Research Article Cooperative Sequential Sensing of

Hindawi Publishing Corporation International Journal of Distributed Sensor Networks Volume 2015, Article ID 456074, 12 pages http://dx.doi.org/10.1155/2015/456074

Research Article Cooperative Sequential Sensing of Radio Transmissions in 5G with Improved Cost-Delay Tradeoff Xiaoyu Qiao, Weiliang Xie, and Fengyi Yang Technology Innovation Center, China Telecom Corporation Limited, Beijing 102209, China Correspondence should be addressed to Xiaoyu Qiao; [email protected] Received 4 March 2015; Revised 16 June 2015; Accepted 28 July 2015 Academic Editor: Abhishek Roy Copyright © 2015 Xiaoyu Qiao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The fifth-generation (5G) wireless networks are generally anticipated to be heterogeneous, consisting of macro cells, wireless local area networks, device-to-device networks, ad hoc networks, and so forth. The spectrum occupancy varies on spatial and temporal basis. So sensing the variation of spectrum occupancy and informing the resource scheduler can optimize the utilization efficiency of spectrum. Especially for deployment of cognitive radio in 5G, spectrum sensing is regarded as a key technique. In this work, we study the cooperative sensing in 5G heterogeneous wireless networks with a centralized control module. By formulating this cooperative sensing problem as a sequential binary hypothesis test problem, the number of unnecessary data samples and the associated cost is substantially reduced, with guaranteed detection precision. We develop a cooperative sequential detection algorithm, in which multiple geographically diverse sensors are sequentially set up to measure and transmit measurement results on demand. Furthermore, we consider different sensor sampling schemes to address the cost-delay tradeoff problems and then propose a conditional mean activation and sampling algorithm, in which the number of required samples is predicted based on the quality of the collected samples. The performances of different sensor sampling schemes are demonstrated under different sensing environments.

1. Introduction Radio signal detection is a common problem in applications such as wireless sensor networks, radio surveillance, source localization, and cognitive radio. The 5G wireless networks are generally anticipated to be heterogeneous [1, 2], which can consist of macro cells, wireless local area networks (WLAN), vehicular networks, ultra-dense networks (UDN), deviceto-device networks (D2D), ad hoc networks, and so forth. The spectrum occupancy of these networks varies on spatial and temporal basis. Therefore, sensing the radio environment is beneficial for enhancing the utilization of spectrum. Especially for deployment of cognitive radio in 5G [3–5], spectrum sensing is regarded as an essential approach. The cognitive users (CU) need to collect sufficient information about the spectrum occupancy to opportunistically access the spectrum holes and avoid interfering primary users (PU). There are a variety of methods in the literatures on radio signals sensing, including energy detection, feature detection, and coherent detection [6]. Framing the detection problem

into a binary hypothesis test, the signal detection framework typically requires the collection of a number of samples for computing a test statistic against a threshold. Normally, the larger number of samples provides more accuracy in detection outcome. To guarantee sufficient samples, the detector often takes redundant observations, leading to more energy consumption. In addition, when sensors need to report the samples or their processed versions, more transmission cost is caused in terms of both energy and spectrum usage. Besides, processing capabilities for higher sampling rate increase hardware cost of sensors. Hence, there is a clear tradeoff between detection cost and detection precision. We consider the utilization of sequential detection [7] as an effective tool to handle the tradeoff between the cost and the precision in cooperative sensing of radio signals. In sequential detection, the samples are taken sequentially to perform the hypothesis test. Instead of setting the fixed number of samples before sensing, only one more sample is taken at a time. As soon as we reach a confident detection conclusion, we stop collecting additional samples, thereby

2 substantially reducing the number of unnecessary data samples and the cost associated with them. There have already been studies on sequential detection in various scenarios. In [8, 9], a sequential detection framework was applied for the detection of emerging PU in cognitive radio networks, considering single sensor without cooperation. Li et al. extended their work to collaborative spectrum sensing in [10, 11]. Jayaprakasam et al. [12–14] presented a series of works on sequential detection-based cooperative spectrum sensing. The sensors were required to not only take samples but also make calculations, which consumed considerable energy and hardware cost. Hesham et al. [15] proposed a distributed spectrum sensing scheme where all distributed sensors took observations and calculate log likelihood ratios (LLRs) of the observations. Then all the sensors were ranked in descending order of LLR magnitude. Only a few of the sensors ranked in the front transmitted their calculated LLRs to a fusion center for sequential test. Chin and Chuang [16] applied sequential detection in surveillance sensor networks, in which the sensors periodically reported their local decisions to a fusion center. Yilmaz and Wang [17] proposed a novel approach that the random overshoot above or below the sampling thresholds was proportionately encoded into delay, during which the sensors waited to transmit their processed samples. It is noticed that, in the above works, all sensors were required to make observations, although not all of the observations were necessary [15, 17]. In this paper, we study the cooperative sequential sensing problem of radio transmission in 5G wireless networks with a centralized control module (CCM). CCM collects samples from multiple sensors and performs the binary hypothesis test. Once CCM calculates a statistic, it will first decide whether or not the current statistic is reliable before requesting additional samples from one or several sensors. To address the issue of cost-precision tradeoff, we propose a cooperative sequential detection algorithm to determine whether or not to take additional sensing samples and how many additional samples we may wish to acquire based on the cost-precision tradeoff considerations. In the proposed algorithm, the sensors are in hibernation until activated by a control signal from CCM, through which approaching the sensors’ energy consumption is further reduced. Through the sensors cooperation based on hibernation, we can not only combat measurement noise, shadowing, and multipath fading effect, but also improve fairness of energy consumption across the multiple sensors by averaging their run-time. Such fairness is particularly important in networks consisting of low-cost devices with limited energy. We also take into consideration the detection delay as a result of sequentially requesting additional observation and transmission. Because a delayed detection of radio signal may lead to interference of the signal’s transmission and postponement of access, it is a common objective of most sequential detection approaches to minimize detection delay. As detection delay is affected by how sensors are activated to make observations and transmissions, our investigation of cooperative sequential detection algorithms considers various sampling schemes and their impact on the costdelay tradeoff. This cost-delay tradeoff can be improved

International Journal of Distributed Sensor Networks if we can predict the number of required samples and activate corresponding sensors based on the quality of the accumulated samples. Our work proposes a conditional mean activation and sampling algorithm as an option to handle the cost-delay tradeoff. In this work, the main contributions are as follows: (1) a cooperative sequential detection algorithm was proposed for energy efficient cooperative sensing; (2) different sensor activation algorithms and their cost-delay tradeoff were analyzed; (3) a novel sensor activation and signal sampling algorithm by predicting the number of required samples was proposed to improve the cost-delay tradeoff. This paper is organized as follows. Section 2 describes the system model and formulates the cooperative sequential detection problem in radio sensing. Section 3 proposes a cooperative sequential detection algorithm and its detection of radio signal transmissions. The detection cost was analyzed in Section 4 together with the cost-benefit of various sampling schemes, especially a novel sensor activation and signal sampling algorithm. Section 5 presents the numerical results and analysis of the proposed algorithm given various sampling schemes.

2. System Model As shown in Figure 1, the 5G wireless network is heterogeneous based on multi radio access technologies (RAT). To address capacity and data rate challenges in 5G, especially to improve capacity at critical locations, control (C) and user data (U) planes are desirable to be split [1]. In such a C/Uplane split architecture, macro cells provide coverage, while small cells like UDN, WLAN, and so forth provide localized capacity. Centralization of C-plane is implemented in CCM, which can be deployed as a physical entity or a logic function module. It is an option to locate the CCM at the macro cell based station as a physically independent module. The detection enables CCM to construct global resource map and adapt network policies involving issues such as radio access, resource allocation, and multinode cooperation. The sensors are geographically distributed, in the form of dedicated equipment or sensing components integrated in various terminals. Each sensor sets hibernation as its default mode and makes observations upon reception of an activation control signal from CCM. CCM collects samples from multiple sensors and performs the binary hypothesis test. Once CCM calculates a statistic, it will first decide whether or not the current statistic is reliable before requesting additional samples from one or several sensors. Based on the centralized management of the multi-RAT networks, CCM probably possesses prior knowledge of them and performs the hibernation and activation scheme more efficiently. Under the scenario shown in Figure 1, 𝐾 sensors are geographically distributed. Each sensor 𝑘, 𝑘 ∈ {1, . . . , 𝐾}, monitors radio channels after receiving the activation signal and transmits its sample 𝑦𝑘⃗ to CCM. We assume that the sample vector 𝑦𝑘⃗ is independent identically distributed (i.i.d.). Note that the system model is also applicable to the situation when more than 𝐾 samples are required or more than one sample is required from some sensors. With

International Journal of Distributed Sensor Networks

3

Centralized control module

Collected samples Activation signals Sensor

Macro cell

WLAN Femto cell

D2D

Ad hoc UDN

Vehicular networks

C-plane path U-plane path

Figure 1: Sensors deployment and signaling in 5G heterogeneous wireless networks with CCM.

this model, the radio signal detection problem is a binary hypothesis test problem: 𝐻0 : 𝑦𝑘⃗ = 𝑤⃗ 𝑘 ,

(1)

𝐻1 : 𝑦𝑘⃗ = ℎ𝑘 ⋅ 𝑠 ⃗ + 𝑤⃗ 𝑘 ,

where 𝑠 ⃗ is the 𝑛 × 1 signal vector, 𝑛 is the number of antennas per sensor, 𝑤⃗ 𝑘 is the additive white Gaussian noise vector of the 𝑘th radio receiver, and ℎ𝑘 is the channel gain from signal transmitter to the 𝑘th radio receiver. The prior source is assumed as 𝜋0 = 𝑃[𝐻0 ] and 𝜋1 = 𝑃[𝐻1 ]. We model the noise 𝑤⃗ 𝑘 entries and the signal 𝑠 ⃗ entries as zero-mean i.i.d. Gaussian with variances 𝜎𝑤2 and 𝜎𝑠2 , respectively. Then the conditional probability density functions are 𝑓 (𝑦𝑘⃗ | 𝐻0 ) =

−𝑛/2 (2𝜋𝜎𝑤2 )

󵄩󵄩 󵄩󵄩⃗ 2 󵄩𝑦󵄩 ⋅ exp (− 󵄩 󵄩2 ) , 2𝜎𝑤

⃗ ) = 1 indicates whether more samples are where 𝜙(𝑦1⃗ , . . . , 𝑦𝑚 ⃗ ) denotes the detection decision. required, and 𝛿(𝑦1⃗ , . . . , 𝑦𝑚

3. Cooperative Sequential Detection In this section, a cooperative sequential detection algorithm (CSDA) is proposed for sensing radio transmission. First, we introduce the decision rule of CSDA. Then CSDA is represented in detail. 3.1. Decision Rule of Cooperative Sequential Detection Algorithm. Based on the system model, the observed sample 𝑦𝑘⃗ has probability density 𝑓(𝑦𝑘⃗ | 𝐻𝑖 ) under hypothesis 𝐻𝑖 , 𝑖 = 0, 1. If 𝐿(𝑦𝑘⃗ ) denotes the likelihood ratio function for 𝑦𝑘⃗ , then

(2a)

󵄨 󵄨2 𝑓 (𝑦𝑘⃗ | 𝐻1 ) = (2𝜋 (󵄨󵄨󵄨ℎ𝑘 󵄨󵄨󵄨 𝜎𝑠2 + 𝜎𝑤2 ))

𝐿 (𝑦𝑘⃗ ) =

−𝑛/2

󵄩󵄩 󵄩󵄩⃗ 2 󵄩𝑦󵄩 ⋅ exp (− 󵄨 󵄨󵄩2 󵄩 ). 2 (󵄨󵄨󵄨ℎ𝑘 󵄨󵄨󵄨 𝜎𝑠2 + 𝜎𝑤2 )

take more samples take no more samples

decide 𝐻0 decide 𝐻1 ,

𝑛/2

𝜎𝑤2

(3a)

⃗ ) = 1, only if 𝜙 (𝑦1⃗ , . . . , 𝑦𝑚 {0, ⃗ )={ 𝛿 (𝑦1⃗ , . . . , 𝑦𝑚 1, {

(4)

where (2b)

CCM sequentially collects a sequence of random vectors to make decisions. If the obtained samples are not sufficient to make final decision, CCM will wake up some more sensors to get more samples. Once CCM can make decision of the binary hypothesis test, CCM will stop collecting samples. As in standard texts such as [7], a sequential decision rule is formed as follows: {0, ⃗ )={ 𝜙 (𝑦1⃗ , . . . , 𝑦𝑚 1, {

𝑓 (𝑦𝑘⃗ | 𝐻1 ) 󵄩 󵄩2 = 𝛼𝑘 exp (𝛽𝑘 󵄩󵄩󵄩𝑦󵄩󵄩󵄩⃗ ) , 𝑓 (𝑦𝑘⃗ | 𝐻0 )

𝛼𝑘 = ( 󵄨 󵄨2 ) 󵄨󵄨ℎ𝑘 󵄨󵄨 𝜎2 + 𝜎2 𝑤 󵄨 󵄨 𝑠 𝛽𝑘 =

, (5)

1 1 1 − ). ( 2 𝜎𝑤2 󵄨󵄨󵄨ℎ𝑘 󵄨󵄨󵄨2 𝜎2 + 𝜎2 𝑤 󵄨 󵄨 𝑠

With ℓ samples, the likelihood ratio ∏ℓ𝑘=1 𝐿(𝑦𝑘⃗ ) is conveniently expressed in logarithmic form as ℓ

ℓ

𝑘=1

𝑘=1

Λ ℓ = ln (∏𝐿 (𝑦𝑘⃗ )) = ∑ ln (𝐿 (𝑦𝑘⃗ )) .

(6)

Hence the sufficient statistic Λ ℓ is adopted as (3b)

ℓ

ℓ

𝑘=1

𝑘=1

󵄩 󵄩2 Λ ℓ = ∑ ln 𝛼𝑘 + ∑ 𝛽𝑘 󵄩󵄩󵄩𝑦󵄩󵄩󵄩⃗ .

(7)

4


CCM adopts the sequential probability ratio test (SPRT) [7] approach. The decision rule can be expressed in logarithmic form as follows:

The sufficient statistic Λ ℓ is

⃗ ) 𝜙 (𝑦1⃗ , . . . , 𝑦𝑚 if 𝑎 < Λ ℓ < 𝑏 {0 (take more samples) , ={ 1 (take no more samples) , otherwise, { {0 (decide 𝐻0 ) , if Λ ℓ ≤ 𝑎 ⃗ )={ 𝛿 (𝑦1⃗ , . . . , 𝑦𝑚 1 (decide 𝐻1 ) , if Λ ℓ ≥ 𝑏, {

(8a)

(8b)

where 𝑎 and 𝑏 are the lower and upper threshold, respectively. Assuming the system requirement is that missing probability 𝑝𝑚 ≤ 𝑃𝑀𝐷 and false alarm probability 𝑝𝑓 ≤ 𝑃𝐹𝐴. According to [7],

𝑎 = ln

𝑃𝑀𝐷 , 1 − 𝑃𝐹𝐴

𝑏 = ln

1 − 𝑃𝑀𝐷 . 𝑃𝐹𝐴

(9)

If 𝑒𝑘 denotes the amplitude squares of the received 𝑛×1 vector 𝑦𝑘⃗ , 𝑛

󵄨 󵄩 󵄩2 󵄨2 𝑒𝑘 = 󵄩󵄩󵄩𝑦𝑘⃗ 󵄩󵄩󵄩 = ∑ 󵄨󵄨󵄨𝑦𝑘⃗ (𝑗)󵄨󵄨󵄨 .

(10)

𝑗=1

Then 𝑒𝑘 is the sum of Gaussian random variables and 𝐻0 :

𝑒𝑘 ∼ 𝜒2 (𝑛) ; 𝜎𝑤2

(11)

where 𝜒2 (𝑛) denotes 𝜒2 distribution with 𝑛 degree of freedom. The energy probability density can be obtained as

𝑓 (𝑒𝑘 | 𝐻0 ) =

(1/2)𝑛/2 ⋅ Γ (𝑛/2)

𝑓 (𝑒𝑘 | 𝐻1 ) =

𝑒𝑘𝑛/2−1 (1/2)𝑛/2 ⋅ Γ (𝑛/2) (󵄨󵄨ℎ 󵄨󵄨2 𝜎2 + 𝜎2 )𝑛/2 󵄨󵄨 𝑘 󵄨󵄨 𝑠 𝑤 ⋅ exp (−

⋅ exp (−

ℓ

𝑘=1

𝑘=1

(13)

which is essentially an energy detection problem. In other words, even if we only consider the likelihood ratio based on energy reception, we have 𝐿 (𝑒𝑘 ) =

𝑓 (𝑒𝑘 | 𝐻1 ) = 𝛼𝑘 exp (𝛽𝑘 𝑒𝑘 ) , 𝑓 (𝑒𝑘 | 𝐻0 )

(14)

which is identical to the SPRT sufficient statistic. We therefore can see that energy detection is optimum in this problem scenario and does not require separate treatment. 3.2. Cooperative Sequential Detection Algorithm. In CSDA, all sensors hibernate initially. CCM calculates a lower threshold 𝑎 and an upper threshold 𝑏 according to the system requirement, that is, the missing probability 𝑝𝑚 ≤ 𝑃𝑀𝐷 and false alarm probability 𝑝𝑓 ≤ 𝑃𝐹𝐴. Then CCM decides the number of required samples and which sensors to be activated. Once the decision is made, CCM transmits control signals to activate the sensors. The activated sensors make observations and transmit their samples to CCM. Then CCM processes the samples into a statistic and compares it to the thresholds 𝑎 and 𝑏. If the statistic is smaller than 𝑎 or larger than 𝑏, the detection decision is made. If not, additional sensors will be activated to take more samples. The algorithm is summarized in Algorithm 1. In Algorithm 1, 𝑚𝑡 denotes the number of sensors to be activated at time 𝑡. A judicious selection of its value is beneficial for reducing the detection cost in terms of energy, bandwidth, and delay. The schemes for selecting an appropriate value of 𝑚𝑡 will be described in the subsequent discussions. Algorithm 1 (cooperative sequential detection algorithm (CSDA)).

𝑒 𝐻1 : 󵄨 󵄨2 𝑘 ∼ 𝜒2 (𝑛) , 󵄨󵄨ℎ𝑘 󵄨󵄨 𝜎2 + 𝜎2 𝑤 󵄨 󵄨 𝑠

𝑒𝑘𝑛/2−1 𝜎𝑤𝑛

ℓ

Λ ℓ = ∑ ln 𝛼𝑘 + ∑ 𝛽𝑘 𝑒𝑘 ,

Let 𝐶𝑘𝑡 denote the number of times sensor 𝑘 has been activated until time 𝑡. Let 𝑚𝑡 denote the number of sensors to be activated at time 𝑡. Let Λ𝑡 denote the likelihood ratio of all the collected samples from beginning to time 𝑡 in logarithmic form.

𝑒𝑘 ), 2𝜎𝑤2

𝑒𝑘 ). 󵄨󵄨 󵄨󵄨2 2 2 (󵄨󵄨ℎ𝑘 󵄨󵄨 𝜎𝑠 + 𝜎𝑤2 )

All 𝐾 sensors initially hibernate.

Let 𝐾𝑡 denote the set of indices of the previous 𝑚𝑡 sensors. (12)

(1) Initialize 𝑡 = 0, Λ𝑡 = 0; set 𝐶𝑘𝑡 = 0, 𝑘 ∈ {1, . . . , 𝐾}. Rank the 𝐾 sensors according to 𝐶𝑘𝑡 in ascending order. (2) Increment 𝑡 = 𝑡 + 1.


5

Calculate 𝑚𝑡 ; wake up the subset 𝐾𝑡 consisting of 𝑚𝑡 sensors with smallest 𝐶𝑘𝑡 . Let the 𝑚𝑡 sensors take samples and transmit them to CCM. Update 𝐶𝑘𝑡 . Rank the 𝐾 sensors according to the updated 𝐶𝑘𝑡 in ascending order.

(3) CCM receives these samples and updates Λ𝑡 . if 𝑎 < Λ𝑡 < 𝑏 otherwise.

(15)

If 𝜙𝑡 = 1, then 𝑡 is the stoppage time and continue to (5). (5) Make a final decision on the detection (16)

In this section, we analyze the detection cost and define its critical factors. Then we discuss the cost-benefit of various sensing algorithms with different numbers of activated sensors. A novel sensor activation and signal sampling algorithm is proposed, aiming at activating an appropriate number of sensors based on the quality of the samples. The proposed algorithm is described in detail, as well as its derivation. 4.1. Detection Cost. The design of sequential cooperative signal detection is motivated by its ability to reduce the cost associated with unnecessary sensor activation, signal sampling, and sensing data transmission. On the other hand, sequential sensor activation may lead to extra delays in acquiring necessary samples. Furthermore, reducing the number of samples also can result in loss of detection accuracy. Clearly, there is a tradeoff between detection cost and detection accuracy. Firstly, a comprehensive cost function considering the tradeoff is defined. In the function, the detection cost consists of cost resulting from detection error, energy usage during sampling, energy usage during sample transmission, and detection delay. Without loss of generality, we make the following assumptions: (i) 𝐶𝐹 : cost for false alarm; (ii) 𝐶𝑀: cost for missed detection; (iii) 𝐶𝑆 : cost for each additional sample including measurement energy, transmission energy, and bandwidth cost; (v) 𝜏𝑚𝑒 : time for measuring each sample;

(ix) 𝑇𝐷: detection delay, consisting of measure delay and transmission delay: 𝑇𝐷 = 𝜏𝑚𝑒 ⋅ 𝑇 + 𝜏𝑡𝑟 ⋅ 𝑀𝑇 .

𝑅 (𝜙, 𝛿 | 𝐻0 ) = 𝐶𝐹 ⋅ 𝑝𝑓 + 𝐶𝑆 ⋅ 𝑀𝑇 + 𝐶𝐷 ⋅ 𝑇𝐷,

(17)

𝑅 (𝜙, 𝛿 | 𝐻1 ) = 𝐶𝑀 ⋅ 𝑝𝑚 + 𝐶𝑆 ⋅ 𝑀𝑇 + 𝐶𝐷 ⋅ 𝑇𝐷,

(18)

where 𝑝𝑚 = 𝑃[𝛿 = 0 | 𝐻1 ] = 𝑃[Λ𝑡 ≤ 𝑎 | 𝐻1 ]. The detection cost without considering detection error is

4. Activation Algorithms and Considerations

(iv) 𝐶𝐷: unit cost for detection delay;

(viii) 𝑇: stoppage time; that is, no more sample is taken after the 𝑇th iteration: 𝑇 = min{𝑡, 𝜙𝑡 = 1};

where 𝑝𝑓 = 𝑃[𝛿 = 1 | 𝐻0 ] = 𝑃[Λ𝑡 ≥ 𝑏 | 𝐻0 ] and

If 𝜙𝑡 = 0, return to step (2).

𝑡 {0 (decide 𝐻0 ) , if Λ ≤ 𝑎 𝛿={ 1 (decide 𝐻1 ) , if Λ𝑡 ≥ 𝑏. {

(vii) 𝑀𝑡 : the number of measured sample vectors up to time 𝑡, 𝑀𝑡 = ∑𝑡 𝑚𝑡 ;

At the stoppage time 𝑇, the conditional detection cost is

(4) CCM makes a decision according to {0, 𝜙𝑡 = { 1, {

(vi) 𝜏𝑡𝑟 : time for transmitting each sample;

𝑅𝑑 = 𝐶𝑆 ⋅ 𝑀𝑇 + 𝐶𝐷 ⋅ 𝑇𝐷 = (𝐶𝑆 + 𝐶𝐷𝜏𝑡𝑟 ) ⋅ 𝑀𝑇 + (𝐶𝐷𝜏𝑚𝑒 ) ⋅ 𝑇.

(19)

For one specific system with stationary coefficients, 𝐶𝑆 , 𝐶𝐷, 𝜏𝑚𝑒 , 𝜏𝑡𝑟 , 𝑀𝑇 , and 𝑇 are the critical factors of detection cost. In the following subsection, several activation and sampling algorithms focusing on these two factors will be discussed. 4.2. Tradeoffs in Activation and Sampling. The number of samples 𝑚𝑡 at time 𝑡 will affect the tradeoff between energy cost and detection delay. A smaller 𝑚𝑡 leads to more detection delay but less energy by reducing redundant sensors activation, while a larger 𝑚𝑡 reduces the detection delay but may lead to more sensors activation which consumes more energy and bandwidth. For example, in the classic sequential sensing algorithm, the sensors are activated one by one; that is, 𝑚𝑡 = 1. The activated sensor takes one sample for transmission to CCM. From the perspective of reducing cost associated with redundant samples, the one-by-one sequential detection is optimal in [7]. However, it will result in considerable detection delay because of acquiring samples one by one. The stoppage time numerically equals the total number of observed samples; that is, 𝑇 = 𝑀𝑇 . To shorten detection delay, the classic sequential detection algorithm can be improved by activating more than sensors and taking more samples each time. In order to exploit multisensor diversity and improve fairness of energy consumption across the multiple sensors, we prefer activating multiple sensors to take an assigned number of samples simultaneously instead of activating one sensor at a time to take multiple samples. A conditional mean activation and sampling algorithm (CMA) is proposed to solve the problem. The objective of CMA is to activate an appropriate number of sensors each time based on the obtained samples. In CMA,

6


given the knowledge of the acquired samples, the conditional mean of the number of essential samples for reliable sequence detection is calculated. The proposed CMA will be discussed in detail in the following subsection. 4.3. Conditional Mean Activation and Sampling Algorithm (CMA). In CMA, the fusion center calculates the average value of the number of essential samples, which are required to make a final decision. At time 𝑡, the acquired samples ⃗ = {𝑦1⃗ , 𝑦2⃗ , . . . , 𝑦𝑡−1 ⃗ }; the number of sensors that will be 𝑌𝑡−1 ⃗ ]. Initially, because 𝑌0⃗ is activated depends on 𝐸[𝑀𝑇 | 𝑌𝑡−1 𝑇 𝑇 ⃗ empty, 𝐸[𝑀 | 𝑌0 ] = 𝐸[𝑀 ]. According to the decision rule, the number of required samples to make the final decision is defined as 𝑇

𝑀 = min {𝑚, Λ 𝑚 ≥ 𝑏 or Λ 𝑚 ≤ 𝑎} .

(20)

Therefore the mean of the number of required samples at 𝑡 ⃗ . According to can be updated based on the knowledge 𝑒𝑡−1 the definition of 𝑀𝑇 in (20), the number of required samples ⃗ is written as based on 𝑒𝑡−1 ⃗ ) 𝑀𝑇 (𝑒𝑡−1 ⃗ } = min {𝑚, Λ 𝑚+𝑀𝑡−1 ≥ 𝑏 or Λ 𝑚+𝑀𝑡−1 ≤ 𝑎 | 𝑒𝑡−1 = min {𝑚, Λ 𝑚 ≥ 𝑏 − Λ𝑡−1 or Λ 𝑚 ≤ 𝑎 − Λ𝑡−1 } . ⃗ enables us to adjust the threshClearly, the condition of 𝑒𝑡−1 ⃗ ) under 𝐻𝑖 can olds. As a result, the average value of 𝑀𝑇 (𝑒𝑡−1 be derived from (22) and (23) as ⃗ , 𝐻0 ] 𝐸 [𝑀𝑇 | 𝑒𝑡−1 =

The process of getting 𝐸[𝑀𝑇 | 𝐻𝑖 ] [7] is elaborated in the Appendix. The result is as follows: 𝐸 [𝑀𝑇 ] = 𝜋0 𝐸 [𝑀𝑇 | 𝐻0 ] + 𝜋1 𝐸 [𝑀𝑇 | 𝐻1 ] ,

(21)

1 (𝑃 𝑏 + (1 − 𝑃𝐹𝐴) 𝑎) , 𝐸 [𝑀 | 𝐻0 ] = 𝐷0 𝐹𝐴 𝑇

(22)

1 (𝑃 𝑎 + (1 − 𝑃𝑀𝐷) 𝑏) , 𝐷1 𝑀𝐷

1 (𝑃 (𝑎 − Λ𝑡−1 ) + (1 − 𝑃𝑀𝐷) (𝑏 − Λ𝑡−1 )) 𝐷1 𝑀𝐷

= 𝐸 [𝑀𝑇 | 𝐻1 ] −

(23)

(24)

⃗ are not sufficient At time 𝑡, if the collected samples 𝑌𝑡−1 to reach a confident detection conclusion, CCM performs another iteration of sensor activation and sample collection. ⃗ may impact the conditional However, the quality of 𝑌𝑡−1 mean of the number of essential samples. Based on the set ⃗ ] ⃗ , the conditional mean 𝐸[𝑀𝑇 | 𝑌𝑡−1 of obtained samples 𝑌𝑡−1 should be updated. As discussed earlier, for the specific problem in the system model, the sufficient statistics of 𝑌𝑡⃗ can be captured by 1

𝑡

𝑒𝑡⃗ = {𝑒𝑘 , 𝑘 ∈ {𝐾 , . . . , 𝐾 }} .

(28)

Λ𝑡−1 . 𝐷1

⃗ ) = 𝑃 [𝐻0 | 𝑒𝑡−1 ⃗ ]= 𝜋0 (𝑒𝑡−1

𝐷1 = 𝐸 [ln (𝐿 (𝑒𝑘 )) | 𝐻1 ] 𝛽𝑘 ]. 1 − 2𝜎𝑤2 𝛽𝑘

Λ𝑡−1 , 𝐷0

⃗ , 𝐻1 ] 𝐸 [𝑀𝑇 | 𝑒𝑡−1

(25)

⃗ | 𝐻0 ) 𝑃 [𝐻0 ] 𝑓 (𝑒𝑡−1 ⃗ | 𝐻𝑖 ) 𝑃 [𝐻𝑖 ] ∑𝑖=0,1 𝑓 (𝑒𝑡−1

=

𝜋0 ∏𝑘 𝑓 (𝑒𝑘 | 𝐻0 ) 𝜋0 ∏𝑘 𝑓 (𝑒𝑘 | 𝐻0 ) + 𝜋1 ∏𝑘 𝑓 (𝑒𝑘 | 𝐻1 )

=

𝜋0 , 𝜋0 + 𝜋1 exp (Λ𝑡−1 )

with

= 𝐸 [ln 𝛼𝑘 ] + 𝑛𝜎𝑤2 𝐸 [

(27)

The source prior 𝜋𝑖 is transformed into conditional prior via

with 𝐷0 = 𝐸[ln(𝐿(𝑒𝑘 )) | 𝐻0 ] = 𝐸[ln 𝛼𝑘 ] + 𝑛𝜎𝑤2 𝐸[𝛽𝑘 ] and 𝐸 [𝑀𝑇 | 𝐻1 ] =

1 (𝑃 (𝑏 − Λ𝑡−1 ) + (1 − 𝑃𝐹𝐴) (𝑎 − Λ𝑡−1 )) 𝐷0 𝐹𝐴

= 𝐸 [𝑀𝑇 | 𝐻0 ] −

=

where

(26)

(29)

where 𝑘 ∈ {𝐾1 , . . . , 𝐾𝑡 }. Similarly, ⃗ ) = 𝑃 [𝐻1 | 𝑒𝑡−1 ⃗ ]= 𝜋1 (𝑒𝑡−1

𝜋1 exp (Λ𝑡−1 ) 𝜋0 + 𝜋1 exp (Λ𝑡−1 )

.

(30)

Therefore, the conditional mean is ⃗ ] = ∑ 𝜋𝑖 (𝑒𝑡−1 ⃗ ) 𝐸 [𝑀𝑇 | 𝑒𝑡−1 ⃗ , 𝐻𝑖 ] 𝐸 [𝑀𝑇 | 𝑒𝑡−1 𝑖=0,1

=

𝜋0 Λ𝑡−1 𝑇 | 𝐻 ] − ) (𝐸 [𝑀 0 𝐷0 𝜋0 + 𝜋1 exp (Λ𝑡−1 ) +

𝜋1 exp (Λ𝑡−1 ) 𝜋0 + 𝜋1 exp (Λ𝑡−1 )

(𝐸 [𝑀𝑇 | 𝐻1 ] −

Λ𝑡−1 ). 𝐷1

(31)

International Journal of Distributed Sensor Networks Our cooperative sequential detection algorithm based on CMA (CMA-CSDA) is summarized in Algorithm 2. Note that the extra computation overhead of CMA is afforded by CCM which is usually infrastructure with adequate computational resource and energy. Besides, there are multiple steps in the process of the proposed algorithm with low complexity of each step. Hence, instead of being separately defined, the additional computation cost and control overhead for CCM is included in the cost for detection delay. In this work, the cost for the sensors is focused on because of their energy and hardware restriction. In the future work, the cost for CCM should be well-defined and analyzed quantitatively. In Algorithm 2, 𝑓𝑙𝑜𝑜𝑟(⋅) denotes rounding a number to the smaller nearest integer and (𝑥)mod(𝑦) denotes the remainder if 𝑥 is divided by 𝑦. Algorithm 2 (CMA-CSDA).

Table 1: Main simulation parameters. Number of sensors Number of sensing antennas per sensor Channels gain ℎ𝑘 SNR Signal 𝑠 System required detection thresholds Source prior

𝐾 = 10 𝑛=4 ℎ𝑘 ∼ 𝑁(1, 1) −6 dB∼3 dB 𝑠 ∼ 𝑁(0, 1), 𝜎𝑠2 = 1 𝑃𝑀𝐷 = 0.1, 𝑃𝐹𝐴 = 0.1 𝜋0 = 𝜋1 = 0.5

ciated with samples number includes energy for making observations, energy for transmission, spectrum usage for transmission, and hardware cost of sensors. In the simulation, three activation and sampling algorithms are compared: (i) classic sequential sensing: 𝑚𝑡 = 1,

All 𝐾 sensors initially hibernate. Let 𝐶𝑘𝑡 denote at time 𝑡 the number of times sensor 𝑘 has been activated. Let 𝑚𝑡 denote the number of sensors to be activated. Let 𝐾𝑡 denote the set of indices of the previous 𝑚𝑡 sensors. (1) Initialize 𝑡 = 0, Λ𝑡 = 0; set 𝐶𝑘𝑡 = 0, 𝑘 ∈ {1, . . . , 𝐾}. Rank the 𝐾 sensors according to 𝐶𝑘𝑡 in ascending order. (2) Increment 𝑡 = 𝑡 + 1. ⃗ ) = 𝜋0 /(𝜋0 + 𝜋1 exp(Λ𝑡−1 )), 𝜋1 (𝑒𝑡−1 ⃗ )= Update 𝜋0 (𝑒𝑡−1 𝑡−1 𝜋1 exp(Λ )/(𝜋0 + 𝜋1 exp(Λ𝑡−1 )). ⃗ ] according to 𝐸[𝑀𝑇 | 𝑒𝑡−1 ⃗ ]= Calculate 𝐸[𝑀𝑇 | 𝑒𝑡−1 𝑇 ⃗ )(𝐸[𝑀 | 𝐻𝑖 ] − Λ𝑡−1 /𝐷𝑖 ). ∑𝑖=0,1 𝜋𝑖 (𝑒𝑡−1 ⃗ ] ≥ 𝐾, wake up all 𝐾 sensors; let each (3) If 𝐸[𝑀𝑇 | 𝑒𝑡−1 ⃗ ]/𝐾) samples and sensor take 𝑓𝑙𝑜𝑜𝑟(𝐸[𝑀𝑇 | 𝑒𝑡−1 transmit them to CCM. ⃗ ])mod(𝐾) and wake up Calculate 𝑚𝑡 = (𝐸[𝑀𝑇 | 𝑒𝑡−1 the subset 𝐾𝑡 of 𝑚𝑡 sensors with the smallest counter 𝐶𝑘𝑡 . Let the 𝑚𝑡 sensors take a sample and transmit to CCM. Update 𝐶𝑘𝑡 . Rank the 𝐾 sensors updated 𝐶𝑘𝑡 in ascending order.

7

according to the

(4) Perform steps (3), (4), and (5) of Algorithm 1 successively.

5. Numerical Results and Analysis In this section, the cost-delay tradeoff and the detection performance of the proposed cooperative sequential detection algorithms with various activation and sampling schemes are evaluated. As mentioned before, the detection cost asso-

(ii) full activation sensing: 𝑚𝑡 = 𝐾, ⃗ ]. (iii) CMA-CSDA: 𝑚𝑡 = 𝐸[𝑀𝑇 | 𝑒𝑡−1 In the three algorithms, the numbers of sensors to be activated each time 𝑚𝑡 are different. In the classic sequential sensing algorithm, 𝑚𝑡 = 1, which means that only one sensor is activated to take one sample and send it to CCM. In the full activation sensing algorithm, 𝑚𝑡 = 𝐾, which means that all sensors are activated to sense and transmit samples to CCM. ⃗ ], In the proposed CMA-CSDA algorithm, 𝑚𝑡 = 𝐸[𝑀𝑇 | 𝑒𝑡−1 which means that the number of activated sensors depends on the conditional mean of the number of required samples based on the knowledge of the acquired samples at time 𝑡 − 1. The three algorithms are compared with respect to detection cost and detection accuracy. 5.1. Parameters Setting. The simulation tests 𝐾 = 10 sensors and antenna case 𝑛 = 4 sensing antennas per sensor. The procedure can directly extend to other antenna cases. We consider channels gain ℎ𝑘 to be i.i.d. random Gaussian ℎ𝑘 ∼ 𝑁(1, 1). The average signal to noise ratio (SNR) in decibels is defined as 10 log10 (SNR) = 10 log10 (𝜎𝑠2 /𝜎𝑤2 ). The maximum constraint for 𝑝𝑚 and 𝑝𝑓 is set to 0.1, as recommended by the IEEE802.22 standard [18]. Other relevant simulation parameters are listed in Table 1. 5.2. Numerical Results and Analysis. The performance of detection algorithms with various activation and sampling schemes can be shown in terms of the number of samples 𝑀𝑇 , the stoppage time 𝑇, and the missing probability 𝑝𝑚 or the false alarm probability 𝑝𝑓 . The detection cost without considering detection error is 𝑅𝑑 = (𝐶𝑆 + 𝐶𝐷𝜏𝑡𝑟 ) ⋅ 𝑀𝑇 + (𝐶𝐷𝜏𝑚𝑒 ) ⋅ 𝑇. For one specific system with fixed coefficients, that is, 𝐶𝑆 , 𝐶𝐷, 𝜏𝑚𝑒 , 𝜏𝑡𝑟 , the critical factors of detection cost are 𝑀𝑇 and 𝑇. The comparison of detection precision and the critical factors of detection cost adopting different algorithms will be illustrated in this subsection. Figure 2 shows the detection probability 𝑝𝑑 under 𝐻1 . Figure 3 shows the missing probability 𝑝𝑚 under 𝐻1 and

8

International Journal of Distributed Sensor Networks 1

Detection probability pd

0.997

0.994

0.991

0.988

0.985 −6

−5

−4

−3

−2 −1 SNRdB

0

1

2

3

Classic sequential sensing Full activation sensing CMA-CSDA

Figure 2: Detection probability 𝑝𝑑 under 𝐻1 , when 𝐾 = 10, 𝑃𝑀𝐷 = 0.1, 𝑃𝐹𝐴 = 0.1, and 𝜋0 = 𝜋1 = 0.5.

the false probability 𝑝𝑓 under 𝐻0 . It is shown that 𝑝𝑑 increases with growing SNR, while 𝑝𝑚 and 𝑝𝑓 decrease with growing SNR. Even SNR is as low as −6 dB; the detection probability is still larger than 0.95 as shown in Figure 2. In Figure 3, it is obvious that the target that detection performance 𝑝𝑚 ≤ 0.1, 𝑝𝑓 ≤ 0.1 is achieved for all cases. When the performance requirement is satisfied, the detection cost of different algorithms can be compared without consideration of detection error. Figure 4 illustrates the number of samples 𝑀𝑇 versus SNR (dB). The left part of Figure 4 is 𝑀𝑇 under 𝐻1 , while the right part is 𝑀𝑇 under 𝐻0 . The graphs of both scenarios are similar. When SNR varies from −6 dB to 3 dB, 𝑀𝑇 clearly decreases because the quality of the sensed signals improves. It can be seen that the classic sequential sensing algorithm requires the least number of samples, which reduces the energy cost associated with unnecessary sensor activation, redundant signal sampling, and transmission. So it is the most energy-efficient activation and sampling scheme. 𝑀𝑇 of the full activation sensing algorithm is the largest among the three algorithms. Although CCM only needs less than 𝐾 samples to make a final conclusion, it will still activate all the 𝐾 sensors. At a higher SNR it is easier to reach a confident conclusion, so the number of required samples is reduced. When SNR ≥ 1 dB, 𝑀𝑇 remains nearly 𝐾. The full activation sensing algorithm consumes most energy and improves limitedly along with the increase of SNR when SNR ≥ 1 dB because of its activation design. 𝑀𝑇 of CMACSDA is slightly larger than the classic sequential sensing algorithm. Along with the SNR growing, CMA-CSDA approaches the classic sequential sensing algorithm. Combined with Figures 2 and 3, it is proved that the larger number of samples improves the detection performance. CMA-CSDA

enhances detection performance at the expense of a few more samples. Figure 5 shows the stoppage time 𝑇, which represents the detection delay. 𝑇 of the classic sequential sensing algorithm is the largest, in which the stoppage time is equivalent to the number of samples 𝑀𝑇 . Besides, its detection delay is obviously larger at low SNR. The full activation sensing algorithm is the fastest with smallest 𝑇. The stoppage time of CMA-CSDA is much less than the classic sequential sensing algorithm, while the difference between CMA-CSDA and the full activation sensing algorithm is not so considerable. As the SNR grows, 𝑇 of CMA-CSDA approaches the full activation sensing algorithm. 𝑇 of CMA-CSDA changes slightly at different SNR, because the number of required samples is updated based on quality of the acquired samples. Combined with Figures 3, 4, and 5, we find that both the detection performance and the detection cost of CMA-CSDA have median values. With satisfied detection precision, the number of samples 𝑀𝑇 of CMA-CSDA is closer to the classic sequential sensing algorithm, while the stoppage iteration 𝑇 of CMA-CSDA is closer to the full activation sensing algorithm. According to the comparison of Figures 4 and 5, there is an energy-delay tradeoff, that is, a tradeoff between 𝑀𝑇 and 𝑇. 𝐶𝑆 , 𝐶𝐷, 𝜏𝑡𝑟 , and 𝜏𝑚𝑒 are preset system parameters. As mentioned above, for one specific system with stationary coefficients, the critical factors of detection cost are 𝑀𝑇 and 𝑇. So it is achievable to adopt the heuristic scheme for less detection cost with different system parameters.

6. Conclusion In this work, we considered the detection of radio signal transmissions in 5G wireless networks with multi-RAT. The C/U-plane split architecture with a CCM was well suited for implementing the proposed cooperative sequential detection algorithm. With detection precision guaranty, the number of redundant samples and the cost associated with them was reduced by adopting sequential detection and hibernation scheme. We also investigated various activation and sampling schemes by analyzing the tradeoff between detection cost and delay. To optimize the cost-delay tradeoff, we proposed a novel sensor activation and signal sampling algorithm, in which we activated the sensors to make observations according to the conditional mean number of necessary samples and compared it with the classic sequential sensing algorithm and the full activation sensing algorithm. Through the analysis, it is concluded that (1) the target detection performance can be satisfied for all of the cases. (2) To measure the detection cost without consideration of detection error, the critical factors of detection cost were the number of samples and the stoppage time. The classic sequential sensing algorithm requires the least amount of samples. Hence, it is the most energy-efficient activation and sampling scheme. The full activation sensing algorithm is fastest with smallest stoppage time. With contented detection precision, both the number of samples and the stoppage iteration of the proposed


9 0.1 False probability pf

Missing probability pm

0.015 0.012 0.009 0.006 0.003 0 −6

−5

−4

−3

−2 −1 SNRdB

0

1

2

0.08 0.06 0.04 0.02 0 −6

3


−5

−4

−3

−2 −1 SNRdB

0

1

2

3


70

70

60

60 Number of samples MT under H0

Number of samples MT under H1

Figure 3: Missing probability 𝑝𝑚 and false probability 𝑝𝑓 , when 𝐾 = 10, 𝑃𝑀𝐷 = 0.1, 𝑃𝐹𝐴 = 0.1, and 𝜋0 = 𝜋1 = 0.5.

50

40

30

20

10

0 −6

50

40

30

20

10

−5

−4

−3

−2 −1 SNRdB

0

1

2

3


0 −6

−5

−4

−3

−2 −1 SNRdB

0

1

2

3


Figure 4: Number of samples 𝑀𝑇 , when 𝐾 = 10, 𝑃𝑀𝐷 = 0.1, 𝑃𝐹𝐴 = 0.1, and 𝜋0 = 𝜋1 = 0.5.

CMA-CSDA have median values. Furthermore, both the above values were closer to the smallest one among the three algorithms. (3) It was feasible to adopt the heuristic scheme to achieve less detection cost with different signal quality and system parameters. In short, the main contributions are (1) development of a cooperative sequential detection algorithm for energy efficient cooperative sensing; (2) analysis of different sensor activation algorithms and their cost-delay tradeoff; (3) proposal of a novel sensor activation and signal sampling algorithm for the cost-delay tradeoff in sequential sensing. However, it is still a challenge to guarantee sensors detection precision in practical, for it depends on the topology of the sensor deployment and the location of the signal source. Therefore, the deployment of sensors and the corresponding

hibernation and activation approach should be designed based on specific network scenarios and real-time spectrum scheduling adapted by CCM. We hope to explore specific methods for these issues in the future work.

Appendix The Process of Getting (22) and (23) CCM collects a sequence of i.i.d. samples 𝑦𝑛⃗ , 𝑛 = 1, . . . , 𝑁, where 𝑁 = 𝑀𝑇 is the number of required samples for simplicity. Note that CCM does not consider the activation and sampling algorithm when it calculates the mean of the number of required samples 𝐸[𝑁 | 𝐻𝑖 ], so the samples are signed by 𝑛 here instead of the sensor index 𝑘. 𝑧𝑛 is defined as

International Journal of Distributed Sensor Networks 50

50

45

45

40

40

35

35

Stoppage time T under H0

Stoppage time T under H1

10

30 25 20 15

30 25 20 15

10

10

5

5

0 −6

−5

−4

−3

−2 −1 SNRdB

0

1

2

3

0 −6

−5


−4

−3

−2 −1 SNRdB

0

1

2

3


Figure 5: Stoppage time 𝑇, when 𝐾 = 10, 𝑃𝑀𝐷 = 0.1, 𝑃𝐹𝐴 = 0.1, and 𝜋0 = 𝜋1 = 0.5.

𝐺1 (𝑢) = 𝐸 [exp (𝑢𝑧) | 𝐻1 ]

the log-likelihood function of the probability distribution of the 𝑛th sample as follows: 𝑓 (𝑦𝑛⃗ | 𝐻1 ) 𝑧𝑛 = ln [𝐿 (𝑦𝑛⃗ )] = ln . 𝑓 (𝑦𝑛⃗ | 𝐻0 )

𝐸 [𝑧𝑛 | 𝐻1 ] = ∫ ln [

𝑓 (𝑦 | 𝐻1 ) ] 𝑓 (𝑦 | 𝐻1 ) 𝑑𝑦 𝑓 (𝑦 | 𝐻0 )

𝐺1 (𝑢) = 𝐺0 (𝑢 + 1) , 𝐺1 (0) = 𝐺0 (1) = 1,

The following equation is deduced by differentiating 𝐺0 (𝑢) with respect to 𝑢:

(A.3)

𝑢

= ∫ 𝑓 (𝑦 | 𝐻1 ) 𝑓 (𝑦 | 𝐻0 )

𝑑𝐺0 (𝑢) 𝑑𝑢 𝑢

= ∫ 𝑓 (𝑦 | 𝐻1 ) 𝑓 (𝑦 | 𝐻0 )

1−𝑢

𝑓 (𝑦 | 𝐻1 ) ln ( ) 𝑑𝑦 𝑓 (𝑦 | 𝐻0 )

1−𝑢

𝑑𝑦,

(A.6)

and then 𝑓 (𝑦 | 𝐻1 ) 𝑑𝐺0 ) 𝑓 (𝑦 | 𝐻0 ) 𝑑𝑦 = 𝐷0 , (0) = ∫ ln ( 𝑑𝑢 𝑓 (𝑦 | 𝐻0 )

𝐺0 (𝑢) = 𝐸 [exp (𝑢𝑧) | 𝐻0 ]

(A.5)

ln 𝐺1 (0) = ln 𝐺0 (1) = 0.

(A.2)

According to the definition in (26), the process to calculate stoppage time in sequential probability ratio test [2] may be adopted to get 𝐸[𝑁 | 𝐻𝑖 ]. Let 𝐺(𝑢) denote the cumulate generating function of 𝑧𝑛 :

By substituting (A.1) into 𝐺(𝑢), we get

𝑑𝑦.

It is obvious that

= 𝐷1 .

𝐺 (𝑢) = 𝐸 [exp (𝑢𝑧)] .

−𝑢

(A.4)

𝑓 (𝑦 | 𝐻1 ) ] 𝑓 (𝑦 | 𝐻0 ) 𝑑𝑦 𝑓 (𝑦 | 𝐻0 )

= 𝐷0 ,

𝑓 (𝑦 | 𝐻0 )

(A.1)

Then 𝑧𝑛 is a sequence of i.i.d. random variables with 𝐸 [𝑧𝑛 | 𝐻0 ] = ∫ ln [

𝑢+1

= ∫ 𝑓 (𝑦 | 𝐻1 )

𝑓 (𝑦 | 𝐻1 ) 𝑑𝐺0 ) 𝑓 (𝑦 | 𝐻1 ) 𝑑𝑦 = 𝐷1 . (1) = ∫ ln ( 𝑑𝑢 𝑓 (𝑦 | 𝐻0 )

(A.7)


11 ≈ 𝐸 [Λ 𝑁 | Λ 𝑁 ≥ 𝑏, 𝐻0 ] 𝑃 [Λ 𝑁 ≥ 𝑏 | 𝐻0 ]

Consider a process as follows:

𝑋ℓ =

exp (𝑢Λ ℓ ) 𝐺 (𝑢)ℓ

+ 𝐸 [Λ 𝑁 | Λ 𝑁 ≤ 𝑎, 𝐻0 ] 𝑃 [Λ 𝑁 ≤ 𝑎 | 𝐻0 ] ,

ℓ ≥ 0,

≈ 𝑏𝑃𝐹𝐴 + 𝑎 (1 − 𝑃𝐹𝐴) ,

(A.8)

𝐸 [Λ 𝑁 | 𝐻1 ] where Λ ℓ = 0 when ℓ = 0, and Λ ℓ = ∑ℓ𝑛=1 𝑧𝑛 , when ℓ ≥ 1 as in (6). Since 𝑧𝑛 are i.i.d. random variables, it can be obtained that

𝐸 [𝑋𝑁] = 𝐸 [

exp (𝑢Λ 𝑁) 𝐺 (𝑢)𝑁

]=

𝐸 [exp (𝑢𝑁𝑧𝑛 )] 𝐺 (𝑢)𝑁

= 𝐸 [Λ 𝑁 | Λ 𝑁 > 𝑎, 𝐻0 ] 𝑃 [Λ 𝑁 > 𝑎 | 𝐻0 ] + 𝐸 [Λ 𝑁 | Λ 𝑁 ≤ 𝑎, 𝐻0 ] 𝑃 [Λ 𝑁 ≤ 𝑎 | 𝐻0 ] ≈ 𝐸 [Λ 𝑁 | Λ 𝑁 ≥ 𝑏, 𝐻1 ] 𝑃 [Λ 𝑁 ≥ 𝑏 | 𝐻1 ]

= 1. (A.9)

+ 𝐸 [Λ 𝑁 | Λ 𝑁 ≤ 𝑎, 𝐻1 ] 𝑃 [Λ 𝑁 ≤ 𝑎 | 𝐻1 ] ≈ 𝑏 (1 − 𝑃𝑀𝐷) + 𝑎𝑃𝑀𝐷.

Differentiating (A.9) with respect to 𝑢 yields

(A.13) Therefore,

𝐸 [Λ 𝑁 ⋅𝑁

exp (𝑢Λ 𝑁) 𝐺 (𝑢)𝑁

−

exp (𝑢Λ 𝑁)

𝐸 [𝑀𝑇 | 𝐻0 ] =

𝐺 (𝑢)𝑁

1 𝑑𝐺 (𝑢) ] = 0, 𝐺 (𝑢) 𝑑𝑢

=

(A.10)

that is, 𝐸 [Λ 𝑁𝑋𝑁 − 𝑋𝑁 ⋅ 𝑁𝐺 (𝑢)−1

1 𝐸 [Λ 𝑁 | 𝐻0 ] 𝐷0 1 (𝑃 𝑏 + (1 − 𝑃𝐹𝐴) 𝑎) , 𝐷0 𝐹𝐴

1 𝐸 [𝑀 | 𝐻1 ] = 𝐸 [Λ 𝑁 | 𝐻1 ] 𝐷1

(A.14)

𝑇

𝑑𝐺 (𝑢) ] = 0. 𝑑𝑢

= Setting 𝑢 = 0, it turns into

1 (𝑃 𝑎 + (1 − 𝑃𝑀𝐷) 𝑏) . 𝐷1 𝑀𝐷

Equations (22) and (23) have been proved. 𝐸 [Λ𝑁 − 𝑁

𝑑𝐺 (𝑢) 󵄨󵄨󵄨󵄨 󵄨 ] = 0. 𝑑𝑢 󵄨󵄨󵄨𝑢=0

(A.11)

The authors declare that there is no conflict of interests regarding the publication of this paper.

We evaluate the equality under 𝐻0 and 𝐻1 and obtain 𝐸 [Λ 𝑁

Acknowledgment

𝑑𝐺0 (0) | 𝐻0 ] = 𝐸 [𝑁 | 𝐻0 ] 𝑑𝑢

This work is supported in part by the National High Technology Development 863 Program of China (2014AA01A707).

= 𝐷0 𝐸 [𝑁 | 𝐻0 ] , 𝐸 [Λ 𝑁 | 𝐻1 ] = 𝐸 [𝑁 | 𝐻1 ]

𝑑𝐺1 (0) 𝑑𝑢

= 𝐸 [𝑁 | 𝐻1 ]

𝑑𝐺0 (1) 𝑑𝑢

(A.12)

= 𝐷1 𝐸 [𝑁 | 𝐻1 ] . Since the zero-overshoot approximation is taken, the approximate values of 𝐸[Λ 𝑁 | 𝐻𝑖 ], 𝑖 = 0, 1, are 𝐸 [Λ 𝑁 | 𝐻0 ] = 𝐸 [Λ 𝑁 | Λ 𝑁 ≥ 𝑏, 𝐻0 ] 𝑃 [Λ 𝑁 ≥ 𝑏 | 𝐻0 ] + 𝐸 [Λ 𝑁 | Λ 𝑁 < 𝑏, 𝐻0 ] 𝑃 [Λ 𝑁 < 𝑏 | 𝐻0 ]

Conflict of Interests

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