(l)
k→j(xj=−1) (24)
Based on he on Neumann model, we assume ha i k=n,
hen E[ln ε(l)
k→j(x)lnε(l)
n→j(x)] = 0 o all x. Consequen ly,
we ha e E[νk→jνn→j]=0. To measu e he collec i e impac
o e o s on he belie o node j,weuse
E(l)
j(xj)˜
b(l)
j(xj)
b(∗)
j(xj)(25)
whe e ˜
b(l)
j(xj)deno es he belie a node j esul ing om a
BP i e a ion wi h e oneous messages as in (22) while b(∗)
j(xj)
deno es he belie o node ja a ixed poin eached by an
e o - ee BP i e a ion. We use (∗)ins ead o (l) o indica e
he messages and belie s a a ixed poin o he e o - ee BP.
By assuming unco ela ed s ochas ic beha io o he mes-
sage e o s, an uppe bound on cumula i e e o s a ec ing he
belie s can be ob ained. Speci ically, assuming Va ν(l)
k→j≤
(ln u)2 o all k,j,l, an uppe bound on he esul ing cumu-
la i e s eng h o e o s a node jis de i ed in [11] as,
Eln dE(l)
j2≤
k∈Njσ(l)
kj 2
(26)
whe e σ(1)
kj =lnd(ψkj )2and
σ(l+1)
kj 2=ln d(ψkj )2ω(l)
kj +1
d(ψkj)2+ω(l)
kj 2
+(lnu)2(27)
while
ln ω(l)
kj 2=
n∈Nj
kσ(l)
nk2
(28)
whe e
dE(l)
jsup
a,b 



E(l)
j(a)
E(l)
j(b)(29)
d(ψkj)2sup
a,b,c,d
ψkj(a, b)
ψkj(c, d)(30)
We use he uppe bound in (26) in he log domain based
on he ac ha (see (16) and (25))
˜
λ(l)
jln
˜
b(l)
j(+1)
˜
b(l)
j(−1) =λ(∗)
j+lnE(l)
j(+1)
E(l)
j(−1) (31)
which leads o
E
˜
λ(l)
j−λ(∗)
j
2=Eln E(l)
j(+1) −ln E(l)
j(−1)
2
=Eln dE(l)
j2≤
k∈Njσ(l)
kj 2
(32)
Hence, in he de ec ion s uc u e discussed, (26) gi es an uppe
bound on he MSE le el obse ed in he decision a iable a
node j.
B. Linea App oxima ions
In ou analysis, we dis inguish be ween he message e o s
and he e o s in he compu a ion o local LLRs o gain u he
insigh in o he beha io o he BP algo i hm. In pa icula ,
we model he e oneous local LLRs as ˜γkγk+kand e e
o k’s as likelihood e o s (LE) while assuming ha LEs a e
unco ela ed as well, i.e., E[kn]=0 o k=n. We e e
o νk→j’s as message e o s (ME) and assume ha LEs and
MEs a e mu ually independen . Mo eo e , we assume ha all
MEs and LEs a e independen o he messages and o he
local LLRs. No e ha he bound in (32) does no ake LEs
in o accoun .
Taking bo h ypes o e o in o accoun , we exp ess he
messages decision a iables as
˜m(l)
k→j=S⎛
⎝Jkj,˜γk+
n∈Nj
k
˜m(l−1)
n→k⎞
⎠+ν(l)
k→j(33)
˜
λ(l)
j=˜γj+
k∈Nj
˜m(l)
k→j(34)
which shows ha he e o s pass h ough he same nonlinea
ans o ma ion (i.e., S) as he messages do. By using (33),
we can analyze he beha io o e o s. The p oposed linea
BP i e a ion in he p esence o message e o s is exp essed
as
˜m(l)
k→j≈cjk ⎛
⎝˜γk+
n∈Nj
k
˜m(l−1)
n→k⎞
⎠+ν(l)
k→j(35)
Consequen ly, simila o he way (19) is de i ed, he esul ing
e oneous decision a iable is o med as
˜
λ(l)
j≈˜γj+
k∈Nj
cjk˜γk+
k∈Nj
n∈Nj
k
cjkckn˜γn+...
+
k∈Nj
ν(l)
k→j(36)
ABDI AND RISTANIEMI: MODELING AND MITIGATING ERRORS IN BELIEF PROPAGATION FOR DISTRIBUTED DETECTION 3291
which can be eo ganized as
˜
λ(l)
j≈λj+ξ(l)
j(37)
whe e
ξ(l)
j
N

i=1
ajii+
k∈Nj
ν(l)
k→j(38)
Eq. (38) shows ha he e o a ec ing he decision a iable
a node jhas wo dis inc componen s. The i s componen
is buil as a linea combina ion o LEs while he second one
is he sum o he MEs ecei ed a node j om i s one-hop
neighbo s. The i s componen is ixed whe eas he second
one exhibi s a new ealiza ion a e e y i e a ion.
Acco ding o (38), de ia ion om he e o - ee decision
a iables, caused by e o s in he BP i e a ions, can app oxi-
ma ely be measu ed by
E
˜
λ(l)
j−λ(∗)
j
2≈Eξ(l)
j
2=aT
jΣaj+ Σνj(39)
whe e Σco ()and Σνjco ν(l)
jwhile 
[1,...,
N]Tand ν(l)
jdeno es an |Mj|-by-1 ec o ha
con ains ν(l)
k→j’s o k∈M
jwhe e MjNj∪{j}while
ν(l)
j→j0. No e ha (36) includes mo e ME e ms han
jus k∈Njν(l)
k→j. Howe e , hey can all be neglec ed since
|cjk|<1 o all j, k.
Eq. (36) shows ha when BP is used o ealize a dis ibu ed
de ec ion, he e oneous local likelihoods in he ne wo k a e
combined linea ly o build he decision a iables. We can
e alua e he impac o he e o s on he sys em pe o mance
by analyzing he s ochas ic beha io o he e oneous decision
a iables ˜
λ(l)
j.Gi enx, he decision a iable a node j
is ob ained as a linea combina ion o independen andom
a iables. Consequen ly, i s condi ional pd is de i ed as
˜
ιj|x(z|b)≈N

i=1
1
aji
˜γi|xz
aji
|b∗
k∈Nj
νk→j(z)(40)
whe e
˜γi|x(z|b)= γi|x(z|b)∗ i(z)(41)
while and ∗deno e he con olu ion ope a o . Consequen ly,
we ha e
g
j(τj, )P {˜
λj>τ
j|xj= }
=
b∈{−1,1}N−1
px(j)|xj(b| )
∞
τj
˜
ιj|x(z|Ej, (b)) dz
(42)
whe e ∈{−1,+1},x(j)
[x1,x
2,...,x
j−1,x
j+1,...,x
N]Tand Ej, (b){x(j)=
b,x
j= }while px(j)|xj(b| )P {x(j)=b|xj= }.
Sol ing
g
j(τj,−1) = αgi es a h eshold alue ha ixes
he alse-ala m a e a α. Simila ly,
g
j(τj,1) = β ixes he
de ec ion a e a β. Recall ha aji’s a e ound by using cjk’s,
see (21).
As a common p ac ical case, when he local LLRs and
he e o s ollow Gaussian dis ibu ions [17]–[21] he decision
a iable ˜
λj ollows a Gaussian dis ibu ion as well and i
is ully cha ac e ized by i s i s - and second-o de s a is ics.
Speci ically, we ha e
∞
τj
˜
ιj|x(z|Ej, (b)) dz =Qτj−μj, (b)
σj, (b)(43)
whe e
μj, (b)E˜
λj|Ej, (b)
=E[γj|xj= ]+
i=j
ajiE[γi|xi=bi](44)
σ2
j, (b)Va ˜
λj|Ej, (b)
=Va [γj|xj= ]+
i=j
a2
jiVa [γi|xi=bi]+E|ξj|2
(45)
In (44) we ha e assumed, wi hou loss o gene ali y, ze o-mean
e o s. No e ha , wi hou he p oposed app oxima ion hese
pe o mance measu es a e no a ailable analy ically due o
he nonlinea i y o (14). In he es o he pape , we assume
ha he local likelihoods, LEs, and MEs a e Gaussian
andom a iables. Eq. (40) shows ha , acco ding o he
CLT, e en i he local LLRs and e o s a e no Gaussian
andom a iables, he s ochas ic beha io o he decision
a iables can s ill be app oxima ely desc ibed by Gaussian
dis ibu ions.
C.Impac o A e aging
In ABP, he message-passing i e a ion is he same as in BP.
Howe e , ins ead o he ac ual message alues, an a e age o
he messages a e used o build he decision a iables. To be
mo e speci ic, in he log domain and o l≥L+1,le
¯m(l)
k→j1
L+1
l

=l−L
˜m( )
k→j(46)
The decision a iable a node jis calcula ed by
¯
λ(l)
jγj+
k∈Nj
¯m(l)
k→j(47)
Simila o ou discussion ega ding (19), we can show ha
when he message-passing i e a ion is e o - ee, ¯
λ(∗)
j
liml→∞ ¯
λ(l)
j=λj. Hence, we can see ha he a e aging
p ocess does no al e he ixed poin s achie ed by he
e o - ee linea BP. This obse a ion is in line wi h he
con e gence analysis p o ided in [15].
The impac o a e aging on LEs and MEs can be cla i ied
by no ing ha
¯
λ(l)
j=λj+¯
ξ(l)
j(48)
whe e, assuming L o be la ge enough, we ha e
¯
ξ(l)
j=
N

i=1
ajii+
k∈Nj
¯ν(l)
k→j≈
N

i=1
ajii(49)
since ¯ν(l)
k→j1
L+1 l
=l−Lν( )
k→j≈0. We can s a e (49) in
he o m o MSE as
E
¯
λ(l)
j−λ(∗)
j
2≈aT
jΣaj+1
L+1 Σνj(50)
Assuming L o be la ge enough and MEs o ha e ze o
mean, (49) shows ha he esul ing decision a iable buil by
3292 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 69, NO. 5, MAY 2021
ABP in (47) is almos clea ed o MEs. Howe e , he a e aging
p ocess has almos no impac on LEs.
No e ha in ABP he message-passing i e a ion is he same
as in BP and he a e aging is only pe o med when compu ing
he decision a iables. Mo eo e , in ABP, ins ead o s o ing
he messages in pas i e a ions sepa a ely, we only need o
s o e he sum o he messages up o he cu en i e a ion. As a
consequence, he numbe o addi ional memo y cells equi ed
can be kep cons an [15]. We will use ABP in Sec. IV-B o
build an o line lea ning-op imiza ion s uc u e o he linea
BP in he p esence o e o s.
IV. MITIGATING ERRORS BY LINEAR FUSION
In his sec ion, we i s p opose a wo-s age linea usion
scheme o ob ain a nea -op imal de ec ion pe o mance by
supp essing he impac o he e o s. Then, we ealize he
p oposed op imiza ion in a blind decen alized se ing whe e
he equi ed s a is ics a e no a ailable a p io i.
A. Linea Fusion
Fi s , since |cjk|<1, we u he app oxima e he decision
a iable λjin (19) as
λj≈
k∈Mj
cjkγk(51)
Due o he symme y o he da a- usion p ocess in (19),
he app oxima ion in (51) is an e ec i e app oach o building
a dis ibu ed compu ing amewo k o sys em pe o mance
op imiza ion. In his amewo k, each node in e ac s only wi h
i s immedia e neighbo s. We ha e cla i ied his symme y in [5,
Sec. III-B]. By aking in o accoun he e o s while analyzing
he linea BP, (19) and (36) lead o
˜
λj≈
k∈Mj
cjk (γk+k)+ 
k∈Nj
νk→j(52)
We see ha he dis u bance on he decision a iable caused
by LEs is buil , app oxima ely, as a linea combina ion o k’s
wi h cjk’s ac ing as weigh s in his combina ion. The e o e,
we use cjk’s as design pa ame e s o mi iga e he impac o
k’s. Mo eo e , MEs a e combined in (52) linea ly and in his
combina ion, all weigh s a e one. We p opose o ex end his
combina ion by using a modi ied e sion o (34) as
ˆ
λ(l)
j˜γj+
k∈Nj
wjk ˜m(l)
k→j(53)
This modi ica ion in he s uc u e o he decision a iable does
no a ec he con e gence o he p oposed linea BP since i
does no al e he message-passing i e a ion. Now, based on
an app oxima ion simila o he one in (52), we ha e
ˆ
λj≈
k∈Mj
wjkcjk (γk+k)+ 
k∈Nj
wjkνk→j(54)
Since ˆ
λjis a Gaussian andom a iable, we only need
i s mean and a iance o cha ac e ize i s s a is ical beha io .
Speci ically, o b∈{−1,1},weha e
P {ˆ
λj>τ
j|xj=b}=Qτj−E[ˆ
λj|xj=b]
Va [ˆ
λj|xj=b](55)
whe e
E[ˆ
λj|xj=b]≈ T
jμb(56)
Va [ˆ
λj|xj=b]≈ T
jΣγj|b+Σj j+wT
jΣνjwj(57)
whe e jwj◦cjin which ◦deno es he Hadama d p oduc
while Σγj|b=co (γj|xj=b)and Σj=co (j). Mo eo e ,
wj,cj,γj,andja e |Mj|-by-1 ec o s con aining wji’s,
cji’s, γi’s, and i’s o i∈M
j, espec i ely. Eq. (55) gi es he
sys em alse-ala m p obabili y o b=−1and he de ec ion
p obabili y o b=1. The alse-ala m p obabili y can be se
o P(j)
=αby
τj=Q−1(α)Va [ˆ
λj|xj=−1] + E[ˆ
λj|xj=−1] (58)
and hen by using (55) – (57), wjand cjcan join ly be
op imized in a Neyman-Pea son se ing.
To a oid he challenges o his op imiza ion, we maximize
he de lec ion coe icien o he de ec o . We al eady know ha
he esul ing de ec o pe o ms well when he decision a i-
ables ollow Gaussian dis ibu ions. In his manne , we mi i-
ga e he join impac o LEs and MEs wi h low compu a ional
complexi y.
The p oposed op imiza ion is conduc ed in wo consecu-
i e s ages based on he ac ha we can decompose he
cons uc ion o ˆ
λjin o wo consecu i e usion p ocesses.
Tha is, we i s op imize cjk’s by conside ing he impac o
k’s on γk’s. Then, we conside he esul ing scaled LLRs,
i.e., cjkγk’s, as new s a is ics o be linea ly combined, while
being weigh ed by wjk’s and dis o ed by νk→j’s, o make he
decision a iable a node j.
Mo e speci ically, i s , we op imize cjin a hypo he ical
linea de ec o wi h i s decision a iable de ined as
ˆ
λ
jcT
jγj+j(59)
The coe icien s esul ing om his op imiza ion scale up he
mo e eliable local LLRs, wi h espec o he ones buil unde
low SNR egimes, o supp ess he e ec o LEs. We deno e
he esul ing usion weigh s by c∗
j. Then, we use c∗
jwi hin
he s uc u e o he ac ual de ec o o op imize wj o mi iga e
he impac o MEs. Tha is, we conside he ollowing linea
de ec o a node j
ˆ
λ
jwT
jχj+νj(60)
whe e χjc∗
j◦(γj+j)con ains χjk’s o k∈M
jwhile
χjk =c∗
jk(γk+k). The ec o νjcon ains νk→j’s wi h
k∈M
j. In his s uc u e, he elemen s o χjk’s a e seen as
he ac ual local LLRs ha a e combined o build he decision
a iable a node jwhile he combina ion akes in o accoun
he join deg ading e ec o MEs and LEs.
Based on he ma e ial p o ided in Sec. II-A, he i s s age
o he p oposed op imiza ion is o mally s a ed as
c∗
j=a gmax
cjΔ
j(cj),s. ., cj=1 (61)
whe e
Δ
j(cj)= cT
jδj2
cT
jΣγj|−1+Σjcj
(62)
whe e δjE[γj|xj=1]−E[γj|xj=−1]. The esul ing
c∗
jis hen used o ealize he second s age o he p oposed
ABDI AND RISTANIEMI: MODELING AND MITIGATING ERRORS IN BELIEF PROPAGATION FOR DISTRIBUTED DETECTION 3293
op imiza ion by sol ing
w∗
j=a gmax
wjΔ
j(wj),s. ., wj=1 (63)
whe e
Δ
j(wj)= wT
jˆ
δj2
wT
jΣχj|−1+Σνjwj
(64)
whe e ˆ
δj=c∗
j◦δjand Σχj|−1=co (χj|xj=−1) =
c∗
jc∗T
j◦Σγj|−1+Σj.Ha ingc∗
jand w∗
j, he de ec ion
h eshold τjis de i ed as τj=Q−1(α)Va [λ
j|xj=−1] +
E[λ
j|xj=−1] o ix he sys em alse-ala m a e a α.
The con e gence condi ion |cj,k|<1
maxn|Nn|−1,∀(j, k)∈
Ecan be ealized by a simple no maliza ion o c∗
j,k’s since he
objec i e unc ion in (61) does no change by no malizing i s
a gumen .
Th ough he p oposed wo-s age op imiza ion, we enhance
he de ec ion pe o mance a node jby supp essing he join
impac o MEs and LEs wi h low compu a ional complexi y.
The s a is ics equi ed in his op imiza ion a e collec ed om
he one-hop neighbo s o node j. This makes he p oposed
me hod a iable app oach in ad-hoc ne wo k con igu a ions
whe e majo ne wo k unc ionali ies a e conduc ed h ough
one-hop links be ween he ne wo k nodes.
B. O line Lea ning and Adap a ion
To ealize he p oposed op imiza ion, we need he mean
and co a iance o he local e oneous LLRs. In a blind se ing
whe e he e is no p io in o ma ion a ailable ega ding he
adio en i onmen , we ha e o es ima e hose pa ame e s based
on he de ec ion ou comes. The main challenge he e is ha he
s a e o xjis equi ed a node jwhile he only in o ma ion
a ailable in p ac ice is he de ec ion ou come ˆxj. Hence,
node jhas o es ima e he condi ional s a is ics equi ed
in (61) and (63) based on ˆxj. The p oblem wi h such an
adap a ion mechanism is ha i makes he de ec ion ou come
ˆxjdepend on hose es ima es. This dependence c ea es an
inhe en de e io a ing loop by eeding he de ec ion e o s back
in o he sys em s uc u e h ough e oneous es ima es o he
equi ed s a is ics.
To o e come his challenge, we p opose an ex ended e sion
o he blind lea ning-adap a ion loop in [5] ha accommoda es
he p oposed e o -mi iga ing s uc u e. The pseudo-code o
his adap a ion is p o ided in Algo i hm 1 whe e he ask o
each node is speci ied in a dis ibu ed compu ing amewo k.
Algo i hm 1 ope a es on a window o s o ed sensing ou comes
and in ol es a seconda y BP ha is un much less equen ly
han he a e a which he dis ibu ed de ec ion is pe o med.
The ou comes o his o line BP a e used in he es ima ion o
he equi ed unknown s a is ics. In his adap a ion, he desi ed
op imiza ions a e ealized i e a i ely while each node in e ac s
only wi h i s one-hop neighbo s. Consequen ly, Algo i hm 1
can be well inco po a ed in a decen alized ne wo k con igu-
a ion.
In he sequel, we p opose a blind adap a ion s uc u e in
which we use κ o deno e he i e a ion index. No e ha we
use las he i e a ion index in he main BP h ough which
Algo i hm 1 Blind Adap a ion o Fusion Weigh s in E oneous
Linea Belie P opaga ion
Inpu : ˜
γT,¯
γT,τ(0),κmax,η
Ou pu : Nea -op imal cjand wj o j=1,...,N
1. Le κ←0and ini ialize ˆ
x(0) by compa ing ˜
γTagains
τ(0);
2. while κ≤κmax
3. o node j∈{1,2,...,N}
4. Calcula e E[¯γi|ˆx(κ)
j]and co (¯γi,¯γk|ˆx(κ)
j) o all i, k ∈
Mj;
5. Sol e (61) o ind c(κ)
jand τ(κ)
j;
6. Se c∗
jby an η- es on c(κ)
j;
7. end
8. Use c∗
j’s and τ(κ)
j’s o un linea ABP on ˜
γT o ind
ˆ
x(κ+1);
9. κ←κ+1;
10. end
11. o node j∈{1,2,...,N}
12. Use c∗
jand ˆ
x(κmax) o calcula e ˆ
δj,Σχj|0and Σνj;
13. Sol e (63) o ind w∗
j;
14. end
15. Ou pu c∗
jand w∗
j o j∈1,2,...N;
he dis ibu ed de ec ion is ealized. The o line adap a ion
upda es he usion weigh s in he p oposed linea BP by
p ocessing Ts o ed samples o ˜
γ. This window o e oneous
local likelihoods is deno ed by ˜
γTand con ains samples o
˜
γ( ) o =1,2,...,T. Recall ha , ˜
γ=γ+whe e 
deno es he ec o o LEs. The o line de ec ion ou comes a
i e a ion κa e deno ed by ˆ
x(κ)[ˆx(κ)
1,...,ˆx(κ)
N]while he
esul ing usion weigh s and de ec ion h esholds a e deno ed
c(κ)
jand τ(κ)
j espec i ely. ˆ
x(κ)deno es a window o s o ed
sensing ou comes ˆ
x(κ)( ) o =1,2,...,T. Fo simplici y,
we do no show he ime index when dealing wi h ˜
γT,and
ˆ
x(κ).
Due o e o s caused by he wi eless links be ween he
sensing nodes, node jdoes no ha e access o ˜γk( ),k∈N
j.
Speci ically, wha node j ecei es om node kis ˜γk( )+νk→j
whe e νk→jdeno es he co esponding link e o . Wi hou
loss o gene ali y, we a ibu e MEs o wi eless link e o s.
To alle ia e he link e o s, be o e s a ing he adap a ion
p ocess node j ecei es Lcopies o ˜γk( ) om node kand
calcula es an a e age o ob ain ¯γk( )˜γk( )+¯νk→jwhe e
¯νk→jdeno es he a e age o Lindependen ealiza ions o
νk→j. The desi ed s a is ics a e hen calcula ed by p ocessing
¯γk’s, which app oxima e ˜γk’s. We use ¯
γT o con ain he
samples o ¯γk( ) o =1,2,...,T o k=1,2,...,N.
In a ealis ic de ec ion scena io, he da a exchanged be ween
he nodes in he p oposed o line adap a ion is impai ed by
bo h ypes o e o s. Since in he i s linea usion (61) we
ake in o accoun he impac o LEs only, we need o isola e
his op imiza ion om he MEs. To his end, we es ima e he
desi ed s a is ics by using linea ABP. As we saw in Sec. III-C,
MEs do no a ec he ABP ou comes signi ican ly. The e o e,
he esul ing o line decision a iables a e almos clea ed o