Censored Truncated Sequential Spectrum Sensing for Cognitive Radio Networks
aa r X i v : . [ c s . S Y ] M a r Censored Truncated Sequential SpectrumSensing for Cognitive Radio Networks
Sina Maleki Geert Leus
Abstract
Reliable spectrum sensing is a key functionality of a cognitive radio network. Cooperative spectrumsensing improves the detection reliability of a cognitive radio system but also increases the system energyconsumption which is a critical factor particularly for low-power wireless technologies. A censoredtruncated sequential spectrum sensing technique is considered as an energy-saving approach. To designthe underlying sensing parameters, the maximum average energy consumption per sensor is minimizedsubject to a lower bounded global probability of detection and an upper bounded false alarm rate. Thisway both the interference to the primary user due to miss detection and the network throughput as a resultof a low false alarm rate are controlled. To solve this problem, it is assumed that the cognitive radios andfusion center are aware of their location and mutual channel properties. We compare the performanceof the proposed scheme with a fixed sample size censoring scheme under different scenarios and showthat for low-power cognitive radios, censored truncated sequential sensing outperforms censoring. It isshown that as the sensing energy per sample of the cognitive radios increases, the energy efficiency ofthe censored truncated sequential approach grows significantly.
Index Terms distributed spectrum sensing, sequential sensing, cognitive radio networks, censoring, energy effi-ciency.
S. Maleki and G. Leus are with the Faculty of Electrical Engineering, Mathematics and Computer Science, Delft Universityof Technology, 2628 CD Delft, The Netherlands (e-mail: [email protected]; [email protected]). Part of this paper has beenpresented at the 17th International Conference on Digital Signal Processing, DSP 2011, July 2011, Corfu, Greece. This work issupported in part by the NWO-STW under the VICI program (project 10382). Manuscript received date: Jan 5, 2012. Manuscriptrevised dates: May 16, 1012 and Jul 19, 2012
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I. I
NTRODUCTION
Dynamic spectrum access based on cognitive radios has been proposed in order to opportunisticallyuse underutilized spectrum portions of the licensed electromagnetic spectrum [1]. Cognitive radiosopportunistically share the spectrum while avoiding any harmful interference to the primary licensedusers. They employ spectrum sensing to detect the empty portions of the radio spectrum, also known asspectrum holes. Upon detection of such a spectrum hole, cognitive radios dynamically share this hole.However, as soon as a primary user appears in the corresponding band, the cognitive radios have to vacatethe band. As such, reliable spectrum sensing becomes a key functionality of a cognitive radio network.The hidden terminal problem and fading effects have been shown to limit the reliability of spec-trum sensing. Distributed cooperative detection has therefore been proposed to improve the detectionperformance of a cognitive radio network [2], [3]. Due to its simplicity and small delay, a paralleldetection configuration [4], is considered in this paper where each secondary radio continuously sensesthe spectrum in periodic sensing slots. A local decision is then made at the radios and sent to the fusioncenter (FC), which makes a global decision about the presence (or absence) of the primary user and feedsit back to the cognitive radios. Several fusion schemes have been proposed in the literature which can becategorized under soft and hard fusion strategies [4], [5]. Hard schemes are more energy efficient than softschemes, and thus a hard fusion scheme is adopted in this paper. More specifically, two popular choicesare employed due to their simple implementation: the OR and the AND rule. The OR rule dictates theprimary user presence to be announced by the FC when at least one cognitive radio reports the presenceof a primary user to the FC. On the other hand, the AND rule asks the FC to vote for the absence ofthe primary user if at least one cognitive radio announces the absence of the primary user. In this paper,energy detection is employed for channel sensing which is a common approach to detect unknown signals[5], [6], and which leads to a comparable detection performance for hard and soft fusion schemes [3].Energy consumption is another critical issue. The maximum energy consumption of a low-power radiois limited by its battery. As a result, energy efficient spectrum sensing limiting the maximum energyconsumption of a cognitive radio in a cooperative sensing framework is the focus of this paper.
A. Contributions
The spectrum sensing module consumes energy in both the sensing and transmission stages. To achievean energy-efficient spectrum sensing scheme the following contributions are presented in this paper. • A combination of censoring and truncated sequential sensing is proposed to save energy. The sensorssequentially sense the spectrum before reaching a truncation point, N , where they are forced to stop September 3, 2018 DRAFT sensing. If the accumulated energy of the collected sample observations is in a certain region (abovean upper threshold, a , or below a lower threshold, b ) before the truncation point, a decision is sentto the FC. Else, a censoring policy is used by the sensor, and no bits will be sent. This way, a largeamount of energy is saved for both sensing and transmission. In our paper, it is assumed that thecognitive radios and fusion center are aware of their location and mutual channel properties. • Our goal is to minimize the maximum average energy consumption per sensor subject to a specificdetection performance constraint which is defined by a lower bound on the global probability ofdetection and an upper bound on the global probability of false alarm. In terms of cognitive radiosystem design, the probability of detection limits the harmful interference to the primary user andthe false alarm rate controls the loss in spectrum utilization. The ideal case yields no interferenceand full spectrum utilization, but it is practically impossible to reach this point. Hence, currentstandards determine a bound on the detection performance to achieve an acceptable interference andutilization level [7]. To the best of our knowledge such a min-max optimization problem consideringthe average energy consumption per sensor has not yet been considered in literature. • Analytical expressions for the underlying parameters are derived and it is shown that the problemcan be solved by a two-dimensional search for both the OR and AND rule. • To reduce the computational complexity for the OR rule, a single-threshold truncated sequential testis proposed where each cognitive radio sends a decision to the FC upon the detection of the primaryuser. • To make a fair comparison of the proposed technique with current energy efficient approaches, a fixedsample size censoring scheme is considered as a benchmark (it is simply called the censoring schemethroughout the rest of the paper) where each sensor employs a censoring policy after collecting afixed number of samples. The censoring policy in this case works based on a lower threshold, λ and an upper threshold, λ . The decision is only being made if the accumulated energy is not in ( λ , λ ) . For this approach, it is shown that a single-threshold censoring policy is optimal in terms ofenergy consumption for both the OR and AND rule. Moreover, a solution of the underlying problemis given for the OR and AND rule. B. Related work to censoring
Censoring has been thoroughly investigated in wireless sensor networks and cognitive radios [8]–[13].It has been shown that censoring is very effective in terms of energy efficiency. In the early works, [8]–[11], the design of censoring parameters including lower and upper thresholds has been considered and
September 3, 2018 DRAFT mainly two problem formulations have been studied. In the Neyman-Pearson (NP) case, the miss-detectionprobability is minimized subject to a constraint on the probability of false alarm and average networkenergy consumption [9]–[11]. In the Bayesian case, on the other hand, the detection error probability isminimized subject to a constraint on the average network energy consumption. Censoring for cognitiveradios is considered in [12], [13]. In [12], a censoring rule similar to the one in this paper is considered inorder to limit the bandwidth occupancy of the cognitive radio network. Our fixed sample size censoringscheme is different in two ways. First, in [12], only the OR rule is considered and the FC makes nodecision in case it does not receive any decision from the cognitive radios which is ambiguous, since theFC has to make a final decision, while in our paper, the FC reports the absence (for the OR rule) or thepresence (for the AND rule) of the primary user, if no local decision is received at the FC. Second, wegive a clear optimization problem and expression for the solution while this is not presented in [12]. Acombined sleeping and censoring scheme is considered in [13]. The censoring scheme in this paper isdifferent in some ways. The optimization problem in the current paper is defined as the minimization ofthe maximum average energy consumption per sensor while in [13], the total network energy consumptionis minimized. For low-power radios, the problem in this paper makes more sense since the energy ofindividual radios is generally limited. In this paper, the received SNRs by the cognitive radios are assumedto be different while in [13], the SNRs are the same. Finally note that the sleeping policy of [13] can beeasily incorporated in our proposed censored truncated sequential sensing leading to even higher energysavings.
C. Related work to sequential sensing
Sequential detection as an approach to reduce the average number of sensors required to reach adecision is also studied comprehensively during the past decades [14]–[19]. In [14], [15], each sensorcollects a sequence of observations, constructs a summary message and passes it on to the FC and allother sensors. A Bayesian problem formulation comprising the minimization of the average error detectionprobability and sampling time cost over all admissible decision policies at the FC and all possible localdecision functions at each sensor is then considered to determine the optimal stopping and decision rule.Further, algorithms to solve the optimization problem for both infinite and finite horizon are given. In[16], an infinite horizon sequential detection scheme based on the sequential probability ratio test (SPRT)at both the sensors and the FC is considered. Wald’s analysis of error probability, [20], is employed todetermine the thresholds at the sensors and the FC. A combination of sequential detection and censoringis considered in [17]. Each sensor computes the LLR of the received sample and sends it to the FC,
September 3, 2018 DRAFT if it is deemed to be in a certain region. The FC then collects the received LLRs and as soon as theirsum is larger than an upper threshold or smaller than a lower threshold, the decision is made and thesensors can stop sensing. The LLRs are transmitted in such a way that the larger LLRs are sent sooner. Itis shown that the number of transmissions considerably reduces and particularly when the transmissionenergy is high, this approach performs very well. However, our paper employs a hard fusion schemeat the FC, our sequential scheme is finite horizon, and further a clear optimization problem is given tooptimize the energy consumption. Since we employ the OR (or the AND) rule in our paper, the FC candecide for the presence (or absence) of the primary user by only receiving a single one (or zero). Hence,ordered transmission can be easily incorporated in our paper by stopping the sensing and transmissionprocedure as soon as one cognitive radio sends a one (or zero) to the FC. [18] proposes a sequentialcensoring scheme where an SPRT is employed by the FC and soft or hard local decisions are sent tothe FC according to a censoring policy. It is depicted that the number of transmissions decreases but onthe other hand the average sample number (ASN) increases. Therefore, [18] ignores the effect of sensingon the energy consumption and focuses only on the transmission energy which for current low-powerradios is comparable to the sensing energy. A truncated sequential sensing technique is employed in[19] to reduce the sensing time of a cognitive radio system. The thresholds are determined such that acertain probability of false alarm and detection are obtained. In this paper, we are employing a similartechnique, except that in [19], after the truncation point, a single threshold scheme is used to make a finaldecision, while in our paper, the sensor decision is censored if no decision is made before the truncationpoint. Further, [19] considers a single sensor detection scheme while we employ a distributed cooperativesensing system and finally, in our paper an explicit optimization problem is given to find the sensingparameters.The remainder of the paper is organized as follows. In Section II, the fixed size censoring scheme forthe OR rule is described, including the optimization problem and the algorithm to solve it. The sequentialcensoring scheme for the OR rule is presented in Section III. Analytical expressions for the underlyingsystem parameters are derived and the optimization problem is analyzed. In Section IV, the censoring andsequential censoring schemes are presented and analyzed for the AND rule. We discuss some numericalresults in Section V. Conclusions and ideas for further work are finally posed in Section VI.II. F
IXED S IZE C ENSORING P ROBLEM F ORMULATION
A fixed size censoring scheme is discussed in this section as a benchmark for the main contributionof the paper in Section III, which studies a combination of sequential sensing and censoring. A network
September 3, 2018 DRAFT (FC) ...
Cognitive Radio 1Cognitive Radio 2Cognitive Radio M FusionCenter ...
Fig. 1: Distributed spectrum sensing configurationof M cognitive radios is considered under a cooperative spectrum sensing scheme. A parallel detectionconfiguration is employed as shown in Fig. 1. Each cognitive radio senses the spectrum and makes alocal decision about the presence or absence of the primary user and informs the FC by employing acensoring policy. The final decision is then made at the FC by employing the OR rule. The AND rulewill be discussed in Section IV. Denoting r ij to be the i -th sample received at the j -th cognitive radio,each radio solves a binary hypothesis testing problem as follows H : r ij = w ij , i = 1 , ..., N, j = 1 , ..., M H : r ij = h ij s i + w ij , i = 1 , ..., N, j = 1 , ..., M (1)where w ij is additive white Gaussian noise with zero mean and variance σ w . h ij and s i are the channelgain between the primary user and the j -th cognitive radio and the transmitted primary user signal,respectively. We assume two models for h ij and s i . In the first model, s i is assumed to be white Gaussianwith zero mean and variance σ s , and h ij is assumed constant during each sensing period and thus h ij = h j , i = 1 , . . . , N . In the second model, s i is assumed to be deterministic and constant modulus | s i | = s, i = 1 , . . . , N, j = 1 , . . . , M and h ij is an i.i.d. Gaussian random process with zero mean andvariance σ hj . Note that the second model actually represents a fast fading scenario. Although each modelrequires a different type of channel estimation, since the received signal is still a zero mean Gaussianrandom process with some variance, namely σ j = h j σ s + σ w for the former model and σ j = sσ hj + σ w for the latter model, the analyses which are given in the following sections are valid for both models. TheSNR of the received primary user signal at the j -th cognitive radio is γ j = | h j | σ s /σ w under the firstmodel and γ j = s σ hj /σ w under the second model. Furthermore, h ij s i and w ij are assumed statisticallyindependent.An energy detector is employed by each cognitive sensor which calculates the accumulated energy over September 3, 2018 DRAFT N observation samples. Note that under our system model parameters, the energy detector is equivalentto the optimal LLR detector [5]. The received energy collected over the N observation samples at the j -th radio is given by E j = N X i =1 | r ij | σ w . (2)When the accumulated energy of the observation samples is calculated, a censoring policy is employedat each radio where the local decisions are sent to the FC only if they are deemed to be informative[13]. Censoring thresholds λ and λ are applied at each of the radios, where the range λ < E j < λ is called the censoring region. At the j -th radio, the local censoring decision rule is given by send 1, declaring H if E j ≥ λ , no decision if λ < E j < λ , send 0, declaring H if E j ≤ λ . (3)It is well known [5] that under such a model, E j follows a central chi-square distribution with N degrees of freedom under H and H . Therefore, the local probabilities of false alarm and detection canbe respectively written as P fj = P r ( E j ≥ λ |H ) = Γ( N, λ )Γ( N ) , (4) P dj = P r ( E j ≥ λ |H ) = Γ( N, λ γ j ) )Γ( N ) , (5)where Γ( a, x ) is the incomplete gamma function given by Γ( a, x ) = R ∞ x t a − e − t dt , with Γ( a,
0) = Γ( a ) .Denoting C sj and C ti to be the energy consumed by the j -th radio in sensing per sample andtransmission per bit, respectively, the average energy consumed for distributed sensing per user is givenby, C j = N C sj + (1 − ρ j ) C tj , (6)where ρ j = P r ( λ < E j < λ ) is denoted to be the average censoring rate. Note that C sj is fixed and onlydepends on the sampling rate and power consumption of the sensing module while C tj depends on thedistance to the FC at the time of the transmission. Therefore, in this paper, it is assumed that the cognitiveradio is aware of its location and the location of the FC as well as their mutual channel properties orat least can estimate them. Defining π = P r ( H ) , π = P r ( H ) , δ j = P r ( λ < E j < λ |H ) and δ j = P r ( λ < E j < λ |H ) , ρ j is given by ρ j = π δ j + π δ j , (7) September 3, 2018 DRAFT with δ j = Γ( N, λ )Γ( N ) − Γ( N, λ )Γ( N ) , (8) δ j = Γ( N, λ γ j ) )Γ( N ) − Γ( N, λ γ j ) )Γ( N ) . (9)Denoting Q c F and Q c D to be the respective global probability of false alarm and detection, the targetdetection performance is then quantified by Q c F ≤ α and Q c D ≥ β , where α and β are pre-specifieddetection design parameters. Our goal is to determine the optimum censoring thresholds λ and λ suchthat the maximum average energy consumption per sensor, i.e., max j C j , is minimized subject to theconstraints Q c F ≤ α and Q c D ≥ β . Hence, our optimization problem can be formulated as min λ ,λ max j C j s.t. Q c F ≤ α, Q c D ≥ β. (10)In this section, the FC employs an OR rule to make the final decision which is denoted by D F C , i.e., D F C = 1 if the FC receives at least one local decision declaring 1, else D F C = 0 . This way, the globalprobability of false alarm and detection can be derived as Q c F = P r ( D F C = 1 |H ) = 1 − M Y j =1 (1 − P fj ) , (11) Q c D = P r ( D F C = 1 |H ) = 1 − M Y j =1 (1 − P dj ) . (12)Note that since all the cognitive radios employ the same upper threshold λ , we can state that P fj = P f defined in (4). As a result, (11) becomes Q c F = 1 − (1 − P f ) M . (13)Since the FC decides about the presence of the primary user only by receiving 1s (receiving no decisionfrom all the sensors is considered as absence of the primary user) and the sensing time does not dependon λ , it is a waste of energy to send zeros to the FC and thus, the optimal solution of (10) is obtainedby λ = 0 . Note that this is only the case for fixed-size censoring, because the energy consumption ofeach sensor only varies by the transmission energy while the sensing energy is constant. This way (8)and (9) can be simplified to δ j = 1 − P f and δ j = 1 − P dj , and we only need to derive the optimal λ .Since there is a one-to-one relationship between P f and λ , by finding the optimal P f , λ can also beeasily derived as λ = 2Γ − [ N, Γ( N ) P f ] (where Γ − is defined over the second argument). Considering September 3, 2018 DRAFT this result and defining Q c D = H ( P f ) , the optimal solution of (10) is given by P f = H − ( β ) as is shownin Appendix A.In the following section, a combination of censoring and sequential sensing approaches is presentedwhich optimizes both the sensing and the transmission energy.III. S EQUENTIAL C ENSORING P ROBLEM F ORMULATION
A. System Model
Unlike Section II, where each user collects a specific number of samples, in this section, each cognitiveradio sequentially senses the spectrum and upon reaching a decision about the presence or absence ofthe primary user, it sends the result to the FC by employing a censoring policy as introduced in SectionII. The final decision is then made at the FC by employing the OR rule. Here, a censored truncatedsequential sensing scheme is employed where each cognitive radio carries on sensing until it reachesa decision while not passing a limit of N samples. We define ζ nj = P ni =1 | r ij | /σ w = P ni =1 x ij and a i = 0 , i = 1 , . . . , p , a i = ¯ a + i ¯Λ , i = p + 1 , ..., N and b i = ¯ b + i ¯Λ , i = 1 , ..., N , where ¯ a = a/σ w , ¯ b = b/σ w , < ¯Λ < γ j is a predetermined constant, a < , b > and p = ⌊− a/σ w ¯Λ ⌋ [19]. Weassume that the SNR γ j is known or can be estimated. This way, the local decision rule in order to makea final decision is as follows send 1, declaring H if ζ nj ≥ b n and n ∈ [1 , N ] , continue sensing if ζ nj ∈ ( a n , b n ) and n ∈ [1 , N ) , no decision if ζ nj ∈ ( a n , b n ) and n = N, send 0, declaring H if ζ nj ≤ a n and n ∈ [1 , N ] . (14)The probability density function of x ij = | r ij | /σ w under H and H is a chi-square distribution with n degrees of freedom. Thus, x ij becomes exponentially distributed under both H and H . Henceforth,we obtain P r ( x ij |H ) = 12 e − x ij / I { x ij ≥ } , (15) P r ( x ij |H ) = 12(1 + γ j ) e − x ij / γ j ) I { x ij ≥ } , (16)where I { x ij ≥ } is the indicator function.Defining ζ j = 0 , the local probability of false alarm at the j -th cognitive radio, P fj , can be writtenas P fj = N X n =1 P r ( ζ j ∈ ( a , b ) , ..., ζ n − j ∈ ( a n − , b n − ) , ζ nj ≥ b n |H ) , (17) September 3, 2018 DRAFT0 whereas the local probability of detection, P dj , is obtained as follows P dj = N X n =1 P r ( ζ j ∈ ( a , b ) , ..., ζ n − j ∈ ( a n − , b n − ) , ζ nj ≥ b n |H ) . (18)Denoting ρ j to be the average censoring rate at the j -th cognitive radio, and δ j and δ j to be therespective average censoring rate under H and H , we have ρ j = π δ j + π δ j , (19)where δ j = P r ( ζ j ∈ ( a , b ) , ..., ζ Nj ∈ ( a N , b N ) |H ) , (20) δ j = P r ( ζ j ∈ ( a , b ) , ..., ζ Nj ∈ ( a N , b N ) |H ) . (21)The other parameter that is important in any sequential detection scheme is the average sample number(ASN) required to reach a decision. Denoting N j to be a random variable representing the number ofsamples required to announce the presence or absence of the primary user, the ASN for the j -th cognitiveradio, denoted as ¯ N j = E ( N j ) , can be defined as ¯ N j = π E ( N j |H ) + π E ( N j |H ) , (22)where E ( N j |H ) = N X n =1 nP r ( N j = n |H )= N − X n =1 n [ P r ( ζ j ∈ ( a , b ) , ..., ζ n − j ∈ ( a n − , b n − ) |H ) − P r ( ζ j ∈ ( a , b ) , ..., ζ nj ∈ ( a n , b n ) |H )]+ N P r ( ζ j ∈ ( a , b ) , ..., ζ N − j ∈ ( a N − , b N − ) |H ) , (23)and E ( N j |H ) = N X n =1 nP r ( N j = n |H )= N − X n =1 n [ P r ( ζ j ∈ ( a , b ) , ..., ζ nj ∈ ( a n − , b n − ) |H ) − P r ( ζ j ∈ ( a , b ) , ..., ζ nj ∈ ( a n , b n ) |H )]+ N P r ( ζ j ∈ ( a , b ) , ..., ζ N − j ∈ ( a N − , b N − ) |H ) . (24) September 3, 2018 DRAFT1
Denoting again C sj to be the sensing energy of one sample and C tj to be the transmission energy of adecision bit at the j -th cognitive radio, the total average energy consumption at the j -th cognitive radionow becomes C j = ¯ N j C sj + (1 − ρ j ) C tj . (25)Denoting Q cs F and Q cs D to be the respective global probabilities of false alarm and detection for thecensored truncated sequential approach, we define our problem as the minimization of the maximumaverage energy consumption per sensor subject to a constraint on the global probabilities of false alarmand detection as follows min ¯ a, ¯ b max j C j s.t. Q cs F ≤ α, Q cs D ≥ β. (26)As in (11) and (12), under the OR rule that is assumed in this section, the global probability of falsealarm is Q cs F = P r ( D FC = 1 |H ) = 1 − M Y j =1 (1 − P fj ) , (27)and the global probability of detection is Q cs D = P r ( D FC = 1 |H ) = 1 − M Y j =1 (1 − P dj ) . (28)Note that since P f = · · · = P fM , it is again assumed that P fj = P f in this section.In the following subsection, analytical expressions for the probability of false alarm and detection aswell as the censoring rate and ASN are extracted. B. Parameter and Problem Analysis
Looking at (17), (18), (19) and (22), we can see that the joint probability distribution function of p ( ζ j , ..., ζ nj ) is the foundation of all the equations. Since x ij = ζ ij − ζ i − j for i = 1 , ..., N , we have, p ( ζ j , ..., ζ nj ) = p ( x nj ) p ( x n − j ) ...p ( x j ) . (29)Therefore, the joint probability distribution function under H and H becomes p ( ζ j , ..., ζ nj |H ) = 12 n e − ζ nj / I { ≤ ζ j ≤ ζ j ... ≤ ζ nj } , (30) p ( ζ j , ..., ζ nj |H ) = 1[2(1 + γ j )] n e − ζ nj / γ j ) I { ≤ ζ j ≤ ζ j ... ≤ ζ nj } , (31)where I { ≤ ζ j ≤ ζ j ... ≤ ζ nj } is again the indicator function. September 3, 2018 DRAFT2
The derivation of the local probability of false alarm and the ASN under H in this work are similar tothe ones considered in [19] and [21]. The difference is that in [19], if the cognitive radio does not reacha decision after N samples, it employs a single threshold decision policy to give a final decision aboutthe presence or absence of the cognitive radio, while in our work, no decision is sent in case none of theupper and lower thresholds are crossed. Hence, to avoid introducing a cumbersome detailed derivation ofeach parameter, we can use the results in [19] for our analysis with a small modification. However, notethat the problem formulation in this work is essentially different from the one in [19]. Further, since inour work the distribution of x ij under H is exponential like the one under H , unlike [19], we can alsouse the same approach to derive analytical expressions for the local probability of detection, the ASNunder H , and the censoring rate.Denoting E n to be the event where a i < ζ ij < b i , i = 1 , ..., n − and ζ nj ≥ b n , (17) becomes P fj = N X n =1 P r ( E n |H ) . (32)where the analytical expression for P r ( E n |H ) is derived in Appendix B.Similarly for the local probability of detection, we have P dj = N X n =1 P r ( E n |H ) , (33)where the analytical expression for P r ( E n |H ) is derived in Appendix C.Defining R nj = { ζ ij | ζ ij ∈ ( a i , b i ) , i = 1 , ..., n } , P r ( R nj |H ) and P r ( R nj |H ) are obtained as follows P r ( R nj |H ) = 12 n J ( n ) a n ,b n (1 / , n = 1 , ..., N, (34) P r ( R nj |H ) = 1[2(1 + γ j )] n J ( n ) a n ,b n (1 / γ j )) , n = 1 , ..., N, (35)where J ( n ) a n ,b n ( θ ) is presented in Appendix D and (23) and (24) become E ( N j |H ) = N − X n =1 n ( P r ( R n − j |H ) − P r ( R nj |H )) + N P r ( R N − j |H ) = 1 + N − X n =1 P r ( R nj |H ) , (36) E ( N j |H ) = N X n =1 n ( P r ( R n − j |H ) − P r ( R nj |H )) + N P r ( R N − j |H ) = 1 + N − X n =1 P r ( R nj |H ) . (37)With (36) and (37), we can calculate (22). This way, (20) and (21) can be derived as follows δ j = P r ( R Nj |H ) = 12 N J ( N ) a N ,b N (1 / , (38) δ j = P r ( R Nj |H ) = 1[2(1 + γ j )] N J ( N ) a N ,b N (1 / γ j )) . (39) September 3, 2018 DRAFT3
We can show that the problem (26) is not convex. Therefore, the standard systematic optimizationalgorithms do not give the global optimum for ¯ a and ¯ b . However, as is shown in the following lines, ¯ a and ¯ b are bounded and therefore, a two-dimensional exhaustive search is possible to find the globaloptimum. First of all, we have a < and ¯ a < . On the other hand, if ¯ a has to play a role in the sensingsystem, at least one a N should be positive, i.e., a N = ¯ a + N ∆ ≥ which gives ¯ a ≥ − N ∆ . Hence, weobtain − N ∆ ≤ ¯ a < . Furthermore, defining Q cs F = F (¯ a, ¯ b ) and Q cs D = G (¯ a, ¯ b ) , for a given ¯ a , it is easyto show that G − (¯ a, β ) ≤ ¯ b ≤ F − (¯ a, α ) (where F − and G − are defined over the second argument).Before introducing a suboptimal problem, the following theorem is presented. Theorem 1 . For a given local probability of detection and false alarm ( P d and P f ) and N , the censoringrate of the optimal censored truncated sequential sensing ( ρ cs ) is less than the one of the censoring scheme( ρ c ). Proof . The proof is provided in Appendix E.We should note that, in censored truncated sequential sensing, a large amount of energy is to be savedon sensing. Therefore, as is shown in Section V, as the sensing energy of each sensor increases, censoredtruncated sequential sensing outperforms censoring in terms of energy efficiency. However, in case thatthe transmission energy is much higher than the sensing energy, it may happen that censoring outperformscensored truncated sequential sensing, because of a higher censoring rate ( ρ cs > ρ c ). Hence, one corollaryof Theorem 1 is that although the optimal solution of (10) for a specific N , i.e., P d = 1 − (1 − β ) /M and P f = H − ( β ) , is in the feasible set of (26) for a resulting ASN less than N , it does not necessarilyguarantee that the resulting average energy consumption per sensor of the censored truncated sequentialsensing approach is less than the one of the censoring scheme, particularly when the transmission energyis much higher than the sensing energy per sample.Solving (26) is complex in terms of the number of computations, and thus a two-dimensional exhaustivesearch is not always a good solution. Therefore, in order to reach a good solution in a reasonable time,we set a < − N ∆ in order to obtain a = · · · = a N = 0 . This way, we can relax one of the argumentsof (26) and only solve the following suboptimal problem min ¯ b max j C j s.t. Q cs F ≤ α, Q cs D ≥ β. (40)Note that unlike Section II, here the zero lower threshold is not necessarily optimal. The reason is thatalthough the maximum censoring rate is achieved with the lowest ¯ a , the minimum ASN is achieved withthe highest ¯ a , and thus there is an inherent trade-off between a high censoring rate and a low ASN and a September 3, 2018 DRAFT4 zero a i is not necessarily the optimal solution. Since the analytical expressions provided earlier are verycomplex, we now try to provide a new set of analytical expressions for different parameters based onthe fact that a = · · · = a N = 0 .To find an analytical expression for P fj , we can derive A ( n ) for the new paradigm as follows A ( n ) = Z ... Z Γ n I { ≤ ζ j ≤ ζ j ... ≤ ζ n − j } dζ j ...dζ n − j . (41)Since ≤ ζ j ≤ ζ j ... ≤ ζ n − j and a = · · · = a N = 0 , the lower bound for each integral is ζ i − andthe upper bound is b i , where i = 1 , ..., n − . Thus we obtain A ( n ) = Z b ζ j Z b ζ j ... Z b n − ζ n − j dζ j dζ j ...dζ n − j , (42)which according to [21] is A ( n ) = b b n − n ( n − , n = 1 , ..., N. (43)Hence, we have P fj = N X n =1 p n A ( n ) , (44)and p n = e − bn/ n − . Similarly, for P dj , we obtain B ( n ) = Z b ζ j Z b ζ j ... Z b n − ζ n − j dζ j dζ j ...dζ n − j = b b n − n ( n − , n = 1 , ..., N, (45)and thus P dj = N X n =1 q n B ( n ) , (46)where q n = e − bn/ γj ) [2(1+ γ j )] n − . Furthermore, we note that for a = · · · = a N = 0 , A ( n ) = B ( n ) = b b n − n ( n − , n =1 , ..., N .It is easy to see that R nj occurs under H , if no false alarm happens until the n -th sample. Therefore,the analytical expression for P r ( R nj |H ) is given by P r ( R nj |H ) = 1 − n X i =1 p i A ( i ) , (47)and in the same way, for P r ( R nj |H ) , we obtain P r ( R nj |H ) = 1 − n X i =1 q i A ( i ) . (48) September 3, 2018 DRAFT5
Putting (47) and (48) in (36) and (37), we obtain E ( N j |H ) = 1 + N − X n =1 (cid:26) − n X i =1 p i A ( i ) (cid:27) , (49) E ( N j |H ) = 1 + N − X n =1 (cid:26) − n X i =1 q i A ( i ) (cid:27) , (50)and inserting (49) and (50) in (22), we obtain ¯ N j = π N − X n =1 (cid:26) − n X i =1 p i A ( i ) (cid:27)! + π N − X n =1 (cid:26) − n X i =1 q i A ( i ) (cid:27)! . (51)Finally, from (47) and (48), the censoring rate can be easily obtained as ρ j = π (cid:18) − N X i =1 p i A ( i ) (cid:19) + π (cid:18) − N X i =1 q i A ( i ) (cid:19) . (52)Having the analytical expressions for (40), we can easily find the optimal maximum average energyconsumption per sensor by a line search over ¯ b . Similar to the censoring problem formulation, here thesensing threshold is also bounded by Q cs F − ( α ) ≤ ¯ b ≤ Q cs D − ( β ) . As we will see in Section V, censoredtruncated sequential sensing performs better than censored spectrum sensing in terms of energy efficiencyfor low-power radios. IV. E XTENSION TO THE
AND
RULE
So far, we have mainly focused on the OR rule. However, another rule which is also simple in termsof implementation is the AND rule. According to the AND rule, D F C = 0 , if at least one cognitiveradio reports a zero, else D F C = 1 . This way the global probabilities of false alarm and detection, canbe written respectively as Q c F,AND = Q cs F,AND = P r ( D F C = 1 |H ) = M Y j =1 ( δ j + P fj ) , (53) Q c D,AND = Q cs D,AND = P r ( D F C = 1 |H ) = M Y j =1 ( δ j + P dj ) . (54)Note that (53) and (54) hold for both the sequential censoring and censoring schemes. Similar to the casefor the OR rule, the problem is defined so as to minimize the maximum average energy consumptionper sensor subject to a lower bound on the global probability of detection and an upper bound on theglobal probability of false alarm. In the following two subsections, we are going to analyze the problemfor censoring and sequential censoring. September 3, 2018 DRAFT6
A. AND rule for fixed-sample size censoring
The optimization problem for the censoring scheme considering the AND rule at the FC, becomes min λ ,λ max j C j s.t. Q c F,AND ≤ α, Q c D,AND ≥ β. (55)where C j is defined in (6). Since the FC decides for the absence of the primary user by receiving at leastone zero and the fact that the sensing energy per sample is constant, the optimal upper threshold λ is λ → ∞ . This way, cognitive radios censor all the results for which E j > λ , and as a result (53) and(54) become Q c F,AND = P r ( D F C = 1 |H ) = M Y j =1 δ j , (56) Q c D,AND = P r ( D F C = 1 |H ) = M Y j =1 δ j . (57)where δ j = P r ( E j > λ |H ) and δ j = P r ( E j > λ |H ) . Since the thresholds are the same among thecognitive radios, we have δ = δ = · · · = δ M = δ . Since there is a one-to-one relationship between λ and δ , by finding the optimal δ , the optimal λ can be easily derived. As shown in Appendix F,we can derive the optimal δ as δ = α /M . This result is very important in the sense that as far as thefeasible set of (55) is not empty, the optimal solution of (55) is independent from the SNR. Note thatthe maximum average energy consumption per sensor still depends on the SNR via δ j and is reducingas the SNR grows. B. AND rule for censored truncated sequential sensing
The optimization problem for the censored truncated sequential sensing scheme with the AND rule,becomes min ¯ a, ¯ b max j C j s.t. Q cs F,AND ≤ α, Q cs D,AND ≥ β. (58)where C j is defined in (25). Similar to the OR rule, we have − N ∆ ≤ ¯ a < . Defining Q cs F,AND = F AND (¯ a, ¯ b ) and Q cs D,AND = G AND (¯ a, ¯ b ) , for a given ¯ a , we can show that G − AND (¯ a, β ) ≤ ¯ b ≤ F − AND (¯ a, α ) (where F − AND and G − AND are defined over the second argument). Therefore, the optimal ¯ a and ¯ b can againbe derived by a bounded two-dimensional search, in a similar way as for the OR rule. September 3, 2018 DRAFT7
V. N
UMERICAL R ESULTS
A network of cognitive radios is considered for the numerical results. In some of the scenarios, forthe sake of simplicity, it is assumed that all the sensors experience the same SNR. This way, it iseasier to show how the main performance indicators including the optimal maximum average energyconsumption per sensor, ASN and censoring rate changes when one of the underlying parameter of thesystem changes. However, to comply with the general idea of the paper, which is based on differentreceived SNRs by cognitive radios, in other scenarios, the different cognitive radios experience differentSNRs. Unless otherwise mentioned, the results are based on the single-threshold strategy for censoredtruncated sequential sensing in case of the OR rule.Fig. 2a depicts the optimal maximum average energy consumption per sensor versus the number ofcognitive radios for the OR rule. The SNR is assumed to be dB, N = 10 , C s = 1 and C t = 10 .Furthermore, the probability of false alarm and detection constraints are assumed to be α = 0 . and β = 0 . as determined by the IEEE 802.15.4 standard for cognitive radios [7]. It is shown for both highand low values of π that censored sequential sensing outperforms the censoring scheme. Looking atFig. 2b and Fig. 2c, where the respective optimal censoring rate and optimal ASN are shown versus thenumber of cognitive radios, we can deduce that the lower ASN is playing a key role in a lower energyconsumption of the censored sequential sensing. Fig. 2a also shows that as the number of cooperatingcognitive radios increases, the optimal maximum average energy consumption per sensor decreases andsaturates, while as shown in Fig. 2b and Fig. 2c, the optimal censoring rate and optimal ASN increase.This way, the energy consumption tends to increase as a result of ASN growth and on the other handinclines to decrease due to the censoring rate growth and that is the reason for saturation after a number ofcognitive radios. Therefore, we can see that as the number of cognitive radios increases, a higher energyefficiency per sensor can be achieved. However, after a number of cognitive radios, the maximum averageenergy consumption per sensor remains almost at a constant level and by adding more cognitive radiosno significant energy saving per sensor can be achieved while the total network energy consumption alsoincreases.Figures 3a, 3b and 3c consider a scenario where M = 5 , N = 30 , C sj = 1 , C tj = 10 , α = 0 . , β = 0 . and π can take a value of . or . . The performance of the system versus SNR is analyzedin this scenario for the OR rule. The maximum average energy consumption per sensor is depicted inFig. 3a. As for the earlier scenario, censored sequential sensing gives a higher energy efficiency comparedto censoring. While the optimal energy variation for the censoring scheme is almost the same for all September 3, 2018 DRAFT8 E ne r g y sequential censoring, π =0.8censoring, π =0.8sequential censoring, π =0.2censoring, π =0.2 (a) O p t i m a l c en s o r i ng r a t e sequential censoring, π =0.8censoring, π =0.8sequential censoring, π =0.2censoring, π =0.2 (b) O p t i m a l AS N π =0.8 π =0.2 (c) Fig. 2: a) Optimal maximum average energy consumption per sensor versus number of cognitiveradios, b) Optimal censoring rate versus number of cognitive radios, c) Optimal ASN versus numberof cognitive radios for the OR rulethe considered SNRs, the censored sequential scheme’s average energy consumption per sensor reducessignificantly as the SNR increases. The reason is that as the SNR increases, the optimal ASN dramaticallydecreases (almost for γ = 2 dB and π = 0 . ). This shows that as the SNR increases, censoredsequential sensing becomes even more valuable and a significant energy saving per sensor can be achievedcompared with the one that is achieved by censoring. Since the SNR changes with the channel gain ( | h j | September 3, 2018 DRAFT9 −4 −3 −2 −1 0 1 222242628303234 SNR E ne r g y sequential censoring, π =0.8censoring, π =0.8sequential censoring, π =0.2censoring, π =0.2 (a) −4 −3 −2 −1 0 1 20.10.20.30.40.50.60.70.80.91 SNR O p t i m a l c en s o r i ng r a t e sequential censoring, π =0.8censoring, π =0.8sequential censoring, π =0.2censoring, π =0.2 (b) −4 −3 −2 −1 0 1 2141618202224262830 SNR AS N π =0.8 π =0.2 (c) Fig. 3: a) Optimal maximum average energy consumption per sensor versus SNR, b) Optimal censoringrate versus SNR, c) Optimal ASN versus SNR for the OR ruleunder the first model or σ hj under the second model), from Fig. 3a, the behavior of the system withvarying | h j | or σ hj can be derived, if the distribution of | h j | or σ hj is known.Figures 4a and 4b compare the performance of the single threshold censored truncated sequentialscheme with the one assuming two thresholds, i.e, ¯ a and ¯ b for the OR rule. The idea is to find when thedouble threshold scheme with its higher complexity becomes valuable. In these figures, M = 5 , N = 10 , γ = 0 dB, C t = 10 , π = 0 . , . , and α = 0 . , while β changes from . to . . The sensing energy September 3, 2018 DRAFT0 β E ne r g y Double threshold, π =0.2Single threshold, π =0.2Double threshold, π =0.8Single threshold, π =0.8 (a) β E ne r g y Double threshold, π =0.2Single threshold, π =0.2Double threshold, π =0.8Single threshold, π =0.8 (b) Fig. 4: Optimal maximum average energy consumption per sensor versus probability of detectionconstraint, β , for the OR rule, a) C s = 1 , b) C s = 3 per sample, C s in Fig. 4a is assumed , while in Fig. 4b it is . It is shown that as the sensing energyper sample increases, the energy efficiency of the double threshold scheme also increases compared tothe one of the single threshold scheme, particularly when π is high. The reason is that when π is high,a much lower ASN can be achieved by the double threshold scheme compared to the single thresholdone. This gain in performance comes at the cost of a higher computational complexity because of thetwo-dimensional search. September 3, 2018 DRAFT1
15 20 25 30161820222426283032 Number of samples E ne r g y sequential censoring, π =0.5censoring, π =0.5 Fig. 5: Optimal maximum average energy consumption per sensor versus number of samples for the ORruleFig. 5 depicts the optimal maximum average energy consumption per sensor versus the number ofsamples for the OR rule and for a network of M = 5 cognitive radios where each radio experiences adifferent channel gain and thus a different SNR. Arranging the SNRs in a vector γ = [ γ , . . . , γ ] , wehave γ = [1dB, 2dB, 3dB, 4dB, 5dB]. The other parameters are C s = 1 , C t = 10 , π = 0 . , α = 0 . and β = 0 . . As shown in Fig. 5, by increasing the number of samples and thus the total sensing energy, thesequential censoring energy efficiency also increases compared to the censoring scheme. For example,if we define the efficiency of the censored truncated sequential sensing scheme as the difference of theoptimal maximum average energy consumption per sensor of sequential censoring and censoring dividedby the optimal maximum average energy consumption per sensor of censoring, the efficiency increasesapproximately three times from 0.06 (for N = 15 ) to 0.19 (for N = 30 ).In Fig. 6, the sensing energy per sample is C s = 10 while the transmission energy C t changes from 0to 1000. The goal is to see how the optimal maximum average energy consumption per sensor changeswith C t for the or rule and for a network of M = 5 cognitive radios with γ = [1dB, 2dB, 3dB, 4dB,5dB]. The other parameters of the network are N = 30 , π = 0 . , α = 0 . and β = 0 . . The bestsaving for sequential censoring is achieved when the transmission energy is zero. Indeed, we can see thatas the transmission energy increases the performance gain of sequential censoring reduces compared tocensoring. However, in low-power radios where the sensing energy per sample and transmission energyare usually in the same range, sequential censoring performs much better than censoring in terms of September 3, 2018 DRAFT2 t E ne r g y sequential censoring, π =0.5censoring, π =0.5 Fig. 6: Optimal maximum average energy consumption per sensor versus transmission energy for the ORruleenergy efficiency as we can see in Fig. 6.Fig. 7 depicts the optimal maximum average energy consumption per sensor versus the sensing energyper sample for both the AND and OR rule. For the sake of simplicity and tractability, the SNRs areassumed the same for M = 50 cognitive radios. The other parameters are assumed to be N = 10 , C t = 10 , π = 0 . , γ = 0 dB, α = 0 . and β = 0 . . For both fusion rules, the double threshold schemeis employed. We can see that the OR rule performs better for the low values of C s . However, as C s increases the AND rule dominates and outperforms the OR rule, particularly for high values of C s . Thereason that the OR rule performs better than the AND rule at very low values of C s is that the optimalcensoring rate for the OR rule is higher than the optimal censoring rate for the AND rule. However as C s increases, the AND rule dominates the OR rule in terms of energy efficiency due to the lower ASN.The optimal maximum average energy consumption per sensor versus π is investigated in Fig. 8 forthe AND and the OR rule. The underlying parameters are assumed to be C s = 2 , C t = 10 , N = 10 , M = 50 , γ = 0 dB, α = 0 . and β = 0 . . It is shown that as the probability of the primary user absenceincreases, the optimal maximum average energy consumption per sensor reduces for the OR rule while itincreases for the AND rule. This is mainly due to the fact that for the OR rule, we are mainly interestedto receive a ”1” from the cognitive radios. Therefore, as π increases, the probability of receiving a ”1”decreases, since the optimal censoring rate increases. The opposite happens for the AND rule, since forthe AND rule, receiving a ”0” from the cognitive radios is considered to be informative. September 3, 2018 DRAFT3 s E ne r g y sequential censoring, π =0.5, ANDsequential censoring, π =0.5, OR Fig. 7: Optimal maximum average energy consumption per sensor versus sensing energy per sample forAND and OR rule π E ne r g y sequential censoring, ANDsequential censoring, OR Fig. 8: Optimal maximum average energy consumption per sensor versus π for AND and OR ruleVI. S UMMARY AND C ONCLUSIONS
We presented two energy efficient techniques for a cognitive sensor network. First, a censoring schemehas been discussed where each sensor employs a censoring policy to reduce the energy consumption.Then a censored truncated sequential approach has been proposed based on the combination of censoringand sequential sensing policies. We defined our problem as the minimization of the maximum averageenergy consumption per sensor subject to a global probability of false alarm and detection constraint for
September 3, 2018 DRAFT4 the AND and the OR rules. The optimal lower threshold is shown to be zero for the censoring scheme incase of the OR rule while for the AND rule the optimal upper threshold is shown to be infinity. Further,an explicit expression was given to find the optimal solution for the OR rule and in case of the AND rulea closed for solution is derived. We have further derived the analytical expressions for the underlyingparameters in the censored sequential scheme and have shown that although the problem is not convex,a bounded two-dimensional search is possible for both the OR rule and the AND rule. Further, in caseof the OR rule, we relaxed the lower threshold to obtain a line search problem in order to reduce thecomputational complexity.Different scenarios regarding transmission and sensing energy per sample as well as SNR, numberof cognitive radios, number of samples and detection performance constraints were simulated for lowand high values of π and for both the OR rule and the AND rule. It has been shown that under thepractical assumption of low-power radios, sequential censoring outperforms censoring. We conclude thatfor high values of the sensing energy per sample, despite its high computational complexity, the doublethreshold scheme developed for the OR rule becomes more attractive. Further, it is shown that as thesensing energy per sample increases compared to the transmission energy, the AND rule performs betterthan the OR rule, while for very low values of the sensing energy per sample, the OR rule outperformsthe AND rule.Note that a systematic solution for the censored sequential problem formulation was not given in thispaper, and thus it is valuable to investigate a better algorithm to solve the problem. We also did notconsider a combination of the proposed scheme with sleeping as in [13], which can generate furtherenergy savings. Our analysis was based on the OR rule and the AND rule, and thus extensions to otherhard fusion rules could be interesting. A PPENDIX AO PTIMAL SOLUTION OF (10)Since the optimal λ = 0 , (8) and (9) can be simplified to δ j = 1 − P f and δ j = 1 − P dj and so(10) becomes, min λ max j (cid:2) N C sj + ( π P f + π P dj ) C tj (cid:3) s.t. − (1 − P f ) M ≤ α, − M Y j =1 (1 − P dj ) ≥ β. (59) September 3, 2018 DRAFT5
Since there is a one-to-one relationship between λ and P f , i.e., λ = 2Γ − [ N, Γ( N ) P f ] (where Γ − is defined over the second argument), (59) can be formulated as [22, p.130], min P f max j (cid:2) N C sj + ( π P f + π P dj ) C tj (cid:3) s.t. − (1 − P f ) M ≤ α, − Q Mj =1 (1 − P dj ) ≥ β. (60)Defining P f = F ( λ ) = Γ( N, λ )Γ( N ) and P dj = G j ( λ ) = Γ( N, λ γj ) )Γ( N ) , we can write P dj as P dj = G j ( F − ( P f )) . Calculating the derivative of C j with respect to P f , we find that ∂C j ∂P f = ∂ (cid:2) C tj ( π P f + π P dj ) (cid:3) ∂P f = C tj π + ∂P dj ∂P f ≥ , (61)where we use the fact that ∂P dj ∂P f = − N Γ( N ) − [ N, Γ( N ) P f ] N − e − [ N, Γ( N ) P f ] / γ j ) I { − [ N, Γ( N ) P f ] ≥ } − N Γ( N ) − [ N, Γ( N ) P f ] N − e − [ N, Γ( N ) P f ] / I { − [ N, Γ( N ) P f ] ≥ } = e − [ N, Γ( N ) P f ](1 / γ j ) − / ≥ . (62)Therefore, we can simplify (60) as min P f P f s.t. − (1 − P f ) M ≤ α, − Q Mj =1 (1 − P dj ) ≥ β. (63)which can be easily solved by a line search over P f . However, since Q c D is a monotonically increasingfunction of P f , i.e., Q c D = H ( P f ) = 1 − Q Mj =1 (1 − G j ( F − ( P f ))) and thus ∂Q c D ∂P f = ∂Q c D ∂P dj ∂P dj ∂P f = Q l = Ml =1 ,l = j (1 − P dl ) ∂P dj ∂P f ≥ , we can further simplify the constraints in (63) as P f ≤ − (1 − α ) /M and P f ≥ H − ( β ) . Thus, we obtain min P f P f s.t. P f ≤ − (1 − α ) /M , P f ≥ H − ( β ) . (64)Therefore, if the feasible set of (64) is not empty, then the optimal solution is given by P f = H − ( β ) .A PPENDIX BD ERIVATION OF
P r ( E n |H ) Introducing Γ n = { a i < ζ ij < b i , i = 1 , ..., n − } and p n = n − e − b n / , we can write P r ( E n |H ) = Z ... Z Γ n Z ∞ b n n e − ζ nj / I { ≤ ζ j ≤ ζ j ... ≤ ζ nj } dζ j ...dζ nj = p n Z ... Z Γ n I { ≤ ζ j ≤ ζ j ... ≤ ζ n − j } dζ j ...dζ n − j . (65) September 3, 2018 DRAFT6
Denoting A ( n ) = R ... R Γ n I { ≤ ζ j ≤ ζ j ... ≤ ζ n − j } dζ j ...dζ n − j , we obtain A ( n ) = b b n − n ( n − , n = 1 , ..., p + 1 (cid:2) f ( n − a n − ( b n − ) − I { n ≥ } P n − i =0 ( b n − − b i +1 ) n − i − ( n − i − i e bi +12 P r ( E i +1 |H ) (cid:3) , n = p + 2 , ..., q + 1 (cid:2) f ( n − a n − ( b n − ) − P ni =0 f ( n − − i ) ψ n − i,an − ( b n − )2 i e bi +12 P r ( E i +1 |H ) (cid:3) , n = q + 2 , ..., N , (66)where a n − = [ a , . . . , a n − ] . Denoting q to be the smallest integer for which a q ≤ b < b q , and c and d to be two non-negative real numbers satisfying ≤ c < d , a n − ≤ c ≤ b n and a n ≤ d , η = 0 , η k = [ η , ..., η k ] , ≤ η ≤ ... ≤ η k , the functions f ( k ) η k ( ζ ) and the vector ψ ni,c in (66) are asfollows f ( k ) η k ( ζ ) = P k − i =0 f ( k ) i ( ζ − η i +1 ) k − i ( k − i )! + f ( k ) k f ( k ) i = f ( k − i , i = 0 , ..., k − , k ≥ , f ( k ) k = − P k − i =0 f ( k − i ( k − i )! ( η k − η i +1 ) k − i , f (0)0 = 1 , (67) ψ ni,c = [ b i +1 , ..., b i +1 | {z } q , a q + i +1 , ..., a n − , c | {z } n − q − i ] , i ∈ [0 , n − q − b i +1 , ..., b i +1 , c | {z } n − i ] , i ∈ [ n − q − , s − b i +1 n − i , i ∈ [ s, n − , (68)with s denoting the integer for which b s < c ≤ b s +1 and f (0) η k ( ζ ) = 1 .A PPENDIX CD ERIVATION OF
P r ( E n |H ) Introducing q n = γ j )] n − e − b n / γ j ) , we can write P r ( E n |H ) = Z ... Z Γ n Z ∞ b n γ j )] n e − ζ nj / γ j ) I { ≤ ζ j ≤ ζ j ... ≤ ζ nj } dζ j ...dζ nj = q n Z ... Z Γ n I { ≤ ζ j ≤ ζ j ... ≤ ζ n − j } dζ j ...dζ n − j . (69) September 3, 2018 DRAFT7
Denoting B ( n ) = R ... R Γ n I { ≤ ζ j ≤ ζ j ... ≤ ζ n − j } dζ j ...dζ n − j , and using the notations of Appendix B, we obtain B ( n ) = b b n − n ( n − , n = 1 , ..., p + 1 (cid:2) f ( n − a n − ( b n − ) − I { n ≥ } P n − i =0 ( b n − − b i +1 ) n − i − ( n − i − [2(1 + γ j )] i e bi +12(1+ γj ) P r ( E i +1 |H ) (cid:3) , n = p + 2 , ..., q + 1 (cid:2) f ( n − a n − ( b n − ) − P n − i =0 f ( n − − i ) ψ n − i,an − ( b n − )[2(1 + γ j )] i e bi +12(1+ γj ) P r ( E i +1 |H ) (cid:3) , n = q + 2 , ..., N . (70)A PPENDIX DA NALYTICAL EXPRESSION FOR J ( n ) a n ,b n ( θ ) Under θ > , n ≥ and ≤ ζ j ≤ ... ≤ ζ nj , ζ ij ∈ ( a i , b i ) , i = 1 , ..., n , the function J ( n ) a n ,b n ( θ ) isdefined as [19] J ( n ) a n ,b n ( θ ) = n X i =1 θ − i (cid:2) f ( n − i ) a n − i ( a n ) e − θa n − f ( n − i ) a n − i ( b n ) e − θb n (cid:3) − I { n ≥ } n − X k =0 g ( k ) a n ,b n ( θ ) , (71)where using the notations of Appendix B, we have [19] g ( k ) c,d = I ( k ) (cid:2) θ k − n e − θb k +1 − P n − ki =1 θ − i f ( n − k − i ) b k +1 n − k − i ( d ) e − θd (cid:3) , c ≤ b , k ∈ [0 , n − I ( k ) P n − ki =1 θ − i (cid:2) f ( n − k − i ) ψ n − ik,c ( c ) e − θc − f ( n − k − i ) ψ n − ik,d ( d ) e − θd (cid:3) , c > b , k ∈ [0 , s − I ( k ) (cid:2) θ k − n e − θb k +1 − P n − ki =1 θ − i f ( n − k − i ) b k +1 n − k − i ( d ) e − θd (cid:3) , c > b , k ∈ [ s, n − , (72)with I (0) = 1 and I ( n ) = f ( n ) a n ( b n ) − I { n ≥ } P n − i =0 ( b n − b i +1 ) n − i ( n − i )! I ( i ) , n ∈ [1 , q ] f ( n ) a n ( b n ) − P n − i =0 f ( n − i ) ψ ni,an ( b n ) I ( i ) , n ∈ [ q + 1 , ∞ ) . (73)A PPENDIX EP ROOF OF T HEOREM P f and P d are the respective given local probability of false alarm and detection. Denoting ρ c as the censoring rate for the optimal censoring scheme (64), we obtain − ρ c = π P f + π P d , anddenoting ρ cs as the censoring rate for the optimal censored truncated sequential sensing (26), based onwhat we have discussed in Section II, we obtain − ρ cs = π ( P f + L (¯ a, ¯ b )) + π ( P d + L (¯ a, ¯ b )) .Note that L k (¯ a, ¯ b ) , k = 0 , , represents the probability that ζ n ≤ a n , n = 1 , . . . , N under H k which isnon-negative. Hence, we can conclude that − ρ cs ≥ − ρ c and thus ρ c ≥ ρ cs . September 3, 2018 DRAFT8 A PPENDIX FO PTIMAL SOLUTION OF (55)Since the optimal λ → ∞ , (53) and (54) can be simplified to Q c F,AND = δ M and Q c D,AND = Q Mj =1 δ j and so (55) becomes, min λ max j (cid:2) N C sj + ( π (1 − δ ) + π (1 − δ j )) C tj (cid:3) s.t. δ M ≤ α, M Y j =1 δ j ≥ β. (74)Since there is a one-to-one relationship between λ and δ , i.e., λ = 2Γ − [ N, Γ( N ) δ ] (where Γ − is defined over the second argument), (74) can be formulated as [22, p.130], min δ max j (cid:2) N C sj + ( π (1 − δ ) + π (1 − δ j )) C tj (cid:3) s.t. δ M ≤ α, Q Mj =1 δ j ≥ β. (75)Defining δ = F AND ( λ ) = Γ( N, λ )Γ( N ) and δ j = G AND,j ( λ ) = Γ( N, λ γj ) )Γ( N ) , we can write δ j as δ j = G AND ,j ( F − ( δ )) . Calculating the derivative of C j with respect to δ , we find that ∂C j ∂δ = ∂ (cid:2) C tj ( π (1 − δ ) + π (1 − δ j )) (cid:3) ∂δ = − C tj π + ∂ (1 − δ j ) ∂δ ≤ , (76)where we use the fact that ∂δ j ∂δ = − N Γ( N ) − [ N, Γ( N ) δ ] N − e − [ N, Γ( N ) δ ] / γ j ) I { − [ N, Γ( N ) δ ] ≥ } − N Γ( N ) − [ N, Γ( N ) δ ] N − e − [ N, Γ( N ) δ ] / I { − [ N, Γ( N ) δ ] ≥ } = e − [ N, Γ( N ) δ ](1 / γ j ) − / ≥ . (77)Therefore, we can simplify (75) as max δ δ s.t. δ M ≤ α, Q Mj =1 δ j ≥ β. (78)Since Q c D,AND is a monotonically increasing function of δ , i.e., Q c D,AND = H AND ( δ ) = Q Mj =1 ( G AND ,j ( F − AND ( δ ))) and thus ∂Q c D,AND ∂δ = ∂Q c D,AND ∂δ j ∂δ j ∂δ = Q l = Ml =1 ,l = j ( δ l ) ∂δ j ∂δ ≥ , we can furthersimplify the constraints in (78) as δ ≤ α /M and δ j ≥ H − ( β ) . Thus, we obtain max δ δ s.t. δ ≤ α /M , δ j ≥ H − ( β ) . (79)Therefore, if the feasible set of (79) is not empty, then the optimal solution is given by δ = α /M ( β ) . September 3, 2018 DRAFT9 R EFERENCES [1] Q. Zhao and B. M. Sadler, “A Survey of Dynamic Spectrum Access,”
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Sina Maleki received his B.Sc. degree in electrical engineering from Iran University of Science andTechnology, Tehran, Iran, in 2006, and his M.S. degree in electrical engineering from Delft Universityof Technology, Delft, The Netherlands, in 2009. From July 2008 to April 2009, he was an intern studentat the Philips Research Center, Eindhoven, The Netherlands, working on spectrum sensing for cognitiveradio networks. He then joined the Circuits and Systems Group at the Delft University of Technology,where he is currently a Ph.D. student. He has served as a reviewer for several journals and conferences.
Geert Leus was born in Leuven, Belgium, in 1973. He received the electrical engineering degree and thePhD degree in applied sciences from the Katholieke Universiteit Leuven, Belgium, in June 1996 and May2000, respectively. He has been a Research Assistant and a Postdoctoral Fellow of the Fund for ScientificResearch - Flanders, Belgium, from October 1996 till September 2003. During that period, Geert Leuswas affiliated with the Electrical Engineering Department of the Katholieke Universiteit Leuven, Belgium.Currently, Geert Leus is an Associate Professor at the Faculty of Electrical Engineering, Mathematicsand Computer Science of the Delft University of Technology, The Netherlands. His research interests are in the area of signalprocessing for communications. Geert Leus received a 2002 IEEE Signal Processing Society Young Author Best Paper Award anda 2005 IEEE Signal Processing Society Best Paper Award. He was the Chair of the IEEE Signal Processing for Communicationsand Networking Technical Committee, and an Associate Editor for the IEEE Transactions on Signal Processing, the IEEETransactions on Wireless Communications, and the IEEE Signal Processing Letters. Currently, he is a member of the IEEESensor Array and Multichannel Technical Committee and serves on the Editorial Board of the EURASIP Journal on Advancesin Signal Processing. Geert Leus has been elevated to IEEE Fellow.
September 3, 2018 DRAFT r X i v : . [ c s . S Y ] M a r Censored Truncated Sequential Spectrum Sensingfor Cognitive Radio Networks
Sina Maleki Geert Leus
Abstract —Reliable spectrum sensing is a key functionality of acognitive radio network. Cooperative spectrum sensing improvesthe detection reliability of a cognitive radio system but alsoincreases the system energy consumption which is a critical factorparticularly for low-power wireless technologies. A censoredtruncated sequential spectrum sensing technique is consideredas an energy-saving approach. To design the underlying sensingparameters, the maximum average energy consumption persensor is minimized subject to a lower bounded global probabilityof detection and an upper bounded false alarm rate. This wayboth the interference to the primary user due to miss detectionand the network throughput as a result of a low false alarmrate are controlled. To solve this problem, it is assumed that thecognitive radios and fusion center are aware of their location andmutual channel properties. We compare the performance of theproposed scheme with a fixed sample size censoring scheme underdifferent scenarios and show that for low-power cognitive radios,censored truncated sequential sensing outperforms censoring. Itis shown that as the sensing energy per sample of the cognitiveradios increases, the energy efficiency of the censored truncatedsequential approach grows significantly.
Index Terms —distributed spectrum sensing, sequential sensing,cognitive radio networks, censoring, energy efficiency.
I. I
NTRODUCTION
Dynamic spectrum access based on cognitive radios hasbeen proposed in order to opportunistically use underutilizedspectrum portions of the licensed electromagnetic spectrum[1]. Cognitive radios opportunistically share the spectrumwhile avoiding any harmful interference to the primary li-censed users. They employ spectrum sensing to detect theempty portions of the radio spectrum, also known as spectrumholes. Upon detection of such a spectrum hole, cognitive radiosdynamically share this hole. However, as soon as a primaryuser appears in the corresponding band, the cognitive radioshave to vacate the band. As such, reliable spectrum sensingbecomes a key functionality of a cognitive radio network.The hidden terminal problem and fading effects have beenshown to limit the reliability of spectrum sensing. Distributedcooperative detection has therefore been proposed to improvethe detection performance of a cognitive radio network [2],[3]. Due to its simplicity and small delay, a parallel detectionconfiguration [4], is considered in this paper where eachsecondary radio continuously senses the spectrum in periodicsensing slots. A local decision is then made at the radios and
S. Maleki and G. Leus are with the Faculty of Electrical Engineering,Mathematics and Computer Science, Delft University of Technology, 2628CD Delft, The Netherlands (e-mail: [email protected]; [email protected]).Part of this paper has been presented at the 17th International Conference onDigital Signal Processing, DSP 2011, July 2011, Corfu, Greece. This work issupported in part by the NWO-STW under the VICI program (project 10382).Manuscript received date: Jan 5, 2012. Manuscript revised dates: May 16,1012 and Jul 19, 2012 sent to the fusion center (FC), which makes a global decisionabout the presence (or absence) of the primary user and feedsit back to the cognitive radios. Several fusion schemes havebeen proposed in the literature which can be categorized undersoft and hard fusion strategies [4], [5]. Hard schemes are moreenergy efficient than soft schemes, and thus a hard fusionscheme is adopted in this paper. More specifically, two popularchoices are employed due to their simple implementation: theOR and the AND rule. The OR rule dictates the primaryuser presence to be announced by the FC when at least onecognitive radio reports the presence of a primary user to theFC. On the other hand, the AND rule asks the FC to votefor the absence of the primary user if at least one cognitiveradio announces the absence of the primary user. In this paper,energy detection is employed for channel sensing which isa common approach to detect unknown signals [5], [6], andwhich leads to a comparable detection performance for hardand soft fusion schemes [3].Energy consumption is another critical issue. The maximumenergy consumption of a low-power radio is limited by itsbattery. As a result, energy efficient spectrum sensing limitingthe maximum energy consumption of a cognitive radio in acooperative sensing framework is the focus of this paper.
A. Contributions
The spectrum sensing module consumes energy in boththe sensing and transmission stages. To achieve an energy-efficient spectrum sensing scheme the following contributionsare presented in this paper. • A combination of censoring and truncated sequentialsensing is proposed to save energy. The sensors sequen-tially sense the spectrum before reaching a truncationpoint, N , where they are forced to stop sensing. If theaccumulated energy of the collected sample observationsis in a certain region (above an upper threshold, a , orbelow a lower threshold, b ) before the truncation point,a decision is sent to the FC. Else, a censoring policy isused by the sensor, and no bits will be sent. This way,a large amount of energy is saved for both sensing andtransmission. In our paper, it is assumed that the cognitiveradios and fusion center are aware of their location andmutual channel properties. • Our goal is to minimize the maximum average energyconsumption per sensor subject to a specific detectionperformance constraint which is defined by a lowerbound on the global probability of detection and anupper bound on the global probability of false alarm. Interms of cognitive radio system design, the probability of detection limits the harmful interference to the primaryuser and the false alarm rate controls the loss in spectrumutilization. The ideal case yields no interference andfull spectrum utilization, but it is practically impossibleto reach this point. Hence, current standards determinea bound on the detection performance to achieve anacceptable interference and utilization level [7]. To thebest of our knowledge such a min-max optimizationproblem considering the average energy consumption persensor has not yet been considered in literature. • Analytical expressions for the underlying parameters arederived and it is shown that the problem can be solvedby a two-dimensional search for both the OR and ANDrule. • To reduce the computational complexity for the OR rule,a single-threshold truncated sequential test is proposedwhere each cognitive radio sends a decision to the FCupon the detection of the primary user. • To make a fair comparison of the proposed techniquewith current energy efficient approaches, a fixed samplesize censoring scheme is considered as a benchmark (itis simply called the censoring scheme throughout the restof the paper) where each sensor employs a censoringpolicy after collecting a fixed number of samples. Thecensoring policy in this case works based on a lowerthreshold, λ and an upper threshold, λ . The decisionis only being made if the accumulated energy is not in ( λ , λ ) . For this approach, it is shown that a single-threshold censoring policy is optimal in terms of energyconsumption for both the OR and AND rule. Moreover,a solution of the underlying problem is given for the ORand AND rule. B. Related work to censoring
Censoring has been thoroughly investigated in wireless sen-sor networks and cognitive radios [8]–[13]. It has been shownthat censoring is very effective in terms of energy efficiency. Inthe early works, [8]–[11], the design of censoring parametersincluding lower and upper thresholds has been considered andmainly two problem formulations have been studied. In theNeyman-Pearson (NP) case, the miss-detection probability isminimized subject to a constraint on the probability of falsealarm and average network energy consumption [9]–[11]. Inthe Bayesian case, on the other hand, the detection errorprobability is minimized subject to a constraint on the averagenetwork energy consumption. Censoring for cognitive radios isconsidered in [12], [13]. In [12], a censoring rule similar to theone in this paper is considered in order to limit the bandwidthoccupancy of the cognitive radio network. Our fixed samplesize censoring scheme is different in two ways. First, in [12],only the OR rule is considered and the FC makes no decisionin case it does not receive any decision from the cognitiveradios which is ambiguous, since the FC has to make a finaldecision, while in our paper, the FC reports the absence (forthe OR rule) or the presence (for the AND rule) of the primaryuser, if no local decision is received at the FC. Second, we givea clear optimization problem and expression for the solution while this is not presented in [12]. A combined sleeping andcensoring scheme is considered in [13]. The censoring schemein this paper is different in some ways. The optimizationproblem in the current paper is defined as the minimization ofthe maximum average energy consumption per sensor whilein [13], the total network energy consumption is minimized.For low-power radios, the problem in this paper makes moresense since the energy of individual radios is generally limited.In this paper, the received SNRs by the cognitive radios areassumed to be different while in [13], the SNRs are the same.Finally note that the sleeping policy of [13] can be easilyincorporated in our proposed censored truncated sequentialsensing leading to even higher energy savings.
C. Related work to sequential sensing
Sequential detection as an approach to reduce the averagenumber of sensors required to reach a decision is also studiedcomprehensively during the past decades [14]–[19]. In [14],[15], each sensor collects a sequence of observations, con-structs a summary message and passes it on to the FC andall other sensors. A Bayesian problem formulation comprisingthe minimization of the average error detection probabilityand sampling time cost over all admissible decision policiesat the FC and all possible local decision functions at eachsensor is then considered to determine the optimal stoppingand decision rule. Further, algorithms to solve the optimizationproblem for both infinite and finite horizon are given. In [16],an infinite horizon sequential detection scheme based on thesequential probability ratio test (SPRT) at both the sensors andthe FC is considered. Wald’s analysis of error probability, [20],is employed to determine the thresholds at the sensors andthe FC. A combination of sequential detection and censoringis considered in [17]. Each sensor computes the LLR of thereceived sample and sends it to the FC, if it is deemed to bein a certain region. The FC then collects the received LLRsand as soon as their sum is larger than an upper threshold orsmaller than a lower threshold, the decision is made and thesensors can stop sensing. The LLRs are transmitted in such away that the larger LLRs are sent sooner. It is shown that thenumber of transmissions considerably reduces and particularlywhen the transmission energy is high, this approach performsvery well. However, our paper employs a hard fusion schemeat the FC, our sequential scheme is finite horizon, and furthera clear optimization problem is given to optimize the energyconsumption. Since we employ the OR (or the AND) rulein our paper, the FC can decide for the presence (or absence)of the primary user by only receiving a single one (or zero).Hence, ordered transmission can be easily incorporated in ourpaper by stopping the sensing and transmission procedure assoon as one cognitive radio sends a one (or zero) to the FC.[18] proposes a sequential censoring scheme where an SPRTis employed by the FC and soft or hard local decisions are sentto the FC according to a censoring policy. It is depicted thatthe number of transmissions decreases but on the other handthe average sample number (ASN) increases. Therefore, [18]ignores the effect of sensing on the energy consumption andfocuses only on the transmission energy which for current low-power radios is comparable to the sensing energy. A truncated sequential sensing technique is employed in [19] to reducethe sensing time of a cognitive radio system. The thresholdsare determined such that a certain probability of false alarmand detection are obtained. In this paper, we are employing asimilar technique, except that in [19], after the truncation point,a single threshold scheme is used to make a final decision,while in our paper, the sensor decision is censored if nodecision is made before the truncation point. Further, [19]considers a single sensor detection scheme while we employ adistributed cooperative sensing system and finally, in our paperan explicit optimization problem is given to find the sensingparameters.The remainder of the paper is organized as follows. InSection II, the fixed size censoring scheme for the OR ruleis described, including the optimization problem and thealgorithm to solve it. The sequential censoring scheme for theOR rule is presented in Section III. Analytical expressionsfor the underlying system parameters are derived and theoptimization problem is analyzed. In Section IV, the censoringand sequential censoring schemes are presented and analyzedfor the AND rule. We discuss some numerical results inSection V. Conclusions and ideas for further work are finallyposed in Section VI.II. F
IXED S IZE C ENSORING P ROBLEM F ORMULATION
A fixed size censoring scheme is discussed in this section asa benchmark for the main contribution of the paper in SectionIII, which studies a combination of sequential sensing andcensoring. A network of M cognitive radios is consideredunder a cooperative spectrum sensing scheme. A paralleldetection configuration is employed as shown in Fig. 1. Eachcognitive radio senses the spectrum and makes a local decisionabout the presence or absence of the primary user and informsthe FC by employing a censoring policy. The final decision isthen made at the FC by employing the OR rule. The AND rulewill be discussed in Section IV. Denoting r ij to be the i -thsample received at the j -th cognitive radio, each radio solvesa binary hypothesis testing problem as follows H : r ij = w ij , i = 1 , ..., N, j = 1 , ..., M H : r ij = h ij s i + w ij , i = 1 , ..., N, j = 1 , ..., M (1)where w ij is additive white Gaussian noise with zero meanand variance σ w . h ij and s i are the channel gain between theprimary user and the j -th cognitive radio and the transmittedprimary user signal, respectively. We assume two modelsfor h ij and s i . In the first model, s i is assumed to bewhite Gaussian with zero mean and variance σ s , and h ij is assumed constant during each sensing period and thus h ij = h j , i = 1 , . . . , N . In the second model, s i is assumedto be deterministic and constant modulus | s i | = s, i =1 , . . . , N, j = 1 , . . . , M and h ij is an i.i.d. Gaussian randomprocess with zero mean and variance σ hj . Note that the secondmodel actually represents a fast fading scenario. Although eachmodel requires a different type of channel estimation, since thereceived signal is still a zero mean Gaussian random processwith some variance, namely σ j = h j σ s + σ w for the formermodel and σ j = sσ hj + σ w for the latter model, the analyses (FC) ... Cognitive Radio 1Cognitive Radio 2Cognitive Radio M FusionCenter ...
Fig. 1: Distributed spectrum sensing configurationwhich are given in the following sections are valid for bothmodels. The SNR of the received primary user signal at the j -th cognitive radio is γ j = | h j | σ s /σ w under the first modeland γ j = s σ hj /σ w under the second model. Furthermore, h ij s i and w ij are assumed statistically independent.An energy detector is employed by each cognitive sensorwhich calculates the accumulated energy over N observationsamples. Note that under our system model parameters, theenergy detector is equivalent to the optimal LLR detector [5].The received energy collected over the N observation samplesat the j -th radio is given by E j = N X i =1 | r ij | σ w . (2)When the accumulated energy of the observation samples iscalculated, a censoring policy is employed at each radio wherethe local decisions are sent to the FC only if they are deemedto be informative [13]. Censoring thresholds λ and λ areapplied at each of the radios, where the range λ < E j < λ is called the censoring region. At the j -th radio, the localcensoring decision rule is given by send 1, declaring H if E j ≥ λ , no decision if λ < E j < λ , send 0, declaring H if E j ≤ λ . (3)It is well known [5] that under such a model, E j followsa central chi-square distribution with N degrees of freedomunder H and H . Therefore, the local probabilities of falsealarm and detection can be respectively written as P fj = P r ( E j ≥ λ |H ) = Γ( N, λ )Γ( N ) , (4) P dj = P r ( E j ≥ λ |H ) = Γ( N, λ γ j ) )Γ( N ) , (5)where Γ( a, x ) is the incomplete gamma function given by Γ( a, x ) = R ∞ x t a − e − t dt , with Γ( a,
0) = Γ( a ) .Denoting C sj and C ti to be the energy consumed bythe j -th radio in sensing per sample and transmission perbit, respectively, the average energy consumed for distributedsensing per user is given by, C j = N C sj + (1 − ρ j ) C tj , (6)where ρ j = P r ( λ < E j < λ ) is denoted to be the averagecensoring rate. Note that C sj is fixed and only depends on the sampling rate and power consumption of the sensing modulewhile C tj depends on the distance to the FC at the time of thetransmission. Therefore, in this paper, it is assumed that thecognitive radio is aware of its location and the location of theFC as well as their mutual channel properties or at least canestimate them. Defining π = P r ( H ) , π = P r ( H ) , δ j = P r ( λ < E j < λ |H ) and δ j = P r ( λ < E j < λ |H ) , ρ j is given by ρ j = π δ j + π δ j , (7)with δ j = Γ( N, λ )Γ( N ) − Γ( N, λ )Γ( N ) , (8) δ j = Γ( N, λ γ j ) )Γ( N ) − Γ( N, λ γ j ) )Γ( N ) . (9)Denoting Q c F and Q c D to be the respective global probabilityof false alarm and detection, the target detection performanceis then quantified by Q c F ≤ α and Q c D ≥ β , where α and β are pre-specified detection design parameters. Our goal is todetermine the optimum censoring thresholds λ and λ suchthat the maximum average energy consumption per sensor, i.e., max j C j , is minimized subject to the constraints Q c F ≤ α and Q c D ≥ β . Hence, our optimization problem can be formulatedas min λ ,λ max j C j s.t. Q c F ≤ α, Q c D ≥ β. (10)In this section, the FC employs an OR rule to make the finaldecision which is denoted by D F C , i.e., D F C = 1 if the FCreceives at least one local decision declaring 1, else D F C = 0 .This way, the global probability of false alarm and detectioncan be derived as Q c F = P r ( D F C = 1 |H ) = 1 − M Y j =1 (1 − P fj ) , (11) Q c D = P r ( D F C = 1 |H ) = 1 − M Y j =1 (1 − P dj ) . (12)Note that since all the cognitive radios employ the same upperthreshold λ , we can state that P fj = P f defined in (4). As aresult, (11) becomes Q c F = 1 − (1 − P f ) M . (13)Since the FC decides about the presence of the primaryuser only by receiving 1s (receiving no decision from all thesensors is considered as absence of the primary user) and thesensing time does not depend on λ , it is a waste of energyto send zeros to the FC and thus, the optimal solution of (10)is obtained by λ = 0 . Note that this is only the case forfixed-size censoring, because the energy consumption of eachsensor only varies by the transmission energy while the sensingenergy is constant. This way (8) and (9) can be simplifiedto δ j = 1 − P f and δ j = 1 − P dj , and we only need toderive the optimal λ . Since there is a one-to-one relationshipbetween P f and λ , by finding the optimal P f , λ can also be easily derived as λ = 2Γ − [ N, Γ( N ) P f ] (where Γ − isdefined over the second argument). Considering this result anddefining Q c D = H ( P f ) , the optimal solution of (10) is givenby P f = H − ( β ) as is shown in Appendix A.In the following section, a combination of censoring andsequential sensing approaches is presented which optimizesboth the sensing and the transmission energy.III. S EQUENTIAL C ENSORING P ROBLEM F ORMULATION
A. System Model
Unlike Section II, where each user collects a specificnumber of samples, in this section, each cognitive radiosequentially senses the spectrum and upon reaching a decisionabout the presence or absence of the primary user, it sends theresult to the FC by employing a censoring policy as introducedin Section II. The final decision is then made at the FC byemploying the OR rule. Here, a censored truncated sequentialsensing scheme is employed where each cognitive radio carrieson sensing until it reaches a decision while not passing a limitof N samples. We define ζ nj = P ni =1 | r ij | /σ w = P ni =1 x ij and a i = 0 , i = 1 , . . . , p , a i = ¯ a + i ¯Λ , i = p + 1 , ..., N and b i = ¯ b + i ¯Λ , i = 1 , ..., N , where ¯ a = a/σ w , ¯ b = b/σ w , < ¯Λ < γ j is a predetermined constant, a < , b > and p = ⌊− a/σ w ¯Λ ⌋ [19]. We assume that the SNR γ j is knownor can be estimated. This way, the local decision rule in orderto make a final decision is as follows send 1, declaring H if ζ nj ≥ b n and n ∈ [1 , N ] , continue sensing if ζ nj ∈ ( a n , b n ) and n ∈ [1 , N ) , no decision if ζ nj ∈ ( a n , b n ) and n = N, send 0, declaring H if ζ nj ≤ a n and n ∈ [1 , N ] . (14)The probability density function of x ij = | r ij | /σ w under H and H is a chi-square distribution with n degrees offreedom. Thus, x ij becomes exponentially distributed underboth H and H . Henceforth, we obtain P r ( x ij |H ) = 12 e − x ij / I { x ij ≥ } , (15) P r ( x ij |H ) = 12(1 + γ j ) e − x ij / γ j ) I { x ij ≥ } , (16)where I { x ij ≥ } is the indicator function.Defining ζ j = 0 , the local probability of false alarm at the j -th cognitive radio, P fj , can be written as P fj = N X n =1 P r ( ζ j ∈ ( a , b ) , ..., ζ n − j ∈ ( a n − , b n − ) , ζ nj ≥ b n |H ) , (17)whereas the local probability of detection, P dj , is obtained asfollows P dj = N X n =1 P r ( ζ j ∈ ( a , b ) , ..., ζ n − j ∈ ( a n − , b n − ) , ζ nj ≥ b n |H ) . (18)Denoting ρ j to be the average censoring rate at the j -thcognitive radio, and δ j and δ j to be the respective averagecensoring rate under H and H , we have ρ j = π δ j + π δ j , (19) where δ j = P r ( ζ j ∈ ( a , b ) , ..., ζ Nj ∈ ( a N , b N ) |H ) , (20) δ j = P r ( ζ j ∈ ( a , b ) , ..., ζ Nj ∈ ( a N , b N ) |H ) . (21)The other parameter that is important in any sequential de-tection scheme is the average sample number (ASN) requiredto reach a decision. Denoting N j to be a random variablerepresenting the number of samples required to announce thepresence or absence of the primary user, the ASN for the j -thcognitive radio, denoted as ¯ N j = E ( N j ) , can be defined as ¯ N j = π E ( N j |H ) + π E ( N j |H ) , (22)where E ( N j |H ) = N X n =1 nP r ( N j = n |H )= N − X n =1 n [ P r ( ζ j ∈ ( a , b ) , ..., ζ n − j ∈ ( a n − , b n − ) |H ) − P r ( ζ j ∈ ( a , b ) , ..., ζ nj ∈ ( a n , b n ) |H )]+ N P r ( ζ j ∈ ( a , b ) , ..., ζ N − j ∈ ( a N − , b N − ) |H ) , (23)and E ( N j |H ) = N X n =1 nP r ( N j = n |H )= N − X n =1 n [ P r ( ζ j ∈ ( a , b ) , ..., ζ nj ∈ ( a n − , b n − ) |H ) − P r ( ζ j ∈ ( a , b ) , ..., ζ nj ∈ ( a n , b n ) |H )]+ N P r ( ζ j ∈ ( a , b ) , ..., ζ N − j ∈ ( a N − , b N − ) |H ) . (24)Denoting again C sj to be the sensing energy of one sampleand C tj to be the transmission energy of a decision bit at the j -th cognitive radio, the total average energy consumption atthe j -th cognitive radio now becomes C j = ¯ N j C sj + (1 − ρ j ) C tj . (25)Denoting Q cs F and Q cs D to be the respective global probabil-ities of false alarm and detection for the censored truncatedsequential approach, we define our problem as the minimiza-tion of the maximum average energy consumption per sensorsubject to a constraint on the global probabilities of false alarmand detection as follows min ¯ a, ¯ b max j C j s.t. Q cs F ≤ α, Q cs D ≥ β. (26)As in (11) and (12), under the OR rule that is assumed inthis section, the global probability of false alarm is Q cs F = P r ( D FC = 1 |H ) = 1 − M Y j =1 (1 − P fj ) , (27)and the global probability of detection is Q cs D = P r ( D FC = 1 |H ) = 1 − M Y j =1 (1 − P dj ) . (28) Note that since P f = · · · = P fM , it is again assumed that P fj = P f in this section.In the following subsection, analytical expressions for theprobability of false alarm and detection as well as the censor-ing rate and ASN are extracted. B. Parameter and Problem Analysis
Looking at (17), (18), (19) and (22), we can see that thejoint probability distribution function of p ( ζ j , ..., ζ nj ) is thefoundation of all the equations. Since x ij = ζ ij − ζ i − j for i = 1 , ..., N , we have, p ( ζ j , ..., ζ nj ) = p ( x nj ) p ( x n − j ) ...p ( x j ) . (29)Therefore, the joint probability distribution function under H and H becomes p ( ζ j , ..., ζ nj |H ) = 12 n e − ζ nj / I { ≤ ζ j ≤ ζ j ... ≤ ζ nj } , (30) p ( ζ j , ..., ζ nj |H ) = 1[2(1 + γ j )] n e − ζ nj / γ j ) I { ≤ ζ j ≤ ζ j ... ≤ ζ nj } , (31)where I { ≤ ζ j ≤ ζ j ... ≤ ζ nj } is again the indicator function.The derivation of the local probability of false alarm and theASN under H in this work are similar to the ones consideredin [19] and [21]. The difference is that in [19], if the cognitiveradio does not reach a decision after N samples, it employsa single threshold decision policy to give a final decisionabout the presence or absence of the cognitive radio, whilein our work, no decision is sent in case none of the upper andlower thresholds are crossed. Hence, to avoid introducing acumbersome detailed derivation of each parameter, we can usethe results in [19] for our analysis with a small modification.However, note that the problem formulation in this work isessentially different from the one in [19]. Further, since in ourwork the distribution of x ij under H is exponential like theone under H , unlike [19], we can also use the same approachto derive analytical expressions for the local probability ofdetection, the ASN under H , and the censoring rate.Denoting E n to be the event where a i < ζ ij < b i , i =1 , ..., n − and ζ nj ≥ b n , (17) becomes P fj = N X n =1 P r ( E n |H ) . (32)where the analytical expression for P r ( E n |H ) is derived inAppendix B.Similarly for the local probability of detection, we have P dj = N X n =1 P r ( E n |H ) , (33)where the analytical expression for P r ( E n |H ) is derived inAppendix C.Defining R nj = { ζ ij | ζ ij ∈ ( a i , b i ) , i = 1 , ..., n } , P r ( R nj |H ) and P r ( R nj |H ) are obtained as follows P r ( R nj |H ) = 12 n J ( n ) a n ,b n (1 / , n = 1 , ..., N, (34) P r ( R nj |H ) = 1[2(1 + γ j )] n J ( n ) a n ,b n (1 / γ j )) , n = 1 , ..., N, (35) where J ( n ) a n ,b n ( θ ) is presented in Appendix D and (23) and (24)become E ( N j |H ) = N − X n =1 n ( P r ( R n − j |H ) − P r ( R nj |H ))+ N P r ( R N − j |H ) = 1+ N − X n =1 P r ( R nj |H ) , (36) E ( N j |H ) = N X n =1 n ( P r ( R n − j |H ) − P r ( R nj |H ))+ N P r ( R N − j |H ) = 1+ N − X n =1 P r ( R nj |H ) . (37)With (36) and (37), we can calculate (22). This way, (20)and (21) can be derived as follows δ j = P r ( R Nj |H ) = 12 N J ( N ) a N ,b N (1 / , (38) δ j = P r ( R Nj |H ) = 1[2(1 + γ j )] N J ( N ) a N ,b N (1 / γ j )) . (39)We can show that the problem (26) is not convex. Therefore,the standard systematic optimization algorithms do not givethe global optimum for ¯ a and ¯ b . However, as is shown in thefollowing lines, ¯ a and ¯ b are bounded and therefore, a two-dimensional exhaustive search is possible to find the globaloptimum. First of all, we have a < and ¯ a < . On the otherhand, if ¯ a has to play a role in the sensing system, at least one a N should be positive, i.e., a N = ¯ a + N ∆ ≥ which gives ¯ a ≥ − N ∆ . Hence, we obtain − N ∆ ≤ ¯ a < . Furthermore,defining Q cs F = F (¯ a, ¯ b ) and Q cs D = G (¯ a, ¯ b ) , for a given ¯ a , itis easy to show that G − (¯ a, β ) ≤ ¯ b ≤ F − (¯ a, α ) (where F − and G − are defined over the second argument).Before introducing a suboptimal problem, the followingtheorem is presented. Theorem 1 . For a given local probability of detection andfalse alarm ( P d and P f ) and N , the censoring rate of theoptimal censored truncated sequential sensing ( ρ cs ) is less thanthe one of the censoring scheme ( ρ c ). Proof . The proof is provided in Appendix E.We should note that, in censored truncated sequential sens-ing, a large amount of energy is to be saved on sensing.Therefore, as is shown in Section V, as the sensing energyof each sensor increases, censored truncated sequential sensingoutperforms censoring in terms of energy efficiency. However,in case that the transmission energy is much higher than thesensing energy, it may happen that censoring outperformscensored truncated sequential sensing, because of a highercensoring rate ( ρ cs > ρ c ). Hence, one corollary of Theorem 1is that although the optimal solution of (10) for a specific N , i.e., P d = 1 − (1 − β ) /M and P f = H − ( β ) , is in thefeasible set of (26) for a resulting ASN less than N , it doesnot necessarily guarantee that the resulting average energyconsumption per sensor of the censored truncated sequentialsensing approach is less than the one of the censoring scheme,particularly when the transmission energy is much higher thanthe sensing energy per sample.Solving (26) is complex in terms of the number of compu-tations, and thus a two-dimensional exhaustive search is notalways a good solution. Therefore, in order to reach a goodsolution in a reasonable time, we set a < − N ∆ in order toobtain a = · · · = a N = 0 . This way, we can relax one of the arguments of (26) and only solve the following suboptimalproblem min ¯ b max j C j s.t. Q cs F ≤ α, Q cs D ≥ β. (40)Note that unlike Section II, here the zero lower thresholdis not necessarily optimal. The reason is that although themaximum censoring rate is achieved with the lowest ¯ a , theminimum ASN is achieved with the highest ¯ a , and thus thereis an inherent trade-off between a high censoring rate anda low ASN and a zero a i is not necessarily the optimalsolution. Since the analytical expressions provided earlier arevery complex, we now try to provide a new set of analyticalexpressions for different parameters based on the fact that a = · · · = a N = 0 .To find an analytical expression for P fj , we can derive A ( n ) for the new paradigm as follows A ( n ) = Z ... Z Γ n I { ≤ ζ j ≤ ζ j ... ≤ ζ n − j } dζ j ...dζ n − j . (41)Since ≤ ζ j ≤ ζ j ... ≤ ζ n − j and a = · · · = a N = 0 ,the lower bound for each integral is ζ i − and the upper boundis b i , where i = 1 , ..., n − . Thus we obtain A ( n ) = Z b ζ j Z b ζ j ... Z b n − ζ n − j dζ j dζ j ...dζ n − j , (42)which according to [21] is A ( n ) = b b n − n ( n − , n = 1 , ..., N. (43)Hence, we have P fj = N X n =1 p n A ( n ) , (44)and p n = e − bn/ n − . Similarly, for P dj , we obtain B ( n ) = Z b ζ j Z b ζ j ... Z b n − ζ n − j dζ j dζ j ...dζ n − j = b b n − n ( n − , n = 1 , ..., N, (45)and thus P dj = N X n =1 q n B ( n ) , (46)where q n = e − bn/ γj ) [2(1+ γ j )] n − . Furthermore, we note that for a = · · · = a N = 0 , A ( n ) = B ( n ) = b b n − n ( n − , n = 1 , ..., N .It is easy to see that R nj occurs under H , if no falsealarm happens until the n -th sample. Therefore, the analyticalexpression for P r ( R nj |H ) is given by P r ( R nj |H ) = 1 − n X i =1 p i A ( i ) , (47)and in the same way, for P r ( R nj |H ) , we obtain P r ( R nj |H ) = 1 − n X i =1 q i A ( i ) . (48) Putting (47) and (48) in (36) and (37), we obtain E ( N j |H ) = 1 + N − X n =1 (cid:26) − n X i =1 p i A ( i ) (cid:27) , (49) E ( N j |H ) = 1 + N − X n =1 (cid:26) − n X i =1 q i A ( i ) (cid:27) , (50)and inserting (49) and (50) in (22), we obtain ¯ N j = π N − X n =1 (cid:26) − n X i =1 p i A ( i ) (cid:27)! + π N − X n =1 (cid:26) − n X i =1 q i A ( i ) (cid:27)! . (51)Finally, from (47) and (48), the censoring rate can be easilyobtained as ρ j = π (cid:18) − N X i =1 p i A ( i ) (cid:19) + π (cid:18) − N X i =1 q i A ( i ) (cid:19) . (52)Having the analytical expressions for (40), we can easilyfind the optimal maximum average energy consumption persensor by a line search over ¯ b . Similar to the censoring problemformulation, here the sensing threshold is also bounded by Q cs F − ( α ) ≤ ¯ b ≤ Q cs D − ( β ) . As we will see in Section V,censored truncated sequential sensing performs better thancensored spectrum sensing in terms of energy efficiency forlow-power radios.IV. E XTENSION TO THE
AND
RULE
So far, we have mainly focused on the OR rule. However,another rule which is also simple in terms of implementationis the AND rule. According to the AND rule, D F C = 0 , if atleast one cognitive radio reports a zero, else D F C = 1 . Thisway the global probabilities of false alarm and detection, canbe written respectively as Q c F,AND = Q cs F,AND = P r ( D F C = 1 |H ) = M Y j =1 ( δ j + P fj ) , (53) Q c D,AND = Q cs D,AND = P r ( D F C = 1 |H ) = M Y j =1 ( δ j + P dj ) . (54)Note that (53) and (54) hold for both the sequential censoringand censoring schemes. Similar to the case for the OR rule, theproblem is defined so as to minimize the maximum averageenergy consumption per sensor subject to a lower bound onthe global probability of detection and an upper bound onthe global probability of false alarm. In the following twosubsections, we are going to analyze the problem for censoringand sequential censoring. A. AND rule for fixed-sample size censoring
The optimization problem for the censoring scheme consid-ering the AND rule at the FC, becomes min λ ,λ max j C j s.t. Q c F,AND ≤ α, Q c D,AND ≥ β. (55) where C j is defined in (6). Since the FC decides for theabsence of the primary user by receiving at least one zero andthe fact that the sensing energy per sample is constant, theoptimal upper threshold λ is λ → ∞ . This way, cognitiveradios censor all the results for which E j > λ , and as a result(53) and (54) become Q c F,AND = P r ( D F C = 1 |H ) = M Y j =1 δ j , (56) Q c D,AND = P r ( D F C = 1 |H ) = M Y j =1 δ j . (57)where δ j = P r ( E j > λ |H ) and δ j = P r ( E j > λ |H ) .Since the thresholds are the same among the cognitive radios,we have δ = δ = · · · = δ M = δ . Since there is a one-to-one relationship between λ and δ , by finding the optimal δ ,the optimal λ can be easily derived. As shown in Appendix F,we can derive the optimal δ as δ = α /M . This result is veryimportant in the sense that as far as the feasible set of (55) isnot empty, the optimal solution of (55) is independent fromthe SNR. Note that the maximum average energy consumptionper sensor still depends on the SNR via δ j and is reducingas the SNR grows. B. AND rule for censored truncated sequential sensing
The optimization problem for the censored truncated se-quential sensing scheme with the AND rule, becomes min ¯ a, ¯ b max j C j s.t. Q cs F,AND ≤ α, Q cs D,AND ≥ β. (58)where C j is defined in (25). Similar to the OR rule, we have − N ∆ ≤ ¯ a < . Defining Q cs F,AND = F AND (¯ a, ¯ b ) and Q cs D,AND = G AND (¯ a, ¯ b ) , for a given ¯ a , we can show that G − AND (¯ a, β ) ≤ ¯ b ≤ F − AND (¯ a, α ) (where F − AND and G − AND are defined over thesecond argument). Therefore, the optimal ¯ a and ¯ b can againbe derived by a bounded two-dimensional search, in a similarway as for the OR rule.V. N UMERICAL R ESULTS
A network of cognitive radios is considered for the numeri-cal results. In some of the scenarios, for the sake of simplicity,it is assumed that all the sensors experience the same SNR.This way, it is easier to show how the main performanceindicators including the optimal maximum average energyconsumption per sensor, ASN and censoring rate changeswhen one of the underlying parameter of the system changes.However, to comply with the general idea of the paper, whichis based on different received SNRs by cognitive radios,in other scenarios, the different cognitive radios experiencedifferent SNRs. Unless otherwise mentioned, the results arebased on the single-threshold strategy for censored truncatedsequential sensing in case of the OR rule.Fig. 2a depicts the optimal maximum average energy con-sumption per sensor versus the number of cognitive radiosfor the OR rule. The SNR is assumed to be dB, N = 10 , C s = 1 and C t = 10 . Furthermore, the probability of falsealarm and detection constraints are assumed to be α = 0 . and β = 0 . as determined by the IEEE 802.15.4 standard forcognitive radios [7]. It is shown for both high and low values of π that censored sequential sensing outperforms the censoringscheme. Looking at Fig. 2b and Fig. 2c, where the respectiveoptimal censoring rate and optimal ASN are shown versusthe number of cognitive radios, we can deduce that the lowerASN is playing a key role in a lower energy consumption ofthe censored sequential sensing. Fig. 2a also shows that as thenumber of cooperating cognitive radios increases, the optimalmaximum average energy consumption per sensor decreasesand saturates, while as shown in Fig. 2b and Fig. 2c, theoptimal censoring rate and optimal ASN increase. This way,the energy consumption tends to increase as a result of ASNgrowth and on the other hand inclines to decrease due to thecensoring rate growth and that is the reason for saturationafter a number of cognitive radios. Therefore, we can see thatas the number of cognitive radios increases, a higher energyefficiency per sensor can be achieved. However, after a numberof cognitive radios, the maximum average energy consumptionper sensor remains almost at a constant level and by addingmore cognitive radios no significant energy saving per sensorcan be achieved while the total network energy consumptionalso increases.Figures 3a, 3b and 3c consider a scenario where M = 5 , N = 30 , C sj = 1 , C tj = 10 , α = 0 . , β = 0 . and π cantake a value of . or . . The performance of the systemversus SNR is analyzed in this scenario for the OR rule. Themaximum average energy consumption per sensor is depictedin Fig. 3a. As for the earlier scenario, censored sequential sens-ing gives a higher energy efficiency compared to censoring.While the optimal energy variation for the censoring schemeis almost the same for all the considered SNRs, the censoredsequential scheme’s average energy consumption per sensorreduces significantly as the SNR increases. The reason is thatas the SNR increases, the optimal ASN dramatically decreases(almost for γ = 2 dB and π = 0 . ). This shows thatas the SNR increases, censored sequential sensing becomeseven more valuable and a significant energy saving per sensorcan be achieved compared with the one that is achieved bycensoring. Since the SNR changes with the channel gain ( | h j | under the first model or σ hj under the second model), fromFig. 3a, the behavior of the system with varying | h j | or σ hj can be derived, if the distribution of | h j | or σ hj is known.Figures 4a and 4b compare the performance of the singlethreshold censored truncated sequential scheme with the oneassuming two thresholds, i.e, ¯ a and ¯ b for the OR rule. Theidea is to find when the double threshold scheme with itshigher complexity becomes valuable. In these figures, M = 5 , N = 10 , γ = 0 dB, C t = 10 , π = 0 . , . , and α = 0 . ,while β changes from . to . . The sensing energy persample, C s in Fig. 4a is assumed , while in Fig. 4b it is .It is shown that as the sensing energy per sample increases,the energy efficiency of the double threshold scheme alsoincreases compared to the one of the single threshold scheme,particularly when π is high. The reason is that when π ishigh, a much lower ASN can be achieved by the double thresh- E ne r g y sequential censoring, π =0.8censoring, π =0.8sequential censoring, π =0.2censoring, π =0.2 (a) O p t i m a l c en s o r i ng r a t e sequential censoring, π =0.8censoring, π =0.8sequential censoring, π =0.2censoring, π =0.2 (b) O p t i m a l AS N π =0.8 π =0.2 (c) Fig. 2: a) Optimal maximum average energy consumption persensor versus number of cognitive radios, b) Optimal censoringrate versus number of cognitive radios, c) Optimal ASN versusnumber of cognitive radios for the OR rule −4 −3 −2 −1 0 1 222242628303234 SNR E ne r g y sequential censoring, π =0.8censoring, π =0.8sequential censoring, π =0.2censoring, π =0.2 (a) −4 −3 −2 −1 0 1 20.10.20.30.40.50.60.70.80.91 SNR O p t i m a l c en s o r i ng r a t e sequential censoring, π =0.8censoring, π =0.8sequential censoring, π =0.2censoring, π =0.2 (b) −4 −3 −2 −1 0 1 2141618202224262830 SNR AS N π =0.8 π =0.2 (c) Fig. 3: a) Optimal maximum average energy consumptionper sensor versus SNR, b) Optimal censoring rate versusSNR, c) Optimal ASN versus SNR for the OR rule β E ne r g y Double threshold, π =0.2Single threshold, π =0.2Double threshold, π =0.8Single threshold, π =0.8 (a) β E ne r g y Double threshold, π =0.2Single threshold, π =0.2Double threshold, π =0.8Single threshold, π =0.8 (b) Fig. 4: Optimal maximum average energy consumption persensor versus probability of detection constraint, β , for theOR rule, a) C s = 1 , b) C s = 3 old scheme compared to the single threshold one. This gainin performance comes at the cost of a higher computationalcomplexity because of the two-dimensional search.Fig. 5 depicts the optimal maximum average energy con-sumption per sensor versus the number of samples for the ORrule and for a network of M = 5 cognitive radios where eachradio experiences a different channel gain and thus a differentSNR. Arranging the SNRs in a vector γ = [ γ , . . . , γ ] , wehave γ = [1dB, 2dB, 3dB, 4dB, 5dB]. The other parametersare C s = 1 , C t = 10 , π = 0 . , α = 0 . and β = 0 . .As shown in Fig. 5, by increasing the number of samplesand thus the total sensing energy, the sequential censoringenergy efficiency also increases compared to the censoringscheme. For example, if we define the efficiency of thecensored truncated sequential sensing scheme as the differenceof the optimal maximum average energy consumption persensor of sequential censoring and censoring divided by theoptimal maximum average energy consumption per sensor of
15 20 25 30161820222426283032 Number of samples E ne r g y sequential censoring, π =0.5censoring, π =0.5 Fig. 5: Optimal maximum average energy consumption persensor versus number of samples for the OR rule t E ne r g y sequential censoring, π =0.5censoring, π =0.5 Fig. 6: Optimal maximum average energy consumption persensor versus transmission energy for the OR rulecensoring, the efficiency increases approximately three timesfrom 0.06 (for N = 15 ) to 0.19 (for N = 30 ).In Fig. 6, the sensing energy per sample is C s = 10 while the transmission energy C t changes from 0 to 1000.The goal is to see how the optimal maximum average energyconsumption per sensor changes with C t for the or rule andfor a network of M = 5 cognitive radios with γ = [1dB,2dB, 3dB, 4dB, 5dB]. The other parameters of the networkare N = 30 , π = 0 . , α = 0 . and β = 0 . . The best savingfor sequential censoring is achieved when the transmissionenergy is zero. Indeed, we can see that as the transmissionenergy increases the performance gain of sequential censoringreduces compared to censoring. However, in low-power radioswhere the sensing energy per sample and transmission energyare usually in the same range, sequential censoring performsmuch better than censoring in terms of energy efficiency aswe can see in Fig. 6.Fig. 7 depicts the optimal maximum average energy con-sumption per sensor versus the sensing energy per sample for s E ne r g y sequential censoring, π =0.5, ANDsequential censoring, π =0.5, OR Fig. 7: Optimal maximum average energy consumption persensor versus sensing energy per sample for AND and ORruleboth the AND and OR rule. For the sake of simplicity andtractability, the SNRs are assumed the same for M = 50 cognitive radios. The other parameters are assumed to be N = 10 , C t = 10 , π = 0 . , γ = 0 dB, α = 0 . and β = 0 . . For both fusion rules, the double threshold schemeis employed. We can see that the OR rule performs better forthe low values of C s . However, as C s increases the AND ruledominates and outperforms the OR rule, particularly for highvalues of C s . The reason that the OR rule performs better thanthe AND rule at very low values of C s is that the optimalcensoring rate for the OR rule is higher than the optimalcensoring rate for the AND rule. However as C s increases, theAND rule dominates the OR rule in terms of energy efficiencydue to the lower ASN.The optimal maximum average energy consumption persensor versus π is investigated in Fig. 8 for the AND and theOR rule. The underlying parameters are assumed to be C s = 2 , C t = 10 , N = 10 , M = 50 , γ = 0 dB, α = 0 . and β = 0 . .It is shown that as the probability of the primary user absenceincreases, the optimal maximum average energy consumptionper sensor reduces for the OR rule while it increases for theAND rule. This is mainly due to the fact that for the OR rule,we are mainly interested to receive a ”1” from the cognitiveradios. Therefore, as π increases, the probability of receivinga ”1” decreases, since the optimal censoring rate increases.The opposite happens for the AND rule, since for the ANDrule, receiving a ”0” from the cognitive radios is consideredto be informative.VI. S UMMARY AND C ONCLUSIONS
We presented two energy efficient techniques for a cognitivesensor network. First, a censoring scheme has been discussedwhere each sensor employs a censoring policy to reduce theenergy consumption. Then a censored truncated sequentialapproach has been proposed based on the combination ofcensoring and sequential sensing policies. We defined ourproblem as the minimization of the maximum average energy π E ne r g y sequential censoring, ANDsequential censoring, OR Fig. 8: Optimal maximum average energy consumption persensor versus π for AND and OR ruleconsumption per sensor subject to a global probability offalse alarm and detection constraint for the AND and theOR rules. The optimal lower threshold is shown to be zerofor the censoring scheme in case of the OR rule while forthe AND rule the optimal upper threshold is shown to beinfinity. Further, an explicit expression was given to find theoptimal solution for the OR rule and in case of the ANDrule a closed for solution is derived. We have further derivedthe analytical expressions for the underlying parameters in thecensored sequential scheme and have shown that although theproblem is not convex, a bounded two-dimensional search ispossible for both the OR rule and the AND rule. Further, incase of the OR rule, we relaxed the lower threshold to obtaina line search problem in order to reduce the computationalcomplexity.Different scenarios regarding transmission and sensing en-ergy per sample as well as SNR, number of cognitive radios,number of samples and detection performance constraints weresimulated for low and high values of π and for both theOR rule and the AND rule. It has been shown that under thepractical assumption of low-power radios, sequential censoringoutperforms censoring. We conclude that for high values ofthe sensing energy per sample, despite its high computationalcomplexity, the double threshold scheme developed for theOR rule becomes more attractive. Further, it is shown thatas the sensing energy per sample increases compared to thetransmission energy, the AND rule performs better than theOR rule, while for very low values of the sensing energy persample, the OR rule outperforms the AND rule.Note that a systematic solution for the censored sequentialproblem formulation was not given in this paper, and thus it isvaluable to investigate a better algorithm to solve the problem.We also did not consider a combination of the proposedscheme with sleeping as in [13], which can generate furtherenergy savings. Our analysis was based on the OR rule andthe AND rule, and thus extensions to other hard fusion rulescould be interesting. A PPENDIX AO PTIMAL SOLUTION OF (10)Since the optimal λ = 0 , (8) and (9) can be simplified to δ j = 1 − P f and δ j = 1 − P dj and so (10) becomes, min λ max j (cid:2) N C sj + ( π P f + π P dj ) C tj (cid:3) s.t. − (1 − P f ) M ≤ α, − M Y j =1 (1 − P dj ) ≥ β. (59)Since there is a one-to-one relationship between λ and P f ,i.e., λ = 2Γ − [ N, Γ( N ) P f ] (where Γ − is defined over thesecond argument), (59) can be formulated as [22, p.130], min P f max j (cid:2) N C sj + ( π P f + π P dj ) C tj (cid:3) s.t. − (1 − P f ) M ≤ α, − Q Mj =1 (1 − P dj ) ≥ β. (60)Defining P f = F ( λ ) = Γ( N, λ )Γ( N ) and P dj = G j ( λ ) = Γ( N, λ γj ) )Γ( N ) , we can write P dj as P dj = G j ( F − ( P f )) .Calculating the derivative of C j with respect to P f , we findthat ∂C j ∂P f = ∂ (cid:2) C tj ( π P f + π P dj ) (cid:3) ∂P f = C tj π + ∂P dj ∂P f ≥ , (61)where we use the fact that ∂P dj ∂P f = − N Γ( N ) − [ N, Γ( N ) P f ] N − e − [ N, Γ( N ) P f ] / γ j ) I { − [ N, Γ( N ) P f ] ≥ } − N Γ( N ) − [ N, Γ( N ) P f ] N − e − [ N, Γ( N ) P f ] / I { − [ N, Γ( N ) P f ] ≥ } = e − [ N, Γ( N ) P f ](1 / γ j ) − / ≥ . (62)Therefore, we can simplify (60) as min P f P f s.t. − (1 − P f ) M ≤ α, − Q Mj =1 (1 − P dj ) ≥ β. (63)which can be easily solved by a line search over P f . However,since Q c D is a monotonically increasing function of P f , i.e., Q c D = H ( P f ) = 1 − Q Mj =1 (1 − G j ( F − ( P f ))) and thus ∂Q c D ∂P f = ∂Q c D ∂P dj ∂P dj ∂P f = Q l = Ml =1 ,l = j (1 − P dl ) ∂P dj ∂P f ≥ , we can fur-ther simplify the constraints in (63) as P f ≤ − (1 − α ) /M and P f ≥ H − ( β ) . Thus, we obtain min P f P f s.t. P f ≤ − (1 − α ) /M , P f ≥ H − ( β ) . (64)Therefore, if the feasible set of (64) is not empty, then theoptimal solution is given by P f = H − ( β ) .A PPENDIX BD ERIVATION OF
P r ( E n |H ) Introducing Γ n = { a i < ζ ij < b i , i = 1 , ..., n − } and p n = n − e − b n / , we can write P r ( E n |H ) = Z ... Z Γ n Z ∞ b n n e − ζ nj / I { ≤ ζ j ≤ ζ j ... ≤ ζ nj } dζ j ...dζ nj = p n Z ... Z Γ n I { ≤ ζ j ≤ ζ j ... ≤ ζ n − j } dζ j ...dζ n − j . (65) Denoting A ( n ) = R ... R Γ n I { ≤ ζ j ≤ ζ j ... ≤ ζ n − j } dζ j ...dζ n − j , weobtain A ( n ) = b b n − n ( n − , n = 1 , ..., p + 1 (cid:2) f ( n − a n − ( b n − ) − I { n ≥ } P n − i =0 ( b n − − b i +1 ) n − i − ( n − i − i e bi +12 P r ( E i +1 |H ) (cid:3) , n = p + 2 , ..., q + 1 (cid:2) f ( n − a n − ( b n − ) − P ni =0 f ( n − − i ) ψ n − i,an − ( b n − )2 i e bi +12 P r ( E i +1 |H ) (cid:3) , n = q + 2 , ..., N , (66)where a n − = [ a , . . . , a n − ] . Denoting q to be the smallestinteger for which a q ≤ b < b q , and c and d to be two non-negative real numbers satisfying ≤ c < d , a n − ≤ c ≤ b n and a n ≤ d , η = 0 , η k = [ η , ..., η k ] , ≤ η ≤ ... ≤ η k , thefunctions f ( k ) η k ( ζ ) and the vector ψ ni,c in (66) are as follows f ( k ) η k ( ζ ) = P k − i =0 f ( k ) i ( ζ − η i +1 ) k − i ( k − i )! + f ( k ) k f ( k ) i = f ( k − i , i = 0 , ..., k − , k ≥ , f ( k ) k = − P k − i =0 f ( k − i ( k − i )! ( η k − η i +1 ) k − i , f (0)0 = 1 , (67) ψ ni,c = [ b i +1 , ..., b i +1 | {z } q , a q + i +1 , ..., a n − , c | {z } n − q − i ] , i ∈ [0 , n − q − b i +1 , ..., b i +1 , c | {z } n − i ] , i ∈ [ n − q − , s − b i +1 n − i , i ∈ [ s, n − , (68)with s denoting the integer for which b s < c ≤ b s +1 and f (0) η k ( ζ ) = 1 . A PPENDIX CD ERIVATION OF
P r ( E n |H ) Introducing q n = γ j )] n − e − b n / γ j ) , we can write P r ( E n |H ) = Z ... Z Γ n Z ∞ b n γ j )] n e − ζ nj / γ j ) I { ≤ ζ j ≤ ζ j ... ≤ ζ nj } dζ j ...dζ nj = q n Z ... Z Γ n I { ≤ ζ j ≤ ζ j ... ≤ ζ n − j } dζ j ...dζ n − j . (69)Denoting B ( n ) = R ... R Γ n I { ≤ ζ j ≤ ζ j ... ≤ ζ n − j } dζ j ...dζ n − j , andusing the notations of Appendix B, we obtain B ( n ) = b b n − n ( n − , n = 1 , ..., p + 1 (cid:2) f ( n − a n − ( b n − ) − I { n ≥ } P n − i =0 ( b n − − b i +1 ) n − i − ( n − i − [2(1 + γ j )] i e bi +12(1+ γj ) P r ( E i +1 |H ) (cid:3) , n = p + 2 , ..., q + 1 (cid:2) f ( n − a n − ( b n − ) − P n − i =0 f ( n − − i ) ψ n − i,an − ( b n − )[2(1 + γ j )] i e bi +12(1+ γj ) P r ( E i +1 |H ) (cid:3) , n = q + 2 , ..., N . (70)A PPENDIX DA NALYTICAL EXPRESSION FOR J ( n ) a n ,b n ( θ ) Under θ > , n ≥ and ≤ ζ j ≤ ... ≤ ζ nj , ζ ij ∈ ( a i , b i ) , i = 1 , ..., n , the function J ( n ) a n ,b n ( θ ) is defined as [19] J ( n ) a n ,b n ( θ ) = n X i =1 θ − i (cid:2) f ( n − i ) a n − i ( a n ) e − θa n − f ( n − i ) a n − i ( b n ) e − θb n (cid:3) − I { n ≥ } n − X k =0 g ( k ) a n ,b n ( θ ) , (71) where using the notations of Appendix B, we have [19] g ( k ) c,d = I ( k ) (cid:2) θ k − n e − θb k +1 − P n − ki =1 θ − i f ( n − k − i ) b k +1 n − k − i ( d ) e − θd (cid:3) , c ≤ b , k ∈ [0 , n − I ( k ) P n − ki =1 θ − i (cid:2) f ( n − k − i ) ψ n − ik,c ( c ) e − θc − f ( n − k − i ) ψ n − ik,d ( d ) e − θd (cid:3) , c > b , k ∈ [0 , s − I ( k ) (cid:2) θ k − n e − θb k +1 − P n − ki =1 θ − i f ( n − k − i ) b k +1 n − k − i ( d ) e − θd (cid:3) , c > b , k ∈ [ s, n − , (72)with I (0) = 1 and I ( n ) = ( f ( n ) a n ( b n ) − I { n ≥ } P n − i =0 ( b n − b i +1 ) n − i ( n − i )! I ( i ) , n ∈ [1 , q ] f ( n ) a n ( b n ) − P n − i =0 f ( n − i ) ψ ni,an ( b n ) I ( i ) , n ∈ [ q + 1 , ∞ ) . (73)A PPENDIX EP ROOF OF T HEOREM P f and P d are the respective given localprobability of false alarm and detection. Denoting ρ c as thecensoring rate for the optimal censoring scheme (64), weobtain − ρ c = π P f + π P d , and denoting ρ cs as thecensoring rate for the optimal censored truncated sequentialsensing (26), based on what we have discussed in Section II,we obtain − ρ cs = π ( P f + L (¯ a, ¯ b )) + π ( P d + L (¯ a, ¯ b )) .Note that L k (¯ a, ¯ b ) , k = 0 , , represents the probability that ζ n ≤ a n , n = 1 , . . . , N under H k which is non-negative.Hence, we can conclude that − ρ cs ≥ − ρ c and thus ρ c ≥ ρ cs . A PPENDIX FO PTIMAL SOLUTION OF (55)Since the optimal λ → ∞ , (53) and (54) can be simplifiedto Q c F,AND = δ M and Q c D,AND = Q Mj =1 δ j and so (55)becomes, min λ max j (cid:2) N C sj + ( π (1 − δ ) + π (1 − δ j )) C tj (cid:3) s.t. δ M ≤ α, M Y j =1 δ j ≥ β. (74)Since there is a one-to-one relationship between λ and δ ,i.e., λ = 2Γ − [ N, Γ( N ) δ ] (where Γ − is defined over thesecond argument), (74) can be formulated as [22, p.130], min δ max j (cid:2) N C sj + ( π (1 − δ ) + π (1 − δ j )) C tj (cid:3) s.t. δ M ≤ α, Q Mj =1 δ j ≥ β. (75)Defining δ = F AND ( λ ) = Γ( N, λ )Γ( N ) and δ j = G AND,j ( λ ) = Γ( N, λ γj ) )Γ( N ) , we can write δ j as δ j = G AND ,j ( F − ( δ )) .Calculating the derivative of C j with respect to δ , we findthat ∂C j ∂δ = ∂ (cid:2) C tj ( π (1 − δ ) + π (1 − δ j )) (cid:3) ∂δ = − C tj π + ∂ (1 − δ j ) ∂δ ≤ , (76)where we use the fact that ∂δ j ∂δ = − N Γ( N ) − [ N, Γ( N ) δ ] N − e − [ N, Γ( N ) δ ] / γ j ) I { − [ N, Γ( N ) δ ] ≥ } − N Γ( N ) − [ N, Γ( N ) δ ] N − e − [ N, Γ( N ) δ ] / I { − [ N, Γ( N ) δ ] ≥ } = e − [ N, Γ( N ) δ ](1 / γ j ) − / ≥ . (77) Therefore, we can simplify (75) as max δ δ s.t. δ M ≤ α, Q Mj =1 δ j ≥ β. (78)Since Q c D,AND is a monotonically increasing function of δ ,i.e., Q c D,AND = H AND ( δ ) = Q Mj =1 ( G AND ,j ( F − AND ( δ ))) andthus ∂Q c D,AND ∂δ = ∂Q c D,AND ∂δ j ∂δ j ∂δ = Q l = Ml =1 ,l = j ( δ l ) ∂δ j ∂δ ≥ , wecan further simplify the constraints in (78) as δ ≤ α /M and δ j ≥ H − ( β ) . Thus, we obtain max δ δ s.t. δ ≤ α /M , δ j ≥ H − ( β ) . (79)Therefore, if the feasible set of (79) is not empty, then theoptimal solution is given by δ = α /M ( β ) .R EFERENCES[1] Q. Zhao and B. M. Sadler, “A Survey of Dynamic Spectrum Access,”
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Sina Maleki received his B.Sc. degree in electricalengineering from Iran University of Science andTechnology, Tehran, Iran, in 2006, and his M.S.degree in electrical engineering from Delft Univer-sity of Technology, Delft, The Netherlands, in 2009.From July 2008 to April 2009, he was an internstudent at the Philips Research Center, Eindhoven,The Netherlands, working on spectrum sensing forcognitive radio networks. He then joined the Circuitsand Systems Group at the Delft University of Tech-nology, where he is currently a Ph.D. student. Hehas served as a reviewer for several journals and conferences.
Geert Leus was born in Leuven, Belgium, in 1973.He received the electrical engineering degree and thePhD degree in applied sciences from the KatholiekeUniversiteit Leuven, Belgium, in June 1996 and May2000, respectively. He has been a Research Assistantand a Postdoctoral Fellow of the Fund for Scien-tific Research - Flanders, Belgium, from October1996 till September 2003. During that period, GeertLeus was affiliated with the Electrical EngineeringDepartment of the Katholieke Universiteit Leuven,Belgium. Currently, Geert Leus is an AssociateProfessor at the Faculty of Electrical Engineering, Mathematics and ComputerScience of the Delft University of Technology, The Netherlands. His researchinterests are in the area of signal processing for communications. Geert Leusreceived a 2002 IEEE Signal Processing Society Young Author Best PaperAward and a 2005 IEEE Signal Processing Society Best Paper Award. He wasthe Chair of the IEEE Signal Processing for Communications and NetworkingTechnical Committee, and an Associate Editor for the IEEE Transactionson Signal Processing, the IEEE Transactions on Wireless Communications,and the IEEE Signal Processing Letters. Currently, he is a member of theIEEE Sensor Array and Multichannel Technical Committee and serves on theEditorial Board of the EURASIP Journal on Advances in Signal Processing.Geert Leus has been elevated to IEEE Fellow.was born in Leuven, Belgium, in 1973.He received the electrical engineering degree and thePhD degree in applied sciences from the KatholiekeUniversiteit Leuven, Belgium, in June 1996 and May2000, respectively. He has been a Research Assistantand a Postdoctoral Fellow of the Fund for Scien-tific Research - Flanders, Belgium, from October1996 till September 2003. During that period, GeertLeus was affiliated with the Electrical EngineeringDepartment of the Katholieke Universiteit Leuven,Belgium. Currently, Geert Leus is an AssociateProfessor at the Faculty of Electrical Engineering, Mathematics and ComputerScience of the Delft University of Technology, The Netherlands. His researchinterests are in the area of signal processing for communications. Geert Leusreceived a 2002 IEEE Signal Processing Society Young Author Best PaperAward and a 2005 IEEE Signal Processing Society Best Paper Award. He wasthe Chair of the IEEE Signal Processing for Communications and NetworkingTechnical Committee, and an Associate Editor for the IEEE Transactionson Signal Processing, the IEEE Transactions on Wireless Communications,and the IEEE Signal Processing Letters. Currently, he is a member of theIEEE Sensor Array and Multichannel Technical Committee and serves on theEditorial Board of the EURASIP Journal on Advances in Signal Processing.Geert Leus has been elevated to IEEE Fellow.