Computer Communications 34 (2011) 1510–1517
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Computer Communications journal homepage: www.elsevier.com/locate/comcom
A weighted cooperative spectrum sensing framework for infrastructure-based cognitive radio networks Yanxiao Zhao a,⇑, Min Song a,⇑⇑, Chunsheng Xin b a b
Electrical and Computer Engineering Department, Old Dominion University, Norfolk, VA 23529, USA Computer Science Department, Norfolk State University, Norfolk, VA 23504, USA
a r t i c l e
i n f o
Article history: Received 16 February 2010 Received in revised form 13 February 2011 Accepted 14 February 2011 Available online 17 February 2011 Keywords: Cognitive radio Spectrum sensing Weighted cooperative sensing
a b s t r a c t Spectrum sensing plays a critical role in cognitive radio networks. A good sensing scheme can reduce the false alarm probability and the miss detection probability, and thus improves spectrum utilization. This paper presents a weighted cooperative spectrum sensing framework for infrastructure-based cognitive radio networks, to increase the spectrum sensing accuracy. The framework contains two modules. In the first module, each cognitive radio performs local spectrum sensing and computes the total error probability, which combines the false alarm probability and the miss detection probability. The total error probability and the energy signal from the primary user are then sent to the base station. In the second module, the base station makes a final decision after combining the weighted energy signals from all cognitive radios. The final decision is then broadcasted back to all cognitive radios. To reduce the computation complexity and communication overhead, the base station also instructs the cognitive radios that have large total error probabilities not to report their local sensing results. We have developed a theoretical model for the proposed framework, and derived the optimal detection threshold, as well as the minimum number of cognitive radios required to participate in cooperative sensing, subject to a given total error probability. Numerical results verify that the proposed weighted cooperative spectrum sensing framework significantly improves the sensing accuracy. Ó 2011 Elsevier B.V. All rights reserved.
1. Introduction Today’s static spectrum access (SSA) policy grants a fixed spectrum band to each licensed user for exclusive access [1]. With the rapidly proliferated wireless services, SSA is exhausting the radio spectrum and leaves little spectrum for future demands, a problem known as spectrum scarcity. On the other hand, a large number of licensed spectrum bands are considerably under-utilized in both time and spatial domains. According to the report from Federal Communications Commission (FCC), the utilization of the assigned spectrum is limited from 15% to 85% [2]. These issues have motivated the development of dynamic spectrum access (DSA) policy, allowing secondary users (SUs) to dynamically detect idle licensed bands and temporarily access them. In other words, SUs are allowed to utilize an idle licensed spectrum band provided that they withdraw from the band when primary users (PUs) start using it. A key technology for DSA is cognitive radio (CR), which is capable of sensing the spectrum environment, and changing the ⇑ Corresponding author. Tel.: +1 757 683 3470. ⇑⇑ Principal corresponding author. E-mail addresses:
[email protected] (Y. Zhao),
[email protected] (M. Song),
[email protected] (C. Xin). 0140-3664/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.comcom.2011.02.007
transmission and reception parameters accordingly for efficient communication, while avoiding interference to PUs [3]. Spectrum sensing is quite a challenging problem and plays a critical role in the overall system performance for DSA. This is because a CR cannot have a direct measurement of the spectrum band between a PU receiver and a PU transmitter. In fact, a CR cannot even measure if a PU receiver exists, e.g., a TV terminal. Therefore a CR usually has to make its decision based on its local measurement of a spectrum band, which we refer to as a channel hereafter. This type of detection is referred to as local spectrum sensing [4]. One well-known technique for local spectrum sensing is energy detection, in which the SU measures the energy from the PU transmitter to determine the presence of PU signal [5]. Due to the multipath fading and the uncertainty of noise power, the local spectrum sensing from a single SU often cannot accurately detect PU activities and thus leads to miss detection. One solution to this problem is to utilize the spectrum sensing information from other SUs to improve the detection accuracy. This is referred to as cooperative sensing [6]. In cooperative sensing, each SU independently performs local spectrum sensing for a channel and makes a binary decision (idle/occupied) independently from other SUs. A channel is determined as idle only if all SUs detect that there is no PU activity on the channel. However, although this decreases the miss detection probability, it increases the false alarm probability and results in lower spectrum
Y. Zhao et al. / Computer Communications 34 (2011) 1510–1517
utilization. In other words, there is a trade-off between miss detection and false alarm. In this paper, we propose a new metric, total error probability, to measure the accuracy of spectrum sensing. A unique feature of the total error probability is that it combines both the false alarm probability and the miss detection probability, as well as takes the PU activity into account. We then propose a weighted cooperative spectrum sensing framework for infrastructure-based cognitive radio networks. Two modules are included in the framework. In the first module, each SU performs local spectrum sensing and computes the total error probability. Instead of simply providing a binary decision, the total error probability and the energy signal from the PU are sent to the base station. In the second module, the base station makes a final decision after combining the information from all SUs and notifies them with the final decision. To take the fading and shadowing into account, the sensing information from a SU that has a higher total error probability is assigned with a lower weight. To minimize the detection error probability, an optimal threshold is derived. For SUs that have large total error probabilities, their local sensing results have negligible contribution to the decision making at the BS. To reduce the computation complexity and communication overhead, we do not require every SU to participate in cooperative sensing. The minimum number of SUs required to participate in cooperative sensing is also studied. The rest of the paper is organized as follows. In Section 2, we briefly introduce the related work. Section 3 presents the network model and the main idea. In Section 4, we present the detailed design of the proposed spectrum sensing framework along with indepth analysis. The numerical results are presented in Section 5. Concluding remarks are drawn in Section 6. 2. Related work Spectrum sensing in cognitive radio networks can be generally classified into two categories: local sensing and cooperative sensing. In local spectrum sensing, each SU independently makes a decision on channel availability based on the information collected. It will then attempt to access a selected channel if there are idle channels; otherwise it keeps sensing. As discussed earlier, a widely used technique for local sensing is energy detection. Energy detection has low computation complexity and is easy to be implemented. However, it is susceptible to the multipath fading and the uncertainty of noise power [7]. In cooperative sensing, each SU independently performs local spectrum sensing and makes a binary decision (idle or occupied) for a channel. This channel is determined as idle by cooperative sensing if all SUs (or a certain number of SUs) find that it is idle. The reader is referred to [8,9] for the benefits of cooperative sensing. Cooperative sensing can be implemented in a centralized or a distributed mode. In the centralized mode (e.g., see [10]), a base station (BS) collects the local sensing information from all SUs, and then makes the final decision. In the distributed mode (e.g., see [11]), SUs exchange local sensing information with each other and vote for a final decision. Many cooperative sensing schemes have been proposed recently. The detection accuracy and sensing efficiency have been studied in [12], where a theoretical framework is developed to optimize sensing parameters to maximize the sensing efficiency, subject to the interference avoidance constraint. In [13], a coalitional game strategy is proposed to analyze the behavior of SUs in distributed cooperative sensing. The relation of the detection accuracy and sensing efficiency is modeled as a non-transferable coalitional game. In [14], the authors propose a technique that uses a varying number of samples, and introduce a reputation-based mechanism to the sequential probability ratio test. In [15], SUs are divided into a few groups and each group chooses a head with the highest signal-
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to-noise ratio (SNR). Each SU reports its local sensing result to its head which makes a preliminary decision based on the reports from its members. Afterwards, group heads report their decisions to the BS. The BS makes a final decision using an ‘‘OR’’ rule, i.e., the result is 1 (channel is occupied by PU) as long as one group head report is 1. In [16], a weighted cooperative sensing is proposed based on the SNR, with the objective to maximize the detection sensitivity while meeting a given requirement on the false alarm probability. A SU’s weight is decreased if it experiences a lower SNR. The authors in [17] propose another weighted cooperative sensing scheme, with the motivation of equal probabilities of false alarm and miss detection. The idea is that SUs with large SNR are assigned with large weights and thus yield more contributions to the global decision. However, as pointed out in [18], in the presence of noise uncertainty, SUs below a certain SNR cannot improve their performance even with infinite sensing time. In other words, SNR is not a suitable parameter to be used for selecting weights, because of its uncertainty in noise power. In addition, it is well acknowledged that there is a trade-off between the false alarm probability and the miss detection probability. Thus, optimizing one of them as in [16], or simply constraining them to be equal and then optimizing one of them as in [17], does not fully consider this trade-off. In our paper, we propose a weighted cooperative sensing scheme that assigns weights to SUs based on the local detection accuracy of each SU, instead of SNR. Specifically, we optimize the total error probability introduced in Section 3, which not only combines the false alarm probability and the miss detection probability, but also takes the primary user activity into account. We consider infrastructure-based cognitive radio networks, and hence assume the centralized mode for cooperative sensing. With the centralized mode, a simple solution for the BS to determine the status of a channel is to use the ‘‘voting’’ rule, which counts the number of SUs that vote in favor of the occupied status and compares it with a threshold [19,20]. In [10], a ‘‘half-voting’’ rule is introduced, where a channel is deemed as occupied when more than half of SUs have detected the PU signal on the channel. Instead of simply sending a binary decision (idle or occupied) as in such approaches, in our approach, each SU sends the total error probability, and the energy signal from PU to the BS. The BS assigns a weight to each SU based on the total error probability, which indicates the local detection accuracy of the SU. The SU that has a higher total error probability is assigned with a lower weight. Instead of using the voting rule as in [19,20,10], our scheme compares the weighted summation of energy signals from all SUs with a threshold to determine if a channel is occupied. With the transmission of the total error probability and the energy signal, the signaling load in our scheme is higher than in the ‘‘voting’’ rule schemes that send only a binary decision. Fortunately, the increment of signaling load is not significant. This is because the error probability and the energy signal need to be sent within a packet or frame, and for such a small payload, the packet/frame overhead dominates the payload. In fact, without considering the overhead, the control channel needs only a small data rate, 64M bps, assuming that the error probability and the energy d signal are each sent as a 4-byte float, where M denotes the number of SUs and d is the time for reporting. For instance, if M = 10 and d = 1 ms, then the required data rate is only 0.64 Mbps. At last, with our scheme, not all SUs are required to report the sensing results, but only a fraction of SUs need to report (to be discussed). This would further reduce the signaling load. 3. System model and main idea We consider an infrastructure-based cognitive radio network consisting of one PU, one BS, and M SUs. SUs opportunistically share a licensed channel with the PU for data transmission. From
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the perspective of SUs, the channel simply alternates between idle (no PU activity) and occupied (with PU activity) status. Note that the status of idle or occupied is observed at the session level, rather than at the packet transmission level. The short quiet intervals between packets transmissions should not be counted as ‘usable’ idle periods by a good channel selection algorithm in DSA. The channel selection algorithms should select the channels that have been idling for a relatively long period of time which happens between two consecutive communication session. The channel idle/occupied durations are assumed as independent random variables, and the channel activity is modeled as a semi-Markov process. Let H0 and H1 denote the event that the channel is in idle or occupied status, respectively. Let a and b denote the mean occupied and idle durations of the channel, respectively. Then the probabilities of the channel being idle and occupied are given as:
PðH0 Þ ¼
b
aþb
;
PðH1 Þ ¼
a : aþb
SUs use energy detection for local spectrum sensing. We assume that the primary signal is reasonably higher than the noise level. A basic hypothesis model for the local spectrum sensing can be defined as follows:
xi ¼
ni ;
the final sensing decision. To differentiate the local sensing accuracy, a SU that has a higher total error probability is assigned with a lower weight. Mathematically, the output from the BS can be written as:
Y¼
wi yi ;
i ¼ 1; 2; . . . ; M;
i¼1
where wi is the weight of the ith SU. The output of the BS is then compared with a decision threshold c. If Y P c, the channel is determined to be occupied by the PU; otherwise, the channel is determined to be idle. This decision is then broadcasted to all SUs. Before presenting the details of each module in the next section, we first analyze the performance of the generic cooperative spectrum sensing. In the generic cooperative spectrum sensing (e.g., [19,20]), a channel is determined to be idle only if all SUs report the channel idle status to the BS. As a result, the false alarm probability Pf and the miss detection probability Pm can be derived as follows:
Pf ¼ 1
M Y
ð1 Pif Þ;
i¼1
Pm ¼
M Y
Pim :
i¼1
if H0 ;
Correspondingly, the total error probability Pe is:
s þ ni ; if H1 ;
where xi is the signal that SU i received, ni is the zero-mean additive white Gaussian noise (AWGN), i.e., ni Nð0; r2i Þ; s is the signal that the PU transmits. We ignore the channel gain because it is often assumed to be constant during the detection interval. Our weighted cooperative sensing framework has two modules as shown in Fig. 1. In the first module, each SU performs local spectrum sensing and computes the total error probability. The total error probability and the energy signal are then sent to the BS through a common control channel. For the local sensing process of SU i, the received signal xi is first pre-filtered by an ideal band-pass filter. The output of the band-pass filter is then squared and integrated over the observation period. The output of the ith integrator representing the energy signal from the ith SU, denoted by yi, is compared with a local threshold ci to make a local sensing decision. The accuracy of the local sensing decision is characterized by a total error probability, defined as follows,
Pie ¼ PðH0 Þ P if þ PðH1 Þ Pim ;
M X
ð1Þ
where Pif ¼ Pðyi > ci jH0 Þ represents the false alarm probability, and Pim ¼ Pðyi < ci jH1 Þ represents the miss detection probability. Notice that both Pim and Pif are conditional probabilities. Thus the measurement of the total error probability takes the PU activity into account. In the second module, the BS makes a final decision after combining the weighted energy signals and then notifies all SUs with
Pe ¼ PðH0 Þ Pf þ PðH1 Þ P m " # M M Y b a Y i ¼ Pim : 1 ð1 Pf Þ þ a aþb þ b i¼1 i¼1
ð2Þ
Since Pim is in the range (0, 1), we have Pm P im , which means that the generic cooperative sensing substantially decreases the miss detection probability. The price paid, however, is the considerable increment of false alarm probability, since Pf Pif . As a result, the generic cooperative sensing does not perform well in terms of the total error probability (which will be verified in Section 5). 4. Analysis 4.1. Local spectrum sensing Let the bandwidth of the ideal band-pass filter be W. Then the number of samples during the sensing period T is N = 2T W. Let N i0 denote the two-sided noise power density spectrum at SU i. Then the noise variance r2i ¼ 2N i0 W. The output of the ith integrator with N samples can be represented as:
yi ¼
N X
jxi ðkÞj2 ;
i ¼ 1; 2; . . . ; M:
k¼1
Fig. 1. Weighted cooperative spectrum sensing framework.
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Here yi is the sum of the square of N independent Gaussian distributed random variables. As such yi follows the chi-square distribution, i.e.,
(
yi
r
2 i
v2N if H0 ðchi-squareÞ; v2N ðki Þ if H1 ðnoncentral chi-squareÞ;
i¼1
yi
r2i
NðN; 2NÞ
H0
NðN þ ki ; 2ðN þ ki ÞÞ H1
(
) yi
N N r2i ; 2N r4i H0 ; 2 N ðN þ ki Þ ri ; 2ðN þ 2ki Þ r4i H1 :
ð7Þ
M X
w2i Varðyi jH0 Þ;
i¼1 M X
wi Eðyi jH1 Þ;
ð8Þ
M X
w2i Varðyi jH1 Þ:
i¼1
ð4Þ
As discussed in last section, we use a threshold-based approach in cooperative sensing to determine the channel status. Specifically, for the ith SU, if yi is larger than a threshold ci, the channel is seen as occupied. The false alarm probability P if and miss detection probability Pim can be calculated as:
ci Eðyi jH0 Þ
!
¼ Prðyi > c jH0 Þ ¼ Q pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; Varðyi jH0 Þ i
VarðYjH0 Þ ¼
VarðYjH1 Þ ¼
Varðyi jH1 Þ ¼ 2ðN þ 2ki Þ r4i :
Pim
wi Eðyi jH0 Þ;
i¼1
ð3Þ
Eðyi jH1 Þ ¼ ðN þ ki Þ r2i ;
i
M X
i¼1
Varðyi jH0 Þ ¼ 2N r4i ;
Pif
EðYjH0 Þ ¼
EðYjH1 Þ ¼
We denote the (conditional) means and variances of yi as:
Eðyi jH0 Þ ¼ N r2i ;
i¼1
We denote the (conditional) means and variances of Y as:
where ki ¼ 2Ers 2W , and Es is the signal energy [21]. i According to the central limit theorem [22], the chi-square distribution approaches a normal distribution when the degree of freedom, N, increases. Specifically,
8 M M P P > 2 > N w Eðy jH Þ; w Varðy jH Þ H0 ; > i 0 0 i i i < i¼1 i¼1 Y M M > P P > > wi Eðyi jH1 Þ; w2i Varðyi jH1 Þ H1 : :N
ci Eðyi jH1 Þ
ð5Þ !
¼ Prðyi 6 c jH1 Þ ¼ 1 Q pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; Varðyi jH1 Þ
ð6Þ
where Q is the Q-function or tail probability of Gaussian distribution. Substituting Eqs. (5) and (6) into Eq. (1), it is straightforward to get the total error probability. We defer the derivation of the optimal threshold ci⁄ for the ith SU, because it follows the same rationale as the derivation of the optimal threshold c⁄ for the BS which will be described in the next subsection. 4.2. Optimal decision making at the BS
Let c denote the threshold to determine channel status by the BS, i.e., if Y > c, then BS determines that the channel is occupied. Then the false alarm probability Pf and miss detection probability Pm of the weighted cooperative spectrum sensing can be derived as:
! c EðYjH0 Þ Pf ¼ PrðY > cjH0 Þ ¼ Q pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; VarðYjH0 Þ
! c EðYjH1 Þ Pm ¼ PrðY 6 cjH1 Þ ¼ 1 Q pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : VarðYjH1 Þ
Accordingly, the total error probability Pe can be computed as: ! b c EðYjH0 Þ P e ¼ PðH0 Þ P f þ PðH1 Þ P m ¼ Q pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi aþb VarðYjH0 Þ ( !) Z þ1 a c EðYjH1 Þ b t2 cEðYjH Þ e 2 dt þ 1 Q pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ pffiffiffiffiffiffiffi 0 ffi aþb VarðYjH1 Þ 2p ða þ bÞ pffiffiffiffiffiffiffiffiffiffiffi VarðYjH0 Þ 0 1 Z þ1 a t2 B C @1 cEðYjH Þ e 2 dtA: þ pffiffiffiffiffiffiffi 1 ffi 2p ða þ bÞ pffiffiffiffiffiffiffiffiffiffiffi VarðYjH1 Þ
Two issues need to be addressed for the BS to make a decision on the channel status. First, how to assign an appropriate weight to each SU in order to alleviate the fading and shadowing effects? Second, what is the optimal threshold to be used by the BS in order to minimize the total error probability? Our basic idea for weight assignment is that a SU with a higher error probability is assigned with a lower weight, as illustrated in Algorithm 1. First, the weight of each SU is computed based on the error probability. Then, all weights are normalized to satisfy PM i¼1 wi ðkÞ ¼ 1. Algorithm 1: Weight assignment Input: P ie for i ¼ 1; . . . ; MÞ 1: for i = 1 to M do 2: wi ¼ 1=Pie 3: end for 4: for i = 1 to M do P 5: wi ¼ wi = M k¼1 wk 6: end for With the assigned weight for each SU, the output signal at the P BS is Y ¼ M i¼1 wi yi . Due to the fact that all yi (1 6 i 6 M) are independent random variables following Gaussian distribution and wi can be viewed as a constant in each sensing period, Y should also follow a Gaussian distribution, i.e.,
To minimize the total error probability Pe, its derivative is computed as:
@ðPe Þ b pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e ¼ pffiffiffiffiffiffiffi @ðcÞ 2p ða þ bÞ VarðYjH0 Þ
a
12
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e þ pffiffiffiffiffiffiffi 2p ða þ bÞ VarðYjH1 Þ
2 0Þ ffi pcEðYjH ffiffiffiffiffiffiffiffiffiffiffi VarðYjH0 Þ
12
2 1Þ ffi pcEðYjH ffiffiffiffiffiffiffiffiffiffiffi VarðYjH1 Þ
:
Let
@ðPe Þ ¼ 0: @ðcÞ
ð9Þ
The optimal threshold c⁄ to obtain the minimum total error probability (Pe) is as follows
c ¼
pffiffiffiffi EðYjH1 ÞVarðYjH0 Þ EðYjH0 ÞVarðYjH1 Þ 12 D ; ½VarðYjH0 Þ VarðYjH1 Þ
ð10Þ
where
D ¼ ½2EðYjH0 Þ VarðYjH1 Þ 2EðYjH1 Þ VarðYjH0 Þ2 4½VarðYjH0 Þ n VarðYjH1 Þ VarðYjH0 Þ EðYjH1 Þ2 VarðYjH1 Þ EðYjH0 Þ2 " pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi#) b VarðYjH1 Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : ð11Þ þ2VarðYjH0 Þ VarðYjH1 Þ ln a VarðYjH0 Þ
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The shape of Pe and the existence of the minimum of Pe are shown in Figs. 2 and 3 (to be discussed in Section 5). With the optimal threshold c⁄ available, the BS then compares its output Y with this threshold to determine if the channel is occupied (if Y > c⁄). The BS then broadcasts this decision to all SUs. Note that Eq. (10) can also be used by each SU to calculate the optimal local threshold ci⁄ by replacing Y with yi.
4.3. Minimum number of SUs required for cooperative sensing One way to reduce the computation complexity and communication overhead of our weighted cooperative spectrum sensing is to reduce the number of SUs involved in the decision-making. For SUs that have large total error probabilities and thus low weights, their local sensing results have negligible contribution to the decision making at the BS. As such a sound decision can still be made at the BS even though these SUs do not participate in
1 1
0.8
0.9
0.7
0.8
Total Error Probability
Total Error Probability
0.9
0.6 0.5 0.4 0.3 0.2 0.1
Pe Pf Pm
← (70,0.1643)
0 0
20
40
60
80
0.7 0.6 0.5 0.4 0.3 0.2
100
Threshold
0 30
(a) P(H0)=0.2,P(H1)=0.8
40
50
60
80
90
100
110
120
130
(a) P(H0)=0.2,P(H1)=0.8 Pe Pf Pm
0.9 0.8
1 0.9
0.7
0.8
0.6
0.7
Probability
Probability
70
Threshold
1
0.5 0.4 0.3
0.6 0.5 0.4 0.3
0.2
0.2
0.1 0
Pe Pf Pm
← (70,0.1643)
0.1
← (54,0.0907)
0
20
40
60
80
100
0 20
Threshold
Pe Pf Pm
← (62,0.1418)
0.1 30
40
50
60
70
80
90
100
Threshold
(b) P(H0)=0.5,P(H1)=0.5
(b) P(H0)=0.5,P(H1)=0.5
1 1 Pe Pf Pm
0.9 0.8
Pe Pf Pm
0.9 0.8 0.7
0.6
Probability
Probability
0.7
0.5 0.4
0.6 0.5 0.4
0.3
0.3
0.2
0.2
0.1 0 0
0.1 ← (90,0.0771)
← (61,0.0383)
20
40
60
80
100
0 30
40
50
60
70
80
90
100
Threshold
Threshold
(c) P(H0)=0.9,P(H1)=0.1
(c) P(H0)=0.9,P(H1)=0.1
Fig. 2. Pf, Pm and Pe under the moderately noisy environment.
110
120
Fig. 3. Pf, Pm and Pe under the heavily noisy environment.
130
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Y. Zhao et al. / Computer Communications 34 (2011) 1510–1517
i
ð12Þ
where Pe(i) is the total error probability when the BS makes the decision based on the local sensing information of the first i SUs. By the time the BS broadcasts its decision to all SUs, it will also announce the cut-off total error probability f = Pe(m) (the total error probability of the mth SU in the sorted list). In the next sensing period, each SU will utilize the cut-off probability f as a guide to determine if it needs to send its local sensing result to the BS. Specifically, after the SU performs the local spectrum sensing, if the computed total error probability is larger than f, then this SU does not need to report the local sensing result to the BS. Thus, both the computation complexity at the BS and the communication overhead between SUs and the BS are reduced. One may note that f has actually been determined based on the sensing results in the current sensing period, and may not be exactly the same as the one for the next sensing period. This issue can be resolved by two techniques. First, we can add a margin to the cut-off probability to accommodate the sensing variations in the next period, i.e., mþj using a larger value than Pm (j > 0) ine , e.g., announcing f ¼ P e m stead of f ¼ P e (note that with the sorted list, we have P mþj P Pm e e ). Second, if in a sensing period, the total error probability Pe computed by the BS based on the sensing information of all reported SUs is larger than e, this means that the cut-off probability f announced in the last sensing period was too small, and based on the difference between e and Pe, a properly adjusted cut-off probability f + D will be announced for the next sensing period, so that more SUs will send their sensing results to the BS, to obtain a smaller Pe 6 e. We will further develop these two techniques in our future study. 5. Numerical results In this section, we evaluate the performance of our proposed weighted cooperative spectrum sensing. We consider 10 SUs if not otherwise noted. Without loss of generality, the transmitted PU signal is assumed to be s(k) = 1. First we consider a moderately noisy environment, where for each SU, the noise variances ri are randomly generated as Gaussian distribution with mean 0 and variance 1. Fig. 2 illustrates the false alarm probability Pf, the miss detection probability Pm, and the total error probability Pe as a function of the decision threshold c, when P(H0) = 0.2, 0.5 and 0.9, respectively. It can be seen that in Fig. 2, the false alarm probability decreases when the threshold increases. This fits well with the physical meaning of the false alarm. That is, when the decision threshold is low, it is prone to incorrectly detect the presence of PU. With the threshold increasing, the false alarm probability decreases, and drops to a small value at the optimal threshold. The miss detection probability, however, increases along with the threshold. The figure also shows the inherent trade-off relationship between the false alarm probability and the miss detection probability. The total error probability also changes with varying thresholds. The minimum total error probability is obtained at the optimal threshold. Table 1 compares the
c
P(H0)
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
P ie
Theoretical
Sim
Theoretical
Sim
49.7119 51.8787 52.4793 53.2524 54.1804 55.2552 56.5407 58.2112 60.8281
50 52 52 53 54 55 57 58 61
0.0303 0.0546 0.0767 0.0858 0.0908 0.0846 0.0745 0.0597 0.0381
0.0305 0.0556 0.0769 0.0861 0.0907 0.0848 0.0749 0.0598 0.0383
0.5 Generic Cooperative Half Voting Cooperative Equal−Weighted Cooperative SNR−based Weighted Cooperative Our Scheme
0.45 0.4
Total Error Probability
m ¼ arg minfP e ðiÞ 6 eg;
Table 1 The comparison of c and Pie between theoretical and simulation (denoted as ‘Sim’) results in Fig. 2.
0.35 0.3 0.25 0.2 0.15 0.1 0.05 0
0
0.2
0.4
0.6
0.8
1
P(H0)
(a) Performance under the moderately noisy environment Generic Cooperative Half Voting Cooperative Equal−Weighted Cooperative SNR−based Weighted Cooperative Our Scheme
0.6
0.5 Total Error Probability
cooperative sensing. On the other hand, if these SUs do not need to participate, i.e., transmitting their local sensing information to the BS, the computation complexity and communication overhead can be reduced, particularly in large networks. Let e denote the maximum tolerable error probability, and m denote the minimum number of SUs required to participate in cooperating sensing, under the constraint that the total error probability is not larger than e. To find m, the BS first sorts all SUs by their weights at the descending order. Noticing that the total error probability Pe is a non-increasing function with respect to the number of SUs, we have
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(b) Performance under the heavily noisy environment Fig. 4. Comparison of the total error probability.
theoretical results obtained by Eq. (10) and the simulation results with regard to the optimal threshold and the corresponding total error probability. Clearly they match very well. Note that the use of Pe as the performance metric is a good trade-off between the miss detection and the false alarm probabilities, although other combinations of miss detection and false alarm probabilities as performance metrics are also possible. This is because the minimum of Pe is usually obtained at a point where the miss detection and false alarm probabilities are at the expected values for the corresponding user activity. For instance, we have examined the
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0.4 0.35
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0.3 0.25 0.2 0.15
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P(Ho)=0.5 P(Ho)=0.3 P(Ho)=0.1
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Fig. 6. The minimum number of SUs (m) required to the maximum tolerable error probability ().
Fig. 5. The total error probability with different number of SUs.
scenario with P(H0) = 0.9 and P(H1) = 0.1, which indicates that the PU has a low duty cycle. The system obtains the minimum Pe when Pf = 0.05 and Pm = 0.34, i.e., the false alarm probability is low and the miss detection probability is relatively high. This is expected based on the user activity, since when the PU is at low duty cycle, the false alarm probability should be low while a relatively higher miss detection probability can be tolerated (note that it is not possible to let both be low). Next, we want to find out how the level of noise affects the total error probability. The configurations are the same as in the first scenario, except that the noise variances ri are randomly generated as Gaussian distribution with variance 2, referred to as the heavily noisy environment. Fig. 3 illustrates three probabilities, Pf, Pm, and Pe. It can be seen that all probabilities in a heavily noisy environment are bigger than those in a moderately noisy environment. To illustrate the shape of Pe and the existence of the minimum of Pe, we have conducted extensive simulations. Figs. 2 and 3 have shown six representative scenarios. The results in other scenarios are similar to those in Figs. 2 and 3 with regard to the shape of Pe and the existence of the minimum of Pe, i.e., the Pe always first decreases until it reaches a minimum point, and then increases. Due to the space limit, we omit the illustration of the results for other scenarios. For the comparative purpose, the performance of the generic cooperative spectrum sensing, the Half Voting sensing [10], the equal weighted cooperative sensing (a special case of our scheme in which all SUs are assumed the same weight), and the SNR-based weighted cooperative sensing from [17] have also been evaluated. Fig. 4 shows the total error probabilities of these schemes compared with our proposed scheme. The generic cooperative spectrum sensing has the highest error probability. This is because its false alarm probability dramatically increases while its miss detection probability smoothly decreases. Our scheme outperforms all other schemes, including the SNR-based weighted cooperative scheme in [17]. The performance gain is more significant in Fig. 4(b). This implies that our scheme is more robust to, or less impacted by, the heavily noisy environment than other schemes. Fig. 4 also demonstrates the impact of the PU activity on the total error probability. Intuitively, the total error probability is expected to be high at P(H0) = 0.5. This is because it is prone to make an incorrect decision when the channel alternates the status equally. On the other hand, a correct decision is likely to be made when the channel stays idle (or occupied) with a high probability, which is shown on the figure where the total error probability approaches 0 when P(H0) is close to 0 or 1.
At last, we examine the impact of the number of SUs on the total error probability. Fig. 5 shows that the total error probability Pe decreases when the number of SUs participating in the cooperative sensing increases from 1 to 15 for a given P(H0). That is, Pe is a non-increasing function with respect to the number of SUs. Therefore, for a given maximum tolerable error probability, we may let only a fraction of SUs to participate in cooperative sensing. Fig. 6 shows the minimum number of SUs required to participate in cooperative sensing, for a given maximum tolerable error probability (). In this scenario, we assume 20 SUs in the network. It can be seen that for a moderately tolerable error probability, we need much less than 20 SUs to participate in cooperative sensing. For example, if = 0.08, then the number of SUs required to participate in cooperative sensing are 5, 8, and 10 when P(H0) = 0.1, 0.3, and 0.5, respectively. 6. Conclusion In this paper, we have proposed a weighted cooperative spectrum sensing framework. We also propose to use the total error probability, which combines the false alarm probability and miss detection probability, to measure the detection accuracy. We have developed a theoretical model for the proposed weighted cooperative spectrum sensing, and derived the optimal detection threshold, as well as the number of SUs required to participate in cooperative sensing, subject to a given total error probability. Disclaimer Any opinion, finding, and conclusions or recommendations expressed in this material, are those of the authors and do not necessarily reflect the views of the National Science Foundation. Acknowledgement The research of Min Song is supported in part by NSF CAREER Award CNS-0644247 and NSF IPA Independent Research and Development (IR/D) Program. The research of Chunsheng Xin is supported in part by NSF under grants CNS-0721313 and CNS1017172. All the authors are grateful to the anonymous reviewers for their valuable comments to improve the paper. References [1] S. Haykin, Cognitive radio: brain-empowered wireless communications, IEEE Journal on Selected Areas in Communications 23 (2) (2005) 201–220.
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