Distributed Spectrum Sensing with Sequential Ordered Transmissions to a Cognitive Fusion Center
aa r X i v : . [ c s . I T ] M a y Distributed Spectrum Sensing with SequentialOrdered Transmissions to a Cognitive Fusion Center
Laila Hesham, Ahmed Sultan and Mohammed Nafie
Wireless Intelligent Networks Center (WINC)Nile University, Cairo, Egypt.E-mail: [email protected], { asultan, mnafie } @nileuniversity.edu.eg Abstract —Cooperative spectrum sensing is a robust strategythat enhances the detection probability of primary licensed users.However, a large number of detectors reporting to a fusion centerfor a final decision causes significant delay and also presumesthe availability of unreasonable communication resources atthe disposal of a network searching for spectral opportunities.In this work, we employ the idea of sequential detection toobtain a quick, yet reliable, decision regarding primary activity.Local detectors take measurements, and only a few of themtransmit the log likelihood ratios (LLR) to a fusion center indescending order of LLR magnitude. The fusion center runsa sequential test with a maximum imposed on the number ofsensors that can report their LLR measurements. We calculatethe detection thresholds using two methods. The first achievesthe same probability of error as the optimal block detector. Inthe second, an objective function is constructed and decisionthresholds are obtained via backward induction to optimizethis function. The objective function is related directly to theprimary and secondary throughputs with inbuilt privilege forprimary operation. Simulation results demonstrate the enhancedperformance of the approaches proposed in this paper. We alsoinvestigate the case of fading channels between the local sensorsand the fusion center, and the situation in which the sensing costis negligible. ∗ Index Terms —Cognitive Radios, Cooperative Spectrum Sens-ing, Detection delay, Sequential detection.
I. I
NTRODUCTION
Cognitive radio is an emerging technology aimed at solvingthe problem of spectrum under-utilization caused by staticspectrum allocation [1]. The technology allows a cognitive,also called unlicensed or secondary, terminal to make efficientuse of any available spectrum at any given time. This involvesthe detection of the activity of licensed, or primary, usersso that the secondary operation does not interfere with theprimary networks and disrupt their services. This detectiontask may be extremely difficult due to a large array of factorsincluding, inter alia, the wide variety of primary networks andthe uncertainty about the propagation conditions between anactive primary terminal and a secondary sensor attempting todetect primary activity. If the channel between a primary trans-mitter and a secondary sensor is in deep fade, the sensor wouldfalsely decide that the probed channel is vacant. Secondarytransmission on this channel would then cause interference on ∗ Part of this work was presented in ICASSP, May 2011.This work was supported in part by a grant from the Egyptian NTRA(National Telecommunications Regulatory Authority). the primary link. Relying on one sensor to detect the presenceof primary operation is therefore highly unreliable [2].Enhancement of sensing reliability requires a system ofspatially distributed multiple sensors cooperating together inorder to mitigate the impact of channel uncertainty providedthat the channels between them and the primary transmitterare independent. Many papers have shown improvement inspectrum sensing capabilities through cooperation betweenindividual cognitive users (see, e.g., [3], [4], [5], [6], [7]).However, in order to reap the benefits of distributed detectionand cooperative diversity, efficient schemes are needed tocombine the data obtained by the local sensors [8]. Thereis typically a fusion center at which the final decision ismade. Whether the fusion center receives the raw unprocesseddata from local sensors or summary messages obtained aftersome local processing, there are challenges concerning thetransfer of data to the fusion center. It may be assumed, forinstance, that there are orthogonal channels available for thetransmission from the local detectors to the fusion center.In the context of cognitive radio, this assumption might notbe practical given that the main objective of the cognitiveusers is to find communication resources. Moreover, if theduration of data transmission between the local sensors andthe fusion center is large, then the primary user may switchback to activity, thereby depriving the secondary network of acommunication opportunity. In the cognitive radio setting, notonly many terminals are needed for robust detection of primaryactivity, but also this detection should be done in the quickestway possible and under very strict constraints regarding thecommunication between the cognitive detectors and the fusioncenter.Sequential detection is known to reduce the time of de-tection for a certain specified detection reliability (see, e.g.,[9], [10], [11]). In contrast with block detection that operateson a fixed predetermined number of samples, the numberof samples in sequential detection is a random variable. Inaddition, for the case of binary hypothesis testing, sequen-tial detection requires two thresholds and not only one foroperation. This allows for the decision to continue acquiringmore samples to satisfy some detection performance measures.There has been considerable interest in applying sequentialanalysis to distributed or decentralized detection problems(see, e.g., [12], [13], [14], [15], [16], [17]). According to[18], distributed implementation of sequential detectors canave several forms. For instance, the sensors may forwardtheir local decisions to the fusion center, which then runsa sequential test to obtain a global decision. On the otherhand, the sensors themselves may perform sequential tests andcooperate together with or without a fusion center. In [19],the problem of distributed sequential detection in the presenceof communication constraints is studied. An algorithm isdeveloped for optimal rate or bandwidth distribution amongdetectors under a fixed bandwidth constraint. The work isbased on the sequential probability ratio test (SPRT) of Wald[9].In the context of cognitive radio, distributed sequentialdetection has been investigated, for example, in [20], [21],[22], [23]. In the scheme proposed in [21], each sensorcomputes the log-likelihood ratio (LLR) for its observationsand reports it to the fusion center over a perfect reportingchannel. The LLR’s are accumulated sequentially at the fusioncenter till their sum is found sufficient to cross either of twopredefined thresholds. The thresholds are obtained on the basisof the desired global detection and false alarm probabilities.The authors also investigate the case of sequential detectionwith model uncertainties. In [22], the detectors are divided intodifferent sets according to the local signal-to-noise (SNR) ofthe sensing channel. The set with highest SNR sends to thefusion center first, followed by lower SNR until the fusioncenter can make a final decision. A doubly sequential testis proposed in [23] in which both the local detectors andthe fusion center operate sequential tests. The detectors havethresholds computed using Wald’s approximations [9]. Whenthe first sensor reaches a decision, it conveys its decision tothe fusion center which is running a second sequential test fora global decision on channel availability. The authors makeseveral suggestions pertaining to the termination criterion atthe fusion center. For example, the fusion center stops whenit has received a specified number of decisions in favor of aparticular hypothesis.In addition to distributed sequential detection, we make useof the idea of ordered transmissions provided in [24] for thepurpose of energy efficient signal detection in wireless sensornetworks (see also [25]). By ordered transmissions, it is meantthat the local sensor with the most reliable current measure-ment sends first its LLR value to the fusion center. In orderfor a sensor to know that it has the most reliable LLR value,there is no need for extensive information exchange. A timerbackoff mechanism is suggested in [24] to resolve this issue.Note that the sensor selection criterion is its instantaneousmeasurement quality and not the average SNR as in [22] forexample. We discuss the scheme in detail in this paper. Sufficeit now to mention that the proposed scheme in [24] achievesthe minimum average probability of error attained when allthe sensor LLR’s are used for a decision, and at the sametime lowers the average number of transmissions needed toreach a decision. The scheme in [24] is utilized in [26] forsequential fusion in wireless sensor networks. It is certainlybeneficial for a cognitive radio setting because it enhances thesecondary throughput by quickly seizing the available spectral opportunities.In this paper, in order to reach a global decision at thesecondary fusion center as quickly and reliably as possible,we adopt the ordered transmission scheme of [24]. However,we impose a strict constraint on the maximum number oftransmissions to the fusion center. We therefore extend thescheme to work for the case when the fusion center is tomake a decision after receiving a maximum of K observationsfrom a total of M local sensors. Assuming a slotted primarynetwork where the primary user may switch activity everyfixed amount of time, only few observations can be usedwithin a time slot to make a decision regarding the availabilityof a transmission opportunity. In other words, the decisionregarding primary activity must be made soon enough to allowfor channel access, and under the reasonable assumption of anarrowband cognitive control channel, only a few sensors canactually participate in the detection process.Having imposed this constraint of a maximum number ofreports from the local sensors, we devise a scheme wherethe decision thresholds employed at the secondary fusion cen-ter are computed using a dynamic programming framework.Given the LLR observations sent to the fusion center, theposterior probability of the channel being free is calculatedand compared to the thresholds for a decision. We deriveexpressions for the likelihood functions of ordered LLR’s anduse these in both proposed schemes. For the dynamic pro-gramming scheme, a weighted sum of primary and secondarythroughput objective function is constructed that also accountsfor transmission costs, lost transmission opportunities and apenalty for collision with primary transmission. It is importantto note that, in [27], dynamic programming is also employedto get the decision thresholds. However, it is assumed thatall cognitive sensors would be able to report their energymeasurements to the fusion center, where each detector sensesone of the multi-band channels.In summary we make the following contributions in thispaper. We use a cooperative sequential detection frameworkto address the tradeoff between throughput and detectionreliability in cognitive radio systems taking into account thestrict time limitation during which the decision must bemade, which also imposes a limit on the number of sensorsinvolved in the decision process. We employ an ordered LLRtransmission scheme modifying the method of threshold deter-mination in [24] to account for the constraint on the numberof reporting sensors. We also use dynamic programming toobtain the decision thresholds. The proposed method improvesthe throughput performance, which is the desired goal in thecontext of cognitive radios. We prove that if sensing has nocost, the optimal sequential decision is either to declare that theprimary is idle or to continue sampling without ever decidingthat the primary is active except possibly at the last decisionstage. Finally, we address the impact of fading on the reportingchannels linking the cognitive detectors to the fusion center.The rest of the paper is organized as follows: Section IIpresents the system model of cooperative spectrum sensing ina cognitive radio network. Section III discusses the previousork on ordered transmissions scheme, then discusses ourproposed extension derivation. Section IV and V describethe proposed dynamic programming scheme. The effect offading on the channels between the local detectors and thefusion center is explained in Section VI. In Section VII weprovide simulation results and elaborate on the difference inperformance when both schemes are used. We conclude thepaper in Section VIII.II. S YSTEM M ODEL
We consider a slotted primary system as shown in Figure1, where the primary activity, whether on or off, does notchange during the time slot duration, τ s . Primary activityswitches independently from one slot to the next. There are M cognitive sensors which take a number of measurementsat the beginning of each time slot and compute a function ofthese measurements. A maximum of K sensors among the M sensors forward the results sequentially to a fusion center atwhich the final decision regarding primary activity is taken.We consider binary hypothesis testing at the fusion center withthe following two hypotheses: H : Sensed channel is free ; H : Sensed channel is busyThe prior probabilities of each, denoted by π and (1 − π ) ,respectively, are assumed to be known. Observations fromdifferent sensors are conditionally independent given eitherhypothesis but can be non-identical. Let X i ( n ) be the receivedsignal at the i th sensor at instant n , where i = 1 , , ..., M . Ateach sensor i , X i ( n ) are independent given each hypothesisand are identically distributed. Assume that a total of N samples are taken over a time duration τ N . Under the twohypotheses, X i ( n ) is given by H : X i ( n ) = W i ( n ) , n = 1 , , ..., NH : X i ( n ) = S i ( n ) + W i ( n ) , n = 1 , , ..., N (1)where W i is additive white Gaussian noise (AWGN) havingthe same noise power, σ , at all sensors. Without loss ofgenerality, the received primary signal S i ( n ) is assumed tobe real zero-mean Gaussian random variable. The conditionalprobability distributions of X i ( n ) given H and H are de-scribed by f X i ( n ) ( x n | H ) and f X i ( n ) ( x n | H ) , respectively,such that f X i ( n ) ( x n | H ) ∼ N (0 , σ ) f X i ( n ) ( x n | H ) ∼ N (0 , σ s i + σ ) (2)where σ s i is defined as the average received primary signalpower at the i th local sensor, and is assumed to be fixed over atime slot and to change relatively slowly over time. The valuesof σ s i are assumed to be known at the local sensors and aretransmitted periodically after being quantized and coded to thefusion center on a low-rate common control channel that isnot on the band being sensed. The quantization noise effect isassumed to be negligible. The LLR at the i th sensor is definedas Y i = N X n =1 log (cid:20) f X i ( n ) ( x n | H ) f X i ( n ) ( x n | H ) (cid:21) (3)This is the quantity that is computed by the local detector andreported to the fusion center sequentially as explained below.Defining the local signal-to-noise ratio (SNR) as γ i = σ s i σ ,we can compute the LLR at the i th sensor as Y i = 12 σ · γ i γ i + 1 N X n =1 | X i ( n ) | − log (1 + γ i ) N (4)Note that under H the summation P Nn =1 | X i ( n ) | σ is a chi-square distribution with N degrees of freedom. This is thesame for the summation P Nn =1 | X i ( n ) | σ (1+ γ i ) under H . That is,the likelihood functions of Y i given H or H are shifted andscaled chi-square distributions with N degrees of freedom.The LLR values are quantized and coded using a fixednumber of bits. As in the case of the local signal power, thequantization noise is assumed to be negligible. The informa-tion exchange between the cognitive sensors and the fusioncenter occurs on the common control channel. We initiallyassume that the transmission of the LLR values to the fusioncenter is perfect, then we consider the effect of fading inthe channels between the local sensors and the fusion centerin Section VI. We assume that the procedure of seizing thecontrol channel and sending the observation of one sensorrequires an amount of time which we denote as τ . Figure1 illustrates the time durations τ N , τ and τ s . . . . . . . Fig. 1. The primary time slot has a duration of τ s units of time. Over τ N ,each one of M cognitive sensors takes N samples at the beginning of thetime slot. Afterwards, there are K mini-slots, each of duration τ , over whichone LLR observation is transmitted from one cognitive sensor to the fusioncenter over a low-rate common control channel. Since the transmission of each computed LLR value takessome time τ , eliciting another measurement from the sensors,though improving the reliability of detection, wastes a durationof τ from the potential secondary transmission opportunity incase the primary is off. This causes a decrease in secondarythroughput. In other words, we have a reliability-throughputtradeoff [28] which we can control by allowing only the subsetof the cognitive detectors with the most reliable observations totransmit their LLR’s to the fusion center. This is implementedy having two thresholds at the fusion center. The LLR withmaximum magnitude is transmitted first to the fusion center.A decision can be made, but if the metric is between the twothresholds, the second highest LLR in magnitude is transmittedto the fusion center and is combined with the first, and then adecision is attempted again. This continues until a decision infavor of H or H is made, or K LLR’s are accumulated atthe fusion center. If k sensors, ≤ k ≤ K , are probed beforea decision is reached, the time to make a decision is τ N + kτ .Since the slot duration is τ s , this leaves τ s − τ N − kτ fortransmission. Given the durations τ N , τ and τ s , it is obviousthat K must satisfy the inequality τ s − τ N − Kτ ≥ .Some final notes are in order regarding the operation of thesecondary network.(a) The LLR values can be quantized to a few bits only withno considerable performance degradation when comparedwith the case where unquantized LLR’s are used [29].(b) We assume a single hop network in which any broadcastby a secondary node is heard by the fusion center and allother nodes. This assumption is necessary such that whena node seizes the common channel to transmit its LLR,all other nodes become aware of its transmission.(c) We assume that the sensors have nearly synchronizedclocks. This issue and its practical aspects are discussedextensively in [30].III. O RDERED T RANSMISSIONS S CHEME
In this section we describe the scheme in [24]. A backofftimer is set at each of the M sensors according to themagnitude of the locally computed LLR. Specifically, thetimer is decreased as the absolute value of the sensor’s LLRincreases. Consequently, transmissions proceed such that themost confident observations and informative measurements aresent first to the fusion center. The fusion center then comparesthe accumulated sum of the received LLR’s to two thresholdsand makes one of three decisions accordingly; declare H ,continue taking the next ranked observations, or declare H .Let Y [ m ] denote the LLR of rank m = 1 , , ..., K receivedduring the m th reporting mini-slot, where m = 1 is the highestrank. For ≤ k ≤ K , the decision strategy is expressed as If k X m =1 Y [ m ] < t ( L ) k Declare H ∈ ( t ( L ) k , t ( H ) k ) Continue > t ( H ) k Declare H (5)Note that the thresholds are generally time-dependent, i.e, theirvalues depend on the stage index k . In order to force a stoppingat k = K , t (L) K = t (H) K .When the statistics evidently favor one hypothesis over theother, the decision is made and LLR transmissions from thelocal sensors stop. Otherwise, more observations are takenpossibly till the end of the primary time slot. In [24], thedata-dependent thresholds t (L) k and t (H) k are given by t (L) k = log ( π − π ) − ( M − k ) (cid:12)(cid:12) Y [ k ] (cid:12)(cid:12) t (H) k = log ( π − π ) + ( M − k ) (cid:12)(cid:12) Y [ k ] (cid:12)(cid:12) (6)where M − k is the number of sensors that have not transmittedtheir LLR’s yet at the k th stage, and | Y [ k ] | is the absolutevalue of the LLR received at stage k . However, if this schemeis forced to take a decision in favor of either H or H at stage K , then it should work only for the case K = M , as can beseen by the fact that only t ( L ) M = t ( H ) M . The transmission isordered in terms of the absolute LLR value. That is, | Y [ m ] | < | Y [ k ] | , ∀ m > k . Hence, | Y [ k ] | is greater than any LLR valuetransmitted later to the fusion center. It can be shown thatthis choice of thresholds ensure that this scheme has the sameaverage probability of error as maximum a posteriori (MAP)procedure where all the LLR’s are summed and compared to log ( π / (1 − π )) [24]. The average probability of error whenthe guessed hypothesis, b H , is not the true one is given by P e = (1 − π ) P r (cid:16) b H = H | H (cid:17) + π P r (cid:16) b H = H | H (cid:17) (7)The advantage of this system is that the average number oftransmissions needed to reach a decision is about half the totalnumber of the sensors communicating with the fusion center[24].Here, and in our ICASSP paper [31], we consider extendingthe scheme in [24] to work for K ≤ M case such that we onlyprocess the K LLR’s with highest magnitudes out of the M LLR’s. Given the sequence of observations Y [1] = y , Y [2] = y , ..Y [ K ] = y K , the optimal MAP block detector when thehighest in magnitude K out of M LLR values are used hasthe following decision rule: log f Y [1] ..Y [ K ] (cid:12)(cid:12) H (cid:16) y , y , ..y K (cid:12)(cid:12)(cid:12) H (cid:17) f Y [1] ..Y [ K ] (cid:12)(cid:12) H (cid:16) y , y , ..y K (cid:12)(cid:12)(cid:12) H (cid:17) H > In this section, we use dynamic programming to calculatethe thresholds for sequential detection. Specifically, we employthe backward induction technique which provides the optimalaction to be taken in order to minimize the overall decisioncost [33]. In the analysis below we assume that the fusioncenter knows the statistics of all detectors, which are updatedperiodically over the common control channel and changerelatively slowly over time as mentioned in Section II.Since we adopt the ordered transmission scheme and assumeperfect reporting, if an LLR value is received, all subsequentLLR’s would have values with a lesser magnitude. In otherwords, if the value of the LLR received at stage k − is equalto α , then the probability Pr (cid:0) | Y [ m ] | > | α | (cid:1) = 0 , ∀ m ≥ k .Recall that Y m is the LLR from the m th sensor, ≤ m ≤ M ,whereas Y [ m ] is the LLR with the m th highest magnitudethat is transmitted to the fusion center at the m th stage with m = 1 , , ..K .et π k be the probability of the channel being idle at stage k given the sequence of observations Y [1] = y , Y [2] = y , ..Y [ k ] = y k . Probability π k can be obtained recursivelyas follows: π k = Pr ( H | y , y , ..y k ) = f Y [ k ] | Y [1] ..Y [ k − ,H ( y k | y ..y k − , H ) Pr ( H | y ..y k − ) P r =0 , f Y [ k ] | Y [1] ..Y [ k − ,H r ( y k | y ..y k − , H r ) Pr ( H r | y ..y k − )= f Y [ k ] | Y [ k − ,H ( y k | y k − , H ) π k − P r =0 , f Y [ k ] | Y [ k − ,H r ( y k | y k − , H r ) ( r + (1 − r ) π k − ) (14)where in the last step, we use the Markovian property in-duced by ordered transmissions that Y [ k ] is independent of Y [1] , Y [2] , ..Y [ k − given Y [ k − and either hypothesis (Theo-rem 2.4.3, [32]).What is needed for backward induction is the conditionalprobability of Y [ m ] given Y [ m − under both H and H .Assume that the probability density function of LLR value Y k under hypothesis H (either H or H ) is given by f Y k ( y | H ) ,which is a scaled and shifted chi-square distribution with N degrees of freedom as mentioned in Section II. Define β k,b,H = Pr { [ | Y k | > | b | ] | H } . The value of β k,b,H can bereadily computed knowing the likelihood functions of Y k .For non-identical sensors, the joint distribution of Y [ m ] and Y [ m − conditioned on hypothesis H when −| γ | < α < | γ | and m ≥ is given by: f Y [ m ] ,Y [ m − ( α, γ | H ) = M X k =1 M X j = k f Y k ( α | H ) f Y j ( γ | H ) · X S m − k,j Y v = k,j ( β v,γ,H ) x v (1 − β v,α,H ) (1 − x v ) (15)where S m − k,j is a set of m − sensors chosen from the M sensors with sensors k and j excluded. The number ofsets S m − k,j is (cid:0) M − m − (cid:1) . Parameter x v is equal to unity when v ∈ S m − k,j and zero otherwise. f Y [ m ] ,Y [ m − ( α, γ | H ) = 0 if | α | > | γ | . Note that (15) is derived as follows. The jointdistribution between Y [ m ] and Y [ m − involves a summationover all possible sensor pairs. Since, given a particular hypoth-esis, the LLR’s from the local sensors are independent, weget the product f Y k ( α | H ) f Y j ( γ | H ) . For m ≥ and dueto the ordered nature of LLR transmission, we have m − sensors with absolute LLR values exceeding Y [ m − = γ .The rest of the sensors should have absolute LLR values thatare less than Y [ m ] = α . In the case of identical sensors, f Y k ( y | H ) = f Y ( y | H ) and β k,b,H = β b,H , f Y [ m ] ,Y [ m − ( α, γ | H ) = M ( M − · f Y ( α | H ) f Y ( γ | H ) (cid:18) M − m − (cid:19) ( β γ,H ) m − (1 − β α,H ) M − m (16)In order to obtain the conditional distribution f Y [ m ] | Y [ m − ( α | γ, H ) , we need the distribution f Y [ m − ( γ | H ) because f Y [ m ] | Y [ m − ( α | γ, H ) = f Y [ m ] ,Y [ m − ( α, γ | H ) f Y [ m − ( γ | H ) (17)Distribution f Y [ m − ( γ | H ) can be readily obtained as f Y [ m − ( γ | H ) = M X k =1 f Y k ( γ | H ) X S m − k Y v = k ( β v,γ,H ) x v (1 − β v,γ,H ) (1 − x v ) (18)where S m − k is a set of m − sensors chosen from the M sensors with sensor k excluded. The number of sets S m − k is (cid:0) M − m − (cid:1) . Parameter x v is equal to unity when v ∈ S m − k and zero otherwise. In the case of identical sensors, β k,γ,H = β γ,H , f Y [ m − ( γ | H ) = M f Y ( γ | H ) (cid:18) M − m − (cid:19) ( β γ,H ) m − (1 − β γ,H ) M − m +1 (19)We construct now the cost-to-go function which is used toobtain the optimal policy in optimal stopping problems [34].The optimal policy here is comprised of the optimal thresholdsto be used at each stage to determine whether to stop andmake a decision regarding the channel occupancy, or continueobtaining more LLR measurements. Assume that the cost ofdeciding i at stage k when j is the true hypothesis is λ kij , andthe cost to continue taking observations is c ≥ . Define J ( K ) k as the minimum cost-to-go at stage k of the finite horizon K . J ( K ) k ( π k , Y [ k ] ) = min (cid:26) λ k π k + λ k (1 − π k ) ,λ k π k + λ k (1 − π k ) ,c + E Y [ k +1] | Y [ k ] h J ( K ) k +1 ( π k +1 , Y [ k +1] ) (cid:12)(cid:12) Y [ k ] i (cid:27) (20)where π k and π k +1 are related through the expression in (14)with k replaced by k + 1 . The first term in (20) is the cost ofdeciding H , whereas the second term is the cost of deciding H . These correspond to the channel decided to be free orbusy, respectively. The third term in (20) is interpreted as theexpected cost when the fusion center decides that it shouldcontinue taking more observations from local sensors. Theconditional expectation is given by E Y [ k +1] | Y [ k ] h J ( K ) k +1 ( π k +1 , Y [ k +1] ) (cid:12)(cid:12) Y [ k ] i = Z | Y [ k ] |−| Y [ k ] | J ( K ) k +1 ( π k +1 , y ) f Y [ k +1] (cid:12)(cid:12) Y [ k ] (cid:16) y (cid:12)(cid:12) Y [ k ] (cid:17) dy (21)and f Y [ k +1] | Y [ k ] (cid:16) y (cid:12)(cid:12) Y [ k ] (cid:17) = f Y [ k +1] | Y [ k ] ,H (cid:16) y (cid:12)(cid:12) Y [ k ] , H (cid:17) π k + f Y [ k +1] | Y [ k ] ,H (cid:16) y (cid:12)(cid:12) Y [ k ] , H (cid:17) (1 − π k ) (22)he parameter c represents the tradeoff between the timetaken till a decision is made and the average probability oferror. As c increases, the fusion center becomes more likelyto favor one of the two hypotheses in a shorter time using onlya few LLR’s from the cognitive detectors. At the last stage,i.e., when k = K , the fusion center has two choices only; todeclare H or H . This allows backward induction from thelast stage J ( K ) K ( π K , Y [ K ] ) = min (cid:26) λ K π K + λ K (1 − π K ) ,λ K π K + λ K (1 − π K ) (cid:27) (23)Note that there is no actual dependence of J ( K ) K on Y [ K ] because there is no more sampling after the K th stage. J ( K ) K − ( π K − , Y [ K − ) can be obtained using (20) for all valuesof Y [ K − . The process can then be repeated to obtain thethresholds and optimal decisions. It is clear that this is acomputationally extensive task due to the dependence onlast observation. In particular, the thresholds at each stagewould be a function of the value of the LLR sent in theprevious stage. To reduce the complexity, we propose hereto use the distributions of Y [ k ] without conditioning. Thismakes the problem considerably more manageable. We haveverified through simulations that at least for a few number ofsensors dropping this dependency has a negligible effect onthe objective function. Dropping the explicit dependence on Y [ k ] , the dynamic program now acquires the form J ( K ) k ( π k ) = min (cid:26) λ k π k + λ k (1 − π k ) ,λ k π k + λ k (1 − π k ) ,c + E Y [ k +1] h J ( K ) k +1 ( π k +1 ) i (cid:27) (24)such that E Y [ k +1] h J ( K ) k +1 ( π k +1 ) i = Z J ( K ) k +1 ( π k +1 ) f Y [ k +1] ( y ) dy (25)with f Y [ k +1] ( y ) = π k f Y [ k +1] ( y | H ) + (1 − π k ) f Y [ k +1] ( y | H ) (26) π k +1 = π k f Y [ k +1] ( y | H ) π k f Y [ k +1] ( y | H ) + (1 − π k ) f Y [ k +1] ( y | H ) (27)Finally, f Y [ k +1] ( y | H ) can be computed using an expressionsimilar to (28) f Y [ k +1] ( y | H ) = M X r =1 f Y r ( y | H ) X S kr Y v = r ( β v,y,H ) x v (1 − β v,y,H ) (1 − x v ) (28)Note that the optimal thresholds and decisions can becalculated offline so long as the relevant system parametersare fixed. When the fusion center gets the LLR observations, ituses the pre-computed thresholds to decide whether to decidein favor of H or H , or to request the transmission of one more LLR value from sensors. The calculation should beredone when any of the parameters, such as the local SNR’sat the cognitive detectors, change significantly.V. P ERFORMANCE O PTIMIZATION We now provide expressions for costs λ kij introduced inSection IV. Recall that λ kij is the cost to decide i when j is true at the k th mini-slot of duration τ . Our objectiveis to optimize the system performance with a focus on theachievable primary and secondary throughput. Specifically,when we deal with throughput, we mean a weighted sumof primary and secondary throughputs where a factor ω , ≤ ω ≤ , is multiplied by primary throughput and − ω is used to weight the secondary throughput. The closer ω to unity, the more emphasis we put on the primary rate.This implies more protection for the primary link againstinterference and service interruption. Parameter ω should bechosen by the primary network to satisfy the transmission andrate requirements of primary users.Define η p and η s to be the probability of correct reception ofthe primary and secondary signals in the presence of receivernoise only, respectively. Similarly, define δ p and δ s to be theprobability of correct reception of the primary and secondarysignals in the presence of noise in addition to interference fromthe other user due to concurrent transmission. Given that R p and R s are the primary and secondary rates of transmission,respectively, the costs of different decisions are then given by λ k = − (1 − ω ) R s η s ( τ s − τ N − kτ ) τ s + e st ( τ s − τ N − kτ ) τ s (29)where e st is the transmission cost or expended energy for thesecondary terminal if it transmits for the whole slot duration.This cost is given in rate units to be compatible with the firstterm in (29). λ k = − ωR p δ p − (1 − ω ) R s δ s ( τ s − τ N − kτ ) τ s + e pt + e st ( τ s − τ N − kτ ) τ s + P (30)where e pt is the transmission cost or expended energy for theprimary terminal expressed in units of rate, and P is a penaltyterm for collision with primary user. λ k = L f (31)where L f is the cost of losing the opportunity to access thespectrum when the channel is idle. This term can be positiveto account for other negative consequences for remainingidle than the throughput being zero. The failure to transmitmay increase the transmission delay for a delay-sensitiveapplication, or cause packet loss due to queue overflow, etc. λ k = − ωR p η p + e pt + L b (32)where L b is the cost of losing the opportunity to access thespectrum when the channel is busy. All the costs e st , e pt , L f , L b and P are nonnegative.he first term of expression (29) shows the gain of thesecondary user when an idle channel is correctly detected. Itdepends on the weight assigned to the secondary user, the timeremaining from the time slot τ s after making the decision, andthe probability of correct detection in the absence of primaryinterference. Similarly, the first term of (32) is the primarythroughput in the case of correct detection by the secondaryusers. The first two terms in (30) represent the possibility ofcorrect reception in spite of the occurrence of a collision dueto mis-detection. The interference survival probabilities δ p and δ s would be typically small in value.Recall that parameter c ≥ is the cost of taking one moreobservation. Given the aforementioned performance metrics, c can be understood as the cost, in units of rate, of thetransmission phase in which the secondary fusion centersolicits another LLR measurement and receives it from theappropriate local sensor. A. Throughput Maximization If we adopt the weighted sum throughput maximizationobjective, then setting all the costs, including c , to zero exceptfor the throughput terms, declaring H at any stage in the slotexcept for the last stage might cause losing the opportunity toaccess the spectrum by the cognitive radio device. Instead,the fusion center should solicit more observations, for apossible declaration of a free channel before the end of theslot, which consequently increases the normalized secondarythroughput. Hence, rather than two thresholds as in classicalsequential detection, we can just have one threshold and twodecisions: to declare H and assign one of the cognitiveterminals to transmit over the probed channel, or to ask forone more observation if less than K LLR observations havebeen collected. Consequently, the objective cost-to-go at stage k can be expressed as J ( K ) k ( π k ) = min (cid:26) λ k π k + λ k (1 − π k ) , E Y k +1 h J ( K ) k +1 ( π k +1 ) i (cid:27) (33)where λ k and λ k are given by (29) and (30) with e st = e pt = 0 . The above expression is valid for k < K . At the laststage where k = K , we have J ( K ) K ( π K ) = min (cid:26) λ K π K + λ K (1 − π K ) ,λ K π K + λ K (1 − π K ) (cid:27) (34)where λ K and λ K are given by (31) and (32) with k = K , e st = e pt = 0 and L b = L f = 0 . We actually can showthat starting with the two-threshold case, if we set c = 0 , e st = e pt = 0 and L b = L f = 0 , the two-thresholdcase would converge to the one-threshold case as the lowerthreshold would always be set to except at the very laststage. The proof is given in Appendix B. Note that if there aremultiple channels the two-threshold case would not necessarily converge to the the one-threshold case because declaring H would be equivalent to switching to another channel in orderto search for a transmission opportunity.VI. F ADING C HANNELS B ETWEEN S ENSORS AND F USION C ENTER In this section, and instead of the previously assumed perfectreporting channels, we consider fading channels between thelocal detectors and the fusion center. The statistical distributionof the channel gain between the i th sensor and the fusioncenter is given by f g i ( x ) , possibly different for differentsensors. We assume that the instantaneous values of thechannel gains are known by the sensors and the fusion center.These values are fixed over a duration T c which is multiples ofthe primary slot duration τ s . Note that if the channel betweena sensor and a fusion center is in deep fade, then even if thesensor has a reliable observation, it may not be able to transferit correctly to the fusion center within the mini-slot duration τ . For the transmission from the i th sensor to the fusion centerto be decodable, the rate of transmission should be lower thanthe link capacity which, for a fixed and known channel gain g i at the transmitter and additive white Gaussian noise at thereceiver, is given by C i = W log (cid:18) P i g i σ (cid:19) (35)where W is the reporting channel bandwidth, σ is the noisevariance at the receiver of the fusion center, and P i is thetransmitted power by the i th sensor. We consider the casewhere a local sensor transmits with a rate r i given by r i = W log (cid:18) P i g i Γ i σ (cid:19) (36)where Γ i > is the signal-to-noise ratio gap to capacity[35]. This factor accounts for the practical limitations thatforces transmission at a rate below link capacity for correctreception. Assume that a number b of information bits needsto be transferred to the fusion center. This would includethe quantized LLR values with high resolution to justify theneglect of quantization noise. The amount of time needed forthis transfer conditioned that the fusion center would be ableto decode the transmission correctly is given by b/r i . Weconsider that this transmission should take no longer than τ b which is obviously smaller than τ . In other words, the i th sensor would be able to send its LLR value to the fusioncenter in time τ b or less if bW log (cid:16) P i g i Γ i σ (cid:17) ≤ τ b (37)This can be written as g i ≥ Γ i σ P i (cid:16) bWτb − (cid:17) (38)Let ¯ g i be the right-hand-side of inequality (38). The probabilitythen that a sensor would report its LLR measurement to theusion center is given by δ i = Z ∞ ¯ g i f g i ( x ) dx (39)Over a period of T c only a fraction of the local detectorswould participate in the reporting process to the fusion center.Since the channels are assumed to be known at the sensors,each sensor can decide whether it should participate in the nextsensing epoch. The fusion center also, knowing the channels,can know the subset of sensors that would be involved inthe next sensing events that span a time duration of T c . Ouranalysis in the previous sections applies but with a numberof sensors equal to those whose channels satisfy (38). Theaverage number of sensors participating in the sensing issimply P Mi =1 δ i . The probability that the number of sensorsparticipating in sensing is equal to ¯ M , where ≤ ¯ M ≤ M ,is given by X S ¯ M M Y i =1 δ x i i (1 − δ i ) − x i (40)where S ¯ M is a set of ¯ M sensors from M sensors. The numberof elements in set S ¯ M is (cid:0) M ¯ M (cid:1) . Factor x i = 1 if sensor i belongsto the set S ¯ M and zero otherwise. For symmetric channelsbetween the sensors and the fusion center, δ i = δ for all i andthe probability of ¯ M sensors participating is (cid:18) M ¯ M (cid:19) δ ¯ M (1 − δ ) M − ¯ M (41)The average number of sensors participating in the sensingand reporting process is, hence, equal to M δ . This means thatthe performance of the network with M sensors would, onaverage, be equal to the performance of a network with non-fading reporting channels, albeit with M δ sensors.VII. S IMULATION R ESULTS In this section, we investigate via numerical simulationsthe performance of our proposed schemes. Unless otherwisestated, the simulation parameters are as follows. The noisevariance at each detector σ = 1 and the transmission of eachobservation takes τ = 0 . units of time. The time taken atthe beginning of a time slot with duration τ s = 1 to collect N observations is τ N = 2 τ . This leaves K = 8 stages inthe slot in which the fusion center takes the ordered LLRobservations. We consider the case of equal prior probabilitiesof both hypotheses, i.e., π = 0 . , and equal local signal tonoise ratio over the channels between the local sensors andthe primary user, σ s i = 2 , i = 1 , , ..., M . Each sensor takes N = 3 samples in the duration τ N . The cost, c , to continuewithout deciding on one of the two hypotheses is set to . .In order to minimize the average probability of error inequation (24), we use the decision costs: λ kij = 1 when i = j and λ kij = 0 when i = j . Cost λ kii = 0 means that no costis incurred if the decision at stage k is the true hypothesis.Moreover, to study the objective of maximizing the weightedsum throughput, we use the expressions (29)-(32) for the costs.In the simulations, the transmission costs for both the primary and secondary users, e pt and e st , are set to , as well asthe penalty factor P . Furthermore, the costs of losing theopportunity to access the spectrum when it is idle or busy, L f and L b , are assumed to be . The transmission successprobabilities are η p = 1 , η s = 1 , δ p = 0 and δ s = 0 . A. Two-Threshold Based Method Figure 2 shows the normalized primary and secondarythroughputs for a weight ω = 0 . and a weight ω = 0 . .The latter case is the case of interest in the context of cognitiveradios given the privileges of the licensed primary users. Thenormalized throughput is obtained via simulating the systemusing the optimal thresholds. If the simulation is run over Q time slots, the normalized secondary throughput is given by Q Q X q =1 I (S) q R s (cid:18) − τ N + k q ττ s (cid:19) (42)where I (S) q is equal to unity if the one of the secondaryusers transmits successfully over the q th time slot and zerootherwise, and k q is the number of probed sensors in the q th time slot. The normalized primary throughput is equalto Q P Qq =1 I (P) q R p , where I (P) q is equal to unity when theprimary user transmits successfully during the q th time slotand zero otherwise. It is clear from the figure that increasingthe weight puts more emphasis on the primary throughput. Thedifference is, however, small in the case of a large number ofsensors. For the rest of the results in this section and just fordemonstration purposes, we use ω = 0 . . 10 15 20 25 3000.10.20.30.40.50.60.70.80.91 Number of CRs in the network N o r m a li z ed T h r oughpu t ω =0.9992ry− ω =0.9991ry− ω =0.52ry− ω =0.5 Fig. 2. Normalized primary and secondary throughputs vs. M , with N = 3 , σ s i = 2 , K = 8 , η p = 1 = η s = 1 , R p = R s = 1 , δ p = δ s = 0 and c = 0 . for ω = 0 . and ω = 0 . . In the legend, “1ry” refersto primary and “2ry” refers to secondary. The scheme used to generate thisfigure is dynamic programming for throughout optimization. In Figure 3, it is clear that the modified scheme of [24],which we hereafter refer to as Modified B&S scheme, achievesa lower probability of error than the dynamic programmingscheme, denoted as DP scheme, since it achieves the same −4 −3 −2 −1 Number of CRs in the network A v e r age P r obab ili t y o f E rr o r DP (Pe Opt.)Modified B&S Fig. 3. Average probability of error vs. M , N = 3 , σ s i = 2 , K = 8 , and c = 0 . . probability of error as the MAP block detector. However, asillustrated in Figure 4, the number of sensors involved in thesensing process in the dynamic programming scheme is lowerthan that in the Modified B&S scheme. The number of sensorsactually converges slowly to unity using dynamic program-ming. Increasing the number of sensors in the network causesthe ranked LLR observations to be more informative about thestatus of the channel. Therefore, when the DP scheme is used,the decision can be made after taking observations from onesensor on average, given that it has the maximum absoluteLLR. However, for the Modified B&S scheme, the sensingtime increases as the number of CR’s in the network increases.This follows from the fact that the greater the number of sen-sors in the network, the greater the distance between the twothresholds of comparison in (10) causing more observations tobe taken. Thus, the dynamic programming scheme represents atradeoff of the sensing time with the error probability, via the c parameter. The performance enhancement increases whenthe objective becomes to maximize the achievable weightedsum throughput as is evident from Figure 4. Figure 5 demon-strates these results on the normalized secondary throughput,assuming unity primary and secondary transmission rates R p and R s . For the normalized secondary throughput it is obviousfrom the simulation results that it converges to approximately π (cid:16) − τ N + ττ s (cid:17) in the DP scheme. This is the throughput forthe hypothetical case of using just one “perfect” sensor thatreveals that true state of the channel.Using Modified B&S scheme, it is noted that when the localsensors’ SNR is high, the number of probed sensors is almostupperbounded by K as shown in Figure 6. For low SNRregime, the number of probed sensors is close to K , thoughmay exceed it when the SNR is low depending on the value of K . If we use conditional distributions for the measurements atlocal sensors for which the correction term ρ ( y ) in equation(10) is zero, it can be shown analytically following a proof 10 15 20 25 30 35 40012345678 Number of CRs in the network A v e r age N u m be r o f S en s o r s i n v o l v ed i n s en s i ng DP (Throughput Optimization)DP (Pe Optimization)Modified B&S Fig. 4. Average number of probed sensors for both DP schemes and theModified B&S scheme, vs. M , N = 3 , σ s i = 2 , K = 8 , and c = 0 . . in [24] that the maximum number of probed sensors is K .This is the case for example when the local measurementshave conditional distributions that, in contrast with those in(2), are Gaussian with different means and the same variance.The simulation results for such “shift in mean” case are alsoprovided in Figure 6.Taking into consideration the fading effect in the channelsbetween the sensors and the fusion center as explained inSection VI, only a subset of the total number of local de-tectors becomes involved in the sensing and reporting processaccording to the fading coefficients. This means that the fusioncenter takes longer time to detect the presence or absence ofthe primary user. Figure 7 shows the effect of fading, which isto slightly increase the average number of probed sensors. Theparameters used to produce this figure are b = 20 , W = 50 kHz, τ b = τ = 0 . msec, Γ i = 2 , and P i σ = 5 . Assuming thatthe channel gains between the sensors and the fusion center arei.i.d and are exponentially distributed with unity mean, we caneasily find that δ is equal to . . Therefore, slightly less thatthree quarters of the number of sensors participate, on average,in the process of spectrum sensing. The observation that fadinghas a minor effect is just because of the simulation parameters.Note that in Figure 4 the average number of sensors is alreadyless than even for a relatively small M . B. One-Threshold Based Method As mentioned in Section V, instead of applying the two-threshold based scheme to the accumulated LLR observationsat the fusion center, it can be sufficient to perform thecomparison using one threshold only, when the sensing cost, c ,is set to . In fact, this requires consuming more mini-slots insensing the channel, till a decision is made about the channelbeing idle. Otherwise, sensing continues till the end of theslot, declaring a busy channel.Figure 8 shows the convergence of the two-thresholdscheme to the one-threshold scheme at zero sensing cost, for N o r m a li z ed T h r oughpu t DP (Throughput Optimization)DP (Pe Optimization)Modified B&S Fig. 5. Normalized Weighted Sum Throughputs for both DP schemes and theModified B&S scheme, vs. M , N = 3 , σ s i = 2 , K = 8 , and c = 0 . . 10 20 30 40 50 60 70 80 90 10005101520253035404550 K N u m be r o f S en s o r s High SNRK/2Low SNRShift in mean Fig. 6. Number of sensors involved in sensing, for high SNR and low SNRversus different values for K , with M = 100 , N = 3 and σ s i = 2 for lowSNR and σ s i = 50 for high SNR. The “shift in mean” case refers to usingconditional distributions for local sensor measurements that, in contrast with(2), are Gaussian with the same variance and different means. the M = 10 case. As c decreases, the lower threshold isreduced at all stages. When c = 0 , the lower threshold is at all stages except for the last one at which a decisionmust be made. The fusion center therefore continues to takeobservations till the end of the slot if there is no sufficientstatistics to declare H . At any stage up to k = 8 , the fusioncenter has only two choices; to declare H or to probe moresensors, which is the main idea of the one-threshold scheme.To investigate the effect of the cost parameter c on thesensing time, we consider the dynamic programming schemewith decision costs set for minimizing the average probabilityof error. Figure 9 shows that at c = 0 the fusion center doesnot make a decision till the end of the slot. Increasing the cost 15 20 25 30 35 40012345678 Number of CRs in the network A v e r age N u m be r o f S en s o r s i n v o l v ed i n s en s i ng DP−No FadingDP−Fading Fig. 7. Average sensing time vs. M , with N = 3 , σ s i = 2 , K = 8 , usingthe DP scheme for error optimization under fading between the local sensorsand the fusion center. T h r e s ho l d s Upper Threshold c=0Lower Threshold c=0Upper Threshold c=0.01Lower Threshold c=0.01Upper Threshold c=0.1Lower Threshold c=0.1One Threshold Scheme Fig. 8. Thresholds of comparison for different costs vs. stage index k , with K = 8 , M = 10 , σ s i = 2 and N = 3 . gives the fusion center the chance to make a decision beforethe end of the slot allowing for secondary transmission.VIII. C ONCLUSION In this paper, we considered ordering the transmissions fromcognitive users to the decision fusion center according to theinformation they carry about the status of the primary user.Restricting the transmissions to the highly informative setof observations, i.e., the LLR’s with the highest magnitude,increases the time remaining in the time slot after making thedecision, thereby boosting the throughput while protecting theprimary signal. We devised a sequential scheme that achievesthe optimum error probability of the MAP block detector, butmakes the decision faster. We also devised a scheme basedon dynamic programming that allows a trades-off between A v e r age S en s i ng T i m e Fig. 9. Average sensing time vs. c , with N = 3 , σ s i = 2 , K = M = 8 ,using the DP scheme for error optimization. the average probability of error and the average sensingtime. Simulation results have demonstrated that using dynamicprogramming approach to compute the thresholds required tomake the decision at the fusion center yields a considerableenhancement in the performance in terms of reduced delayand increased secondary throughput.Future work needs to address the situation of multipleprimary channels. Also of interest is the design of the multipleaccess scheme that allows the sensor with the highest LLR inmagnitude to seize the common control channel and transmit.Another front that needs further investigation is the channelaccess scheme and how the discovered spectrum opportunitiesare to be distributed among the secondary users.A PPENDIX AFor the optimal probability of error MAP detector, usingthe best K out of M LLR magnitudes, the joint distributionof K observations under hypothesis H (either H or H ) isgiven by f Y ..Y K | H ( y , y , ..y K | H ) . Since the sequence ofobservations forms a Markov chain (Theorem 2.4.3, [32]) f Y ,Y ...Y K | H ( y , y , ..y K | H ) = f Y [1] (cid:16) Y [1] = y (cid:17) · K Y j =2 f Y [ j ] | Y [ j − ,H ( Y [ j ] = y j | Y [ j − = y j − , H ) (43)For symmetric sensors, the conditional distribution of thesensor of rank j receiving an LLR value Y [ j ] = α given thatthe sensor of rank j − has already received an LLR value Y [ j − = γ is given by f Y [ j ] | Y [ j − ,H ( Y [ j ] = α | Y [ j − = γ, H ) = ( β γ,H ) j − ( β α,H ) j − · (1 − β α,H ) M − j M ( M − · f Y ( α | H ) f Y ( γ | H ) (cid:0) M − j − (cid:1) (1 − β α,H ) M − j +1 M · f Y ( γ | H ) (cid:0) M − j − (cid:1) =( M + 1 − j ) f Y ( α | H ) · (1 − β α,H ) M − j (1 − β γ,H ) M − j +1 (44)with β x,H defined as Pr {| Y | ≥ | x | (cid:12)(cid:12)(cid:12) H } , where randomvariable Y is an LLR observation. Hence, f Y ,Y ...Y K | H ( y , y , ..y K | H ) = M f Y (cid:0) y (cid:12)(cid:12) H (cid:1) (1 − β y ,H ) M − · ( M − f Y ( y (cid:12)(cid:12) H ) (1 − β y ,H ) M − (1 − β y ,H ) M − · ( M − f Y ( y (cid:12)(cid:12) H ) (1 − β y ,H ) M − (1 − β y ,H ) M − ·· · ( M + 1 − K ) f Y ( y K (cid:12)(cid:12) H ) (1 − β y K ,H ) M − K (cid:0) − β y K − ,H (cid:1) M − K +1 (45)Then, the likelihood ratio of the best K out of M LLR’s fora block detector can be written as f Y ..Y K | H ( y , y , ..y K | H ) f Y ..Y K | H ( y , y , ..y K | H ) = f Y ( y | H ) ..f Y ( y K | H ) (1 − β y K ,H ) M − K f Y ( y | H ) ..f Y ( y K | H ) (1 − β y K ,H ) M − K (46)We now show that the sequential algorithm may operate onthe received LLR’s directly rather than computing their LLR’s.This requires that the LLR is a strictly monotonic function ofthe observation. Consider observation y with llr given byllr = log f Y ( y | H ) f Y ( y | H ) = g ( y ) (47)For a monotonic g function, conditioned on H (either H or H ), f LLR ( llr | H ) = f Y ( y | H ) (cid:12)(cid:12)(cid:12) y = g − ( llr ) · (cid:12)(cid:12)(cid:12)(cid:12) dg − ( y ) d llr (cid:12)(cid:12)(cid:12)(cid:12) (48)Then, log f LLR ( llr | H ) f LLR ( llr | H ) = log f Y ( y | H ) (cid:12)(cid:12)(cid:12) y = g − ( llr ) f Y ( y | H ) (cid:12)(cid:12)(cid:12) y = g − ( llr ) (49)But the right-hand-side of (49) is given by log f Y ( y | H ) (cid:12)(cid:12)(cid:12) y = g − ( llr ) f Y ( y | H ) (cid:12)(cid:12)(cid:12) y = g − ( llr ) = log f Y (cid:0) g − ( llr ) | H (cid:1) f Y ( g − ( llr ) | H )= g (cid:0) g − ( llr ) (cid:1) = llr (50)herefore, log f LLR ( llr | H ) f LLR ( llr | H ) = llr (51)If the LLR monotonicity assumption is not satisfied, thethresholds in (10) can be modified as follows: ˆ t (H) k = log π − π − min ≤ y ≤| y k | ρ ( y )ˆ t (L) k = log π − π − max ≤ y ≤| y k | ρ ( y ) (52)where ρ ( y ) = ( M − K ) ρ ( y ) + ( K − k ) log f Y (cid:0) y (cid:12)(cid:12) H (cid:1) f Y (cid:0) y (cid:12)(cid:12) H (cid:1) (53)The proof that these thresholds achieve the same averageprobability of error if the best K out of M sensors are usedfollows the same outline as that presented in Section (III). Wefocus here on the lower threshold. A decision in favor of H is made if the metric P km =1 log f Y ( y m | H ) f Y ( y m | H ) gets below ˆ t (L) k . If P km =1 log f Y ( y m | H ) f Y ( y m | H ) < ˆ t (L) k , k X m =1 log f Y ( y m | H ) f Y ( y m | H ) < log π − π − max ≤ y ≤| y k | ρ ( y ) < log π − π − K X m = k +1 log f Y ( y m | H ) f Y ( y m | H ) − ( M − K ) max ≤ y ≤| y k | ρ ( y ) Then K X m =1 log f Y ( y m | H ) f Y ( y m | H ) < log π − π − ( M − K ) max ≤ y ≤| y k | ρ ( y ) < log π − π − ( M − K ) max ≤ y ≤| y K | ρ ( y ) < log π − π − ( M − K ) log 1 − β y K ,H − β y K ,H (54)which is the MAP rule for a decision based on a block of the K most reliable measurements among M observations.A PPENDIX BIn this appendix we prove that when the throughput termsare preserved while all other costs, including parameter c , areset to zero, the lower detection threshold for the sequentialprocedure becomes always zero except for the last stage. Thismeans that, from a throughput point of view, the optimaldecisions based on the observations are either to declare H and then select a cognitive terminal for channel access, or tocontinue taking more samples so long as less than K LLRvalues have been gathered.We start by proving the concavity of function J Kk ( π k ) usinginduction. At the last stage J ( K ) K ( π K ) = min (cid:26) − r s (cid:18) − τ N + Kττ s (cid:19) π K , − r p (1 − π K ) (cid:27) (55) where r s = (1 − ω ) R s and r p = ωR p . For k < K , J ( K ) k ( π k ) = min (cid:26) − r s (cid:18) − τ N + kττ s (cid:19) π k , − r p (1 − π k ) , c + Ψ k ( π k ) (cid:27) (56)where Ψ k ( π k ) is given by Ψ k ( π k ) = Z J Kk +1 (cid:18) π k f π k f + (1 − π k ) f (cid:19) · h π k f + (1 − π k ) f i dY [ k +1] (57)Note that we use f in place of f Y [ k +1] ( y | H ) and f inplace of f Y [ k +1] ( y | H ) to simplify notation. Parameter c iskept in expression (56) and is set to zero after the concavityof J Kk ( π k ) is established.If Ψ k ( π k ) is concave, then c + Ψ k ( π k ) is concave and,consequently, J Kk ( π k ) is concave because J Kk ( π k ) is theminimum of three concave terms [36]. We now prove theconcavity of Ψ k ( π k ) under the assumption that J Kk +1 ( π k +1 ) is concave. Function Ψ k ( π k ) is concave if it satisfies thefollowing inequality κ Ψ k ( x ) + (1 − κ ) Ψ k ( z ) ≤ Ψ k ( κx + (1 − κ ) z ) (58) ∀ κ ∈ [0 , [36]. Let ǫ = κ [ xf + (1 − x ) f ] + (1 − κ ) [ zf + (1 − z ) f ] (59)The right-hand-side of (58) then can be written as Z (cid:26) κǫ h xf + (1 − x ) f i J Kk +1 (cid:18) xf xf + (1 − x ) f (cid:19) +1 − κǫ h zf + (1 − z ) f i J Kk +1 (cid:18) zf zf + (1 − z ) f (cid:19)(cid:27) ǫdY [ k +1] (60)Noting that κǫ [ xf + (1 − x ) f ]+ − κǫ [ zf + (1 − z ) f ] = 1 and using the assumption that J Kk +1 is concave, we obtain κ Ψ k ( x ) + (1 − κ ) Ψ k ( z ) ≤ Z J Kk +1 (cid:18) κǫ xf + 1 − κǫ zf (cid:19) ǫdY [ k +1] (61)Factor ǫ can be expressed as ǫ = ¯ ǫf + (1 − ¯ ǫ ) f , where ¯ ǫ = κx + (1 − κ ) z . Using this, inequality (61) becomes κ Ψ k ( x ) + (1 − κ ) Ψ k ( z ) ≤ Z J Kk +1 (cid:18) ¯ ǫf ¯ ǫf + (1 − ¯ ǫ ) f (cid:19) h ¯ ǫf + (1 − ¯ ǫ ) f i dY [ k +1] = Ψ k (¯ ǫ ) = Ψ k ( κx + (1 − κ ) z ) (62)Therefore, Ψ k ( π k ) is concave in π k assuming that J Kk +1 ( π k +1 ) is concave. It is evident from (55) that J KK ( π K ) is concave since it is the minimum of two affine terms. Byinduction both Ψ k ( π k ) and J Kk ( π k ) are concave in π k for k < K .We now study the value of J Kk ( π k ) at π k = 0 . It is clear that J KK (0) = − r p . Since Ψ K − (0) = R J KK (0) f dY [ K ] = − r p ,hen J KK − (0) = min { , − r p , c − r p } = − r p for c ≥ . Itis straightforward to show that Ψ k (0) = J Kk (0) = − r p for ≤ k < K .When π k = 0 , the minimum of the three terms of (56) is − r p obtained from the second term − r p (1 − π k ) for c > .The value of the derivative of J Kk ( π k ) at π k = 0 is equal tothe derivative of the term − r p (1 − π k ) at π k = 0 , which isequal to r p . The derivative of Ψ k ( π k ) is given by d Ψ k dπ k = Z dJ Kk +1 dπ k +1 dπ k +1 dπ k h π k f + (1 − π k ) f i dY [ k +1] + J Kk +1 ( π k +1 ) h f − f i dY [ k +1] (63)since π k +1 = π k f π k f +(1 − π k ) f , π k +1 = 0 when π k = 0 and itis straightforward to show the dπ k +1 dπ k at π k = 0 is equal to f f .Hence, d Ψ k dπ k (cid:12)(cid:12)(cid:12)(cid:12) π k =0 = Z (cid:18) r p f f f − r p [ f − f ] (cid:19) dY [ k +1] = r p (64)Since Ψ k is concave, it satisfies the following inequality [36] Ψ k ( z ) ≤ Ψ k ( x ) + d Ψ k dx (cid:12)(cid:12)(cid:12)(cid:12) x ( z − x ) (65)For x = 0 and z = π k , Ψ k ( π k ) ≤ − r p + r p π k = − r p (1 − π k ) . This means that when c = 0 , the third term in(56) is always less than or equal the second term − r p (1 − π k ) .Hence, the cost of getting one more sample is always less thanor equal to the cost of declaring H . 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