Spectrum sensing is a key component of cognitive radio. The prediction of the primary user status in a low signaltonoise ratio is an important factor in spectrum sensing. However, because of noise uncertainty, secondary users have difficulty distinguishing between the primary signal and an unauthorized signal when an unauthorized user exists in a cognitive radio network. To resolve the sensitivity to the noise uncertainty problem, we propose an entropybased spectrum sensing scheme to detect the primary signal accurately in the presence of an unauthorized signal. The proposed spectrum sensing uses the conditional entropy between the primary signal and the unauthorized signal. The ability to detect the primary signal is thus robust against noise uncertainty, which leads to superior sensing performance in a low signaltonoise ratio. Simulation results show that the proposed spectrum sensing scheme outperforms the conventional entropybased spectrum sensing schemes in terms of the primary user detection probability.
1. Introduction
C
ognitive Radio (CR) enables efficient use of a limited spectrum by allowing secondary users (SUs) to access licensed frequency bands of primary users (PUs)
[1]
,
[2]
. Spectrum sensing is a key element to allow SUs to use a vacant frequency band in a CR network. Because of noise uncertainty, however, the performance of the traditional detectors is rapidly deteriorated at a low signaltonoise ratio (SNR). The current prevailing spectrum sensing schemes are an energy detectionbased scheme, a cyclostationarybased scheme, a matched filterbased scheme, and an entropybased scheme
[3]
,
[4]
.
Many researchers have endeavored to increase the sensing performance of the detectors in CR networks. Some researchers have increased the spectrum sensing performance of the energy detector by adjusting parameters such as decision thresholds, sensing frequency, and the number of sensing operations
[5]

[7]
. Cooperative spectrum sensing can increase the PU detection performance by combining the sensing information from several SUs
[2]
,
[8]
. The newly developed entropybased spectrum sensing scheme generally outperforms the other spectrum sensing schemes
[9]

[13]
. In information theory, entropy is a measure of the uncertainty associated with a discrete random variable. The term usually refers to Shannon entropy
[14]
. The authors of
[9]
introduced an entropybased approach for PU detection with uncertainty in noise and presented a likelihood ratio test for detecting a PU signal. To counteract the effect of noise uncertainty at a low SNR, the authors of
[10]

[12]
investigated an entropybased spectrum sensing scheme in the frequencydomain. Estimating entropy in the time domain does not provide good performance under a low SNR, because the estimated entropy value is a constant regardless of the existence of PUs at a low SNR while the entropy can be estimated in the frequencydomain even at a low SNR. These studies identified the state of a PU solely from the current detected data set. The authors of
[13]
presented a new cross entropybased spectrum sensing scheme that has two timeadjacent detected data sets of the PU. This scheme showed an enhanced discriminating ability due to the consideration of more information of the PU signal. The previous works approaches described in
[9]

[13]
, on the other hand, showed the sensing performance in the CR network without the presence of an unauthorized user (UU), where the UU is known as a PU emulation attacker (PUEA). The PUEA emits a signal with a similar form to that of the PU so as to deter access to vacant channels by other SUs
[15]
. Several approaches have been studied to combat PUEAs. A locationbased defence technique was employed in which a number of sensing nodes are deployed to pinpoint PUE attacks
[16]
. A cooperative spectrum sensing technique, where the existence of a PUEA in a CR network is considered, has been proposed wherein several SUs report the detected signal to the fusion center and the fusion center then calculates the decision statistic
[17]
,
[18]
. However, the works of
[15]

[18]
fail to increase the detection performance of each SU because they employ the conventional energy detector.
This paper proposes a conditional entropybased spectrum sensing scheme to detect the PU in a CR network with an UU. The proposed spectrum sensing scheme uses the mutual information between the expected primary signal and the unauthorized signal; it thereupon enhances the sensing performance by reducing the noise uncertainty. In particular, the proposed spectrum sensing scheme substantially increases the sensing performance at a low SNR in comparison with the previous entropybased spectrum sensing schemes. This paper is organized as follows. In Section 2, the system model is introduced. In Section 3, a conditional entropybased spectrum sensing scheme is proposed and analyzed. Simulation results are presented in Section 4, and conclusions are drawn in Section 5.
2. System Model
We consider a CR network with PUs and SUs along with an UU that influences the CR network, as shown in
Fig. 1
. The UU is assumed to be an attacker who is malicious and does not belong to the CR network. In the spectrum sensing period of each time slot, the UU may generate a PU emulation signal in order to deceive SUs. An UU has ability to mimic the behavior of the PU and therefore the unauthorized signal shows similar properties to a primary signal.
A CR network with an UU
We consider a frequency bandwidth
B_{W}
, with central frequency
f_{c}
and sampling frequency
f_{s}
. Each SU senses the signal during
N
samples. The signal received by SUs at the
n
th sample is
where
s
(
n
) is the primary signal,
z
(
n
) is the unauthorized signal, and
w
(
n
) represents background noise, which follows a Gaussian distribution
and
N
is the sample size.
α
and
β
are binary indicators, where
α
= 1 or
β
= 1 indicates the presence of the PU or UU and
α
= 0 or
β
= 0 implies their absence.
The spectrum sensing problem can be formulated as the following hypotheses:
H
_{0}
denotes the absence of a primary and unauthorized signal;
H
_{1}
denotes the presence of a primary signal when there is no UU; and
H
_{2}
denotes the presence of an unauthorized signal when there is no PU, i.e., the detected signal was transmitted by UUs. The observed signal of a SU can then be expressed as
When we consider two hypotheses,
H
_{1}
that a PU transmits a signal and
H
_{2}
that an UU transmits a signal, two kinds of risks are incurred in the hypothesis test:

False alarm: Although the actual transmission is made by the UU, the SU decides that the transmission is due to the PU. In other words, a PUEA occurs.

Miss detection: Although the actual transmission is made by the PU, the SU decides that the transmission is due to the UU. In other words, the SU unintentionally violates the spectrum sensing rule.
From the Wald’s sequential probability ratio test (SPRT), we can specify the desired thresholds
λ
_{1}
and
λ
_{2}
for the false alarm and miss detection probabilities, respectively. The space of all observations is the sample space of the received power measured at the SU. Let the sequence of the measured power at the SU for
N
samples be denoted by {
x
_{1}
,
x
_{2}
, ⋯,
x_{N}
}, where
x
_{n}
is the measured power at the
n
th sample. According to the Wald’s SPRT, we can decide which hypothesis is correct
[19]
. The SPRT is based on considering the likelihood ratio as a function of the number of observations. After
N
samples, the likelihood ratio (LR) is given by
[20]
where
f
^{(PU)}
(‧) is the probability density function (pdf) of the received power at a SU from the PU and
f
^{(UU)}
(‧) is the pdf of the received power at a SU from the UU. As shown in
Fig. 2
, if Λ
_{N}
≤
T
_{1}
, we decide
H
_{1}
, if Λ
_{N}
≥
T
_{2}
, we then decide
H
_{2}
, otherwise, we decide
H
_{0}
. The decision criteria can then be expressed as follows
[19]
,
[20]
:
Decision criteria in SPRT
In (4), as two thresholds,
λ
_{1}
and
λ
_{2}
, decrease, the threshold
T
_{1}
decreases and the threshold
T
_{2}
increases. Hence, as shown in
Fig. 2
, it is more likely that a SU makes another decision
H
_{0}
that there is no signal although there is a PU or UU. To correctly detect the presence of the PU, when
H
_{1}
is true, the LR of (3) should be small enough that it is less than or equal to
T
_{1}
. Similarly, when
H
_{2}
is true, the LR of (3) should be large enough that it is greater than or equal to
T
_{2}
. However, in the SPRT, there is a tradeoff between a reliable decision and the time to detect.
3. Proposed Conditional Entropybased Spectrum Sensing
 3.1 FrequencyDomain Entropy
The structure of the proposed conditional entropybased detector is shown in
Fig. 3
, where the detector consists of three blocks: a frequencydomain converter, a conditional entropy estimator, and a test statistic. The frequencydomain detector is generally superior to the time domain detector
[10]
,
[11]
. Applying the discrete Fourier transform (DFT) to (1), we have
where
K
, which is the DFT size, is equal to the sample size
N
; the parameters,
represent the complex spectrum of the received signal, the primary signal, and the background noise, respectively. Hence, in the frequencydomain, we have the following hypotheses:
The complex spectrum of the received signal can be expressed as follows
[11]
:
where
Y_{r}
(
k
) and
Y_{i}
(
k
) represent the real part and the imaginary part of
respectively. The spectrum magnitude can then be expressed as
A block diagram of the conditional entropybased frequencydomain detector
In information theory, the conditional entropy quantifies the amount of information needed to describe the outcome of a random variable
Y
given that the value of another random variable
Z
is known. The distribution of
Z
is assumed to be estimated by observing UUs
[17]
. The conditional entropy is
where
H
(
Y

Z
) is estimated from the probability mass function of
Y
and
Z
, and is compared with a threshold to decide the current knowledge of the PU.
p
(
y
) and
p
(
z
) denote the probability mass function of discrete random variables,
Y
and
Z
, respectively.
p
(
y
,
z
) denotes the joint probability mass function of
Y
and
Z
.
A number of schemes have been proposed by earlier researchers for estimating the entropy of a continuous random variable based on a finite number of observations
[10]
,
[11]
,
[14]
. To reduce the computational complexity, we use the simplest approach, histogrambased estimation of the density function. The histogrambased method estimates the probability of each state. Let
Y
and
Z
represent the distribution of the spectrum magnitude of the measured signal in the presence of a primary signal and in the presence of an unauthorized signal, respectively. We divide the range of
Y
and
Z
into
L_{y}
bins and
L_{z}
bins, respectively. Hence, the bin widths are Δ
_{y}
=
Y
_{max}
/
L_{y}
and Δ
_{z}
=
Z
_{max}
/
L_{z}
, where
Y
_{max}
and
Z
_{max}
denote the maximum value of the random variables,
Y
and
Z
, respectively. The probability mass function can then be approximated as the frequency of occurrences in each bin width. Hence, we have
p
(
y
) ≈
k_{y}
/
N
and
p
(
z
) ≈
k_{z}
/
N
, where
k_{y}
and
k_{z}
are respectively the total number of occurrences in the
y
th bin of
Y
and in the
z
th bin of
Z
;
N
is the number of observations. As described above, a SU can estimate the distribution of
Y
based on the observations of the received signals. However, in practice, it is difficult to estimate the distribution of
Z
without the help of the primary system or without the location information of UUs. Some works analytically derived a distribution of unauthorized signals when UUs are uniformly located in a cell
[20]
, while other works assumed that the channel information, both for the PU and for the UU, can be obtained
[17]
. In this paper, we assume that a primary system has a periodic duration for the pilot transmission or the silence. A SU can then receive unauthorized signals for every the periodic duration and it may estimate the distribution of
Z
based on the cumulative observations of the unauthorized signals.
 3.2 Spectrum Statistics of the Received Signal
In hypothesis
H
_{0}
, the received signal,
y
(
n
) =
w
(
n
), consists of noise. Both the real part
W_{r}
and the imaginary part
W_{i}
of the spectrum follow a Gaussian distribution. Hence,
follows a Rayleigh distribution with the parameter
σ_{y}
, and the differential entropy of
Y
can be expressed as
[10]
,
[14]
where
γ
is the EulerMascheroni constant.
In hypothesis
H
_{1}
, the received signal,
y
(
n
) =
s
(
n
) +
w
(
n
), consists of both the primary signal and the noise. The entropy of the spectrum amplitude in the presence of the primary signal is much smaller than in the absence of the primary signal. Let
D
be the distance of the estimated entropies between hypothesis
H
_{0}
and hypothesis
H
_{1}
. If
D
≥
δ
, we decide
H
_{1}
and otherwise, we decide
H
_{0}
, where
δ
is the threshold determined by false alarm and miss detection probabilities
[11]
. Because the entropy of the noise signal has almost a constant value, we can do the test statistic with the estimated entropy of the received signal in hypothesis
H
_{1}
; i.e., if
H
(
Y

Z
) ≤
δ
', we decide
H
_{1}
and otherwise, we decide
H
_{0}
, where
δ'
is the difference between the entropy of the noise signal and the value of
δ
.
 3.3 Reduction of Noise Uncertainty
The proposed spectrum sensing uses the conditional entropy with mutual information between the primary signal and the unauthorized signal. On the basis of the mutual information, the conditional entropy,
H
(
Y

Z
) is obtained as
[14]
where
H
(
Y
) is the entropy of
Y
, and
I
(
Y
;
Z
) is the mutual information between
Y
and
Z
. The mutual information is equal to the relative entropy which measures the distance between probability distributions of
Y
and
Z
. The mutual information can be expressed as
where
D
(
p
(
y
,
z
) ║
p
(
y
)
p
(
z
)) is the KullbackLeiber divergence of the product,
p(y)p(z)
, of the two marginal probability distributions from the joint probability distribution of
Y
and
Z
, i.e., the expected number of extra bits that must be transmitted to identify
Y
and
Z
if they are coded using only their marginal distributions instead of the joint distribution. In the following, we derive how to reduce the noise uncertainty with the conditional entropy between the primary and unauthorized signals.
Proposition 1.
The mutual information of entropy reduces the uncertainty of the primary signal due to the knowledge of the unauthorized signal. With a fixed bin number, L_{z}, the entropy of the spectrum of WGN can be approximated by a constant; the proposed spectrum sensing technique on the basis of the mutual information is hence intrinsically robust to noise uncertainty
.
Proof
. The maximum values of
Y
and
Z
,
Y
_{max}
and
Z
_{max}
, can be expressed as
Y
_{max}
=
C_{y}σ_{y}
and
Z
_{max}
=
C_{z}σ_{z}
, respectively, where
C_{y}
and
C_{z}
are constant values
[10]
. The bin width can then be expressed as Δ
_{y}
=
Y
_{max}
/
L_{y}
=
C_{y}σ_{y}
/
L_{y}
and Δ
_{z}
=
Z
_{max}
/
L_{z}
=
C_{z}σ_{z}
/
L_{z}
, respectively. If the density
f
(
y
) of the random variable
Y
is Riemann integrable, then the entropy of the quantized version
H
(
Y
^{Δ}
) is
[14]
Hence, from (9), (10), and (12), with the natural logarithm, the conditional entropy
H
(
Y

Z
) can be expressed as
where
i
(
Y
;
Z
) represents the differential entropy of the mutual entropy
I
(
Y
;
Z
). From (13), it is seen that the conditional entropy is approximated by a constant for a given bin number
L_{z}
, which implies that the proposed conditional entropybased detection is robust against noise uncertainty.
4. Simulation Results
We evaluate the performance of the spectrum sensing schemes in a cognitive radio network with the existence of UUs. The performance of the proposed spectrum sensing has been compared with that of
[11]
and
[13]
in the frequencydomain. MATLAB is used as a tool for evaluating the performance of the proposed spectrum sensing scheme though extensive simulations in Gaussian and Rayleigh fading channel environments. The simulation parameters are identical to those in the work of
[11]
. A single sideband signal is selected as a candidate PU signal. The UU is assumed to mimic the PU signal with a half power of the PU signal. The nominal noise power is 90 dBm with ±5 dB fluctuation. In the simulation, all channel information is assumed to be known to the SUs
[17]
. To estimate the probability mass functions, the probability space is partitioned into equal bin numbers,
L_{y}
= 15 and
L_{z}
= 10. The probabilities for hypotheses in SPRT are assumed to be
λ
_{1}
= 0.1 and
λ
_{2}
= 0.2. Moreover, the threshold for the test statistic is assumed to be
δ
= 0.3. The false alarm probability is no more than 0.1. The other simulation parameters used in this paper are summarized in
Table 1
. The terms “Ebased scheme” and “CEbased scheme” in the figures denote the entropybased and cross entropybased spectrum sensing scheme, respectively. In the conventional Ebased scheme, SUs decide the presence of a primary signal based on the estimated entropy
H
(
Y
) and in the previous CEbased scheme, SUs decide the presence of a primary signal based on the estimated cross entropy
H
(
Y_{i}
,
Y
_{i}
_{1}
) between the current detected data set
Y_{i}
and the previous detected data set
Y
_{i}
_{1}
[11]
,
[13]
. In the proposed spectrum sensing scheme, SUs decide the presence of a primary signal based on the estimated conditional entropy
H
(
Y

Z
).
Simulation parameters
 4.1 Gaussian Channel
Fig. 4
,
5
, and
6
shows the performance of the spectrum sensing schemes under Gaussian channel environments. As shown in
Fig. 4
, the distances between the estimated entropies of the noise and signal in
[11]
and
[13]
are smaller than that of the proposed spectrum sensing scheme. A higher gap between the noise and the signal ensures better performance in distinguishing the signal from the noise regardless of the absolute value of the estimated entropy. For example, the distance between the noise and the signal of the estimated entropy in the proposed spectrum sensing is about ten times greater than that of
[11]
and twice that of
[13]
at a SNR of 10 dB. The entropy detector is based on the characteristic that the entropy of a stochastic signal is maximized if the signal is Gaussian. If the received signal contains the PU signal, the entropy is reduced. Hence, the signal in the estimated entropy degrades smoothly as the value of the SNR increases. From
Fig. 4
it can be concluded that the proposed detector can better distinguish the signal from the noise even at a low SNR as compared with the conventional entropy detectors.
Estimated entropy in the Gaussian channel
Comparison of detection probability in the Gaussian channel
Comparison of ROC curves in the Gaussian channel when SNR = 10 dB
Fig. 5
shows the detection probabilities of the spectrum sensing schemes. The distance between the estimated entropies of the noise and signal in
Fig. 4
results in the difference of the detection performance seen in
Fig. 5
. For example, to satisfy the detection probability over 0.9, the detectors of the Ebased scheme, CEbased scheme, and proposed scheme require a SNR of 0.9 dB, 4.1 dB, and 5.8 dB, respectively. When the SNR is 5 dB, the detection probability of the proposed spectrum sensing scheme is about 1304% greater than that of the Ebased scheme of
[11]
and about 28% greater than that of the CEbased scheme of
[13]
. Consequently, the proposed spectrum sensing can detect the primary signal even at a low SNR.
Fig. 6
shows the receiver operating characteristic (ROC) curves of each sensing scheme when SNR = 10 dB. When the false alarm probability
P_{f}
= 0.2, the detection probability of the proposed scheme outperforms the Ebased scheme of
[11]
by about 150% and the CEbased scheme of
[13]
by about 10%.
 4.2 Rayleigh Fading Channel
Fig. 7
,
8
, and
9
show the performance of the spectrum sensing schemes under the Rayleigh fading channel environments. The primary signal is a single sideband signal, which is assumed to experience deep fading such that the magnitude follows a Rayleigh distribution when the delay time of each path is 0.01 seconds and the number of the paths is 15.
Estimated entropy in the Rayleigh fading channel
Comparison of detection probability in the Rayleigh fading channel
Comparison of ROC curves in the Rayleigh fading channel when SNR = 10 dB
As shown in
Fig. 7
, the proposed spectrum sensing scheme shows better performance than the previous spectrum sensing schemes of
[11]
,
[13]
under a Rayleigh fading channel. Superior performance in discerning signals from noise requires a greater gap between the noise and the signal regardless of the value of the estimated entropy. At any SNR value, the proposed scheme possesses higher gaps than others. For example, the distance between the noise and the signal of the estimated entropy is about three times greater than that of
[11]
and twice that of
[13]
at a SNR of 10 dB.
Fig. 8
shows that the detection performance of the proposed scheme outperforms that of the conventional spectrum sensing schemes of
[11]
and
[13]
. For example, to satisfy detection probability over 0.9, the proposed spectrum sensing scheme has a SNR gain of about 7 dB and 1.4 dB in comparison with
[11]
and
[13]
, respectively. Moreover, at a SNR of approximately 10 dB, the detector of
[11]
is unable to detect PU signals while the detector of
[13]
and that of the proposed spectrum sensing scheme show detection probability of 0.40 and 0.65, respectively.
Fig. 9
shows that the detection ability of the proposed scheme is more robust than that of the conventional entropybased schemes. By selecting a SNR = 10 dB, we have simulated the ROC curves in a Rayleigh fading channel. This figure shows that the detection probability of the proposed spectrum sensing scheme is better than that of
[11]
,
[13]
when the false alarm probability is identical.
5. Conclusion
A conditional entropybased spectrum sensing scheme has been proposed for a cognitive radio network with an unauthorized signal. The proposed spectrum sensing scheme uses the mutual information and exploits the difference between the primary signal and the unauthorized signal. The proposed spectrum sensing scheme in the frequencydomain is shown to be robust to noise uncertainty and presents good primary user detection performance at a low signaltonoise ratio. Under the Gaussian channel, when the signaltonoise ratio is 5 dB, the proposed spectrum sensing scheme increases the detection probability by more than 28% as compared with the conventional entropybased sensing schemes. However, the proposed spectrum sensing has a limitation that the characteristic of the unauthorized signal can be estimated from the observations due to the help of the primary system. Our future work will study an entropybased spectrum sensing scheme with the partial or blind information of the UU signal characteristic. Future work will also include comparisons with other detection schemes designed to combat PUEAs, where the proposed conditional entropybased spectrum sensing will be extended into cognitive radio networks with cooperative spectrum sensing.
BIO
Jaewoo So received the B.S. degree in electronic engineering from Yonsei University, Seoul, Korea, in 1997, and received the M.S. and Ph.D. degrees in electrical engineering from the Korea Advanced Institute of Science and Technology (KAIST), Daejeon, Korea, in 1999 and 2002, respectively.
From 2001 to 2005, he was with IP One, Seoul, Korea, where he led several research projects and developed IEEE 802.11a/b/g products and heterogeneous network solutions. From 2005 to 2007, he was a Senior Engineer at Samsung Electronics, Suwon, Korea, where he involved in the design, performance evaluation, and development of mobile WiMAX systems and B3G wireless systems. From 2007 to 2008, he was a Postdoctoral Fellow in the Department of Electrical Engineering, Stanford University, Stanford, CA, USA. Since September 2008, he has been with the Department of Electronic Engineering, Sogang University, Seoul, Korea, where he is currently an Associate Professor. His current research interests include radio resource management, multiple antenna systems, cognitive radio networks, and M2M systems. He is a Senior Member of IEEE, a Life Member of KICS, a Member of IEEK, a Member of KSII, and a Member of IEICE.
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