Detection efficiency of cooperative spectrum sensing in cognitive radio network

The Journal of China Universities of Posts and Telecommunications September 2008, 15(3): 1–7 www.buptjournal.cn/xben

Detection efficiency of cooperative spectrum sensing in cognitive radio network CHEN Xing, BIE Zhi-song, WU Wei-ling Key Laboratory of Information Processing and Intelligent Technology, Beijing University of Posts and Telecommunications, Beijing 100876, China

Abstract

Sensing the spectrum in a reliable and efficient manner is crucial to cognitive radio. To combat the channel fading suffered by the single radio, cooperative spectrum sensing is employed, to associate the detection of multiple radios. In this article, the optimization problem of detection efficiency under the constraint of detection probability is investigated, and an algorithm to evaluate the required radio number and sensing time for maximal detection efficiency is presented. To show the effect of cooperation on the detection efficiency, the proposed algorithm is applied to cooperative sensing using the spectral correlation detector under the Rayleigh flat fading channel. Keywords cognitive radio, spectrum sensing, cooperative sensing, cyclostationarity

1

Introduction

According to actual measurement, wide ranges of already licensed spectrum are underutilized [1], although the frequency allocation charts are very crowded in most countries around the world. To satisfy the dramatic increasing demand for radio spectrum, cognitive radio [2] is proposed to realize the dynamic spectrum allocation, by enabling a rental system to get access to the spare spectrum of the licensed system. This technology will enhance spectrum usage without sacrificing the transmission quality of the licensed system, and it does not require any change to the licensed system. Spectrum sensing is a key of cognitive radio. Cognitive radio should demonstrate that it does not interfere with the transmission of the licensed system, otherwise the licensees will not permit renting out their spectrum. As the amount of interference is directly linked to the detection probability, a high detection probability must be achieved. As it shall detect the available spectrum periodically at time intervals, short enough, and vacant spectrum bands, in time, if the licensed system reaccesses the spectrum, cognitive radio will spend considerable resources for spectrum sensing. Thus, the efficiency of spectrum sensing is significant to the transmission quality of cognitive radio network. Reliability and efficiency are two sides of the performance of spectrum sensing. There are two approaches to improve the Received date: 06-10-2007 Corresponding author: CHEN Xing, E-mail: [email protected]

detection performance. One is to employ detection techniques with high performance at individual radios. For example, the spectral correlation detector [3], which utilizes the cyclostationary characters of modulated signals for spectrum sensing, can outperform the traditional energy detector [4]. The other approach is to conduct cooperative spectrum sensing [5], which combines the detection results of multiple radios to obtain a more detailed and correct sensing. As a single radio may face deep fading with respect to the licensed system, its detection result is unreliable. Exploiting the diversity gain provided by associated radios, cooperative sensing will improve the detection probability and mitigate the sensing requirements on individual radios. Several Refs. [6, 7] have discussed cooperative sensing, but the selection of sensing parameters are still open, which is essential to realize spectrum sensing, in practice. The sensing parameters include the probability of detection and false alarm, the decision threshold, the number of detection radios to be involved, and the sensing time, which denotes the time consumed by an individual radio for once sensing. In this article, the optimization problem of the detection efficiency in cooperative spectrum sensing is investigated and an algorithm to deal with the parameter selection is proposed. The algorithm is also applied to cooperative sensing with each radio performing a spectral correlation detector. This article is structured as follows: Section 2 describes the system model. In Sect. 3, the optimization problem of cooperative sensing is discussed, and the algorithm to

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compute the sensing parameters is also presented. Section 4 applies the algorithm to the sensing, by using the spectral correlation detector. Finally, this article is concluded in Sect. 5.

2

System model

The setup of cognitive radio network is shown in Fig. 1. In a rental system based on cognitive radio, a central controller chooses multiple radios for cooperative spectrum sensing. Each selected radio conducts its local detection and transmits its decision to the central controller. At the central controller, the final decision will be made by combining these radio’s decisions, and then the information about the spare spectrum bands will be broadcasted to all the radios.

2008

spectrum environment can share the results of detection radios. Furthermore, as some radios may present correlation caused by the shadowing effect, it will not make much sense to associate correlated radios. Reference [7] shows that the gain from increasing the number of correlated radios is asymptotically approximated to a limitation. Therefore, the cooperative scheme would like to select less, but independent radios, to meet the detection requirement. For the decision combination, there are several schemes in Ref. [4] that can be chosen. As the resource for signaling overhead is limited, especially before the spare spectrum is obtained, the OR-rule [2, 6] is preferred to be implemented, which requires the least overhead for radios transmitting their decisions. By OR-rule, the radios only transmit 1 bit decisions (1 or 0) for each observed frequency band to the central controller. Then the controller considers the licensed signal being present if any radio has detected it. The combination yields dc

Nd

*d

(2)

i

i 1

where d c is the final decision for certain frequency bands.

Fig. 1 Cognitive radio network setup

2.1

Local spectrum detection

For the ith radio, the goal of local detection is to distinguish between the two hypotheses, H 0 , for licensed signal absent, and H1 for licensed signal present.

H0 ­ni (t ); (1) ® h ( t ) s ( t )  n ( t ); H1 ¯ i i where xi (t ) is the signal received by the ith radio, xi (t )

i 1, 2,..., N d , t  (0, Td ) , N d stands for the number of radios selected by the central controller, and Td is the time consumed by the radio for once sensing. s (t ) is the licensed signal, hi (t ) represents the amplitude gain of the channel from licensed system to the radio, and ni (t ) denotes complex-valued additive white Gaussian noise (AWGN). The ith radio uses certain techniques to get the detection variable Yi about the licensed signal being present or not. Then, Yi is compared with threshold Oi to get the local decision d i , which is 1 when it represents H1 if Yi ! Oi , otherwise it is 0 for H 0 . 2.2

Cooperation scheme

It is unnecessary to involve all radios of the rental system in cooperative spectrum sensing. Other radios within the same

3

Cooperative spectrum sensing

In this section, the metrics for the performance of cooperative spectrum sensing are discussed. On the basis of the definition of detection efficiency, the optimization problem is described. Then an algorithm is provided to compute the sensing parameters to maximize the detection efficiency. 3.1

Metrics for cooperative spectrum sensing

Reliability and efficiency are two sides of the performance of cooperation sensing. Reliability can be measured by the probabilities of detection and false alarm for the whole network (denoted by Qd and Qf respectively), which have the following connections with the probabilities of detection and false alarm for individual radio (denoted by Pd, i and

Pf , i for ith radio, respectively) under the OR-rule, Qd

Pr( d c

Nd

1 | H 1 ) 1  – (1  Pd, i )

(3)

i 1

Qf

Pr( d c

Nd

1 | H 0 ) 1  – (1  Pf , i )

(4)

i 1

where Pd, i and Pf , i can be generally computed by Pd, i

Pr(Yi ! Oi | H 1 )

(5)

Pf , i

Pr(Yi ! Oi | H 0 )

(6)

To focus on the issue concerned, it is assumed that each detection radio is independent and has the same detection and

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CHEN Xing, et al. / Detection efficiency of cooperative spectrum sensing in cognitive radio network

false alarm probability. Let Pd

Pd, i , Pf

Pf , i , and O

Oi

for all i . Under the fading channel, Pd and Pf should

3

model is adapted to describe the requirement of the licensed system about the spectrum sensing, which is illustrated in Fig. 2.

represent the average probabilities of detection and false alarm. This assumption is valid when the coverage of the rental system is small compared to the distance between the licensed system and the rental system, and all radios are considered in the same region and equipped with identical spectrum sensing detectors with the same parameters, such as, the sensing time Td and decision threshold O . Thus the values of path loss experienced by the radios are equal, which also means the radios have equal average signal-to-noise ratio (SNR). Moreover, the channel corresponding to each radio is independent and follows identical distribution. The noise is modeled with the same variance V 02 . For efficiency, the detection efficiency K is chosen as the metric, which is defined by the proportion of the remaining resources that can be used for data transmission after the sensing process § N dTd · (7) K ¨1  ¸ (1  Qf ) ¨ N shareTcycle ¸¹ © where N share denotes the number of radios sharing the sensing result, from which the N d radios are selected to conduct the cooperative sensing. Td is the sensing time, and Tcycle denotes the period of detection cycle.

Equation (7) only considers the resources in time dimension. Actually, the frequency and space dimensions can also be exploited. It should be noted that Eq. (7) implies ( N share  N d )

Fig. 2

The model of the spectrum sensing requirement

In this model, when the rental system is at a certain distance from the licensed system, the probability of detection should not be less than Qdo and transmission power should not exceed Po . The more close to the licensed system the rental system is, the higher Qdo or lower Po will be demanded. As the distance is equivalent to the average SNR, J , of the licensed signal, the requirement is described as a triple data J , Qdo , Po which will be obtained in advance. Pd and Pf can be represented as: Pd

f d (J , Td , O )

(8)

Pf

f f (Td , O )

(9)

f d (˜) and

where

f f (˜) indicate the sensing parameters’

relations of some kind of detector. As Pf is considered for the case of signal absent, it is independent of J . Substituting the Eq. (8) into Eq. (3) yields, Oİf d1 J , Td , 1  N 1  Qdo





d

(10)

radios are able to work for a transmission, whereas, the other N d radios are in the sensing process, otherwise N d should

where f d1 ( a, b, ˜) is the inverse function of f d ( a , b, ˜) .

be equal to N share . If the detection radio is capable of

Pf ıf f Td , f d1 J , Td , 1  N d 1  Qdo

transmitting at the same time as spectrum sensing, Td shall be zero. In this article, N share and Tcycle are assumed to be some constants, and Td ! 0 . 3.2

Optimal detection efficiency

To meet its own transmission requirement, the rental system needs to minimize the false alarm probability and maximize the detection efficiency. However, as the licensed system has the highest priority for the spectrum usage, it will require the rental system to fulfill some mandatory requirements first. To avoid sacrificing the transmission quality of the licensed system, it demands that the interference from the rental system does not exceed certain upper bounds. Abstracting from the complicated interference relationship between licensed system and rental system, a simple but reasonable

Thus,







(11)

and the optimization problem becomes ­° § N dTd · 1 K max ® ¨1  ¸¸ 1  f f Td , f d  J , Td , 1  N d , Td ¨ °¯ © N shareTcycle ¹





Nd

½ ¾ ¿ where 0  N d İN share and 0  Td İTcycle . Nd

3.3

1  Qdo

(12)

Optimization algorithm

Usually, it is difficult to get the exact closed-form expression of maximal K and corresponding N d and Td from Eq. (12), especially for the case of fading channel. To utilize as much spare spectrum as possible for the transmission of the rental system, it is reasonable to require the probability of false alarm to be less than a certain Qfo . The optimization problem is considered under the constraint

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Qf İQfo besides Qd ıQdo . Thus, the maximal K can be

achieved by the following algorithm: 1) Set N d, n n ( n 1, 2,..., N share ) . 2) For each N d, n , compute the lower bound of Pd and the upper bound of

Pf

(denoted by

Pd, n

and

Pf , n ,

respectively) Pd, n 1  Nd , n 1  Qdo

(13)

Pf , n 1  Nd , n 1  Qfo

(14)

3) For each N d, n , evaluate the corresponding Td, n and

4.1

SCF detector

Supposing a radio in a rental system receives the signal vector X (t ) [ x(t ), x(t  Ts ),..., x(t  (2TdW  1)Ts )] at time t where 2TdW is the number of samples, Ts is the time

interval of samples, and one-side bandwidth W

efficiency ( N d , Td ) arg

min

0  N d ,n İN share 0 Td ,n İTcycle

( N d, nTd, n )

(16)

5) Compute the maximal K associated with N d and Td by Eq. (7). This algorithm makes it easy to get the maximal K and the required N d and Td . Even if f d (˜) and f f (˜) are very complex or have no close-form expression, Eq. (15) can still be solved by numerical analysis.

4 Spectral correlation detection In traditional spectral analysis, the energy detector detects the spectrum by measuring the energy of the received signal in a certain frequency band, whose performance is poor especially in severe noise and interference environments. Although the spectral correlation detector is complicated when compared with the energy detector, it is worth it because of its outstanding performance in discriminating the licensed signal in the background of noise and interference signals from other rental systems. The spectral correlation detector is the most popular of all kinds of detectors, based on the cyclostationary theory. It first generates the spectral correlation function (SCF) of the received signal, then searches for the cyclostationary characters (namely cyclic-feature), which are only generated by the licensed signal, and finally makes a decision on whether the licensed signal is present or not. In this section, the probability distribution of the SCF is presented, so that the probabilities of detection and false alarm of the detector can be calculated. Then the optimization algorithm is applied to cooperative sensing using the spectral correlation detector.

1/(2Ts ) .

Using a discrete time-averaging cyclic periodogram [8, 9], the SCF of a received signal can be estimated by the vector X (t ) , which is expressed as S xD (t , f )

On by solving the equations array as follows: ­° Pd, n f d (J , Td, n , On ) (15) ® °¯ Pf , n f f (Td, n , On ) 4) Choose the N d and Td , which maximize the detection

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X (t , f )

1 JM

JM 1

1

§

¦ N X ¨© t  u 0

N 1

¦ x(t  nT )e s

uNTs · § uNTs · , f1 ¸ X * ¨ t  , f 2 ¸ (17) J J ¹ © ¹

 j2ʌf ( t  nTs )

(18)

n 0

where S xa (t , f ) is the SCF of x (t ) at frequency f and cycle frequency D , f1 f  D / 2, f 2 f  D / 2, and D z 0. X (t , f ) is the discrete Fourier transform (DFT) of x (t ) frequencyshifted to baseband, and (˜)* for the conjugate operation. J, M, and N are the parameters with 2TdW (1  M  1/ J ) N . The SCF has the frequency resolution 'f cycle frequency resolution 'D 'f / M .

1/( NTs ) and

Assuming the spectral correlation detector finds out the licensed signal s (t ) has a cyclic-feature at point ( f , D ) , it is proved in the Appendix A that the distribution of S xD (t , f ) is approximate to a complex normal distribution when M is large enough, which is given by ­ § H 0V 04 · H0 °& ¨ 0, ¸; JM ¹ © ° D S x (t , f )  ® (19) 4 °& § S D ( f ), H 0V 0 §1  Ss ( f1 )  Ss ( f 2 ) · · ; H ¨¨ s ¨ ¸ ¸¸ 1 ° JM © V 02 ¹¹ © ¯ where & denotes the complex normal distribution, SsD ( f ) is the SCF of s (t ) at point ( f , D ) , Ss ( f ) is the power spectrum at f , H 0 and H1 are correction factors , which are related to J and M. Hence, the detection variable Yi of the spectral correlation detector, which is the magnitude of S xD (t , f ) , has a Rayleigh distribution for H 0 , and Rice distribution for H1 . To validate the distribution of Yi

S xD (t , f ) , Figure 3 compares

the probability density function (PDF) of approximate distribution with the exact PDF obtained from a simulation using 1 000 snapshots. It is obvious that the approximate distribution coincides with the exact distribution. In Fig. 3, x(t ) is normalized by the noise standard deviation V 0 , SNR

J

3 dB, SsD ( f )

S s ( f1 )

Ss ( f 2 )

N / 4, N

64, J

4,

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CHEN Xing, et al. / Detection efficiency of cooperative spectrum sensing in cognitive radio network

64, H 0

2.6 and H1

3.9 H1

3.9 . There is one issue

that must be noted. As the detection is based on only one cyclic-feature of the licensed signal, the SNR mentioned here is not the SNR of the whole signal, but the SNR of the two spectral components at frequency f r D / 2 , which generate this cyclic-feature.

5

the Rayleigh flat fading channel can be formulated as f 1 J Pd ³ Pd (J ) exp dJ 

J

0

(23)

J

The sample number 2TdW

and threshold O

of the

spectral correlation detector over Rayleigh channel can be evaluated by solving Eqs. (23) and (20). As the mean of S xD (t , f ) for H1 , SsD ( f ) , is in direct ratio to N , and the variance for H 0 and H1 is in inverse ratio to M . When 2TdW is fixed, higher N will benefit to separate the PDFs of H1 and H 0 . Therefore, M is fixed and N changed to increase 2TdW . Figure 4 shows the plots of the required sample number and threshold to achieve Pm 1  Pd (probability of missed

(a) H 0

detection) and Pf , with average SNR to be 5 dB, in which x (t ) is normalized by the noise standard deviation V 0 . It

can be found in the graph that if both Pm and Pf are required to be less than 102 , the sample number must be 639 at least, and threshold, 0.216. In Fig. 4, SsD ( f ) (b) H1 Fig. 3

4.2

Ss ( f1 )

Ss ( f 2 )

N / 2, J

Fig. 4

2TdW and O of SCF detector under flat Rayleigh channel

4, M 64, H 0

2.6 and H1

3.9 .

Comparison between exact and approximate PDF

Optimal detection efficiency of SCF detector

According to the PDF of

S xD (t , f ) , the false alarm

probability of the spectral correlation detector is formulated as § O 2 JM · (20) Pf exp ¨ 4 ¸ © H 0V 0 ¹ which will remain unchanged under any channel condition. However, the detection probability is dependent on channel condition. Pd under certain channel realizations is given by § J SsD ( f ) O · Q1 ¨ , ¸ (21) ¨ V V¸ © ¹ where J denotes the instantaneous SNR of the received Pd (J )

signal, O is the threshold normalized by V 02 , and

V

H1 2 JM

1  J Ss ( f1 )  J Ss ( f 2 )

(22)

where SsD ( f ) and Ss ( f ) are corresponding to the s (t ) with unitary energy per symbol. For fading channels, the Pd can be obtained by averaging the Pd (J ) over the corresponding SNR fading distribution.

For example, the SNR of Rayleigh flat fading channel
follows an exponential PDF, thus the average detection probability of

After obtaining the sample number for each possible number of detection radios, the maximum detection efficiency of the rental system can be figured out. Figure 5 shows the curves of the detection efficiency and required sample number for the cooperative sensing under Rayleigh flat fading channel. The sample number decreases sharply with an increase in N d at the beginning, and a maximum of 95% of K can be obtained by 10 detection radios. In Fig. 5, the average SNR J 10 dB , Pm Pf 103 , N share 100 , Tcycle 1 000Ts ,

SsD ( f ) and H1

Ss ( f 1 ) 4.

Ss ( f 2 )

N / 2, J

4, M

64, H 0

2.6

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The Journal of China Universities of Posts and Telecommunications

2008

(1  M  1/ J ) N . Then the SCF is computed by averaging a cross spectrum over these segments. Equation (17) is rewritten as 1 JM 1 D § uNTs · (A.1) S xD (t , f ) ¦ S x, seg ¨© t  J , f ¸¹ JM u 0 where S xD, seg (t , f ) is the cross spectrum for one segment. (a) Rquired sample number

1 § uNTs · § uNTs · (A.2) X ¨t  , f1 ¸ X * ¨ t  , f2 ¸ N © J J ¹ © ¹ X (t , f ) is the down-converted output of the DFTs of x (t ) at this segment. Here, f1 k1 f s N , f 2 k2 f s N ,

S xD, seg (t , f )

f s 1 Ts , and the integers k1 , k2  >  N / 2, ( N / 2)  1@ . In Eqs. (A.1) and (A.2), f

( f1  f 2 ) / 2 , D

f1  f 2 , D z 0 ,

and thus f1 z f 2 . Equation (18) can be rewritten as (b) Detection efficiency Fig. 5

2TdW and K of cooperative spectrum sensing using

SCF detector under Rayleigh flat fading channel

5

Conclusions

As sensing dependent on a single radio could not be trusted for it may suffer deep fading, cooperative spectrum sensing involving multiple radios is significant to fulfill the rigorous requirements. In this article, the optimization problem of detection efficiency with the constraint of detection probability is studied. An algorithm to obtain the maximal detection efficiency by selecting proper radio number and sensing time is proposed and applied to the cooperative sensing using a spectral correlation detector. It shows that the detection efficiency is improved dramatically with increasing the number of detection radios, but it will be degraded beyond a certain point. Acknowledgements

X (t , f ) [ x(t ) x(t  Ts ) ... x(t  ( N  1)Ts )] ˜ [0 e  j2ʌfTs ... e  j2ʌfNTs ]T

(A.3)

T

where [˜] for the transpose operation. 1) SCF for licensed signal absent When the licensed signal s (t ) is absent, the received signal x (t ) has only the component of noise. Then the vector X (t ) follows a complex Gaussian distribution as follows: [ x(t ) x(t  Ts ) ... x(t  (2TdW  1)Ts )] ~ & (0, V 02 I 2TdW ) (A.4) where the I 2TdW denotes the 2TdW u 2TdW identity matrix. Therefore,

X (t , f )

is subject to a complex Gaussian

distribution & (0, N V 02 ) . As [0 e  j2ʌf1Ts ... e  j2ʌf1 NTs ] is orthogonal with [0 e  j2ʌf2Ts ... e  j2ʌf2 NTs ] , X (t , f1 ) and X (t , f 2 ) are statistically independent. Supposing the complex random variable S xD, seg (t , f )

K  j] , it is deduced that K and ] are uncorrelated, and the characteristic function of (K , ] ) is given by, 1

This work is supported by the National Basic Research Program of China (2007CB310604), the National Natural Science Foundation of China (60772168).

Appendix A Derivation of SCF’s distribution The 1u 2TdW

signal vector X (t ) was received over

AWGN channel. Its SCF will be evaluated by the discrete time-averaging cyclic periodogram. This method cuts each vector into JM segments, and each segment has N samples. The last (1  1/ J ) N samples of each previous segment are overlapped with the next segment. J is the overlapping parameter, M represents the number of equivalent nonoverlapping segments in a vector, and 2TdW

§ V 4 (t 2  t 2 ) · º ¨1  0 K ] ¸ (A.5) )x , seg (tK , t] ) E ª¬e ¼ ¨ ¸ 4 © ¹ When J 1 , the signal vector is divided into M nonoverlapping segments, thus S xD (t , f ) is the average of j( tK K  t] ] )

M independent and identically-distributed random variables. Supposing the complex random variable S xD (t , f ) [  j9 , the characteristic function of ([ , 9 ) is expressed as M

§ V 4 (t 2  t 2 ) · º ¨1  0 [ 2 9 ¸ )X (t[ , t] ) E ¬ªe (A.6) ¼ ¨ ¸ 4M © ¹ Moreover, as K and ] are uncorrelated, [ and 9 are j( t[ [  t9 9 )

also uncorrelated. When M is large enough, S xD (t , f ) is approximated to a complex Gaussian distribution which is given by

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CHEN Xing, et al. / Detection efficiency of cooperative spectrum sensing in cognitive radio network

§ V4 · S xD (t , f ) ~ & ¨ 0, 0 ¸ (A.7) © M¹ When J ! 1 , for the reason of the overlapping between adjacent segments, the cross spectrums at different segments may not be independent. For different segments and f1 , f 2 ,

7

S xD (t , f ) follows a complex Gaussian distribution whose variance is the sum of the variances of SsnD (t , f ) , SnsD (t , f ) , and SnD (t , f ) . However, for J ! 1 , the segment overlapping D D also causes the correlation among Ssn, seg ( t1 , f ) , S ns, seg ( t2 , f ) ,

when

and Sn,D seg (t3 , f ) , whereas, the differences of their time are less

t1  t2 ıNTs , and they are correlated when t1  t2  NTs .

than NTs . The variance should be multiplied by another

X ( t , f1 )

and

X (t , f 2 )

are

only

independent

S xD, seg (t , f ) at different segments are also independent when

t1  t2 ıNTs , and correlated when t1  t2  NTs . Therefore,

correction factor H1 to correct the deviation. Then the distribution of S xD (t , f ) will be

the distribution shown as Eq. (A.7) is inaccurate for J ! 1 . However, the SCF still follows a complex Gaussian distribution. After multiplying the variance by a correction factor H 0 , the deviation caused by the segment overlapping is

§ H V 4 § S ( f )  S ( f ) ·· S xD (t , f ) ~ & ¨ SsD ( f ), 1 0 ¨1  s 1 2 s 2 ¸ ¸ ¨ ¸ JM © V0 ¹¹ © where H1 1 for J 1 , and H1 3.9 for J 4 .

corrected. Then the distribution will be § H V4 · S xD (t , f ) ~ & ¨ 0, 0 0 ¸ JM ¹ ©

References

where H 0

1 for J

1 , and H 0

2.6 for J

(A.8) 4.

2) SCF for licensed signal present When the licensed signal is present, the S xD (t , f ) has four components that are produced by licensed signal s (t ) and noise n(t ) ,

S xD (t , f )

SsD (t , f )  SsnD (t , f )  SnsD (t , f )  S nD (t , f )

(A.9)

D

Where Ss (t , f ) is the SCF of s (t ) , and can be regarded as a constant, SsD ( f ) , which is the mean of S xD (t , f ) .

SnD (t , f ) is the SCF of noise, which is equal to the S xD (t , f ) when the licensed signal is absent. The distribution of the cross-SCF of licensed signal and noise ( SsnD (t , f ) and

SnsD (t , f ) ) can be deduced by the similar process presented in Appendix A Sect. 1). Their distributions are shown as follows, respectively, § H V 2 S ( f  D / 2) · (A.10) SsnD (t , f ) ~ & ¨ 0, 0 0 s ¸ JM © ¹ § H V 2 S ( f  D / 2) · SnsD (t , f ) ~ & ¨ 0, 0 0 s (A.11) ¸ JM © ¹ When J 1 the four components are independent, therefore

(A.12)

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