Preliminary Performance Evaluation of Sensor Network with Intermediate Fusion Helper for Cognitive Radios

Research Paper / Jan 2012

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Preliminary Performance Evaluation of Sensor Network with Intermediate Fusion Helper for Cognitive Radios

Research Paper / Jan 2012

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Preliminary Performance Evaluation of Sensor

Network with Intermediate Fusion Helper for

Cognitive Radios

Chunxuan Ye, Alpaslan Demir and Yan Li

InterDigital Communications, LLC.

781 Third Ave, King of Prussia, PA 19406

Email: {chunxuan.ye, alpaslan.demir, yan.li}@interdigital.com

Abstract—In this paper, we consider the problem of using

three sensors to detect whether a certain spectrum is occupied

or not. Each sensor sends its binary decision to the data fusion

center through a wireless fading channel. The data fusion center

combines the outcomes for an overall decision. Our analysis

shows that a basic sensor network does not result in a high

enough correct probability of the overall decision when the

wireless fading channels experience low SNR. Then, we observe

that this probability could be significantly increased with the

deployment of relays in the network. However, a sensor network

with relays still suffers from energy and spectral inefficiency.

The sensor network with an intermediate fusion helper was

recently proposed to reduce the traffic load at the data fusion

center. We examine the correct probability of the overall decision

resulting from a sensor network with an intermediate fusion

helper. Our evaluation establishes that a sensor network with an

intermediate fusion helper performs almost as good as the sensor

network with relays, but with energy and spectral advantages.

I. INTRODUCTION

Cognitive radio (CR) is a potential technology for increasing

spectral efficiency in wireless communications systems. In

a cognitive radio system, secondary users temporarily use

spectrum that is not utilized, as long as negligible impact is

caused to primary licensed users. In order to opportunistically

access temporarily unused spectrum, the spectrum in an area

needs to be sensed from time to time. In a simple scenario,

a secondary user acts as a sensory node; it senses and

uses the available spectrum. The spectrum sensing techniques

include energy detector-based sensing, waveform-based sens-

ing, cyclostationarity-based sensing, radio identification-based

sensing, matched-filtering, etc [21].

Due to noise uncertainty and wireless channel fading, the

sensing decision made by a single sensor is sometimes un-

reliable. Cooperative sensing among multiple sensors is an

efficient approach to addressing this issue, because it provides

multiple measurements and, hence, increases the diversity.

Additionally, having sensors cooperating over a wide area also

provides a possible solution to the hidden-terminal problem.

This is because sensors, separated by a distance larger than

the correlation distance of shadow fading, are unlikely to be

shadowed simultaneously from the primary user.

In cooperative sensing, after performing the spectrum sens-

ing operations, each sensor sends its sensing results to a

data fusion center, which makes an overall decision about

the spectrum occupancy. The process of making an overall

decision based on multiple sensing results is called data fusion

or information combining. Depending on the type of sensing

results sent from the sensors to the data fusion center, the

information combining can be classified into three categories:

hard combining (cf. e.g., [14]), hard combining with side

information (cf. e.g., [2], [6], [14]), and soft combining (cf.

e.g., [6], [12], [13], [15], [18], [22]).

In the above work, the sensing results from all the sensors

are assumed to be delivered to the data fusion center without

error. In other words, the fusion channels, i.e., the channels

from sensors to the data fusion center, are error-free and

bandwidth unlimited. There are many applications (e.g., [17])

for large scale sensor networks, with power limited sensors

wirelessly connected to the data fusion center. The fusion

channels are noisy and experience wireless fading. Much work

[3], [4], [7], [20] has been devoted to examine the informa-

tion combining rules under the condition of rate-constrained

fusion channels. The optimal information combining rules

were extensively studied in [5], [8]–[11], [16], [19], when the

fusion channels are noisy channels or wireless fading channels.

Furthermore, it was proposed in [9], [10] to use relays for

reliable transmissions on the noisy fusion channels. It should

be mentioned that most of the efforts, in the presence of the

noisy or rate-constrained fusion channels, are focused on the

optimal information combining rules.

It was recently proposed in [1] to reduce the traffic load at

the data fusion center by using an intermediate fusion helper in

a sensor network. Specifically, the intermediate fusion helper

combines the decisions it receives from several sensors, and

transmits the (combined) intermediate decision to the data

fusion center. Although the spectral advantage of the sensor

network with an intermediate fusion helper is obvious, its

detection performance, especially in the noisy fusion channel

environment, is unclear.

The contribution of this paper is two-fold:

i). We establish a system model to incorporate the practical

difazira

Sticky Note

Do you really want to use "Preliminary" in the title? There is always more to do on any topic.

Fig. 1. Block diagram for the basic sensor network

situations of wireless fading fusion channels. Within this

model, we analyze the performance of a basic sensor network,

a sensor network with relays, and a sensor network with

an intermediate fusion helper. It is shown that the basic

sensor network does not perform well, in terms of the correct

probability of the overall decision at the data fusion center.

This probability is significantly increased with the deployment

of relays in the network. Our analysis shows that the correct

probability of the overall decision in the sensor network with

an intermediate fusion helper is almost as good as that in the

sensor network with relays, and is much higher than that in

the basic sensor network. Subsequently, to achieve the same

detection probability, the sensor network with an intermediate

fusion helper consumes the least transmission energies. This

is because only a single transmission is needed from the

intermediate fusion helper, compared with one transmission

per relay in the sensor network with relays.

ii). In the sensor networks with relays or an intermediate

fusion helper, we study the locations of the relays or the in-

termediate fusion helper for the optimal network performance.

Specifically, the optimal relay location is the middle of the

sensors and the data fusion center, while the optimal location

of the intermediate fusion helper should be a bit closer to the

data fusion center. Such examination facilitates the design of

sensor networks.

The rest of this paper is organized as follows. The problem

formulation is given in Section II. Section III discusses the

sensor network with relays. The sensor network with an

intermediate fusion helper is introduced in Section IV. The

performance of all these sensor networks is analyzed in the

separate sections. Simulation results are provided in Section

V. Section VI contains conclusions and discussions.

II. PROBLEM FORMULATION

Consider a wireless sensor network (cf. Figure 1) deployed

with three sensors to detect whether a spectrum is occupied

or not. The detection problem can be stated in terms of a

binary hypothesis test: hypothesis H0 is the signal absence or

spectrum unoccupied, and hypothesis H1 is the signal presence

or spectrum occupied. The a priori probabilities of the two

hypotheses are Pr(H0) = pi0 and Pr(H1) = pi1. Suppose

each sensor listens to a certain spectrum and applies some

spectrum sensing technique. Let Si, 1 ≤ i ≤ 3, denote the

decision made by the ith sensor, where

Si =

{ −1, if H0 is declared,

1, if H1 is declared.

The probability Ai that the decision Si is true is given by

Ai = Pr(Si = −1|H0)pi0 + Pr(Si = 1|H1)pi1.

The observations and decisions made by the three sensors are

assumed to be statistically independent conditioned on either

hypothesis, i.e.,

Pr(S1, S2, S3|Hj) =

3∏

i=1

Pr(Si|Hj), j = 0, 1.

After the spectrum sensing operations, each sensor sends its

decision to the data fusion center through its own fusion

channel. The three fusion channels are mutually independent

wireless fading channels. Let Xi and Yi be the input and the

output of the ith fusion channel. Then,

Yi = hiXi +Ni, (1)

where hi is the channel fading and Ni is the additive white

Gaussian noise with distribution N (0, σ2i ). Before transmis-

sion, the ith sensor modulates its decision Si to Xi, using the

BPSK scheme with transmission power Pi. Hence, we have

Xi =

√

PiSi.1

The data fusion center demodulates the received signal Yi

to Ti ∈ {−1, 1}. It then applies the majority combining rule

(cf. e.g., [14]) to make an overall decision. Specifically, if at

least two of the demodulated decisions are 1, then the data

fusion center declares the presence of the signal. Otherwise,

it declares the absence of the signal. The overall decision at

the data fusion center can be expressed as

U =

{ −1, if∑3i=1 Ti < 0,

1, if

∑3

i=1 Ti ≥ 0,

(2)

The probability Pc that the overall decision U matches the

true hypothesis is defined as

Pc = pi0 Pr(U = −1|H0) + pi1 Pr(U = 1|H1). (3)

Next, we shall characterize this probability. To simplify our

calculations, we make the following symmetry assumptions in

the rest of this paper.

i). For each sensor: Pr(Si = −1|H0) = Pr(Si = 1|H1).

Hence, Ai is equal to the detection probability Pr(Si = 1|H1).

ii). All the sensors have the same detection probability:

A1 = A2 = A3 = A.

iii). All the sensors have the same transmission power: P1 =

P2 = P3 = P .

iv). The noise powers of all the fusion channels are identical:

σ21 = σ

2

2 = σ

2

3 = σ

2.

With these simplifications, we define the signal to noise

ratio as SNR = Pσ2 .

1Throughput this paper, we ignore error-correction coding as it would have

the same effects in all of the discussions.

difazira

Sticky Note

energy

difazira

Sticky Note

Is this the same as the Ai definition above?

To further facilitate our computations, we ignore the fast

fading of the fusion channels at this moment, and only take

the path loss into account. Hence, the channel fading hi in (1)

has |hi|2 = d−βi , where di is the distance from the ith sensor

to the data fusion center and β is the path loss exponent. We

assume of the equal distance from all the sensors to the data

fusion center, i.e., d1 = d2 = d3 = d.

Let Pt be the probability that a transmission on a fusion

channel is demodulated correctly at the data fusion center.

Then, it follows from the BPSK modulation scheme that

Pt = Pr(Ti = Si)=1−Q

(√

|hi|2P

σ2

)

=1−Q

(√

d−βSNR

)

,

(4)

where Q(x) = 1√

2pi

∫∞

x

e−

t2

2 dt is the usual Gaussian tail

function. It follows from the Markov chain and (4) that

Pr(Ti = −1|H0) = Pr(Ti = 1|H1)

= Pr(Ti = −1, Si = −1|H0) + Pr(Ti = −1, Si = 1|H0)

= APt + (1−A)(1− Pt) , PB .

According the majority combining rule (2), we have

Pr(U = −1|H0)=Pr(

3∑

i=1

Ti = −1|H0)+Pr(

3∑

i=1

Ti = −3|H0)

= 3P 2B(1− PB) + P 3B = Pr(U = 1|H1).

Hence, it follows from (3) that

Pc = 3P 2B(1− PB) + P 3B .

This probability vs. SNR is illustrated using the square curve

in Figure 2. In plotting this curve, we set A = 0.9, β =

3.5, and d = 10. Even though the correct probability of the

individual decision is as high as 0.9, we observe from the

figure that the correct probability of the overall decision is

quite small in the low SNR region. In order to achieve a correct

probability of the overall decision higher than A = 0.9, the

SNR of each fusion channel needs to be no less than 36 dB.

It follows from the majority combining rule that the cor-

rect probability of the overall decision is upper bounded by

3A2(1 − A) + A3. This upper bound is achieved when the

fusion channels are noiseless. For A = 0.9, this upper bound

is equal to 0.972, as seen in the figure.

III. SENSOR NETWORK WITH RELAYS

As discussed, the basic sensor network does not perform

well at low SNRs. A natural way to increase Pc is via

enhancing the sensors’ transmission power P , and hence the

SNR. This approach may be infeasible due to the power

limitation of the sensors, as well as the potential interference

caused. An approach to improving the transmission reliability

without enhancing the sensors’ transmission power is by

means of relays. The usage of relays for reliable transmissions

and throughput increment has been widely studied, while its

application for reliable transmissions on the fusion channels

has been adopted in [9], [10].

24 26 28 30 32 34 36 38 40 42 44

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

SNR (dB)

Co

rre

ct

p

ro

ba

bi

lity

Basic sensor network

Sensor network with DF relays

Sensor network with AF relays

Sensor network with fusion helper (α = 0.5)

Sensor network with fusion helper (optimized)

Fig. 2. The correct probability of the overall decision in different sensor

networks, with parameters A = 0.9, β = 3.5, d = 10.

Consider the sensor network in Figure 1, but with a relay

located between every sensor and the data fusion center. It is

known that the usual relaying schemes include the demodulate-

and-forward scheme, and the amplify-and-forward scheme. We

shall characterize the correct probability of the overall decision

Pc in the sensor network with relays, using either of these two

relaying schemes.

Here, we assume that the distance from a sensor to its

serving relay is αd, 0 < α < 1, and the distance from a

relay to the data fusion center is (1−α)d. All the relays have

the same transmission power as the sensors.

A. Demodulate-and-Forward Relays

For the demodulate-and-forward scheme, a relay first de-

modulates the transmission from a sensor. It then re-modulates

the binary decision and transmits it to the data fusion center.

Note that all the channels to and from the relays are wireless

fading channels. The data fusion center demodulates the trans-

missions from the relays, and applies the majority combining

rule to make an overall decision.

Denote by Ri the demodulated decision at the ith relay. Let

Pt,1 be the probability that a transmission from a sensor is

demodulated correctly at the corresponding relay. Let Pt,2 be

the probability that a transmission from a relay is demodulated

correctly at the data fusion center. Then, we have

Pt,1 = Pr(Ri = Si) = 1−Q

(√

(αd)−βSNR

)

, (5)

and

Pt,2 = Pr(Ti = Ri) = 1−Q

(√

[(1− α)d]−βSNR

)

. (6)

It can be derived that

Pr(Ti = −1|H0) = Pr(Ti = 1|H1)

= APt,1Pt,2 +A(1− Pt,1)(1− Pt,2)

+(1−A)(1− Pt,1)Pt,2 + (1−A)Pt,1(1− Pt,2)

, PR.

difazira

Sticky Note

I suggest clarifying that this is the SNR fo the fusion channel. The previous time I reviewed this paper, the very large SNR values confused me. I'm still not sure why the fusion SNR needs to be so high, especially with 2/3 detection but I'll assume you gave that some thought after my comments last time.

Therefore,

Pc = 3P 2R(1− PR) + P 3R. (7)

B. Amplify-and-Forward Relays

Let Xi,1 and Yi,1 denote the inputs and the outputs of the

channel from the ith sensor to its serving relay. Let Xi,2 and

Yi,2 denote the inputs and the outputs of the channel from the

ith relay to the data fusion center. Then, we have

Yi,1 = hi,1Xi,1 +Ni,1 = hi,1

√

PSi +Ni,1,

and

Yi,2 = hi,2Xi,2 +Ni,2, (8)

where hi,1 and hi,2 represent the fading on the respective chan-

nel, and Ni,1 and Ni,2 represent the additive white Gaussian

noise with distribution N (0, σ2) on the respective channel. It

follows from the path loss model that |hi,1|2 = (αd)−β and

|hi,2|2 = [(1− α)d]−β .

For the amplify-and-forward scheme, a relay amplifies its

received signal Yi,1 by a factor of K before transmitting it to

the data fusion center, i.e.,

Xi,2 = KYi,1 = Khi,1

√

PSi +KNi,1. (9)

Since the transmission power of a relay is equal to P , we

obtain that

K =

√

P

(αd)−βP + σ2

=

√

SNR

(αd)−βSNR+ 1

. (10)

Denote by ESNR the equivalent SNR for the transmissions

from the sensor to the data fusion center. Then, it follows from

(8), (9) and (10) that

ESNR =

SNR2(αd)−β [(1− α)d]−β

SNR(αd)−β + SNR[(1− α)d]−β + 1 .

Let Pt,A be the probability that a transmission from a sensor

is demodulated correctly at the data fusion center. Then, we

have

Pt,A = 1−Q

(√

SNR2(αd)−β [(1− α)d]−β

SNR(αd)−β + SNR[(1− α)d]−β + 1

)

.

By the similar arguments as in Section II, we derive that

Pc = 3P 2A(1− PA) + P 3A, (11)

where PA = APt,A + (1−A)(1− Pt,A).

The probability (7) is plotted as the circle curve in Figure 2

and the probability (11) is plotted as the star curve in Figure

2. In plotting these curves, we adopt the same parameters

as before, i.e., A = 0.9, β = 3.5, d = 10. Moreover,

the parameter α is set as 0.5. It is seen from the figure

that the sensor network with relays (either demodulate-and-

forward or amplify-and-forward) significantly outperforms the

basic sensor network. Furthermore, the sensor network with

demodulate-and-forward relays performs better than that with

amplify-and-forward relays in the operational SNR region

(though such conclusion may be contrary at lower SNRs).

Hence, we shall focus on the demodulate-and-forward relays

in the remaining discussions of this paper.

The sensor network with relays achieves the desired cor-

rect probability of the overall decision, in the cost of three

additional relays. A simplified version [9] of this network is a

single relay taking all the relaying functionalities. In other

words, the single relay repeats the operations for each of

the sensors. Note that the relay makes three transmissions,

one for each sensor. This consumes much energy, and may

be infeasible for low-power relays. Moreover, the multiple

transmissions may become a communication bottleneck at the

data fusion center if the number of sensors in the network is

large.

IV. SENSOR NETWORK WITH INTERMEDIATE FUSION

HELPER

Consider the sensor network in Figure 1, but with a single

intermediate fusion helper located between all the sensors and

the data fusion center. The intermediate fusion helper receives

and demodulates the transmissions from all the sensors, and

then applies the majority combining rule to make an interme-

diate decision on whether the signal is present or not. It sends

this binary decision to the data fusion center. Subsequently,

the data fusion center simply demodulates this message and

declares the same decision.

The channels to and from the intermediate fusion helper are

wireless fading channels. We assume that the distances from

the sensors to the intermediate fusion helper are αd and the

distance from the intermediate fusion helper to the data fusion

center is (1−α)d. The intermediate fusion helper has the same

transmission power as the sensors.

Let Pt,1 be the probability that a transmission from a

sensor is demodulated correctly at the intermediate fusion

helper. Let Pt,2 be the probability that a transmission from

the intermediate fusion helper is demodulated correctly at the

data fusion center. Then, these probabilities follow from (5)

and (6).

Denote by Fi the demodulated decision from the ith sensor

at the intermediate fusion helper, and denote by UF the

intermediate fusion decision made at the intermediate fusion

helper. Then, by the majority combining rule,

UF =

{ −1, if∑3i=1 Fi < 0,

1, if

∑3

i=1 Fi ≥ 0.

It is not difficult to derive

Pr(Fi = −1|H0) = Pr(Fi = 1|H1)

= APt,1 + (1−A)(1− Pt,1) , PF,i, (12)

and

Pr(UF = −1|H0) = Pr(UF = 1|H1)

= 3P 2F,i(1− PF,i) + P 3F,i , PF . (13)

Therefore, the correct probability of the overall decision is

obtained as

Pc = PFPt,2 + (1− PF )(1− Pt,2). (14)

Using the same parameters as before, we plot the probability

Pc of the sensor network with an intermediate fusion helper

as the diamond curve in Figure 2. It is seen from the figure

that the sensor network with an intermediate fusion helper

performs almost as good as the sensor network with relays.

However, only a single message is transmitted from the

intermediate fusion helper to the data fusion center. This saves

two thirds of the overall bandwidth at the data fusion center,

and the energy consumption by the intermediate fusion helper

is approximately one third of that used by the relays in the

sensor networks with relays.

A. Optimal Location of the Intermediate Fusion Helper

In drawing the diamond curve in Figure 2, we locate the

intermediate fusion helper in the middle of the sensors and

the data fusion center, i.e., α = 0.5. However, such a location

may not be optimal for maximizing Pc. In this sub-section, we

shall examine the optimal location of the intermediate fusion

helper.

It is not difficult to derive from (5) and (6) that

∂Pt,1

∂α

= −

√

β2d−βSNR

8pi

e−

(αd)−βSNR

2 α−

β

2−1, (15)

∂Pt,2

∂α

=

√

β2d−βSNR

8pi

e−

[(1−α)d]−βSNR

2 (1− α)− β2−1. (16)

It follows from (12) and (13) that

∂PF

∂α

= 6PF,i(1− PF,i)(2A− 1)∂Pt,1

∂α

. (17)

Finally, we obtain from (14) that

∂Pc

∂α

= −∂Pt,2

∂α

− ∂PF

∂α

+ 2PF

∂Pt,2

∂α

+ 2Pt,2

∂PF

∂α

. (18)

By inserting (6), (13), (15), (16) and (17) into (18), and setting

it to zero, we obtain the optimal location α∗ as a function of

(A,SNR, β, d)2.

Figure 3 shows the optimal location α∗ vs. SNR for different

values of A and for fixed β = 3.5 and d = 10. It is observed

from the figure that α∗ is, in general, larger than 0.5. This

indicates that for the best performance, the intermediate fusion

helper should be closer to the data fusion center than to the

sensors. With the increase of SNR, the optimal location tends

to α∗ = 0.5. The Pc of the sensor network with the optimally

located intermediate fusion helper is shown as the dot curve

in Figure 2. A few performance gains of the sensor network

with optimized intermediate fusion helper location over that

with a fixed α = 0.5 can be observed from the figure.

Following similar arguments, we find, as expected, that

the optimal relay location for the sensor network with either

demodulate-and-forward relays or amplify-and-forward relays

is always at α∗ = 0.5. Hence, the circle curve and the star

curve in Figure 2 already illustrate the largest achievable Pc

of the sensor network with demodulate-and-forward relays or

amplify-and-forward relays.

2Note that Pc is not a concave function of α at low SNRs. For simplicity,

we focus only on the medium-high SNR range (i.e., SNR ≥ 23 dB), which

insures a concave Pc function.

24 26 28 30 32 34 36 38 40 42 44

0.5

0.52

0.54

0.56

0.58

0.6

0.62

0.64

SNR (dB)

O

pt

im

al

lo

ca

tio

n

(α

*

)

A = 0.8

A = 0.9

A = 0.95

A = 0.99

Fig. 3. The optimal location of the intermediate fusion helper, with

parameters β = 3.5 and d = 10.

0.65 0.7 0.75 0.8 0.85 0.9 0.95 1

0

0.5

1

1.5

2

2.5

3 x 10

4

Correct probability

To

ta

l t

ra

ns

m

is

si

on

p

ow

er

Basic sensor network

Sensor network with relays

Sensor network with fusion helper (optimized)

Fig. 4. The total transmission energies used in different sensor networks

(with the unit noise power)

B. Transmission Energy of the Sensor Networks

As mentioned, the basic sensor network, as well as the sen-

sor network with relays, is not energy and spectrally efficient.

In the basic sensor network, in order to achieve a desired

correct probability of the overall decision, the transmission

energy of each sensor needs to be very large. In the sensor

network with relays, each relay will consume the same amount

of transmission energy as a sensor. Since the network with

intermediate fusion helper requires a single transmission from

the intermediate fusion helper, its total consumed transmission

energy is less than that of the sensor network with relays.

Figure 4 shows the total transmission energy used in the

three networks. It is seen that to achieve the same correct

probability of the overall decision, the network with interme-

diate fusion helper consumes the least transmission energy.

V. SIMULATION RESULTS

In the analytical analysis above, we omit the fast fading of

the fusion channels for simplicity. In this section, we simulate

25 30 35 40 45

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

Average SNR (dB)

Co

rre

ct

p

ro

ba

bi

lity

Basic sensor network

Sensor network with relays

Sensor network with fusion helper

Fig. 5. The performance of different sensor networks with 3 sensors

the performance of different sensor networks, where the fusion

channels experience both path loss and Rayleigh fading. The

other parameters used in our simulations are the same as those

for Figure 2. A total of 5×105 sensing operations is executed,

and the correct probability of the overall decision is averaged

over these trials. The average correct probability of the overall

decision is shown in Figure 5. By comparing Figure 2 and

Figure 5, we observe that our simulation results manifest the

same trend as the analytical conclusions.

VI. CONCLUSIONS AND DISCUSSIONS

We have considered the problem of using three sensors for

cooperative spectrum sensing, in which the fusion channels

from the sensors to the data fusion center are wireless fading

channels. We examined the performance of sensor networks

with or without relays. Then, we considered a sensor network

deploying an intermediate fusion helper to combine the sens-

ing decisions from the sensors before transmitting them to the

data fusion center. The performance of all sensor networks was

compared. Our analysis and simulations establish the perfor-

mance advantage of the sensor network with an intermediate

fusion helper.

Though this paper focused on sensor networks with only

three sensors, we have generalized our performance evaluation

to sensor networks with an arbitrary number of sensors. In

summary, we reached the similar conclusions in general sensor

networks as in the 3-sensor networks, viz., we observed

the performance advantage of the sensor network with an

intermediate fusion helper. Further, to alleviate the potential

communication bottleneck at the intermediate fusion helper

when the number of sensors in a network is large, we have

also proposed to deploy multiple intermediate fusion helpers.

In this paper, we based our analysis on the majority com-

bining rule at either the data fusion center or the intermediate

fusion helper. We have also considered the AND combining

rule and the OR combining rule. Our analysis shows that under

these combining rules, the sensor network with an intermediate

fusion helper even outperforms the sensor network with relays,

in terms of the correct probability of the overall decision.

Moreover, we have attempted to extend our analysis beyond

hard combining rules. Due to the length limitation, we did not

present the details of these results in this paper.

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