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Cooperative Quickest Spectrum Sensing in Cognitive Radios with Unknown Parameters Sepideh Zarrin and Teng Joon Lim Department of Electrical and Computer Engineering University of Toronto 10 King’s College Road, Toronto, Ontario, Canada M5S 3G4 Email: {szarrin, limtj}@comm.utoronto.ca

Abstract—In this paper, cooperative quickest spectrum sensing for cognitive radios is studied. Various cooperative schemes are considered based on the cumulative sum (CUSUM) algorithm, for different memory and communication constraint scenarios. The optimal CUSUM statistics are derived for each of these cooperative sensing schemes in the noisy channel scenario. In practice, due to unknown parameters in the distribution of the observations, the CUSUM-based approaches are not directly applicable to cognitive radios. Therefore, we propose to apply a linear test, which does not require any prior knowledge or estimates of the unknown parameters, for quickest spectrum sensing of cognitive radios. We derive linear-based CUSUM statistics for different cooperative sensing scenarios. The proposed approach results in fast and simple algorithms for cooperative quickest detection with unknown parameters, while maintaining a performance close to that of the perfectly known parameter schemes.

I. I NTRODUCTION Cognitive radio has recently emerged as a potential solution to the problem of spectrum scarcity and under-utilization in wireless communications [1], [2]. The cognitive radio technology relies on detection of white spaces in the allocated spectrum to the licensed (primary) users and opportunistic utilization of these bands by the unlicensed (secondary) users. In order to avoid any interference, the secondary users need to vacate the frequency bands as soon as the primary user starts its transmission. Thus, spectrum sensing plays a key role in cognitive radio. There has been plenty of research on spectrum sensing using classical detection schemes such as matched filtering, feature detection or energy detection schemes [3]–[5]. In such schemes, the secondary users derive their test statistics from observations within a fixed sensing time window. These blockbased detection schemes aim to maximize the probability of detection while maintaining a certain level of false alarm, using classical Neyman-Pearson detection theory. But practically, changes in primary-network activity within the frequency band of interest should be sensed as quickly as possible, in order to maximally utilize any vacated spectrum, as well as to minimize interference to the primary network after it resumes transmissions. Therefore, detection delay is an important criterion in spectrum sensing, especially in bursty applications. However, due to the block-based nature of conventional schemes and randomness of the primary activity time, detection delay has not been addressed in the above methods. Clearly, a change in the activity level of the primary transmitter results in a change in the probability distribution

of the observed signal at the secondary user. Therefore, sequential change-point detection (quickest detection) can be an appropriate framework for spectrum sensing in cognitive radios [8]. The well-known Page’s cumulative sum (CUSUM) algorithm [6] has been shown to be optimal in the sense of minimizing the detection delay while maintaining a certain level of false alarm [7]. [8] and [9] have considered CUSUM algorithm for spectrum sensing by a single cognitive radio. However, in practice, the quickest spectrum sensing based on a single user’s observations may not be reliable in detecting weak primary signals of unknown type due to fading or the hidden terminal problem between the primary and secondary user. In this paper, we present cooperative quickest spectrum sensing schemes based on the CUSUM algorithm to address the above problem in cognitive radios. The study of distributed quickest detection has been based on two different formulations in the detection literature. One is based on a Bayesian formulation in which the change point (= time at which the change occurs) is assumed to have a known prior distribution [10]. In this case, the joint optimization of local users and the fusion center policies over time becomes intractable [11]. The other formulation is the minimax formulation, proposed by Lorden [7], for which Page’s CUSUM procedure is the optimal scheme. The asymptotically optimal decentralized CUSUM approaches for limited and full local memory scenarios have been studied by Mei [11]. In this paper, we apply and compare three CUSUM-based cooperative spectrum sensing schemes in cognitive radios mainly based on Mei’s studies [11]: global CUSUM with soft local decisions, global CUSUM with quantized local decisions and hard fusion of local CUSUM. We derive the local and global test statistics in each scheme assuming noisy channels between the secondary users and the fusion center. In cognitive radios, an important limitation is the lack of knowledge about the primary signal at the secondary users. In particular, when the primary user starts transmission, there are unknown parameters in the distribution of the observed signals, e.g. the variance of the observed signals, due to unknown primary signal statistics and channel gains between the primary and secondary users. Therefore, the CUSUMbased approaches that are based on perfectly known distributions cannot be directly applied to spectrum sensing in cognitive radios. The generalized likelihood ratio test (GLRT) was adopted by Lorden [13] for CUSUM with unknown

978-1-4244-4148-8/09/$25.00 ©2009 This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE "GLOBECOM" 2009 proceedings.

parameters. However, due to the non-recursive expression of the GLRT-based CUSUM and the need to store all the observations and re-estimate the unknown parameters in all time slots, this algorithm turns out to be infeasible and impractical. Alternatively, we propose a linear test [16] for quickest spectrum sensing which does not require the maximum likelihood (ML) estimates of the unknown parameters and has the same asymptotic performance as the GLRT. We derive the test statistics of the linear test version of the CUSUM-based cooperative sensing schemes and show that the CUSUMbased schemes using the linear test perform closely to the optimal CUSUM-based schemes, without requiring knowledge of the primary signal or channel gains. Unlike the successive refinement algorithm in [9], our scheme does not require any iterative algorithm or any parameter estimation and, thus, it is much faster and less complex to implement. It has also a much simpler implementation and much better performance compared to the nonparametric algorithm adopted in [8]. The remainder of this paper is organized as follows. The system model is described in Section II. The CUSUM-based cooperative quickest sensing schemes with known parameters are presented in Section III. The cooperative quickest detection schemes with unknown parameters are presented in Section IV where the linear test is applied to the CUSUM-based schemes. The simulation results are provided in Section V. Finally, Section VI concludes the paper. II. S YSTEM M ODEL We assume L secondary users are monitoring the frequency band allocated to the primary user. Let Xl,i denote the observation at the lth secondary user at time i. When the primary user is not active Xl,i = Nl,i , where Nl,i is zero-mean 2 . When the primary white Gaussian noise with variance σN l user is using the band Xl,i = Zl,i + Nl,i , where Zl,i is the faded received primary signal at the lth secondary user. We assume time-invariant channel gain hl between the primary user and the lth secondary user, and Zl,i = hl Si to be zero2 mean Gaussian with variance σZ . l We assume that the primary user is initially inactive, and at an unknown time τ it becomes active. Thus there is a 2 ) change in the distribution of the observations from N (0, σN l 2 2 to N (0, σZl + σNl ) at unknown time τ . The sequential change detection modeling of this problem is to detect this change as quickly as possible while maintaining a certain false alarm probability. Defining the detection time T as the time at which the change is detected, if T > τ then the detection delay is δ = T − τ . If T < τ , a false alarm has occurred with the average run length (ARL) of false alarm being Tf = E0 [T ], where E0 denotes the average before the change. The detection of the primary user vacating the band can be approached similarly. The non-Bayesian approach to this problem based on Lorden’s formulation [7] minimizes the worst case detection delay, E1 [T ] = sup(ess sup E[T − τ |T ≥ τ, X1τ ]) τ ≥1

(1)

detection delay, Td = sup Eτ [T − τ |T ≥ τ ] τ ≥1

(2)

which is asymptotically equivalent to worst case delay [11]. The algorithm that finds the minimum Td in this problem, is Page’s CUSUM algorithm [6]. The stopping time in this algorithm for a single user detection problem is determined as follows, (3) T (q) = inf{n : Cn ≥ q} with cumulative statistic Cn = max

1≤k≤n

where Li (Xi ) = log

n

Li

(4)

i=k

f1 (Xi ) f0 (Xi )

(5)

and f0 (Xi ) and f1 (Xi ) are the distributions of Xi before and after the change, respectively. The cumulative statistic Cn can be recursively calculated for n ≥ 1 as follows Cn = max(Cn−1 , 0) + Ln (Xn )

(6)

where C0 = 0. In other words, this algorithm finds the first n for which Cn ≥ q > 0. The only analytical results on quickest detection performance bounds and their relations to the threshold q deal with asymptotic scenarios where q → ∞ [11], [12], which are not practical. The analysis in [11] may be used to asymptotically bound the performance of the schemes in the next section. III. C OOPERATIVE CUSUM WITH F ULL K NOWLEDGE OF PARAMETERS Due to fading and shadowing effects in cognitive radio networks, detection performance improves with the number of available sensors that make independent observations. In this section, we introduce cooperative quickest sensing schemes for cognitive radios, given full knowledge of all parameters. We apply CUSUM-based approaches mainly based on [11] to various local memory and transmission rate scenarios, and derive the necessary test statistics. In all these methods, the local decisions are sent to a decision making fusion center, which makes the global final decision. In the next section, we will tackle the problem of unknown parameters. A. Global CUSUM Assuming noiseless error-free channels between the secondary users and the fusion center, the local observations can be completely provided to the fusion center. Thus, the global CUSUM can be applied at the fusion center based on the local observations. In this case, the CUSUM test statistic at the fusion center can be written as

while maintaining the ARL of false alarms larger than a certain threshold, Tf ≥ λ. An alternative formulation is to use average

Cn = max

1≤k≤n

n L

Lli (Xl,i ).

(7)

i=k l=1


In our assumed model, f1l (Xl,i )

=

f0l (Xl,i ) −

1 e 2 +σ 2 ) 2π(σN Z l

=

√ 1 2πσNl

2 Xl,i 2(σ 2 +σ 2 ) Nl Zl

−

e

2 Xl,i 2σ 2 Nl

and

, thus the global

l

CUSUM statistic is derived as n L 2 2 2 σ Zl Xl,i σN 1 l Cn = max log 2 . 2 + 2σ 2 (σ 2 +σ 2 ) (8) 1≤k≤n 2 σNl +σZ Nl Zl Nl l i=k l=1

The fusion center declares that the primary user is active as soon as Cn exceeds threshold q. In practice, due to communication constraints in cognitive radios, the exact local observations can not be completely provided to the fusion center. Therefore, we consider the quantized local observation scenario in the next section.

Thus, lr(Xl,k ) ≥ al,k is equivalent to ⎞ ⎛ 2 + σ2 2 2 al,d σN (σ 2 + σZ )σN Z l l 2 l l ⎠ Nl ≥ log ⎝ Xl,k 2 σNl σZ l

(14)

where the right hand side of the inequality (14) is the equiva2 , which we call al,d . Given the Gaussian lent threshold for Xl,k 2 distribution of Xl,k , Xl,k will be exponentially distributed

with parameters 2σNl and 2 respectively. Therefore, 2 P [al,d−1 ≤ Xl,k

≤

al,d |H0 ]

2 + σ 2 , under H and H σN 0 1 Zl l

−

=e

−

2 P [al,d−1 ≤ Xl,k ≤ al,d |H1 ] = e

al,d−1 2σN l

2

a

− 2σl,d

−e

,

Nl

al,d−1 σ 2 +σ 2 Nl Zl

√

−

−e

2

√

al,d

σ 2 +σ 2 Nl Zl

.

Consequently, the log-likelihood ratio is derived as B. Global CUSUM with Quantized Local Decision The case of unlimited bandwidth between the fusion center and the secondary users have been considered in the previous section. However, bandwidth constraints in wireless communications implies quantization of the local observations before they are sent to the fusion center. For memoryless local users, it has been shown that the monotone likelihood ratio quantizer (MLRQ) is the optimal local mapping [11]. In this mapping, the lth user local decision at time n is Ul,n = d if and only if al,d ≤

f1 (Xl,n ) < al,d+1 f0 (Xl,n )

(9)

where 0 = al,0 . . . ≤ al,Dl −1 ≤ al,Dl = ∞ are the lth user quantization thresholds. These local decisions are sent to the fusion center over noisy channels characterized by p(Yl,k |Ul,k ). The fusion center applies Page’s CUSUM scheme with stopping time T (q) = inf{n : Cn ≥ q}

(10)

with C0 = 0 and Cn = max(Cn−1 , 0) +

L

Lln (Yl,n ),

where Lln (Yl,n ) is the log-likelihood ratio of Yl,n at the fusion center. Using the graphical approach in [15], we derive this log-likelihood ratio as follows Dl p(Yl,k |d)P [al,d−1 ≤ lr(Xl,k ) ≤ al,d |H1 ] Llk (Yl,k )=log Dd=1 l p(Y l,k |d)P [al,d−1 ≤ lr(Xl,k ) ≤ al,d |H0 ] d=1 (12) f (Xn,l ) where lr(Xl,k ) = f01(Xn,1 and H and H represent X ∼ 0 1 l,n ) 2 2 2 ) and X ∼ N (0, σ + σ ) respectively. In order N (0, σN l,n Nl Zl l to calculate P [al,d−1 ≤ lr(Xl,k ) ≤ al,d |Hθ ], for θ = 1, 2, we need to translate the MLRQ thresholds into thresholds for the observations Xl,k . For Gaussian distributed local observations, the likelihood ratio is derived as 2 2 Xl,k σZ σNl l lr(Xl,k ) = . (13) exp 2 + σ 2 )σ 2 (σN Zl Nl σ2 + σ2 l Zl

d=0

−

p(Yl,k |d)[e

Dl −1 d=0

2

al,d

√

σ 2 +σ 2 Nl Zl a − 2σl,d Nl

p(Yl,k |d)[e

−

−e

a

2

√ l,d+1 2 2

σ +σ Nl Zl

a l,d+1 − 2σ Nl

−e

] .

]

(15) For the case of binary local decisions with thresholds al , the likelihood ratio becomes −

Llk (Yl,k )=log

p(Yl,k |0) + [p(Yl,k |1) − p(Yl,k |0)]e

2

al σ 2 +σ 2 Nl Zl

√

a

− 2σ l

p(Yl,k |0) + [p(Yl,k |1) − p(Yl,k |0)]e

.

Nl

(16) The algorithm stops and the presence of the primary user is detected right after Cn ≥ q. C. Hard Fusion of Local CUSUM In this section we assume that each secondary user has sufficient memory to individually perform the CUSUM algorithm and send its local decision to the fusion center. The CUSUM test statistic at the lth user is derived as Cnl

(11)

l=1

Nl

Llk (Yl,k )=log

Dl −1

= max

1≤k≤n

n

Lli (Xl,i ),

(17)

i=k

where the log-likelihood ratio Lli (Xl,i ) is derived as 2 2 2 σ Zl Xl,i σN 1 l l + Li (Xl,i ) = log 2 2 2 2 2 ) . (18) 2 σNl + σZl 2σNl (σZl + σN l After each update the lth user sends its local decision 1 Cnl ≥ ql Ul,n = 0 otherwise,

(19)

to the fusion center. The fusion center can use any of the wellknown hard-decision combining schemes, such as OR, AND, M -out-of-L rules, to make the final decision. The stopping time in the M -out-of-L rule scenario can be written as T (q) = inf{n : Cnl ≥ ql for at least M out of L users} (20) where q = [q1 , . . . , qL ] captures all L users’ local thresholds.


IV. C OOPERATIVE CUSUM WITH U NKNOWN PARAMETERS All the CUSUM-based approaches are based on perfect knowledge of the likelihood functions. In cognitive radios, however, the parameters of the distribution under H1 are not known due to unknown primary signal statistics and channel gains. Therefore, detection schemes that accommodate unknown parameters should be employed. A. The GLRT for Quickest Detection Lorden [7] proposed to apply the GLRT in CUSUM algorithms with uncertain parameters. Let us denote the unknown parameters by β. The GLRT replaces the unknown parameter with its ML estimate. Thus the basic CUSUM statistic in (4) can be replaced with n f1 (Xi |β) Cn = max max . (21) log 1≤k≤n β f0 (Xi ) i=k

Equation (21) implies that we can no longer have a recursive expression for the CUSUM as in (6), since the GLRT requires a re-calculation of β for each k, using all the observations up to the current time n. Thus, in practice, the GLRT approach is hard or even impossible to be implemented in CUSUMbased systems. Therefore, for quickest detection schemes, we propose to use the asymptotically equivalent tests for GLRT that do not require the ML estimate of the parameters and are practical to implement. B. The Rao Test for Quickest Detection Schemes The Rao test is asymptotically equivalent to the GLRT, and does not require the ML estimate of the unknown parameters thereby making it simpler computationally [13]. In classical detection, the Rao test statistic for observation set X = {X1 , . . . , Xn } is given by ∂ log f1 (X|β) T ∂ log f1 (X|β) R(X) = I−1 (0) (22) ∂β ∂β β=0 β=0 where I(0) is the Fisher information matrix (see [14] pg. 40) evaluated at β = 0. However, as the quadratic form implies the Rao test can not result in a recursive expression for the CUSUM as in (6) and thus, is not applicable to CUSUM-based approaches. The calculation of the inverse Fisher information matrix, on the other hand, is computationally demanding in multi-parameter scenarios. Therefore, in next section, we propose a linear test which results in a recursive expression for the test statistic and is simple to implement. C. The Proposed Linear Test for Quickest Detection Schemes The standard Rao test is a two-sided test, which necessitates the quadratic form in (22). In our model, however, the un2 which are nonnegative. As a result, known parameters are σZ l the parameter test is one-sided which eliminates the necessity of the quadratic form. Therefore, we propose a modified test with a linear statistic of L ∂ log f1 (X|βl ) Λ(X) = . (23) ∂βl βl =0 l=1

As (23) shows, the linear test replaces each log-likelihood ratio term depending on βl , with its derivative at βl = 0 which distributes over the summation in CUSUM test statistic unlike the ML estimation in the GLRT approach in (21). Therefore, unlike GLRT-based approaches it does not require storing the observations and re-estimating the parameters at every sampling interval. The linear test is easily applicable to CUSUM-based algorithms, unlike the Rao test which has a quadratic form and does not allow for a recursive expression for the test statistic. We observe that the above linear test for a one-parameter scenario is identical to the locally most powerful (LMP) test [13] which is locally optimal for small departures from zero. We apply the linear test for all ranges of signal power, whereas the LMP is restricted to small signal powers. Moreover, the linear test accommodates multiple 2 ’s in our case), while the LMP unknown parameters (e.g., σZ l is confined to scalar parameter tests. In this section, we apply the linear test (23) to the cooperative CUSUM-based schemes discussed in Section III, when the received signal variances are not known upon primary user activity. All the proposed linear-based schemes in this section, do not require any iterative algorithm and any parameter estimation. As a result, they are faster and less complicated compared to the successive refinement algorithm in [9] and the nonparametric algorithm adopted in [8] which are applied for non-cooperative quickest detection. 1) Global CUSUM with linear test: Although the primary signal and channel statistics may not be known at the fusion center, we can still apply the global CUSUM algorithm by adopting the linear-based approach as follows. By denoting 2 and replacing the lth log-likelihood ratio term in (8) βl = σZ l with its derivative with respect to βl evaluated at βl = 0, we derive the recursive linear-based global CUSUM statistic as follows L 2 X 1 l,n λ Cnλ = max(Cn−1 , 0) + (24) 2 2 −1 2σN σN l l l=1

= 0 and the superscript λ represents the linear test where approach. The algorithm will stop as soon as it finds the first n ≥ 1 that satisfies Cnλ ≥ q λ . 2) Global CUSUM using quantized local decisions with linear test: The linear-based algorithm for global CUSUM with local MLRQ can be developed as follows. We replace the lth log-likelihood ratio term in (11) with its derivative evaluated at βl = 0 and derive the linear-based global CUSUM recursively as C0λ

λ , 0) + Cnλ = max(Cn−1

L

Λln (Yl,n )

(25)

l=1

where Dl −1 Λln (Yl,n ) =

d=0

a

p(Yl,n |d) 4σ13 N

l

Dl −1 d=0

− l,d [al,d e 2σNl a − 2σl,d Nl

p(Yl,n |d)[e

− −al,d+1 e a l,d+1 − 2σ Nl

−e

al,d+1 2σN l

]

]

(26) and C0λ = 0. The primary user is declared to be present right after Cnλ goes beyond threshold q λ .


0

10 1−bit global CUSUM 2−bit global CUSUM OR fusion of local CUSUM 3−out−of−6 fusion of local CUSUM 3−bit global CUSUM global CUSUM

−1

10

1−bit global CUSUM with linear test 1−bit global CUSUM 3−out−of−6 local CUSUM with linear test 3−out−of−6 local CUSUM global CUSUM with linear test global CUSUM

−1

10

−2

Pf

Pf

10

−3

10

−3

10

−4

10

Fig. 1.

−2

10

−4

5

10

15 Td

20

10

25

3) Hard fusion of local CUSUM with linear test: When the secondary users have sufficient memory but no knowledge of the primary signal and channel statistics, they can individually perform the linear-based CUSUM algorithm. The test statistic at the lth user can be recursively derived by taking derivative 2 and evaluating it at βl = 0: from (17) with respect to βl = σZ l 2 Xl,n 1 λ λ Cl,n = max(Cl,n−1 , 0) + 2 (27) 2 −1 2σNl σN l λ = 0. The lth user decision sent to the fusion center where Cl,0 at time n is determined as 1 , C λ ≥ qλ l,n

=

l

(28) 0

5

10

15

20

25

Td

Probability of false alarm, Pf , versus ARL of detection delay Td .

λ Ul,n

0

, otherwise.

The global stopping time depends on the combining rule at the fusion center. For OR combining rule, for example, the λ algorithm stops as soon as Cl,n exceeds qlλ for at least one of the users. V. S IMULATION R ESULTS In this section, we present our simulation results to evaluate the performance of the cooperative quickest detection schemes presented in this paper. We consider both cases of known and unknown parameters and show the performance of the proposed linear-based schemes. We assume that L = 6 secondary users independently sense the primary user’s spectrum. The received signals at the secondary users are assumed to be initially drawn from N (0, 1) and after change time τ , when the primary user becomes active, drawn from N (0, 3). The change time τ is assumed to be geometrically distributed with parameter ρ = 0.0463. Fig. 1 shows the probability of false alarm versus the average detection delay for the CUSUM-based cooperative quickest sensing schemes discussed above for the knownparameter scenario. The local quantization thresholds are set as

Fig. 2.

Probability of false alarm, Pf , versus ARL of detection delay Td .

follows: for the one-bit global CUSUM the local threshold is a = 1/3, for the 2-bit global CUSUM a1 = 1/2, a2 = 1 and a3 = 3/2, and local thresholds of the 3-bit Global CUSUM are a1 = 1/3, a2 = 1/2, a3 = 2/3, a4 = 5/6, a5 = 7/6, a6 = 5/3 and a7 = 13/6. As shown in [10], constant local quantization thresholds would work as the optimal policy. From Fig. 1, we observe that the global CUSUM scheme performs the best, as we expected, due to the assumption of availability of perfect local observations at the fusion center. The global CUSUM with one-bit local decisions performs the worst, as the global CUSUM is applied based on only one bit of information about the local observations. We can also observe that by having only three bits of information about the local observations, the global CUSUM performs better than the hard fusion of local CUSUM, where each user requires sufficient memory to perform CUSUM. Fig. 2 depicts probability of false alarm versus average detection delay curves for cooperative quickest sensing schemes based on linear test under the unknown-parameter scenario. We observe that the linear-based cooperative quickest sensing schemes perform closely to the their known-parameter coun2 . terparts without requiring the knowledge or estimation of σZ l We can also observe that the linear-based 1-bit global CUSUM performs the closest to its known-parameter counterpart compared to the linear-based hard fusion and global CUSUM schemes. This can be intuitively observed from difference in its test statistic (26) from the test statistics of linear-based hard fusion and global CUSUM in (24) and (27) respectively. Once again, we note that unlike the LMP approach, the linear-based schemes used here are not restricted to small signal powers (upon primary user activation, the received signal power at the 2 = 2 compared to the noise variance secondary users are σZ l 2 σNl = 1). Table I shows the ARL of detection delay for different cooperative quickest sensing schemes maintaining certain levels of ARL of false alarm. We observe that all the cooperative


Cooperative quickest sensing scheme Global CUSUM Global CUSUM with linear test 3-out-of-6 fusion of local CUSM 3-out-of-6 fusion of local CUSM with linear test OR fusion of local CUSUM OR fusion of local CUSUM with linear test 3-bit global CUSUM 3-bit global CUSUM with linear test 2-bit global CUSUM 2-bit global CUSUM with linear test 1-bit global CUSUM 1-bit global CUSUM with linear test single CUSUM single CUSUM with linear test

Tf = 20 1.94 2.23 2.49 2.88 2.70 3.15 3.12 3.67 4.35 4.40 5.42 5.72 6.36 6.85

Tf = 25 2.59 2.61 3.27 2.39 3.66 3.84 4.68 4.82 5.23 5.57 7.02 7.63 9.26 9.54

TABLE I ARL OF DETECTION DELAY FOR VARIOUS COOPERATIVE QUICKEST SENSING SCHEMES .

quickest detection schemes outperform the single CUSUM scheme. Once again, we observe that the global CUSUM with available observations at the fusion center outperforms the other schemes and the global CUSUM with one-bit local decision results in the highest detection delay among the discussed cooperative quickest sensing schemes. Moreover, we observe that the linear-based schemes perform closely to their known-parameter counterparts while not requiring any parameter estimation.

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VI. C ONCLUSIONS In this paper, we presented a linear test approach for cooperative quickest sensing in cognitive radios. We presented several cooperative quickest sensing schemes based on the CUSUM algorithm and derived their test statistics under noisy and noiseless channel scenarios. However, due to unknown parameters in the distribution of the observations, such as primary signal variance, the CUSUM-based approaches are not directly applicable to cognitive radios. For this reason, we proposed a linear test for cooperative spectrum sensing, which does not require any prior knowledge or estimation of the primary signal or channel statistics. Unlike other proposed schemes, the linear-based schemes do not lead to any iterative algorithm or any parameter estimation and, thus, are much simpler and faster to implement. We showed that linearbased cooperative quickest sensing schemes perform closely to their known-parameter counterparts, while not relying on prior knowledge or estimation of the uncertain parameters. R EFERENCES [1] J. Mitola and G. Q. Maguire, “Cognitive radios: making software radios more personal,” IEEE Personal Communacions, vol. 6, no. 4, pp. 13-18, Aug. 1999. [2] S. Haykin, “Cognitive radio: Brain-empowered wireless communications,” IEEE Journal on Selected Areas in Communications, vol. 23, no. 2, pp. 201-220, Feb 2005. [3] R. Tandra and A. Salehi, “SNR walls for signal detection,” IEEE Journal on Special Topics in Signal Processing, vol. 2, pp. 4-17, Feb. 2008 [4] A. Ghasemi and E. S. Sousa, “Collaborative spectrum sensing for opportusitic access in fading enviroments,” in Proc. IEEE Int’l Symposium on New Frontiers in Dynamic Spectrum Access Networks (DySPAN), Baltimore, MD, Nov. 2005.
