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

(1.1)
where*𝑁 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 ≤ 𝑛*0*− 1 (idle)*
*ℋ1*:
{
*𝑥[𝑛] = 𝑣[𝑛],* *0 ≤ 𝑛 ≤ 𝑛*0*− 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 *𝑛*0

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 about*𝑛*0for 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

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 hypothesis*ℋ*1 in (1.2) and conditioned

on a fixed*𝑛*0, let us decompose the test statistic *𝑇 into*
*𝑇 =*
*𝑛*∑0*−1*
*𝑛=0*
*∣𝑥[𝑛]∣*2
*:=𝑇*1
+
*𝑁−1*_{∑}
*𝑛=𝑛*0
*∣𝑥[𝑛]∣*2
*:=𝑇*2
*.* (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 to*𝑛*0.

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

zero-mean white sequences with variances given by*𝜎*2

*𝑠* and

*𝜎*2

*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/𝜎*2

*𝑣* *∼ 𝜒*2*𝑛*0 and*𝑧2* *:= 𝑇*2/(𝜎2*𝑣+ 𝜎𝑠*2*) ∼ 𝜒*2*𝑁−𝑛*0, and hence

the associated probability density functions (PDF) are

*𝑓𝑧*1*(𝑥) =*
*𝑥(𝑛*0*/2)−1𝑒−𝑥/2*
*√*
2*𝑛*0*Γ(𝑛*0/2) *𝑢(𝑥) and*
*𝑓𝑧*2*(𝑥) =*
*𝑥[(𝑁−𝑛*0*)/2]−1𝑒−𝑥/2*
*√*
2*(𝑁−𝑛*0)*Γ ((𝑁 − 𝑛*_{0}*)/2)𝑢(𝑥),* (2.3)

where*𝑢(𝑥) is the unit step function. To simplify notation let*

us consider the equivalent test statistic
¯
*𝑇 =* _{𝜎}𝑇_{2}
*𝑣* =
1
*𝜎*2
*𝑣*
*𝑁−1*_{∑}
*𝑛=0*
*∣𝑥[𝑛]∣*2_{=}*𝑇1*
*𝜎*2
*𝑣* +
*𝑇2*
*𝜎*2
*𝑣*
*= 𝑧*1+ (*𝜎*
2
*𝑠+ 𝜎𝑣*2
*𝜎*2
*𝑣* *)𝑧*2*= 𝑧*1*+ (1 + 𝑆𝑁𝑅)𝑧*2,
(2.4)
where*𝑆𝑁𝑅 := 𝜎*2

*𝑠/𝜎𝑣*2. Since*𝑧*1 and*𝑧*2 are independent, the

pdf of ¯*𝑇 is given by*

*𝑓 _{𝑇}*¯

*(𝑥) = 𝑓𝑧*1

*(𝑥) ∗ 𝑓𝑧*2

*((1 + 𝑆𝑁𝑅)𝑥) ,*(2.5)

where *∗ denotes the convolution. In terms of Laplace *

trans-form, (2.5) reads
*𝐹 _{𝑇}*¯

*(𝑠) = 𝐹𝑧*1

*(𝑠) × ℒ {𝑓𝑧*2

*((1 + 𝑆𝑁𝑅)𝑥)}*

*= 𝐹𝑧*1

*(𝑠) ×*1

*1 + 𝑆𝑁𝑅𝐹𝑧*2 (

*𝑠*

*1 + 𝑆𝑁𝑅*)

*,*(2.6)

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

*(𝑎)−1 _{𝐹 (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

*ℒ*{*𝑥𝜆−1 _{𝑒}−𝑎𝑥_{𝑢(𝑥)}*}

_{= Γ(𝜆)(𝑠 + 𝑎)}−𝜆_{.}

_{□}From (2.3) and by means of Lemma 2.1, we immediately have

*𝐹𝑧*1*(𝑠) =*
*Γ (𝑛*0* _{√}/2) (𝑠 + 1/2)−𝑛*0

*/2*2

*𝑛*0

*Γ (𝑛*

_{0}

*/2)*=

*(𝑠 + 1/2)−𝑛*0

*/2*

*√*2

*𝑛*0 (2.7) and

*𝐹𝑧*2

*(𝑠) =*

*Γ ((𝑁 − 𝑛*0

_{√}*)/2) (𝑠 + 1/2)−(𝑁−𝑛*0

*)/2*2

*(𝑁−𝑛*0)

*Γ ((𝑁 − 𝑛*

_{0}

*)/2)*=

*(𝑠 + 1/2)√*

*−(𝑁−𝑛*0

*)/2*2

*(𝑁−𝑛*0)

*.*(2.8)

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

*𝐹 _{𝑇}*¯

*(𝑠) =*1

*(1 + 𝑆𝑁𝑅)√*2

*𝑁*(

*𝑠 +*1

_{2})

*−𝑛*0

*/2*

*×*(

*𝑠*

*1 + 𝑆𝑁𝑅*+ 1 2 )

*−(𝑁−𝑛*0

*)/2*=

*(1 + 𝑆𝑁𝑅)√[(𝑁−𝑛*0

*)/2]−1*2

*𝑁*(

*𝑠 +*1 2 )

_{−𝑛}_{0}

_{/2}*×*(

*𝑠 +1 + 𝑆𝑁𝑅*

_{2})

*−(𝑁−𝑛*0

*)/2*

*.*(2.9) With the aid of (2.9), the pdf

*𝑓𝑇*¯

*(𝑥) is given by*

*𝑓 _{𝑇}*¯

*(𝑥) =*

*(1 + 𝑆𝑁𝑅)*

*[(𝑁−𝑛*0

*)/2]−1*

*√*2

*𝑁*{

*ℒ−1*{

*2 }*

_{(𝑠 + 1/2)}−𝑛0*∗ℒ−1*{

*0*

_{(𝑠 + (1 + 𝑆𝑁𝑅)/2)}−(𝑁−𝑛*)/2*}}

*(𝑎)*=

*(1 + 𝑆𝑁𝑅)√[(𝑁−𝑛*0

*)/2]−1*2

*𝑁*{[

*𝑥𝑛0*2

*−1𝑒−𝑥*2

*𝑢(𝑥)*

*Γ(𝑛*0/2) ]

*∗*[

*𝑥[(𝑁−𝑛*0

*)/2]−1𝑒−(1+𝑆𝑁𝑅)𝑥/2𝑢(𝑥)*

*Γ ((𝑁 − 𝑛*0

*)/2)*]} =

*√(1 + 𝑆𝑁𝑅)[(𝑁−𝑛*0

*)/2]−1*2

*𝑁*

_{Γ(𝑛}_{0}

_{/2)Γ ((𝑁 − 𝑛}_{0}

*∫*

_{)/2)}*0*

_{𝑥}*𝜏*

*[(𝑁−𝑛*0

*)/2]−1*

*× 𝑒−(1+𝑆𝑁𝑅)𝜏/2*0

_{(𝑥 − 𝜏)}𝑛*/2−1𝑒−(𝑥−𝜏)/2𝑑𝜏*=

*√(1 + 𝑆𝑁𝑅)[(𝑁−𝑛*0

*)/2]−1*2

*𝑁*

_{Γ(𝑛}_{0}

_{/2)Γ ((𝑁 − 𝑛}_{0}

*∫*

_{)/2)}𝑒−𝑥/2×*0*

_{𝑥}*𝜏*

*[(𝑁−𝑛*0

*)/2]−1*0

_{(𝑥 − 𝜏)}𝑛*/2−1*

_{𝑒}−𝑆𝑁𝑅𝜏/2_{𝑑𝜏, (2.10)}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(𝑛*0) =
∫ _{∞}*𝛾* *𝑓𝑇*¯*(𝑥)𝑑𝑥*
= *√(1 + 𝑆𝑁𝑅)[(𝑁−𝑛*0*)/2]−1*
2*𝑁 _{Γ(𝑛}*

_{0}

_{/2)Γ ((𝑁 − 𝑛}_{0}

*∫*

_{)/2)}

_{∞}*𝛾*

*𝑑𝑥×*[

*𝑒−𝑥/2*∫

*𝑥*0

*𝜏*

*[(𝑁−𝑛*0

*)/2]−1(𝑥 − 𝜏)𝑛0*2

*−1𝑒−𝑆𝑁𝑅𝜏/2𝑑𝜏*]

*:=𝑝(𝑥)*

*.*(2.11) To find a closed-form expression of

*𝑃𝐷(𝑛*0) 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−1_{𝑒}𝛿𝑡_{𝑑𝑡 = 𝐵(𝜇, 𝜈)𝑥}𝜇+𝜈−1_{Φ(𝜈, 𝜇 + 𝜈; 𝛿𝑥),}*

(2.12)
where*𝐵(⋅, ⋅) is the beta function, and Φ(⋅, ⋅; ⋅) is the confluent*

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

1 + *𝛼 _{𝛾}*

*⋅*

_{1!}

*𝑧*+

*𝛼(𝛼 + 1)*

_{𝛾(𝛾 + 1)}*⋅𝑧*

_{2!}2 +

*𝛼(𝛼 + 1)(𝛼 + 2)*

_{𝛾(𝛾 + 1)(𝛾 + 2)}*⋅𝑧*

_{3!}3

*+ . . . .*(2.13)

*□*

Based on Lemma 2.2, equation (2.11) becomes

*𝑃𝐷(𝑛*0) = *(1 + 𝑆𝑁𝑅)*
*[(𝑁−𝑛*0*)/2]−1𝐵*(*𝑁−𝑛*0
2 *,𝑛*20
)
*√*
2*𝑁 _{Γ(𝑛}*

_{0}

_{/2)Γ ((𝑁 − 𝑛}_{0}

_{)/2)}*×*∫

_{∞}*𝛾*

*𝑒*

*−𝑥/2*[

_{𝑥}(𝑁/2)−1*∑*

_{∞}*𝑖=0*

*𝑎𝑖𝑥𝑖*]

*𝑑𝑥,*(2.14) where

*𝑎*0

*= 1, 𝑎*1=

*(𝑁 − 𝑛*0

_{𝑁/2}*)/2⋅(−𝑆𝑁𝑅/2)*

_{1!}

*,*

*𝑎2*=

*[(𝑁 − 𝑛*0

*0*

_{𝑁/2{[𝑁/2] + 1}})/2]{[(𝑁 − 𝑛*)/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(𝑛*0) = *(1 + 𝑆𝑁𝑅)*
*[(𝑁−𝑛*0*)/2]−1𝐵*(*𝑁−𝑛*0
2 *,𝑛*20
)
*√*
2*𝑁 _{Γ(𝑛}*

_{0}

_{/2)Γ ((𝑁 − 𝑛}_{0}

_{)/2)}*×*[

*∑*

_{∞}*𝑖=0*

*𝑎𝑖*∫

_{∞}*𝛾*

*𝑒*

*−𝑥/2*]

_{𝑥}(𝑁/2)+𝑖−1_{𝑑𝑥}*(𝑏)*=

*(1 + 𝑆𝑁𝑅)[(𝑁−𝑛*0

*)/2]−1𝐵*(

*0 2*

_{𝑁−𝑛}*,𝑛*20 )

*√*2

*𝑁*

_{Γ(𝑛}_{0}

_{/2)Γ ((𝑁 − 𝑛}_{0}

_{)/2)}*×*

*∞*∑

*𝑖=0*

*𝑎𝑖*[ 2

*(𝑁/2)+𝑖*

_{Γ}(

*𝑁*2

*+ 𝑖,*

*𝛾*2 )]

*,*(2.16) 1

_{In the case of}

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

where (b) follows since ∫_{𝛾}∞𝑥𝜈−1_{𝑒}−𝜇𝑥_{𝑑𝑥 = 𝜇}−𝜈_{Γ(𝜈, 𝜇𝛾)}

[6, 346], and *Γ(𝛼 , 𝑦) :=* ∫_{𝑦}∞𝑒−𝑡_{𝑡}𝛼−1_{𝑑𝑡 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*=* _{𝑁}*1

*𝑁−1*

_{∑}

*𝑛*0=0

*𝑃𝐷(𝑛*0) = 1

*𝑁*

*𝑁−1*

_{∑}

*𝑛*0=0 {

*(1 + 𝑆𝑁𝑅)[(𝑁−𝑛*0

*)/2]−1𝐵*(

*𝑁−𝑛*0 2

*,𝑛*20 )

*√*2

*𝑁*

_{Γ(𝑛}_{0}

_{/2)Γ ((𝑁 − 𝑛}_{0}

_{)/2)}*×*

*∞*∑

*𝑖=0*

*𝑎𝑖*[ 2

*(𝑁/2)+𝑖*

_{Γ}(

*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 probability*𝑃𝐷(𝑛*0).

* Lemma 2.4: Let𝑃𝐷(𝑛*0) be defined in (2.16). Then we have

Γ(*𝑁*
2*, 𝛾*
(* _{1+𝑆𝑁𝑅}*
2
))

*(1 + 𝑆𝑁𝑅)(𝑛*0

*/2)+1Γ(𝑁/2)*

*≤ 𝑃𝐷(𝑛*0)

*≤*

*(1 + 𝑆𝑁𝑅)[(𝑁−𝑛*0

*)]/2−1*Γ (

*2*

_{𝑁}*,𝛾*2 )

*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)(𝑛*0*/2)+1*
*≤ 𝑃𝐷(𝑛*0)
*≤ (1 + 𝑆𝑁𝑅)[(𝑁−𝑛*0*)/2]−1 _{𝑒}−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(𝑛*0*) → 𝑒−𝛾/2*
*(𝑁/2)−1*_{∑}
*𝑘=0*
*(𝛾/2)𝑘*
*𝑘!*
*(𝑐)*
*= 𝑄 _{𝜒}*2

*𝑁(𝛾),*(2.20)

where (c) holds directly by definition of the right-tail
probability of the Chi-square random variable *𝜒*2

*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.

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= 𝑄𝜒*2*𝑁(𝛾).* (2.21)

*□*

Recall from [6, Sec. 5.3] that*𝑄 _{𝜒}*2

*𝑁(𝛾) is the detection *

prob-ability for ED when*𝑆𝑁𝑅 = 𝜎*2

*𝑥/𝜎𝑣*2 *→ 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 *𝑛*0 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 hypotheses*ℋ0* and*ℋ1* are

* 𝑝 (x; ℋ0*) = 1

*(2𝜋𝜎*2 0)

*𝑁/2*exp [

*−1*

*2𝜎*2 0

*𝑁−1*

_{∑}

*𝑛=0*

*∣𝑥[𝑛]∣*2 ]

*,*(3.1) and

*1) = 1*

**𝑝 (x; ℋ***(2𝜋𝜎*2 0)

*𝑛*0

*/2*exp [

*−1*

*2𝜎*2 0

*𝑛*∑0

*−1*

*𝑛=0*

*∣𝑥[𝑛]∣*2 ]

*×*1

*(2𝜋(𝜎*2 0

*+ 𝜎*21))

*(𝑁−𝑛*0

*)/2*exp [

*−1*

*2(𝜎*2 0

*+ 𝜎*21)

*𝑁−1*

_{∑}

*𝑛=𝑛*0

*∣𝑥[𝑛]∣*2 ]

*.*(3.2) The Bayesian test decides

*ℋ1*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 probability*𝑃𝐷*versus 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 be*𝑁 = 200, and the signal arrival*

time*𝑛*0is uniformly distributed within*0 ≤ 𝑛*0*≤ 199. Figure*

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

Figure 2 plots the probability of detection*𝑃𝐷* 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, compare*𝑃𝐷*and*1 − 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
improves*𝑃𝐷* but also leads to larger*1 − 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 (ℋ*0*∣ℋ*1*)𝑃 (ℋ*1*) + 𝑃 (ℋ*1*∣ℋ*0*)𝑃 (ℋ*0*),* (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 (ℋ*1) [13, p-20]. In our

simulation, the equally likely hypothesis is assumed, thereby

*𝜏 = 1. Figure 5 compares the achievable 𝑃𝑒*of (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

1
*𝑁*
∑_{𝑁−1}*𝑛*0=0* 𝑝(x; 𝑛0, ℋ1*)

*) = 1*

**𝑝(x; ℋ0***𝑁*∑

_{𝑁−1}*𝑛*0=0(

*21 0)*

_{2𝜋𝜎}*𝑛0/2*exp[

*−1*

*2𝜎*2 0 ∑

_{𝑛}_{0}

_{−1}*𝑛=0*

*∣𝑥[𝑛]∣*2 ]

*×*1 (

*2𝜋(𝜎*2 0

*+𝜎*12))

*(𝑁−𝑛0)/2*exp[

*−1*

*2(𝜎*2 0

*+𝜎*21) ∑

_{𝑁−1}*𝑛=𝑛*0

*∣𝑥[𝑛]∣*2 ] 1 (

*2𝜋𝜎*2 0)

*𝑁/2*exp [

*−1*

*2𝜎*2 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
*−*
P*f*
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/2*∫

*𝑥*0

*𝜏*

*[(𝑁−𝑛*0

*)/2]−1(𝑥 − 𝜏)𝑛*0

*/2−1𝑑𝜏*

*≤ 𝑝(𝑥)*

*≤ 𝑒−𝑥/2*∫

*𝑥*0

*𝜏*

*[(𝑁−𝑛*0

*)/2]−1*0

_{(𝑥 − 𝜏)}𝑛*/2−1*

_{𝑑𝜏.}_{(A.1)}Since ∫

*0*

_{𝑥}*𝜏*

*[(𝑁−𝑛*0

*)/2]−1*0

_{(𝑥 − 𝜏)}𝑛*/2−1*

_{𝑑𝜏}*= 𝑥[(𝑁−𝑛*0

*)/2]−1*0

_{𝑢(𝑥) ∗ 𝑥}𝑛*/2−1*

_{𝑢(𝑥),}_{(A.2)}we have

*ℒ*{∫

*0*

_{𝑥}*𝜏*

*[(𝑁−𝑛*0

*)/2]−1(𝑥 − 𝜏)𝑛*0

*/2−1𝑑𝜏*}

*= ℒ*{

*𝑥[(𝑁−𝑛*0

*)/2]−1*}

_{𝑢(𝑥)}*{*

_{× ℒ}*0*

_{𝑥}𝑛*/2−1*} =

_{𝑢(𝑥)}*Γ ((𝑁 − 𝑛*

_{𝑠}_{(𝑁−𝑛}_{0}

*0*

_{)/2}*)/2)×Γ(𝑛*

_{𝑠}_{𝑛}_{0}0

*=*

_{/2}/2)*Γ ((𝑁 − 𝑛*0

_{𝑠}*0/2)*

_{𝑁/2})/2) Γ(𝑛*.*(A.3)

By taking the inverse Laplace transform of both sides of (A.3)
we have
∫ * _{𝑥}*
0

*𝜏*

*[(𝑁−𝑛*0

*)/2]−1(𝑥 − 𝜏)𝑛*0

*/2−1𝑑𝜏*

*= Γ ((𝑁 − 𝑛*0

*)/2) Γ(𝑛*0

*/2)ℒ−1*{ 1

*𝑠𝑁/2*}

=*Γ ((𝑁 − 𝑛*0*)/2) Γ(𝑛*0*/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 − 𝑛*0*)/2) Γ(𝑛*0/2)

*Γ(𝑁/2)* *𝑥𝑁/2−1𝑒−(1+𝑆𝑁𝑅)𝑥/2*

*≤ 𝑝(𝑥)*

*≤Γ ((𝑁 − 𝑛*0*)/2) Γ(𝑛*0*/2)*

*Γ(𝑁/2)* *𝑥𝑁/2−1𝑒−𝑥/2.* (A.5)

Based on (A.5), we have

*𝑃𝐷*= *(1 + 𝑆𝑁𝑅)*
*[(𝑁−𝑛*0*)/2]−1*
*√*
2*𝑁 _{Γ(𝑛}*

_{0}

_{/2)Γ((𝑁 − 𝑛}_{2}

*∫*

_{)/2)}

_{∞}*𝛾*

*𝑝(𝑥)𝑑𝑥*

*≥*

*(1 + 𝑆𝑁𝑅)√*

*[(𝑁−𝑛*0

*)/2]−1*2

*𝑁*∫

_{Γ(𝑁/2)}

_{∞}*𝛾*

*𝑥*

*𝑁*2

*−1𝑒−(1+𝑆𝑁𝑅)𝑥/2𝑑𝑥*

*(𝑎)*=

*(1 + 𝑆𝑁𝑅)√*

*[(𝑁−𝑛*0

*)/2]−1*2

*𝑁*(

_{Γ(𝑁/2)}*1 + 𝑆𝑁𝑅*2 )

*−𝑁/2*

*×*Γ (

*𝑁*2

*, 𝛾*(

*1 + 𝑆𝑁𝑅*2 )) = Γ (

*2*

_{𝑁}*, 𝛾*(

*2 ))*

_{1+𝑆𝑁𝑅}*(1 + 𝑆𝑁𝑅)(𝑛*0

*/2)+1Γ(𝑁/2),*(A.6)

where (a) follows since∫_{𝛾}∞𝑥𝜈−1_{𝑒}−𝜇𝑥_{𝑑𝑥 = 𝜇}−𝜈_{Γ(𝜈, 𝜇𝛾) [11,}

p-346]. Similarly we have
*𝑃𝐷≤(1 + 𝑆𝑁𝑅)*
*[(𝑁−𝑛*0*)/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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