Performance Analysis of Energy Detection Based Spectrum Sensing with Unknown Primary Signal Arrival Time

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Performance Analysis of Energy Detection Based Spectrum Sensing with

Unknown Primary Signal Arrival Time

Jwo-Yuh Wu, Member, IEEE, Chih-Hsiang Wang, and Tsang-Yi Wang, Member, IEEE

Abstract—Spectrum sensing in next-generation wireless

cog-nitive systems, such as overlay femtocell networks, is typically subject to timing misalignment between the primary transmitter and the secondary receiver. In this paper, we investigate the performance of the energy detector (ED) when the arrival time of the primary signal is modeled as a uniform random variable over the observation interval. The exact formula for the detection probability is derived and corroborated via numerical simulation. To further improve the detection performance, we propose a robust ED based on the Bayesian principle. Computer simulation confirms the effectiveness of the Bayesian based solution when compared with the conventional ED.

Index Terms—Cognitive radio, spectrum sensing, energy

de-tection.

I. INTRODUCTION

C

OGNITIVE radio (CR) is a widely known opportunistic spectrum access technique for enhancing the cell-wide spectrum utilization efficiency [1-2]. In order to detect the idle frequency band so as to gain the channel access, spectrum sensing performed at the CR users is indispensible. In the literature, the detection of idle spectrum is typically considered as a binary hypothesis test, and a commonly used signal model under both hypotheses is [1-2]

0: 𝑥[𝑛] = 𝑣[𝑛], 0 ≤ 𝑛 ≤ 𝑁 − 1 (idle) 1: 𝑥[𝑛] = 𝑠[𝑛] + 𝑣[𝑛], 0 ≤ 𝑛 ≤ 𝑁 − 1 (occupied)

(1.1) where𝑁 is the length of the data record, 𝑠[𝑛], 𝑥[𝑛], 𝑣[𝑛] are,

respectively, the signal of the primary user, the received signal at the CR terminal, and the measurement noise. The hypoth-esis model (1.1) implicitly assumes perfect synchronization between the primary transmitter and the CR receiver. Such an assumption, however, is not valid in many practical situations. For example, in an overlay femto cell network [3], the signal of the macro mobile subscriber, synchronized with the macro base station (BS), will arrive at a femto BS asynchronously. The spectrum detection at the femto BS is typically subject

Paper approved by M. R. Buehrer, the Editor for Cognitive Radio and UWB of the IEEE Communications Society. Manuscript received June 15, 2010; revised November 29, 2010 and January 28, 2011.

This work is sponsored by the National Science Council of Taiwan under grant NSC 97-2221-E-009-101-MY3, NSC 98-2221-E-110-043-MY3, and NSC 99-2628-E-009-004, by the Ministry of Education of Taiwan under the MoE ATU Program, by the Telecommunication Laboratories, Chunghwa Telecom Co., LTD. under grant TL-99-G107, and by MediaTek Research Center at National Chiao Tung University, Taiwan.

J.-Y. Wu and C.-H. Wang are with the Department of Electrical Engineer-ing, National Chiao Tung University, Taiwan (e-mail: jywu@cc.nctu.edu.tw; ccrush@gmail.com).

T.-Y. Wang is with the Institute of Communications Engineering, National Sun Yat-sen University, Taiwan (e-mail: tcwang@faculty.nsysu.edu.tw).

Digital Object Identifier 10.1109/TCOMM.2011.050211.100146

to timing misalignment of the primary signal [4], [5]. Also, in heavy-traffic networks in which primary users may dynam-ically enter the network, time delays observed in the sensing period is unavoidable, especially when a long sensing duration is adopted for obtaining good sensing performance. Thus, in the aforementioned cases, a more reasonable signal model for the binary hypothesis test is thus

ℋ0: 𝑥[𝑛] = 𝑣[𝑛], 0 ≤ 𝑛 ≤ 𝑛0− 1 (idle) ℋ1: { 𝑥[𝑛] = 𝑣[𝑛], 0 ≤ 𝑛 ≤ 𝑛0− 1 𝑥[𝑛] = 𝑠[𝑛] + 𝑣[𝑛], 𝑛0≤ 𝑛 ≤ 𝑁 − 1 (occupied) (1.2) where 𝑛0 accounts for the primary signal arrival time. Therefore, in contrast to the spectrum sensing schemes in the literature focusing on the synchronized signal model (1.1) [1-2], this paper considers the spectrum detection aimed for tackling signal timing uncertainty under the hypothesis (1.2). Among the existing spectrum sensing schemes, the energy detector (ED) [6] is quite popular mainly because it involves only the partial knowledge (the second moment) of the pri-mary signal and is thus cost-effective to implement [1-2]. Even though various performance characteristics of the ED have recently been investigated, e.g., [7-9], the discussions in all these works were based on the idealized model (1.1). In this paper, we study the detection performance of ED under the hypothesis (1.2). As the detection of arrival is the main focus, as in [10], we consider the scenario that the primary user is present only after spectrum sensing is started. Motivated by the fact that, in high-traffic random access networks, the traffic patterns of primary users are typically unknown to the secondary users, the signal arrival time 𝑛0

is assumed to be uniformly distributed over the observation window 0 ≤ 𝑛 ≤ 𝑁 − 1. Specific technical contributions of this paper can be summarized as follows. Firstly, conditioned on a fixed 𝑛0, the exact formula for the conditional detection probability under the hypothesis model (1.2) is derived. The average detection probability can then be accordingly obtained by taking the expectation with respect to 𝑛0. To the best of our knowledge, the performance study shown in this paper is the original contribution in the literature that is tailored for the ED scheme in the realistic sensing environment characterized by the model (1.2). Secondly, to further exploit the prior knowledge about𝑛0for improving the detection performance,

we also propose a robust ED based on the Bayesian formu-lation [6]. Simuformu-lation study shows that the Bayesian based solution improves the receiver operation characteristics (ROC). Moreover, under a prescribed detection probability threshold, the Bayesian scheme does lead to a smaller false-alarm

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probability, thereby enhancing spectrum utilization efficiency of the CR networks. Finally, we would like to remark that the problem of ED based spectrum sensing in the presence of signal arrival timing misalignment was recently addressed in [4] within the OFDMA system framework. Rather than developing robust sensing schemes, the conventional ED was considered in [4] for spectrum detection. In addition, analyses of the associated ROC characteristics therein (conditioned on a fixed set of delays) were not exact, but instead resorted to the Chi-square approximation of the true data distribution [4, p-5305]. Given these facts, the distinctive features of current paper in contrast with [4] are: (i) derivations of the exact conditional and average detection probabilities for the conventional ED under the timing-misaligned signal model (1.2); (ii) development of a Bayesian ED robust against timing uncertainty.

II. PERFORMANCEANALYSIS

A. Exact Detection Probability of ED Test Under (1.2)

The test statistic of the conventional ED is by definition given by

𝑇 =

𝑁−1 𝑛=0

∣𝑥[𝑛]∣2. (2.1)

Under the alternative hypothesis1 in (1.2) and conditioned

on a fixed𝑛0, let us decompose the test statistic 𝑇 into 𝑇 = 𝑛∑0−1 𝑛=0 ∣𝑥[𝑛]∣2    :=𝑇1 + 𝑁−1 𝑛=𝑛0 ∣𝑥[𝑛]∣2    :=𝑇2 . (2.2)

Based on (2.2), we shall first derive the conditional detection probability; the average detection probability can then be easily obtained by taking the expectation with respect to𝑛0.

Let us assume that (i) the signal 𝑠[𝑛] and noise 𝑣[𝑛] are

zero-mean white sequences with variances given by𝜎2

𝑠 and

𝜎2

𝑣, respectively; (ii)𝑠[𝑛] and 𝑣[𝑛] are independent. Note that,

with 𝑇1 and 𝑇2 defined in (2.2), it is easy to verify 𝑧1 :=

𝑇1/𝜎2

𝑣 ∼ 𝜒2𝑛0 and𝑧2 := 𝑇2/(𝜎2𝑣+ 𝜎𝑠2) ∼ 𝜒2𝑁−𝑛0, and hence

the associated probability density functions (PDF) are

𝑓𝑧1(𝑥) = 𝑥(𝑛0/2)−1𝑒−𝑥/2 2𝑛0Γ(𝑛0/2) 𝑢(𝑥) and 𝑓𝑧2(𝑥) = 𝑥[(𝑁−𝑛0)/2]−1𝑒−𝑥/2 2(𝑁−𝑛0)Γ ((𝑁 − 𝑛0)/2)𝑢(𝑥), (2.3)

where𝑢(𝑥) is the unit step function. To simplify notation let

us consider the equivalent test statistic ¯ 𝑇 = 𝜎𝑇2 𝑣 = 1 𝜎2 𝑣 𝑁−1 𝑛=0 ∣𝑥[𝑛]∣2=𝑇1 𝜎2 𝑣 + 𝑇2 𝜎2 𝑣 = 𝑧1+ (𝜎 2 𝑠+ 𝜎𝑣2 𝜎2 𝑣 )𝑧2= 𝑧1+ (1 + 𝑆𝑁𝑅)𝑧2, (2.4) where𝑆𝑁𝑅 := 𝜎2

𝑠/𝜎𝑣2. Since𝑧1 and𝑧2 are independent, the

pdf of ¯𝑇 is given by

𝑓𝑇¯(𝑥) = 𝑓𝑧1(𝑥) ∗ 𝑓𝑧2((1 + 𝑆𝑁𝑅)𝑥) , (2.5)

where ∗ denotes the convolution. In terms of Laplace

trans-form, (2.5) reads 𝐹𝑇¯(𝑠) = 𝐹𝑧1(𝑠) × ℒ {𝑓𝑧2((1 + 𝑆𝑁𝑅)𝑥)} = 𝐹𝑧1(𝑠) × 1 1 + 𝑆𝑁𝑅𝐹𝑧2 ( 𝑠 1 + 𝑆𝑁𝑅 ) , (2.6)

where the second equality follows since ℒ {𝑓(𝑎𝑥)} =

(𝑎)−1𝐹 (𝑠/𝑎) [11]. To derive an explicit expression for 𝐹¯ 𝑇(𝑠)

in (2.6), we need the next lemma.

Lemma 2.1 [11]: For 𝜆 > 0, we have

{𝑥𝜆−1𝑒−𝑎𝑥𝑢(𝑥)}= Γ(𝜆)(𝑠 + 𝑎)−𝜆.

From (2.3) and by means of Lemma 2.1, we immediately have

𝐹𝑧1(𝑠) = Γ (𝑛0/2) (𝑠 + 1/2)−𝑛0/2 2𝑛0Γ (𝑛0/2) = (𝑠 + 1/2)−𝑛0/2 2𝑛0 (2.7) and 𝐹𝑧2(𝑠) = Γ ((𝑁 − 𝑛 0)/2) (𝑠 + 1/2)−(𝑁−𝑛0)/2 2(𝑁−𝑛0)Γ ((𝑁 − 𝑛0)/2) =(𝑠 + 1/2)√ −(𝑁−𝑛0)/2 2(𝑁−𝑛0) . (2.8)

Based on (2.6), (2.7), and (2.8), direct manipulation shows

𝐹𝑇¯(𝑠) = 1 (1 + 𝑆𝑁𝑅)√2𝑁 ( 𝑠 +12 )−𝑛0/2 × ( 𝑠 1 + 𝑆𝑁𝑅+ 1 2 )−(𝑁−𝑛0)/2 =(1 + 𝑆𝑁𝑅)√[(𝑁−𝑛0)/2]−1 2𝑁 ( 𝑠 +1 2 )−𝑛0/2 × ( 𝑠 +1 + 𝑆𝑁𝑅2 )−(𝑁−𝑛0)/2 . (2.9) With the aid of (2.9), the pdf𝑓𝑇¯(𝑥) is given by

𝑓𝑇¯(𝑥) = (1 + 𝑆𝑁𝑅) [(𝑁−𝑛0)/2]−1 2𝑁 { ℒ−1{(𝑠 + 1/2)−𝑛02 } ∗ℒ−1{(𝑠 + (1 + 𝑆𝑁𝑅)/2)−(𝑁−𝑛0)/2}} (𝑎) = (1 + 𝑆𝑁𝑅)√[(𝑁−𝑛0)/2]−1 2𝑁 {[ 𝑥𝑛02 −1𝑒−𝑥2𝑢(𝑥) Γ(𝑛0/2) ] [ 𝑥[(𝑁−𝑛0)/2]−1𝑒−(1+𝑆𝑁𝑅)𝑥/2𝑢(𝑥) Γ ((𝑁 − 𝑛0)/2) ]} =√(1 + 𝑆𝑁𝑅)[(𝑁−𝑛0)/2]−1 2𝑁Γ(𝑛0/2)Γ ((𝑁 − 𝑛0)/2)𝑥 0 𝜏 [(𝑁−𝑛0)/2]−1 × 𝑒−(1+𝑆𝑁𝑅)𝜏/2(𝑥 − 𝜏)𝑛0/2−1𝑒−(𝑥−𝜏)/2𝑑𝜏 =√(1 + 𝑆𝑁𝑅)[(𝑁−𝑛0)/2]−1 2𝑁Γ(𝑛0/2)Γ ((𝑁 − 𝑛0)/2)𝑒−𝑥/2×𝑥 0 𝜏 [(𝑁−𝑛0)/2]−1(𝑥 − 𝜏)𝑛0/2−1𝑒−𝑆𝑁𝑅𝜏/2𝑑𝜏, (2.10)

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where (a) holds by using Lemma 2.1. Hence, for a given threshold 𝛾 determined according to the prescribed

false-alarm probability, the conditional detection probability can be computed based on (2.10) as 𝑃𝐷(𝑛0) = ∫ 𝛾 𝑓𝑇¯(𝑥)𝑑𝑥 = √(1 + 𝑆𝑁𝑅)[(𝑁−𝑛0)/2]−1 2𝑁Γ(𝑛0/2)Γ ((𝑁 − 𝑛0)/2) 𝛾 𝑑𝑥× [ 𝑒−𝑥/2𝑥 0 𝜏 [(𝑁−𝑛0)/2]−1(𝑥 − 𝜏)𝑛02−1𝑒−𝑆𝑁𝑅𝜏/2𝑑𝜏 ]    :=𝑝(𝑥) . (2.11) To find a closed-form expression of 𝑃𝐷(𝑛0) in (2.11), we

need the next lemma.

Lemma 2.2 [11]: For 𝜈 > 0 and 𝜇 > 0, it follows

𝑥

0 𝑡

𝜈−1(𝑥 − 𝑡)𝜇−1𝑒𝛿𝑡𝑑𝑡 = 𝐵(𝜇, 𝜈)𝑥𝜇+𝜈−1Φ(𝜈, 𝜇 + 𝜈; 𝛿𝑥),

(2.12) where𝐵(⋅, ⋅) is the beta function, and Φ(⋅, ⋅; ⋅) is the confluent

hyper-geometric function defined by Φ(𝛼 , 𝛾; 𝑧) =

1 + 𝛼𝛾 1!𝑧 +𝛼(𝛼 + 1)𝛾(𝛾 + 1) ⋅𝑧2!2 +𝛼(𝛼 + 1)(𝛼 + 2)𝛾(𝛾 + 1)(𝛾 + 2) ⋅𝑧3!3 + . . . . (2.13)

Based on Lemma 2.2, equation (2.11) becomes

𝑃𝐷(𝑛0) = (1 + 𝑆𝑁𝑅) [(𝑁−𝑛0)/2]−1𝐵(𝑁−𝑛0 2 ,𝑛20 ) 2𝑁Γ(𝑛0/2)Γ ((𝑁 − 𝑛0)/2) × 𝛾 𝑒 −𝑥/2𝑥(𝑁/2)−1 [𝑖=0 𝑎𝑖𝑥𝑖 ] 𝑑𝑥, (2.14) where 𝑎0= 1, 𝑎1= (𝑁 − 𝑛𝑁/20)/2⋅(−𝑆𝑁𝑅/2)1! , 𝑎2=[(𝑁 − 𝑛0𝑁/2{[𝑁/2] + 1})/2]{[(𝑁 − 𝑛0)/2] + 1}⋅(−𝑆𝑁𝑅/2)2! 2, . . . . (2.15) Based on (2.14), the exact form of the conditional detection probability can be obtained as1

𝑃𝐷(𝑛0) = (1 + 𝑆𝑁𝑅) [(𝑁−𝑛0)/2]−1𝐵(𝑁−𝑛0 2 ,𝑛20 ) 2𝑁Γ(𝑛0/2)Γ ((𝑁 − 𝑛0)/2) × [𝑖=0 𝑎𝑖 𝛾 𝑒 −𝑥/2𝑥(𝑁/2)+𝑖−1𝑑𝑥 ] (𝑏) = (1 + 𝑆𝑁𝑅)[(𝑁−𝑛0)/2]−1𝐵 (𝑁−𝑛 0 2 ,𝑛20 ) 2𝑁Γ(𝑛0/2)Γ ((𝑁 − 𝑛0)/2) × 𝑖=0 𝑎𝑖 [ 2(𝑁/2)+𝑖Γ(𝑁 2 + 𝑖, 𝛾 2 )] , (2.16) 1In the case of𝑛

0= 0, (2.16) reduces to the widely known result in [6, p-144].

where (b) follows since ∫𝛾∞𝑥𝜈−1𝑒−𝜇𝑥𝑑𝑥 = 𝜇−𝜈Γ(𝜈, 𝜇𝛾)

[6, 346], and Γ(𝛼 , 𝑦) :=𝑦∞𝑒−𝑡𝑡𝛼−1𝑑𝑡 is the incomplete

Gamma function. Based on (2.16), we summarize the main result in the following theorem.

Theorem 2.3: The average detection probability of the ED

under the hypothesis test (1.2) is given by

𝑃𝐷=𝑁1 𝑁−1 𝑛0=0 𝑃𝐷(𝑛0) = 1 𝑁 𝑁−1 𝑛0=0 { (1 + 𝑆𝑁𝑅)[(𝑁−𝑛0)/2]−1𝐵(𝑁−𝑛0 2 ,𝑛20 ) 2𝑁Γ(𝑛0/2)Γ ((𝑁 − 𝑛0)/2) × 𝑖=0 𝑎𝑖 [ 2(𝑁/2)+𝑖Γ(𝑁 2 + 𝑖, 𝛾 2 )]} , (2.17)

where 𝛾 is the threshold determined according to the

pre-scribed false-alarm probability.

B. Low-SNR Regime

While the formula (2.17) appears quite involved, in the low-SNR regime it admits a very simple form that is compatible with the existing study of ED [6]. To see this, we need the next lemma, which provides an upper and lower bounds for the conditional detection probability𝑃𝐷(𝑛0).

Lemma 2.4: Let𝑃𝐷(𝑛0) be defined in (2.16). Then we have

Γ(𝑁 2, 𝛾 (1+𝑆𝑁𝑅 2 )) (1 + 𝑆𝑁𝑅)(𝑛0/2)+1Γ(𝑁/2) ≤ 𝑃𝐷(𝑛0) (1 + 𝑆𝑁𝑅)[(𝑁−𝑛0)]/2−1Γ (𝑁 2,𝛾2 ) Γ(𝑁/2) . (2.18)

Proof: See appendix.

To gain further insight based on (2.18), let us assume without loss of generality that the total number of samples

𝑁 is even, so that 𝑁/2 is a positive integer. In this case, we

have Γ(𝑁/2) = [(𝑁/2) − 1]! and Γ(𝑁/2, 𝑦) = [(𝑁/2) − 1]!𝑒−𝑦(𝑁/2)−1 𝑘=0 𝑦 𝑘 𝑘! [11, p-900]. Hence (2.18) becomes 𝑒−𝛾(1+𝑆𝑁𝑅)/2(𝑁/2)−1 𝑘=0 [𝛾(1+𝑆𝑁𝑅)/2] 𝑘 𝑘! (1 + 𝑆𝑁𝑅)(𝑛0/2)+1 ≤ 𝑃𝐷(𝑛0) ≤ (1 + 𝑆𝑁𝑅)[(𝑁−𝑛0)/2]−1𝑒−𝛾/2 (𝑁/2)−1 𝑘=0 [𝛾/2]𝑘 𝑘! . (2.19)

When 𝑆𝑁𝑅 → 0, we have 1 + 𝑆𝑁𝑅 → 1 and (2.19) then

becomes 𝑃𝐷(𝑛0) → 𝑒−𝛾/2 (𝑁/2)−1 𝑘=0 (𝛾/2)𝑘 𝑘! (𝑐) = 𝑄𝜒2 𝑁(𝛾), (2.20)

where (c) holds directly by definition of the right-tail probability of the Chi-square random variable 𝜒2

𝑁 with an

even degree-of-freedom [6, p-25]. With the aid of (2.20) and since the limiting probability is independent of 𝑛0, we have the following asymptotic result.

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0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Pf PD experimental result analytic result (2.17)

Fig. 1. Analytic and experimental ROC curves of energy detector (SNR = -5 dB).

Proposition 2.5: Let𝑃𝐷 be the average detection

probabil-ity defined in (2.17). Then we have lim

𝑆𝑁𝑅→0𝑃𝐷= 𝑄𝜒2𝑁(𝛾). (2.21)

Recall from [6, Sec. 5.3] that𝑄𝜒2

𝑁(𝛾) is the detection

prob-ability for ED when𝑆𝑁𝑅 = 𝜎2

𝑥/𝜎𝑣2 → 0. In this case, the

performance of ED can be very poor since the energy of the received signal in either hypothesis is very close to the noise floor. To further enhance the detection performance when SNR is low and the signal timing mismatch is present, a robust ED scheme based on the Bayesian principle is proposed next.

III. PROPOSEDBAYESIANDECISION RULE

A. The Bayesian Test

To exploit the prior statistical knowledge of 𝑛0 for

en-hancing the detection performance, a typical approach is the Bayesian philosophy [6]. The conditional joint pdf of the data samples under the two hypothesesℋ0 andℋ1 are

𝑝 (x; ℋ0) = 1 (2𝜋𝜎2 0)𝑁/2 exp [ −1 2𝜎2 0 𝑁−1 𝑛=0 ∣𝑥[𝑛]∣2 ] , (3.1) and 𝑝 (x; ℋ1) = 1 (2𝜋𝜎2 0)𝑛0/2 exp [ −1 2𝜎2 0 𝑛∑0−1 𝑛=0 ∣𝑥[𝑛]∣2 ] × 1 (2𝜋(𝜎2 0+ 𝜎21))(𝑁−𝑛0)/2 exp [ −1 2(𝜎2 0+ 𝜎21) 𝑁−1 𝑛=𝑛0 ∣𝑥[𝑛]∣2 ] . (3.2) The Bayesian test decidesℋ1 if (3.3), shown at the top of the next page, holds (see [6, Chap. 6]). The performance advantages of the Bayesian test (3.3) over the conventional ED scheme (2.1) under the considered scenario will be illustrated via computer simulation in the next section.

−150 −10 −5 0 5 10 15 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 SNR (dB) PD experimental result analytic result (2.17)

Fig. 2. Detection probability𝑃𝐷versus SNR (𝑃𝑓 = 0.05).

B. Numerical Simulation

In the following simulations we compare the performance of the conventional ED (2.1) and that of the proposed Bayesian ED (3.3), based on the hypothesis signal model (1.2); the total number of samples is set to be𝑁 = 200, and the signal arrival

time𝑛0is uniformly distributed within0 ≤ 𝑛0≤ 199. Figure

1 plots the ROC curves of ED (2.1), with SNR set to be−5 dB;

Figure 2 plots the probability of detection𝑃𝐷 at various SNR

levels, assuming that the false-alarm probability 𝑃𝑓 = 0.05.

As can be seen from the figures, the derived analytic formula (2.17) closely matches the simulated results. Figures 3 and 4, respectively, compare𝑃𝐷and1 − 𝑃𝑓 curves (as a function of

SNR) of the ED (2.1) and the robust ED solution (3.3); note that large values of 1 − 𝑃𝑓 mean better channel utilization

efficiency of secondary users [12]. The figures show that the Bayesian based solution (3.3), which takes into account the statistical knowledge of the signal arrival time, not only improves𝑃𝐷 but also leads to larger1 − 𝑃𝑓, especially when

SNR is low. Since the proposed Bayesian ED (3.3) is optimal in accordance with the criterion of minimizing average cost function, a natural approach to evaluate the performance is to compare the detectors (2.1) and (3.3) in terms of the average probability of error, i.e.,

𝑃𝑒= 𝑃 (ℋ0∣ℋ1)𝑃 (ℋ1) + 𝑃 (ℋ1∣ℋ0)𝑃 (ℋ0), (3.4)

where 𝑃 (ℋ𝑖∣ℋ𝑗) denotes the probability that ℋ𝑖 is decided

given that ℋ𝑗 is true. The optimal decision threshold of the

Bayesian rule (3.3) for minimizing average error probability is known to be simply 𝜏 = 𝑃 (ℋ0)/𝑃 (ℋ1) [13, p-20]. In our

simulation, the equally likely hypothesis is assumed, thereby

𝜏 = 1. Figure 5 compares the achievable 𝑃𝑒of (2.1) and (3.3)

at various SNR. As expected, the Bayesian test (3.3) is seen to yield a smaller average error probability.

IV. CONCLUSION

Spectrum sensing in the presence of unknown arrival time of the primary signal finds applications in many practical system scenarios and is thus an important issue in the study of CR networks. In this paper we have provided the exact formula

(5)

1 𝑁𝑁−1 𝑛0=0𝑝(x; 𝑛0, ℋ1) 𝑝(x; ℋ0) = 1 𝑁𝑁−1 𝑛0=0(2𝜋𝜎21 0)𝑛0/2 exp[−1 2𝜎2 0 ∑𝑛0−1 𝑛=0 ∣𝑥[𝑛]∣2 ] × 1 (2𝜋(𝜎2 0+𝜎12))(𝑁−𝑛0)/2 exp[ −1 2(𝜎2 0+𝜎21) ∑𝑁−1 𝑛=𝑛0∣𝑥[𝑛]∣2 ] 1 (2𝜋𝜎2 0)𝑁/2exp [ −1 2𝜎2 0 ∑𝑁−1 𝑛=0 ∣𝑥[𝑛]∣2 ] > 𝛾 (3.3) −150 −10 −5 0 5 10 15 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 SNR (dB) PD Bayesian ED (3.3) ED (2.1)

Fig. 3. Detection probabilities of detectors (2.1) and (3.3) versus SNR (𝑃𝑓= 0.05). −150 −10 −5 0 5 10 15 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 SNR (dB) 1 Pf Bayesian ED (3.3) ED (2.1) Fig. 4. 1 − 𝑃𝑓 versus SNR (𝑃𝐷= 0.95).

of the average detection probability for ED when the arrival time of the primary signal is modeled as a uniform random variable over the observation interval. To further improve the detection performance against the timing uncertainty, we have then proposed a Bayesian based detection scheme. Simulation results show that the Bayesian ED reduces the false-alarm probability and thus enhances the spectrum utilization in the considered asynchronous scenario. Future research will be dedicated to extending the current results to the cooperative sensing scenario. −15 −10 −5 0 5 10 15 10−3 10−2 10−1 100 SNR (dB) Pe ED (2.1) Bayesian ED (3.3)

Fig. 5. Average error probabilities of detectors (2.1) and (3.3) versus SNR.

APPENDIX

PROOF OFLEMMA2.4

We first observe that 𝑝(𝑥) in (2.10) satisfies 𝑒−𝑥/2× 𝑒−𝑆𝑁𝑅𝑥/2𝑥 0 𝜏 [(𝑁−𝑛0)/2]−1(𝑥 − 𝜏)𝑛0/2−1𝑑𝜏 ≤ 𝑝(𝑥) ≤ 𝑒−𝑥/2𝑥 0 𝜏 [(𝑁−𝑛0)/2]−1(𝑥 − 𝜏)𝑛0/2−1𝑑𝜏. (A.1) Since ∫ 𝑥 0 𝜏 [(𝑁−𝑛0)/2]−1(𝑥 − 𝜏)𝑛0/2−1𝑑𝜏 = 𝑥[(𝑁−𝑛0)/2]−1𝑢(𝑥) ∗ 𝑥𝑛0/2−1𝑢(𝑥), (A.2) we have {∫ 𝑥 0 𝜏 [(𝑁−𝑛0)/2]−1(𝑥 − 𝜏)𝑛0/2−1𝑑𝜏 } = ℒ{𝑥[(𝑁−𝑛0)/2]−1𝑢(𝑥)}× ℒ{𝑥𝑛0/2−1𝑢(𝑥)} =Γ ((𝑁 − 𝑛𝑠(𝑁−𝑛0)/20)/2)×Γ(𝑛𝑠𝑛00/2/2) =Γ ((𝑁 − 𝑛𝑠0𝑁/2)/2) Γ(𝑛0/2). (A.3)

By taking the inverse Laplace transform of both sides of (A.3) we have ∫ 𝑥 0 𝜏 [(𝑁−𝑛0)/2]−1(𝑥 − 𝜏)𝑛0/2−1𝑑𝜏 = Γ ((𝑁 − 𝑛0)/2) Γ(𝑛0/2)ℒ−1 { 1 𝑠𝑁/2 }

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=Γ ((𝑁 − 𝑛0)/2) Γ(𝑛0/2)

Γ(𝑁/2) 𝑥𝑁/2−1, (A.4)

where the last equality holds due to Lemma 2.1. With the aid of (A.4), (A.1) becomes

Γ ((𝑁 − 𝑛0)/2) Γ(𝑛0/2)

Γ(𝑁/2) 𝑥𝑁/2−1𝑒−(1+𝑆𝑁𝑅)𝑥/2

≤ 𝑝(𝑥)

≤Γ ((𝑁 − 𝑛0)/2) Γ(𝑛0/2)

Γ(𝑁/2) 𝑥𝑁/2−1𝑒−𝑥/2. (A.5)

Based on (A.5), we have

𝑃𝐷= (1 + 𝑆𝑁𝑅) [(𝑁−𝑛0)/2]−1 2𝑁Γ(𝑛0/2)Γ((𝑁 − 𝑛2)/2) 𝛾 𝑝(𝑥)𝑑𝑥 (1 + 𝑆𝑁𝑅)√ [(𝑁−𝑛0)/2]−1 2𝑁Γ(𝑁/2) 𝛾 𝑥 𝑁 2−1𝑒−(1+𝑆𝑁𝑅)𝑥/2𝑑𝑥 (𝑎) = (1 + 𝑆𝑁𝑅)√ [(𝑁−𝑛0)/2]−1 2𝑁Γ(𝑁/2) ( 1 + 𝑆𝑁𝑅 2 )−𝑁/2 × Γ ( 𝑁 2, 𝛾 ( 1 + 𝑆𝑁𝑅 2 )) = Γ (𝑁 2, 𝛾 (1+𝑆𝑁𝑅 2 )) (1 + 𝑆𝑁𝑅)(𝑛0/2)+1Γ(𝑁/2), (A.6)

where (a) follows since∫𝛾∞𝑥𝜈−1𝑒−𝜇𝑥𝑑𝑥 = 𝜇−𝜈Γ(𝜈, 𝜇𝛾) [11,

p-346]. Similarly we have 𝑃𝐷≤(1 + 𝑆𝑁𝑅) [(𝑁−𝑛0)/2]−1 Γ(𝑁/2) Γ ( 𝑁 2, 𝛾 2 ) . (A.7)

The assertion follows from (A.6) and (A.7).

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數據

Fig. 1. Analytic and experimental ROC curves of energy detector (SNR = -5 dB).
Fig. 1. Analytic and experimental ROC curves of energy detector (SNR = -5 dB). p.4
Fig. 2. Detection probability
Fig. 2. Detection probability p.4
Figure 2 plots the probability of detection

Figure 2

plots the probability of detection p.4
Fig. 5. Average error probabilities of detectors (2.1) and (3.3) versus SNR.
Fig. 5. Average error probabilities of detectors (2.1) and (3.3) versus SNR. p.5
Fig. 3. Detection probabilities of detectors (2.1) and (3.3) versus SNR (
Fig. 3. Detection probabilities of detectors (2.1) and (3.3) versus SNR ( p.5

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