Cooperative Spectrum Sensing with Censoring of ...

Cooperative Spectrum Sensing with Censoring of Cognitive Radios with Majority Logic Fusion in Hoyt Fading Srinivas Nallagonda

S. Dhar Roy and S. Kundu

G. Ferrari and R. Raheli

Department of ECE NIT Durgapur, India [email protected]

Department of ECE NIT Durgapur, India [email protected] and [email protected]

Dept. of Information Engineering University of Parma, Italy [email protected] and [email protected]

Abstract— In this paper, the performance of cooperative spectrum sensing (CSS) is assessed in the presence of Hoyt fading. A cognitive radio (CR) user, which senses the primary users (PUs) using an energy detector (ED) in sensing channel (Schannel), is censored on the basis of the quality of the reporting channel (R-channel). A threshold-based censoring scheme is used: CR users, whose estimated R-channel fading coefficients towards the FC exceed a predefined threshold (denoted as censoring threshold), are allowed to transmit. Majority logic fusion is considered at the FC to estimate the performance in terms of average missed detection and total error probabilities for various values of the censoring threshold, the number of CRs, the false alarm probability, the average S-channel and R-channel signal-to-noise ratios (SNRs), under both perfect and imperfect channel coefficient estimations. The impact of the Hoyt fading parameter on the average missed detection probability is highlighted. Furthermore, an analytical expression for the probability of selection of CRs, in terms of the censoring threshold, is derived and validated with simulations. Index Terms— Cooperative spectrum sensing, censoring, majority logic fusion, average missed detection probability

I.

INTRODUCTION

Cognitive radio (CR) techniques have been proposed to overcome spectrum scarcity by exploiting spectrum underutilization [1]. Spectrum sensing is an important aspect of CR technology since a CR needs to detect the presence of primary users (PUs) accurately and quickly when information about PU signaling activity is unavailable. In such scenarios, one appropriate choice consists of using an energy detector (ED), which simply measures the energy of the received waveform, over an observation time window [2]-[3]. The channels between the PUs and CRs are called “sensing” channels (S-channels), while the channels between CRs and the FC are denoted as “reporting” channels (R-channels). The performance of a CR using an ED is sometimes limited due to severe fading or shadowing in the S-channel [4]. Cooperative spectrum sensing (CSS) with CRs employing EDs improves the detection performance where all CRs sense the PU individually and send their sensing information to FC [5]-[7]. Though most works on spectrum sensing assume

noiseless (ideal) R-channel [5]-[7], the presence of fading in R-channels is likely to effect the decisions sent by CRs where the FC is located at afar distance from CRs. If the R-channel is heavily faded, the decision received at FC is likely to be erroneous with respect to that transmitted by the CR. Under such scenario, it is better to stop transmitting decisions from this CR and, thus, censoring is expedient in these scenarios. The CRs whose R-channels are estimated as reliable by the FC are censored, i.e., they are allowed to transmit. The CRs which are not participating in improving the detection performance may be stopped, so that system complexity can be reduced and detection performance can be improved. Therefore, censoring of CRs is necessary to improve the performance of CSS. In [8]-[9], the performance of CSS with censoring of CRs in Rayleigh faded environment has been evaluated, based on the quality of Rayleigh faded R-channel. Using minimum mean square estimation (MMSE)-based estimation of the Rchannels, the FC selects the ‘P’ CRs (out of the total ‘N’ CRs) which have highest (after proper ranking) channel coefficients, i.e., CRs associated with best estimated channel coefficients are selected. In the present work, we consider channel threshold-based censoring, where both R-channels and Schannels are assumed to be Hoyt faded. The Hoyt distribution [10-12], also known as Nakagami-q distribution (q being the fading severity parameter), allows one to span the range of fading distribution from one-sided Gaussian (q=0) to Rayleigh fading (q=1), and is used extensively for modeling more severe than Rayleigh fading wireless links. Though all the CRs detect PUs using EDs, only those CRs censored (on the basis of the censoring threshold in R-channels) are allowed to transmit. In [13], this threshold-based censoring scheme in Rayleigh faded environments is considered and analyzed in the context of distributed detection in a (not cognitive) sensor network, where a number of sensors observe a common binary phenomenon. In this paper, we analyze CSS performance with censored CRs, considering a network of N CRs where each CR makes local observations on the activity of PUs using energy detectors. We consider channel quality-based censoring: a CR is selected to transmit its decision if the estimated R-channel

fading coefficient exceeds a given threshold (called as censoring threshold, Cth). The FC employs coherent reception to fuse binary local decisions received from censored CRs to obtain a final decision regarding the presence or absence of PUs. Low-complexity majority logic fusion of the decisions received from the selected CRs is considered. The average missed detection and average total error probabilities are selected as the key performance metrics and are evaluated, through simulations, under several channel and network conditions. More precisely, our contributions in the present work can be summarized as follows. A simulation testbed is developed for performance evaluation of censoring schemes in Hoyt faded distributed detection scenarios with majority logic fusion at the FC. An analytical expression for the selection of CRs, in terms of the censoring threshold, is derived in the presence of Hoyt fading, followed by the derivation of the probability mass functions (pmf) of the number of censored CRs. An extensive performance analysis, in terms of average missed detection probabilities under several channel and network conditions (including the number of available CRs, false alarm probability, average S-channel and R-channel SNRs) is carried out. The impacts of (i) the Hoyt fading parameter on the average missed detection probability, (ii) the number of available CRs and R-channel SNRs on the average total error performance of the considered CSS scheme are evaluated. Furthermore, the impact of censoring threshold on the average missed detection and total error probability, with estimation of an optimized censoring threshold, is shown. The rest of the paper is organized as follows. In Section II, the system model is described. Results and discussions are presented in Section III. Finally, we conclude in Section IV. II.

SYSTEM MODEL

We consider a network of N CRs each using ED with identical threshold (λ) which makes hard binary decision and transmits it using binary phase shift keying (BPSK) as modulation format to the FC if it is selected to transmit. All CRs are assumed to be relatively close to each other. The distance between any two CRs is less than the distance between a primary user (PU) and a CR or the distance between a CR and the FC. The table 1 shows some important notations which are used in this paper. As already discussed in Section I, both S- and R-channels are modeled as noisy and Hoyt faded. Each CR is having one ED as shown in Fig. 1. It receives a signal x(t) as defined below at input and gives binary decision regarding the presence or the absence of a PU. x(t )

(⋅)

2

BPF

Signal squarer

T

H0

0

H1

∫ (⋅)dt Integrator

or

Decision device

Fig. 1. Block diagram of an energy detector

The received signal x(t) at i-th CR can be represented as:

n (t )  x i (t ) =   h i s (t ) + n ( t )

H0 H1

(1)

FC

Hoyt faded reporting channels

CR 1

CR 2

CR N

Hoyt faded sensing channels PU

not selected to transmit selected to transmit

Fig. 2. Cooperative spectrum sensing network with censoring. TABLE I : DESCRIPTION AND NOTATION Description Notation One-sided noise power spectral density

N 01

Primary signal energy Average S-channel SNR

γS = Es / N01

ES

Average R-channel SNR

γR = Eb /σn

One-sided bandwidth (Hz), i.e., positive bandwidth of low-pass (LP) signal Gaussian variate with mean µ and variance σ2

2

W

N (µ , σ 2 )

where s(t) is the PU signal and n(t) is the sample of AWGN noise present in S-channel. The noise n(t) is modeled as a zero-mean white Gaussian random process. The S-channel fading coefficient hi is modeled as a complex Hoyt process, as explained in detail at the end of this section. H1 and H0 are the two hypotheses associated with the presence and the absence of a PU. The block diagram of considered CSS scheme is shown in Fig. 2. When the PU is absent, each CR receives only noise signal at the input of the ED and the noise energy can be approximated, over the time interval (0, T), as follows [2]-[3]: T

∫n

2

( t ) dt =

0

1 2m 2 ∑nj, 2W j = 1

(2)

where

n j ~ N ( 0 , N 01 W ),

∀

j.

(3)

The decision statistic at i-th CR, denoted as Yi , can be written as [2]-[3] 2m

Y i = ∑ n ′j 2 ; n ′j = j =1

nj

(4) , N 01W The same approach is applied when the primary signal s(t) is present with the replacement of each nj by nj+sj where s j = s( 2Wj ). In non-faded environment (AWGN case i.e., hi=1) the probabilities of detection and false alarm for the i-th CR is expressed as follows [2]-[4]: (5) Pd , i = P (Yi > λ| H 1 ) = Q m ( 2γ S , λ) Pf , i = P (Yi > λ|H 0 ) = Γ ( m , λ / 2) / Γ ( m )

where γs is instantaneous S-channel SNR, Г(.,.) is the

(6)

incomplete gamma function, and Qm(.,.) is the generalized Marcum Q-function. The expression for the probability of false alarm (Pf,i) for the i-th CR, is given in (6), remains the same when fading is considered in the S-channel due to independence of Pf,i from SNR ( γS ). The detection threshold

λ can be set for a chosen Pf,i following equation (6). The ED compares Yi with its preset detection threshold λ and takes a hard binary decision about the presence of a PU. If this CR is censored to transmit, its decision is sent to the FC using BPSK over the corresponding Hoyt faded R-channel. Transmissions between the CRs and the FC are carried out in two phases. In the first transmission phase, each CR sends one training symbol to enable the FC to estimate all N fading channels’ coefficients [16]-[17]. Estimation (MMSE) of the Rchannel coefficients is obtained at the FC using training symbols sent by the CRs to the FC. The signal from k-th CR received at the FC is: y k = sk hk + nk (7) k ∈ {1, 2 , ...,N } where

sk is BPSK signal (

E b and −

E b ), indicating H1

and H0, respectively. R-channel coefficient hk is modeled as a complex Hoyt (discrete time) process as in [11]-[12] and nk ~ CN 0, σ n2 . We assume that the FC estimates the Hoyt fading coefficients hk according to MMSE estimation strategy

(

)

on the basis of the observable y k as follows [14], [15]: hˆ k = E [ h k | y k ] =

Eb Eb + σ =

=

E b + σ n2

(

E b hk + n k

)

Eb Eb hk + nk . E b + σ n2 E b + σ n2

(8)

 Eb  Eb ~ hk = hk −  h + n   Eb + σ n2 k Eb + σ n2 k    2    Eb / σ n   Eb / σ n2 − = hk 1 − nk  2   2  (1 + Eb / σ n )   (1 + Eb / σ n ) Eb    γ R   γR nk  − = hk 1 −  (1 + γ R )   (1 + γ R ) Eb 

where γR is the average R-channel SNR. Censoring is carried out only on the basis of a comparison between the amplitude of estimated channel coefficient and the censoring threshold (Cth). The channel estimation can be perfect with no estimation error (hˆk = hk ) or imperfect with ~ estimation error (hˆ = h − h ) . Accordingly, there are two k

}

is

fading coefficient | hˆk | is above Cth. The complex Gaussian Hoyt process can be expressed as | hk |=| hI + jhQ | where the inphase and quadrature components have the following expressions [11]-[12]: 2 (11) h I ~ N (0, σ 12 ) , hQ ~ N (0, σ 2 ) where

Ω q2 1+ q

2

,σ 2 =

Ω 1+ q 2

,

(12)

and the average fading factor Ω, is normalized to unity while q is the Hoyt fading parameter and ranges from 0 to 1. The PDF of the Hoyt fading coefficient is given in [9, Eq. (45)] as f X ( x) =

 x2  1 1    x2  1 1  exp −  2 + 2   I 0   2 − 2   x ≥ 0. (13)      4 σ σ1σ 2  1 σ 2    4  σ 2 σ1    x

Using (13) and [15, Eq. (58)], the probability of selecting a CR can be expressed as 1 1+ exp −ACth2 I0 BCth2 − p = Pr | hˆk | > Cth =1− 2 2  2σ1σ2 A − B

(

)

 Q1  (A − 

where

(9)

{

)

Censoring Rule for CR In the considered scenario, a CR (says the k-th) is selected for transmission if the amplitude of the estimated R-channel

yk

2 n

Eb

k

(

where channel noise n k , d ~ CN 0, σ n2 and mk ∈ + Eb ,− Eb the BPSK modulated binary decisions.

σ1 =

We model the channel estimation error coefficient as the difference between the actual and the estimated channel ~ coefficient hk = hk - hˆk where hk is actual channel coefficient while hˆk is its estimate. The channel estimation error ~ coefficient hk can be rewritten by using equation (8) as

k

cases of censoring schemes: one case is based on perfect channel estimation, while the other case is based on imperfect channel estimation. After the first phase, ‘K’ number of CRs out of ‘N’ available CRs whose estimated channel coefficients exceeds the Cth is selected. In the second transmission phase, the ‘K’ selected CRs send their local binary BPSK modulated decisions to FC over the corresponding R-channels. The fading coefficients of a R-channel are assumed to be fixed over decision symbol transmission time as the channel is assumed to be slow faded. The signal at the FC received from k-th selected CR is [14]: y k ,d = m k hk + n k ,d k ∈ {1, 2 , ...,K } (10)

(

) (

 (14) A2 − B 2 )Cth 2    1 1 1  and B=  2 − 2   4  σ 2 σ 1 

)

A2 − B 2 )Cth 2 , (A +

A =

1 4

 1 1  + 2 σ 2 σ2  1

   

.

The probability of selecting K number of CRs from N available CRs which follows binomial distribution with probability of selection can then be expressed as follows [13]:  N  K (15) P (K ) = p (1 − p ) ( N − K )    K 

Majority Logic Fusion Rule Since the communication channel is noisy and affected by fading, a decision received by the FC might differ from the one sent by the corresponding CR. The received decision denoted by uk from the k-th selected CR is:

if the received decision is in favor of H1 (16)

1 Cth=0.5, q=0.2, perfect channel, simulation

if the received decision is in favor of H 0

0.9

general majority-like expression [13], [17]:

u 0 = Γ (u 1 , … , u K

 H 1   )= H 0    H 0 or H 1 

K

∑u

if

k =1

>

k

K

if

∑u k =1

∑u k =1

(17)

K < 2

k

K

if

K 2

k

=

Q f = P (false alarms) =

N

K =0

m

(error | K ) P ( K )

N

∑P

K =0

f

(error | K ) P ( K )

Average total error probability = Q m + Q f

(18) (19) (20)

The average missed detection probability Q m and average false alarm probability Q f are the functions of chosen censoring threshold Cth, as the probability mass function (pmf) {P(K)} of the number of censored CRs depends on Cth. III.

RESULTS AND DISCUSSIONS

A simulation testbed has been developed in MATLAB to assess the performance of above censoring scheme. Table-II shows the parameters used in simulation. TABLE II: PARAMETER USED IN SIMULATION Parameter Number of cognitive radios , N Average S-channel SNR Average R-channel SNR Censoring threshold, Cth Time-bandwidth product, m Probability of false alarm at a CR, Pf Hoyt fading parameter, q

Cth=1.5, q=0.2, imperfect channel

0.7

Cth=3.0, q=0.2, perfect channel

0.6 0.5 0.4 0.3

0.1

probability when decisions from K CRs as in (15) are fused. Given P(K), the probability of selecting K CRs, the average probability of missed detection can be expressed as:

∑P

Cth=1.5, q=0.2, perfect channel

0.2

K 2

In other words, if the number of decisions in favor of H1 is larger than the number of decisions in favor of H0, the FC takes a global decision in favor of H1 and vice versa. Sometimes, if the number of decisions in favor of H1 is equal to the number of decisions in favor of H0, then FC flips a coin and takes a decision in favor of either H1 or H0. Let Pm (error | K ) indicates the conditional missed detection

Q m = P ( missed detections ) =

Cth=0.5, q=1.0 (Rayleigh), perfect channel

0.8 Probability of selection P(K)

where k ∈ {1, 2, ...,K} . The FC finally makes a global decision u 0 according to a fusion rule u 0 = Γ (u1 , … , u K ) following

Cth=0.5, q=0.2, perfect channel, analytical

Values 10, 30 20 dB, 25 dB -7 dB, -9 dB 0 to 3.5 5 0.05, 0.0005 0.2, 0.5, 1.0

In Fig. 3, the binomially distributed pmf of the number of selected CRs is shown for various values of Cth and q, under both the cases of perfect and imperfect channel estimation. It can be observed that for small values of Cth, a relatively larger number of CRs is likely to be selected, while the pmf tends to

0

0

5

10 15 20 Number of CRs selected (K)

25

30

Fig. 3. Probability mass function (pmf) of the number of selected CRs for different values of Cth and q under both perfect and imperfect channel estimation. 0

10

N=10, Perfect channel estimation N=30, Perfect channel estimation N=10, Imperfect channel estimation

Average missed detection probability

1 uk =  0

N=30, Imperfect channel estimation

(i) (ii)

(iii)

(iv)

-1

10

0

0.5

1

1.5 2 Threshold (Cth)

2.5

3

3.5

Fig. 4. Average missed detection probability, as a function of Cth for different values of N under both perfect and imperfect channel estimation (S-channel SNR=20 dB, R-channel SNR=-9 dB, Pf=0.05, m=5, and q=0.5).

concentrate, for both censoring schemes, around small values for higher values of Cth. For example, for a censoring threshold of 0.5 it is seen that K= 20 CRs have highest probability (0.15) of being selected under perfect channel estimation. It can also be observed that as Cth increases, the pmf moves towards to origin for both censoring schemes. This occurs as a higher censoring threshold decreases the number of selected CRs. From the results presented in the figure, it can be concluded that the pmf of the number of selected CRs under imperfect channel estimation shifts rightward of the pmf of the number of selected CRs under perfect channel estimation for a particular value of Cth. According to (9), in the case of imperfect channel estimation, a larger number of CRs, for a fixed value of the Rchannel SNR, may be selected with respect to the number of CRs selected with perfect channel estimation. The pmf of the number of selected CRs as obtained based on our simulation testbed matches exactly with the binomially distributed pmf as obtained based on analytical expression in (14), which

validates both the simulation testbed and the analytical derivation.

threshold to finally reach a value equal to 0.5. The optimum censoring threshold is found to be different for the cases with perfect and imperfect channel estimation and it also depends on the number of CRs (N). For example, as can be seen from the obtained results, for N=10 the optimum Cth is around 0.5 with perfect channel estimation [curve (ii)], and around 0.3 with imperfect channel estimation. This behavior of the average missed detection probability is due to the changing pmf of the number of censored CRs for various values of Cth. For very small values of the threshold, even unreliable links tend to be selected, and the average probability of missed detection is rather high. On the other hand, as the censoring threshold is increased to a very high level, no CR is selected to transmit, i.e., P (0) = 1, and the FC takes a decision by flipping a fair coin resulting an average missed detection probability of 0.5. Therefore, there exists an optimal value of the censoring threshold, in correspondence to which the average probability of missed detection is minimized. Moreover, as expected, it can be seen that a larger number of CRs leads to a reduced average missed detection probability in correspondence to the optimized censoring threshold. In Fig. 5 and Fig. 6, the impacts of (i) the Hoyt fading parameter, (ii) the false alarm probability at CRs, (iii) the Schannel and R-channel SNRs on the average missed detection probability, under both perfect and imperfect channel estimation, respectively, are investigated. We observe that, for a fixed value of Cth, the average probability of missed detection decreases, for both perfect channel estimation (Fig. 5) and imperfect channel estimation (Fig. 6), in correspondence to an increase of either of the following: the Hoyt parameter (q), the false alarm probability, the S-channel SNR and/or R-channel SNRs. Due to an increase in false alarm probability (according to equation (6)) the detection threshold (λ) at each CR level decreases, thus improving the performance of CR. When the S-channel or R-channel SNRs increase, the FC receives a larger number of correct decisions and this, in turn, leads to a reduction in the average missed detection probability. When q increases from 0.5 to 1.0, the fading severity in the channel decreases so that the FC receives more correct decisions which leads to reduction in average missed detection probability. In Fig. 7, the impact of censoring threshold, number of available CRs and S-channel SNR on the average total error probability (sum of average missed detection and average


attains a minimum value at an ‘optimal’ Cth level and, from then on Qm increases for a further increase of the censoring

(i) -1

(ii)

10

(iii)

P =0.05, S-ch SNR=20dB, R-ch SNR= -9dB, q=0.5 f


(iv)


P =0.05, S-ch SNR=20dB, R-ch SNR= -7dB, f

q=1.0 (Rayleigh)

(v)


q=0.5

0

0.5

1


2.5

3

3.5

Fig. 5. Average missed detection probability, as a function of Cth for various values of q, Pf, S-channel and R-channel SNRs under perfect channel estimation (N=30, m=5). 0

10


In Fig. 4, the average missed detection probability (equation (18)) is shown as a function of the censoring threshold Cth, considering a direct comparison between perfect and imperfect channel estimation for different number CR users. Two different values of the available number N of CRs (namely, 10 and 30) are considered. It can be seen from the figure that as Cth increases, the average missed detection probability ( Qm )

0

10

(i) (ii)

-1

10

(iii)


P =0.05, S-ch SNR=20dB, R-ch SNR= -7dB, q=0.5

(iv)

f



(v)

q=1.0 (Rayleigh) P =0.00005, S-ch SNR=20dB, R-ch SNR= -7dB, f

q=0.5

0

0.5

1


2.5

3

3.5

Fig. 6. Average missed detection probability, as a function of the censoring threshold for various values of q, Pf , S-channel and R-channel SNRs under imperfect channel estimation (N=30, m=5).

false alarm probabilities) are shown. In Fig. 8, the effects of false alarm probability at CR and R-channel SNR on the average total error probability are shown. In both Fig. 7 and Fig. 8, the performance comparison between perfect and imperfect channel estimation is evaluated. It can be seen from both Fig. 7 and Fig. 8 that as Cth increases, the average total error probability attains a minimum value at an ‘optimal’ Cth level and thereafter increases with further increase in Cth to finally attain a value of 1.0 (average missed detection probability reaches a value of 0.5 and average false alarm probability reaches a value of 0.5). There exists an optimal value of Cth, in correspondence to which the average total error probability is minimized. In Fig. 7, the optimum Cth is found to exist near 0.7 for N=30, S-channel SNR=25 dB, Pf =0.05, and R-channel SNR=-7 dB [curve (v)]. In Fig.8, the optimum Cth is found to exist near 0.6 for N=30, S-channel SNR=20 dB, Pf =0.05, and R-channel SNR=-9 dB [curve (ii)]. It is observed from Fig. 7 and Fig.8 that, the optimum Cth is different for different channel and network parameters.

imperfect channel estimation. The above study is useful in designing a CSS scheme for an energy constrained cognitive radio network.

0

Average total error probability

10

REFERENCES (i)

[1]

[2] [3]

(ii) N=10, S-ch SNR=20dB, imperfect channel (iii)

N=30, S-ch SNR=20dB, perfect channel

(iv)

-1

10

(v)

N=30, S-ch SNR=25dB, perfect channel N=30, S-ch SNR=20dB, imperfect channel N=30, S-ch SNR=25dB, imperfect channel

0

0.5

1


2.5

3

[4] 3.5

Fig. 7. Average total error probability, as a function of Cth for various values of N and S-channel SNRs under both perfect and imperfect channel estimation (R-channel SNR=-7 dB, Pf=0.05, m=5, and q=0.5).

[5]

0

10

Average total error probability

[6]

[7]

[8] (i) P =0.05, R-ch SNR= -9dB, perfect channel f

(ii)

P =0.05, R-ch SNR= -7dB, perfect channel f

(iii)

P =0.0005, R-ch SNR= -7dB, perfect channel

(iv)

f

P =0.05, R-ch SNR= -9dB, imperfect channel P =0.05, R-ch SNR= -7dB, imperfect channel

-1

10

[9]

f

(v)

f

0

0.5

1


2.5

3

3.5

Fig. 8. Average total error probability, as a function of Cth for various values of Pf, and R-channel SNRs under perfect and imperfect channel estimation (Schannel SNR=20 dB, N=30, m=5, and q=0.5).

[10]

[11]

CONCLUSION We have investigated the performance of CSS with CRs censored on the basis of the quality of R-channels in the presence of Hoyt fading. The censoring threshold for the selection of CRs has a significant impact on the average missed detection performance. Depending on the channel and network parameters---such as Hoyt fading parameter, probability of false alarm, S-channel and R-channel SNRs--an optimal censoring threshold can be identified in correspondence to the minimum average missed detection and the minimum average total error probability. The optimal censoring threshold is found to be different for the cases with perfect and imperfect channel estimations---for fixed networking/communication conditions. Our results show that the optimal threshold is an increasing function of the Hoyt fading parameter. The average missed detection probability well as the average total error probability, decreases with increasing of Hoyt fading parameter, number of available CRs, sensing and reporting channel SNRs for both perfect and

[12] [13]

[14]

[15]

[16]

[17]

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