We propose two novel cooperative detection schemes based on the AF (Amplify and Forward) and DF (Decode and Forward) protocols to achieve spatial diversity gains for cognitive radio networks, which are referred to as the AF-CDS, (AF-based Cooperative Detection Scheme) and DF-CDS (DF-based Cooperative Detection Scheme), respectively. Closed-form expressions of detection probabilities for the noncooperation scheme, AND-CDS (AND-based Cooperative Detection Scheme), AF-CDS and DF-CDS, are derived over Rayleigh fading channels. Also, we analyze the overall agility for the proposed cooperative detection schemes and show that our schemes can further reduce the detection time. In addition, we compare the DF-CDS with the AF-CDS in terms of detection probability and agility gain, depicting the advantage of DF-CDS at low SNR region and high false alarm probability region.
1. Introduction
Cognitive radio (CR), built on software-defined radio, has been proposed in [1] as a means to promote the efficient use of the precious radio spectrum resources. It is defined as an intelligent wireless communication system [2] that is aware of the surrounding environment and utilizes the methodology of understanding-by-building to learn from the environment. Spectrum detection technique (also referred to as spectrum sensing) enables CR networks to adapt to the environment by detecting spectrum holes, and the most efficient way to detect the spectrum holes is to detect the presence of primary users [3]. In reality, however, it is difficult for a cognitive radio to have a direct measurement of the channel between a primary receiver and a transmitter. Therefore, the most recent work focuses on the primary transmitter detection based on local observations of secondary users (see [4‐7]). Generally speaking, the spectrum sensing schemes proposed in recent years can be classified as noncooperative detection and cooperative detection.
At present, three noncooperative transmitter detection methods, namely, the matched filter detection, the energy detection and the cyclostationary feature detection, have been presented for CR networks. In [4], Sahai et al. have investigated the matched filter detector that can achieve high processing gain by employing coherent reception. In [5], energy detector has been put forward as an optimal method for the occasion where the secondary users cannot gather sufficient information about the primary user signal such as the modulation type, the pulse shape and so on, but it cannot differentiate signal types, thus inclining to false detection triggered by some unintended signals. The cyclostationary feature detection, as an alternative method, has been further presented in [6, 7], which can differentiate the modulated signal from the additive noise. However, this scheme is computationally complex and requires long observation time.
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As to the cooperative detection, a collaborative spectrum sensing method has been proposed by Ghasemi and Sousa in [8], where the hard decision about the presence of the primary user from each secondary user is pooled together to determine the presence of primary user by utilizing a majority logic rule without the consideration of cooperative technology that has been proven as an effective means to combat Rayleigh fading [9‐12]. More recently, Ganesan and Li [13, 14] have applied the AF protocol to the detection of primary users, and shown that by allowing the secondary users to cooperate with each other the detection time can be reduced. However, this cooperative scheme needs a centralized controller to manage all secondary users, which is some unreasonable for a practical wireless communication system. Besides, how to detect the presence of primary users through cooperation in DF-based CR networks, where the cooperative user has the ability to decode its received signal, is an open challenge.
In this paper, we address the above mentioned issues and present a more practical AF-based cooperative detection scheme without the assumption of centralized controller and a totally new DF-based cooperative detection scheme for the occasion where the cooperative relay has decoding ability. Our main contributions can be described as follows. Firstly, we propose two new cooperative detection schemes, namely, AF-CDS and DF-CDS, to detect the presence of primary users more quickly and accurately. Secondly, we develop the closed-form expressions of detection probability and detection time for both AF-CDS and DF-CDS over Rayleigh fading channels. Thirdly, the performance analysis for the noncooperation scheme and the AND-CDS (AND-based cooperative detection scheme) is also presented for the purpose of comparison with our schemes.
The remainder of this paper is organized as follows. Section 2 describes the system model used throughout this paper and proposes two new cooperative detection schemes (i.e., AF-CDS and DF-CDS) to improve the detection performance of cognitive radio network. In Section 3, we derive the closed-form expressions of detection probabilities and agility gains for the AF-CDS and DF-CDS as well as the traditional noncooperation scheme and AND-CDS, followed by numerical results analysis in Section 4, where we show the superiority of the proposed AF-CDS and DF-CDS schemes in terms of detection performances. Finally, we make some concluding remarks in Section 5.
2. Proposed Cooperative Detection Schemes
In this section, we first describe the system model used in the paper, and then propose AF-CDS and DF-CDS to improve the detection performance of CR networks.
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2.1. System Model
Consider a cognitive radio network with a primary user and two secondary users as shown in Figure 1, where the wireless link between the primary user and the secondary user occurs shadowing fading, and acts as a cooperative relay for All users in the CR network are equipped with a single antenna, and the antenna at any user can be utilized for both transmission and reception. There is independent, additive complex white Gaussian noise with zero-mean and double-sided power spectral density at each receiver.
Figure 1
Cooperative spectrum sensing model in cognitive radio networks.
×
Without loss of generality, let be the primary user indicator, namely, implies the presence of primary user and implies its absence. Therefore, the signals received by secondary users and can be expressed as
(1)
(2)
where the subscripts and denote the transmission from the primary user to and that from the primary user to respectively. Besides, , and are the fading coefficients of the wireless channel from the primary user to , from the primary user to , and from to , respectively. Note that the variances of the three random variables (RVs) , and are and respectively. Throughout the paper, we make following assumptions: All wireless channels are independent from each other in space; The relaying protocols (i.e., the AF and the DF) employ full duplex mode, meaning that the relay can perform signal reception when transmits.
2.2. AF-Based Cooperative Detection Scheme
An important requirement of a cognitive radio network is to detect the presence of primary users as quickly as possible. Suppose that the primary user starts using the spectrum band. Then, the two secondary users need to sense the unavailability of the band as soon as possible to avoid collision with primary user. However, when the wireless link between the primary user and encounters shadowing fading, the signal received by from the primary user is so weak that takes a long time to detect its presence. We show that by cooperation with the detection probability of can be increased, thus reducing the overall detection time of the CR network.
Throughout the paper, we allow the secondary user to act as a cooperative relay for . Figure 1 describes a scenario where two secondary users and are engaged in detecting the presence of primary user in a particular band. The whole implementation process of AF-CDS can be separated into two consecutive phases, that is, in odd time slot , both and receive the signals transmitted from primary user; in even time slot starts relaying its received information to in accordance with AF protocol, and thus, in this time slot, would receive two signal copies simultaneously from the primary user and respectively. In the time slot the signals received by and can be expressed as
(3)
(4)
where and are the additive complex Gaussian noise with zero mean and double-sided power spectral density and independent from each other. In the even slot the cooperative user, relays the message from primary user to According to AF protocol, the received signal as defined in (3) will be multiplied by a relay gain and then forwarded to without any sort of decoding. Meanwhile, in the slot simultaneously receives the signal from primary user. Therefore, the signal received by in even time slot can be written as
(5)
where denotes the instantaneous fading coefficient of the wireless channel from to and denotes the zero-mean additive complex Gaussian noise with double-sided power spectral density Substituting from (4) into (5) gives
(6)
For the convenience of theoretical analysis, consider relay gain to compensate the fading distortion from to . Substituting this result into (6) yields
(7)
Now, the detection problem of under AF-CDS can be stated as follows: given the observation
(8)
in the odd time slotand
(9)
in the even time slot the detector decides on
(10)
This is a standard detection problem for which there are many choices of detector available such as energy detector, matched filter detector and cyclostationary feature detector in the literatures [4‐7]. In this paper, we use the energy detector (ED) [15] to show the advantage of the proposed cooperative scheme. The reasons for choosing ED are two-fold [13]: We want to show the effect of user cooperation on detection of primary user in CR networks. Hence, the choice of detector is not critical; We model the signal as a random variable with known power, and thus ED is optimal [15]. Let and be the output power of the ED of the secondary user in the odd time slot under hypothesis and , respectively. Thus, from (8), we can easily obtain
(11)
Similarly, define and as the output power of 's ED in the even slot under hypothesis and , respectively. Hence, from (9), we can get
(12)
(13)
We will use (11)–(13) listed above to analyze the detection probability and detection time for the proposed AF-CDS in Section 3.
2.3. DF-Based Cooperative Detection Scheme
In this subsection, we present DF-CDS to achieve a better detection performance for CR networks. Consider the same scenario as AF-CDS with two secondary users and operating in a fixed TDMA mode for detecting the presence of primary user (see Figure 1). Here, according to DF protocol, the cooperative user should regenerate the primary user indicator based on its received signals, and then transmit the estimated indicator to . Consequently, the detailed process of DF-CDS can be described as follows: in the odd time slot , both and receive the signal from the primary user; in the even time slot decodes its received information and forwards the decoding result to . Clearly, in odd time slots, the process of DF-CDS is the same as AF-CDS, implying that the output power of energy detector of for DF-CDS in odd time slots can also be expressed as (11). Besides, in the even time slot the signal received by can be given by
(14)
One can see that there are two possible cases for the decoding result of the primary user indicator, namely, the correct and the wrong decisions. Without loss of generality, let cases and denote the estimated indicator and , respectively. Let and be the output power of 's ED in even time slots for case under hypothesis and , respectively. Thus, from (14), it is easy to obtain
(15)
Similarly, define and as the output power of 's ED for case under hypothesis and respectively. Thus, we can easily get
(16)
Now, we have described the system model and formulated the primary user detection problems for AF-CDS and DF-CDS, based on which a detailed performance analysis will be presented in the following.
3. Performance Analysis of the Proposed Schemes over Rayleigh Fading Channels
In this section, we investigate the detection probability and detection time for the proposed AF-CDS and DF-CDS in Rayleigh fading channels. For the purpose of comparison, let us consider first the noncooperative detection scheme and the existing AND-CDS (AND-based cooperative detection scheme) proposed in [8]. Given the transmitted signal with power , the signal received by the user can be expressed as
(17)
Hence, according to the energy detector principles, we can easily calculate the probability of detection of primary user by under noncooperative scheme as
(18)
where is energy detection threshold that is determined by false alarm probability, which will be illustrated in the following. Note that the PDF of RV can be given by
(19)
where is a unit step function. Thus, (18) can be further calculated by
(20)
Furthermore, from (17), the false alarm probability can be given by
(21)
Noting that is zero-mean complex Gaussian noise with double-sided spectral power density , we can easily obtain . Thus, the false alarm probability is calculated as
(22)
which results in
(23)
Substituting the threshold from (23) into (20) yields
(24)
where is the transmitting signal-to-noise ratio (T-SNR). In a similar way, the detection probability of under noncooperative scheme can be given by
(25)
We now investigate the overall detection probability for the cognitive radio network scenario depicted as Figure 1, that is, the probability that the presence of primary user is detected by both and . Since the two users independently detect the primary user in a distributed way, it can be easily shown that the overall detection probability is given by
(26)
where and are defined in (24) and (25), respectively. Besides, let be the number of slots taken by (and ) to detect the presence of primary user under the noncooperation scheme. This detection time can be modeled as a geometric random variable, that is,
(27)
It can be easily shown that the overall detection time is given by
(28)
Thus, from (28), we can calculate the average overall detection time as
(29)
where and Then, we focus on the detection performance analysis for the existing cooperative sensing as presented in [8], where an AND-based cooperative detection scheme (AND-CDS) is proposed in order to combat the fading environments. Similarly to the sense protocols of the AF-CDS and DF-CDS, the implementation process of the AND-CDS is also divided into two phases for the convenience of making a fair comparison with our scheme, that is, in odd time slot and detect independently whether or not the primary user is active; in subsequent even time slot , forwards its detected result to who would then use the AND rule to fuse the detection results of and . Clearly, it is easy to show the probability of detection of primary user by in odd time slots being given by
(30)
where is the false alarm probability. In even time slots, would fuse the decision results by using AND rule, and thus, the corresponding detection probability and false alarm probability are calculated as (see [8] for details)
(31)
(32)
Combining (31) and (32) gives
(33)
Hence, for any given time slot , we can calculate the probability of detection of primary user by using AND-based cooperation scheme as
(34)
where and are the probabilities of time slot belonging to odd and even, respectively. Generally, the probability events follow an equal probability distribution, giving . Substituting this result into (34) yields
(35)
In addition, the detection probability of is easily given by
(36)
Therefore, the overall detection probability of the cognitive system using the AND-CDS can be given by
(37)
where the parameters and are defined in (35) and (36), respectively. Besides, the detection time slots, and , taken by the secondary users and using AND-CDS, respectively, can be modeled as
(38)
where and Thus, the overall detection time of the CR network by using AND-CDS can be given by
(39)
from which the corresponding average detection time can be calculated as (see Appendix A for details)
(40)
where the closed-form expressions of items (I), (II), (III), and (IV) can be found in Appendix A. Based on (40), we define a agility gain of the AND-CDS over the noncooperation scheme as
(41)
where the parameters and are given in (29) and (40), respectively. In the following, we focus on deriving the closed-form expressions of detection probabilities and agility gains for the AF-CDS and DF-CDS.
3.1. Detection Performance of AF-CDS
Obviously, from (11), we can show that the probability of detection of primary user by in odd time slots is given by
(42)
In addition, from (9), we can calculate the probability of detection of primary user by in even slots as
(43)
where is given in (13) and is determined by
(44)
where is given in (12). Thus, substituting from (13) into (43) yields
(45)
Note that RVs and follow the exponential distribution with parameters and , respectively, and independent from each other. Consequently, the joint PDF of can be written as
(46)
combining (45) and (46) yields (see Appendix B for details)
(47)
Besides, from (44), the corresponding false alarm probability is given by
(48)
from which we can obtain (see Appendix C for details)
(49)
where is Lambert's W function. Substituting from (49) into (47) gives
(50)
Similarly to (35), we can calculate the probability of detection of primary user by under AF-CDS as
(51)
Meanwhile, the detection probability of under AF-CDS can be easily given by
(52)
Hence, we can obtain the overall detection probability for the AF-CDS as
(53)
where and are defined in (51) and (52), respectively. Also, the detection time slots, and taken by the secondary users and , respectively, can be modeled as
(54)
where and . Similarly, the overall detection time of the CR network under AF-CDS can be given by
(55)
from which the corresponding average detection time can be calculated as
(56)
where the closed-form expressions of the summations of infinite series can be obtained in a similar way as shown in (40). We can use (56) as a performance evaluation to show the advantage of the proposed AF-CDS. In addition, the computational complexity of (56) is moderate since the corresponding closed-form expressions involve simple arithmetic operations only. Define the agility gain of the AF-CDS over the noncooperation scheme as
(57)
where and are given in (29) and (56), respectively.
3.2. Detection Performance of DF-CDS
As has been said in Section 2.3, in odd time slots, the implementation process of DF-CDS is the same as AF-CDS, and thus the detection probability of in odd time slots can be given by
(58)
Without loss of generality, let case and denote the estimated indicator and respectively.
Case 1 ([ = 0]).
This case corresponds to implying
(59)
Hence, the probability of occurrence of case can be given by
(60)
Also, it can be shown that
(61)
Substituting the threshold from (61) into (60) yields
(62)
From (15), we can easily calculate the corresponding detection probability as
(63)
Performing the probability integration in (63) yields
(64)
Case 2 ().
This case corresponds to , meaning
(65)
from which we can obtain
(66)
Therefore, from (16), it can be shown that
(67)
Hence, by using (62), (64), (66) and (67), the probability of detection of primary user by in even time slots can be obtained as
(68)
According to (58) and (68), the probability of detection of primary user by under DF-CDS is given by
(69)
Besides, the detection probability of under DF-CDS can be easily given by
(70)
Thus, we can obtain the overall detection probability of the CR network under DF-CDS as
(71)
where and are given in (69) and (70), respectively. Also, we can model the detection time slots and (taken by the secondary users and under DF-CDS, resp.) as
(72)
In a similar way, we can obtain the average overall detection time as
(73)
Similarly, the closed-form expressions of the summations of infinite series can be obtained as (40). Also, we will utilize (73) to show the merits of the proposed DF-CDS. Define the agility gain of the DF-CDS over the noncooperation scheme as
(74)
where and are given in (29) and (73), respectively.
4. Numerical Results and Analysis
Figure 2 shows the plots of (26), (37), (53) and (71) as a function of T-SNR with , and , where and . From Figure 2, one can see that, the detection probabilities of the cooperative detection schemes (i.e., AND-CDS, AF-CDS and DF-CDS) are always larger than the noncooperation scheme across the whole range of T-SNR As observed from Figure 2, the proposed AF-CDS and DF-CDS outperform the traditional AND-CDS in terms of detection probability, showing the superiority of our schemes. Besides, it is shown from Figure 2 that the detection probability of DF-CDS is superior to the AF-CDS at low SNR region, but inferior to AF-CDS when .
Figure 2
Detection probability performance versus T-SNR with , and , where and .
×
In Figure 3, we plot (26), (37), (53) and (71) as a function of false alarm probability with and , where and . It is seen from Figure 3 that the detection probability performances of the three cooperative schemes are much better than the noncooperation scheme. Also, Figure 3 shows that the detection performances of our schemes (i.e., AF-CDS and DF-CDS) are superior to the known AND-CDS scheme. In addition, one can see from Figure 3 that DF-CDS can achieve more performance gains than AF-CDS when . Therefore, DF-CDS outperforms AF-CDS at low SNR region or high false alarm probability region.
Figure 3
Detection probability versus false alarm probability with, and , where and .
×
Figure 4 illustrates the agility gain versus T-SNR of the AND-CDS, AF-CDS and DF-CDS with , , and , where the three curves are the plots of (41), (57) and (74), respectively. From the figure, we find that at this scenario, both the AF-CDS and the DF-CDS can achieve more agility gains than the AND-CDS across the whole T-SNR range, implying the advantages of the proposed schemes over the existing AND-CDS. For completeness sake, in Figure 5, we also plot the agility gains of the AND-CDS, AF-CDS and DF-CDS as a function of the false alarm probability As can be seen from Figure 5, the agility gains of the three cooperation schemes are larger than zero, showing the effectiveness of applying cooperation technology to spectrum sensing. In addition, from Figure 5, one can see that the proposed AF-CDS and DF-CDS outperform the known AND-CDS in terms of agility gain, which further confirms the merits of our schemes.
Figure 4
Agility gain versus T-SNR with, , and , where and .
Figure 5
Agility gain versus false alarm probability with, , and , where and .
×
×
5. Conclusion
In this paper, we have presented two novel cooperative detection schemes (i.e., AF-CDS and DF-CDS) to improve the detection performance of cognitive radios. We have developed closed-form expressions of detection probability and agility gain for both AF-CDS and DF-CDS over Rayleigh fading channels. For the purpose of comparison, we have also analyzed the detection performances for the noncooperation and the AND-based cooperation schemes. Through conducting numerical experiments, it has been shown that both AF-CDS and DF-CDS are superior to the noncooperation and the AND-base cooperation schemes in terms of the detection probability and the agility gain. Furthermore, we have shown that DF-CDS outperforms AF-CDS at low SNR region or high false alarm probability region.
Appendices
A. Calculation of (40)
From (39), we can calculate the corresponding average detection time as
(A1)
Since RVs and are independent from each other, (A.1) can be further expressed as
Note that both and are complex white Gaussian noise with zero mean and double-sided power spectral density Without loss of generality, let denote the complex Gaussian noise, which can be expressed as
(C2)
where and . Obviously, and follow the standard normal distribution, giving , where is the chi-square distribution with 2 degrees of freedom. From (C.2), we can find , implying Consequently, it is easily shown that and are distributed as . Noting that and are independent from each other, we obtain . Let , and thus its PDF (probability density function), , can be given by
(C3)
Let , resulting in Hence, the CDF (cumulative distribution function) of , is calculated as
(C4)
Combining (C.3) and (C.4) gives
(C5)
Performing the integration in (C.5) yields
(C6)
From (C.1), we can obtain
(C7)
Combining (C.6) and (C.7) gives
(C8)
Here, we have utilized the positive property of the energy detection threshold to obtain (C.8). Besides, (C.8) can be equivalently written as
(C9)
By using the Lambert's W function that is used to solve the equation
(C10)
for as a function of the parameter can be given by
(C11)
and this is (49).
Acknowledgments
This work was partially supported by the Postgraduate Innovation Programs of Scientific Research of Jiangsu Province (no. CX09B_150Z, CX08B_080Z), the Key Project of Nature Science Funding of Jiangsu Province (no. BK2007729), the National High Technology Research Development Plan (no. 2009AA01Z241), and the Major Development Program of Jiangsu Educational Committee (no. 06KJA51001).
Open Access
This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (
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), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
a more practical AF-based cooperative detection scheme without the assumption of centralized controller and
a totally new DF-based cooperative detection scheme for the occasion where the cooperative relay has decoding ability. Our main contributions can be described as follows. Firstly, we propose two new cooperative detection schemes, namely, AF-CDS and DF-CDS, to detect the presence of primary users more quickly and accurately. Secondly, we develop the closed-form expressions of detection probability and detection time for both AF-CDS and DF-CDS over Rayleigh fading channels. Thirdly, the performance analysis for the noncooperation scheme and the AND-CDS (AND-based cooperative detection scheme) is also presented for the purpose of comparison with our schemes.
occurs shadowing fading, and
acts as a cooperative relay for
All users in the CR network are equipped with a single antenna, and the antenna at any user can be utilized for both transmission and reception. There is independent, additive complex white Gaussian noise with zero-mean and double-sided power spectral density
at each receiver.
be the primary user indicator, namely,
implies the presence of primary user and
implies its absence. Therefore, the signals received by secondary users
and
can be expressed as
and
denote the transmission from the primary user to
and that from the primary user to
respectively. Besides,
,
and
are the fading coefficients of the wireless channel from the primary user to
, from the primary user to
, and from
to
, respectively. Note that the variances of the three random variables (RVs)
,
and
are
and
respectively. Throughout the paper, we make following assumptions:
All wireless channels are independent from each other in space;
The relaying protocols (i.e., the AF and the DF) employ full duplex mode, meaning that the relay can perform signal reception when transmits.
encounters shadowing fading, the signal received by
from the primary user is so weak that
takes a long time to detect its presence. We show that by cooperation with
the detection probability of
can be increased, thus reducing the overall detection time of the CR network.
to act as a cooperative relay for
. Figure 1 describes a scenario where two secondary users
and
are engaged in detecting the presence of primary user in a particular band. The whole implementation process of AF-CDS can be separated into two consecutive phases, that is,
in odd time slot
, both
and
receive the signals transmitted from primary user;
in even time slot
starts relaying its received information to
in accordance with AF protocol, and thus, in this time slot,
would receive two signal copies simultaneously from the primary user and
respectively. In the time slot
the signals received by
and
can be expressed as
and
are the additive complex Gaussian noise with zero mean and double-sided power spectral density
and independent from each other. In the even slot
the cooperative user,
relays the message from primary user to
According to AF protocol, the received signal
as defined in (3) will be multiplied by a relay gain
and then forwarded to
without any sort of decoding. Meanwhile, in the slot
simultaneously receives the signal from primary user. Therefore, the signal received by
in even time slot
can be written as
denotes the instantaneous fading coefficient of the wireless channel from
to
and
denotes the zero-mean additive complex Gaussian noise with double-sided power spectral density
Substituting
from (4) into (5) gives
to compensate the fading distortion from
to
. Substituting this result into (6) yields
under AF-CDS can be stated as follows: given the observation
and
the detector decides on
We want to show the effect of user cooperation on detection of primary user in CR networks. Hence, the choice of detector is not critical;
We model the signal as a random variable with known power, and thus ED is optimal [15]. Let
and
be the output power of the ED of the secondary user
in the odd time slot
under hypothesis
and
, respectively. Thus, from (8), we can easily obtain
and
as the output power of
's ED in the even slot
under hypothesis
and
, respectively. Hence, from (9), we can get
and
operating in a fixed TDMA mode for detecting the presence of primary user (see Figure 1). Here, according to DF protocol, the cooperative user
should regenerate the primary user indicator
based on its received signals, and then transmit the estimated indicator
to
. Consequently, the detailed process of DF-CDS can be described as follows:
in the odd time slot
, both
and
receive the signal from the primary user;
in the even time slot
decodes its received information and forwards the decoding result to
. Clearly, in odd time slots, the process of DF-CDS is the same as AF-CDS, implying that the output power of energy detector of
for DF-CDS in odd time slots can also be expressed as (11). Besides, in the even time slot
the signal received by
can be given by
and
denote the estimated indicator
and
, respectively. Let
and
be the output power of
's ED in even time slots for case
under hypothesis
and
, respectively. Thus, from (14), it is easy to obtain
and
as the output power of
's ED for case
under hypothesis
and
respectively. Thus, we can easily get
with power
, the signal received by the user
can be expressed as
under noncooperative scheme as
is energy detection threshold that is determined by false alarm probability, which will be illustrated in the following. Note that the PDF of RV
can be given by
is a unit step function. Thus, (18) can be further calculated by
can be given by
is zero-mean complex Gaussian noise with double-sided spectral power density
, we can easily obtain
. Thus, the false alarm probability is calculated as
from (23) into (20) yields
is the transmitting signal-to-noise ratio (T-SNR). In a similar way, the detection probability of
under noncooperative scheme can be given by
and
. Since the two users independently detect the primary user in a distributed way, it can be easily shown that the overall detection probability is given by
and
are defined in (24) and (25), respectively. Besides, let
be the number of slots taken by
(and
) to detect the presence of primary user under the noncooperation scheme. This detection time
can be modeled as a geometric random variable, that is,
is given by
as
and
Then, we focus on the detection performance analysis for the existing cooperative sensing as presented in [8], where an AND-based cooperative detection scheme (AND-CDS) is proposed in order to combat the fading environments. Similarly to the sense protocols of the AF-CDS and DF-CDS, the implementation process of the AND-CDS is also divided into two phases for the convenience of making a fair comparison with our scheme, that is,
in odd time slot
and
detect independently whether or not the primary user is active;
in subsequent even time slot
,
forwards its detected result to
who would then use the AND rule to fuse the detection results of
and
. Clearly, it is easy to show the probability of detection of primary user by
in odd time slots being given by
is the false alarm probability. In even time slots,
would fuse the decision results by using AND rule, and thus, the corresponding detection probability and false alarm probability are calculated as (see [8] for details)
, we can calculate the probability of detection of primary user by
using AND-based cooperation scheme as
and
are the probabilities of time slot
belonging to odd and even, respectively. Generally, the probability events
follow an equal probability distribution, giving
. Substituting this result into (34) yields
is easily given by
and
are defined in (35) and (36), respectively. Besides, the detection time slots,
and
, taken by the secondary users
and
using AND-CDS, respectively, can be modeled as
and
Thus, the overall detection time
of the CR network by using AND-CDS can be given by
can be calculated as (see Appendix A for details)
and
are given in (29) and (40), respectively. In the following, we focus on deriving the closed-form expressions of detection probabilities and agility gains for the AF-CDS and DF-CDS.
in odd time slots is given by
in even slots as
is given in (13) and
is determined by
is given in (12). Thus, substituting
from (13) into (43) yields
and
follow the exponential distribution with parameters
and
, respectively, and independent from each other. Consequently, the joint PDF of
can be written as
is given by
is Lambert's W function. Substituting
from (49) into (47) gives
under AF-CDS as
under AF-CDS can be easily given by
and
are defined in (51) and (52), respectively. Also, the detection time slots,
and
taken by the secondary users
and
, respectively, can be modeled as
and
. Similarly, the overall detection time
of the CR network under AF-CDS can be given by
can be calculated as
and
are given in (29) and (56), respectively.
in odd time slots can be given by
and
denote the estimated indicator
and
respectively.
= 0]).
implying
can be given by
from (61) into (60) yields
).
, meaning
in even time slots can be obtained as
under DF-CDS is given by
under DF-CDS can be easily given by
and
are given in (69) and (70), respectively. Also, we can model the detection time slots
and
(taken by the secondary users
and
under DF-CDS, resp.) as
as
and
are given in (29) and (73), respectively.
with
,
and
, where
and
. From Figure 2, one can see that, the detection probabilities of the cooperative detection schemes (i.e., AND-CDS, AF-CDS and DF-CDS) are always larger than the noncooperation scheme across the whole range of T-SNR
As observed from Figure 2, the proposed AF-CDS and DF-CDS outperform the traditional AND-CDS in terms of detection probability, showing the superiority of our schemes. Besides, it is shown from Figure 2 that the detection probability of DF-CDS is superior to the AF-CDS at low SNR region, but inferior to AF-CDS when
.
with
,
and
, where
and
.
with
and
, where
and
. It is seen from Figure 3 that the detection probability performances of the three cooperative schemes are much better than the noncooperation scheme. Also, Figure 3 shows that the detection performances of our schemes (i.e., AF-CDS and DF-CDS) are superior to the known AND-CDS scheme. In addition, one can see from Figure 3 that DF-CDS can achieve more performance gains than AF-CDS when
. Therefore, DF-CDS outperforms AF-CDS at low SNR region or high false alarm probability region.
,
and
, where
and
.
of the AND-CDS, AF-CDS and DF-CDS with
,
,
and
, where the three curves are the plots of (41), (57) and (74), respectively. From the figure, we find that at this scenario, both the AF-CDS and the DF-CDS can achieve more agility gains than the AND-CDS across the whole T-SNR range, implying the advantages of the proposed schemes over the existing AND-CDS. For completeness sake, in Figure 5, we also plot the agility gains of the AND-CDS, AF-CDS and DF-CDS as a function of the false alarm probability
As can be seen from Figure 5, the agility gains of the three cooperation schemes are larger than zero, showing the effectiveness of applying cooperation technology to spectrum sensing. In addition, from Figure 5, one can see that the proposed AF-CDS and DF-CDS outperform the known AND-CDS in terms of agility gain, which further confirms the merits of our schemes.
,
,
and
, where
and
.
,
,
and
, where
and
.
as
and
are independent from each other, (A.1) can be further expressed as
. From (B.1), it is easy to obtain
we can get
,
is given by
into (B.4) and (B.6) yields
and
are complex white Gaussian noise with zero mean and double-sided power spectral density
Without loss of generality, let
denote the complex Gaussian noise, which can be expressed as
and
. Obviously,
and
follow the standard normal distribution, giving
, where
is the chi-square distribution with 2 degrees of freedom. From (C.2), we can find
, implying
Consequently, it is easily shown that
and
are distributed as
. Noting that
and
are independent from each other, we obtain
. Let
, and thus its PDF (probability density function),
, can be given by
, resulting in
Hence, the CDF (cumulative distribution function) of
,
is calculated as
to obtain (C.8). Besides, (C.8) can be equivalently written as
as a function of
the parameter
can be given by
