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Performance Analysis of Sensor Network with Intermediate Fusion Helper for Cognitive Radios
Research Paper / Jan 2011

Performance Analysis 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
multiple 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 [20].

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], [17], [21]).

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. On the other hand, much work [3],
[4], [7], [19] has been devoted to examine the information
combining rules under the condition of rate-constrained fu-
sion channels. The optimal information combining rules were
extensively studied in [5], [8]–[11], [16], [18], 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

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



Fig. 1. Block diagram for the basic sensor network

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 .
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.

1Throughput this paper, we ignore error-correction coding as it would have
the same effects in all of the discussions.



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, Si = −1|H0) + Pr(Ti = −1, Si = 1|H0)
= APt + (1−A)(1− Pt) , PB ,

and
Pr(Ti = 1|H1) = 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 with-
out 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].

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

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

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, 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.

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


*

)





A = 0.8
A = 0.9
A = 0.95
A = 0.99

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

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 Power 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
power of each sensor needs to be very large. In the sensor
network with relays, the multiple relays will consume as
much transmission energy as the sensors. The network with
intermediate fusion helper requires a single transmission from
the intermediate fusion helper. Hence, the total transmission
energy consumed in the sensor network with intermediate
fusion helper is only 43 of that from the sensors.

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 ignore the fast fading
of the fusion channels for simplicity. In this section, we shall



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

simulate 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 multiple 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
3 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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