Wireless Pers Commun DOI 10.1007/s11277-009-9785-1
Detection Probability Estimation of Directional Antennas and Omni-Directional Antennas Xiaofeng Lu · Fletcher D. Wicker · Don Towsley · Zhang Xiong · Pietro Lio’
© Springer Science+Business Media, LLC. 2009
Abstract This paper studies the problem of detection of directional antennas and omnidirectional antennas by hostile detection systems. We present a model for calculating the probability of detecting a transmitter at arbitrary location around the transmitter. Our study shows that, if a directional antenna employs the same transmit power as an omni-directional antenna, the directional antenna can not decrease the probability of being detected. In some scenarios, a directional antenna is more likely to be detected than an omni-directional antenna. However, if a directional antenna provides the same Effective Isotropic Radiated Power in the direction of the receiver as an omni-directional antenna, the transmit power needed by a directional antenna to send data is much less than that of an omni-directional antenna. In this scenario, the probability of detecting a directional antenna is reduced by over 90%. This reveals that directional antennas can be used to build a secure path to send data at low probability of being detected by adversaries. Keywords Detection probability · Directional antenna · Omni-directional antenna · Security · Wireless communication
X. Lu (B) · Z. Xiong School of Computer Science, Beijing University of Aeronautics and Astronautics, 100191 Beijing, China e-mail:
[email protected] Z. Xiong e-mail:
[email protected] F. D. Wicker Communication Network Architectures Subdivision, The Aerospace Corporation, Los Angeles, CA, USA e-mail:
[email protected] D. Towsley Department of Computer Science, University of Massachusetts, Amherst, MA, USA e-mail:
[email protected] P. Lio’ Computer Laboratory, University of Cambridge, Cambridge CB3 0FD, UK e-mail:
[email protected]
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1 Introduction A clear advantage of the ad hoc network is that it obviates the need to build a network infrastructure (e.g. base stations, routers, etc.). In an ad hoc network, nodes communicate with each other from time to time and maintain dynamic and temporary connectivities through peer-to-peer wireless communication. A message is routed from the source node to the destination by a dynamic routing protocol. Typically, there are two kinds of antenna working modes, omni-directional and directional [1,2]. An omni-directional antenna radiates and receives equally in all directions. Only a small faction of the overall energy sent into the environment by omni-directional antenna reaches the desired receiver. A directional antenna can overcome this disadvantage. A directional antenna radiates and receives most of the signal power in one direction. Pointing to the desired receiver, a directional antenna can increase signal strength in that direction and forms a directional beam towards the receiver. Transmitters can use directional antennas to transmit signals farther away than omni-directional antennas with the same transmit power, or transmit signals to a receiver while using less transmit power [1–3]. The ability to be set up fast and operate without the need of any wired infrastructure makes mobile ad hoc networks a promising candidate for military scenarios. In a military network or an untrustworthy network, the security or anonymity of the transmitters is often more important than other. Due to the broadcast nature of radio transmissions, communications in wireless networks are more susceptible to malicious traffic analysis. Hostile nodes observe networks in order to discover when and where nodes transmit data. Our study focuses on the impact of directional transmission on the probability of the transmitter being detected by an adversary. It seems reasonable that directional communication covers a smaller transmission region, and so the transmitter is less likely to be detected by an adversary. Authors of [4–7] mentioned that directional antenna can reduce detection probability. However no study has been conducted to compare the detection probability of directional and omni-directional antennas. Our study shows that a directional antenna does not reduce detection probability if its transmit power is the same as that of an omni-directional antenna. In some conditions, a directional antenna can even increase the probability of being detected if a hostile node is in the direction of its main lobe. Our study shows that a directional antenna has lower detection probability than an omni-directional antenna only when it employs less transmit power to send data than the omni-directional antenna. In this paper, we address the problem of estimating detection probability. This paper is organized as follows. Section 2 introduces the antenna communication model. We introduce our detection probability model in Sect. 3 and compare the detection probabilities of directional and omni-directional antennas in Sect. 4. We perform a parameter study in Sect. 5 and give the conclusion in Sect. 6.
2 Antenna Communication Model 2.1 Preliminary: Radio Propagation Model A radio system transmits information to its antenna, which converts the radio frequency signal into an electromagnetic wave and transmits it [8]. The measure of end-to-end performance is usually bit-error rate (BER), which quantifies the reliability of the entire radio system from “bits in” to “bits out” [9]. BER is related to the ratio of the total received signal power to the total noise which includes thermal and system noise plus total interference, denoted by
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Detection Probability Estimation Table 1 Variable definitions for link budget equations Symbol
Meaning
Value
S Pt Gt f d Gr Ct Cr BW Tr Te T I N
Total received signal level after receive antenna Transmitter power level Transmitter’s antenna gain Radio frequency Distance between the transmitter and receiver Receiver’s antenna gain Transmitter’s cable attenuation Receiver’s cable attenuation Signal Bandwidth Noise Temperature of receiver’s antenna Environment noise temperature at receiver’s antenna Total system noise temperature at receiver’s antenna Total interference power level Total noise level at receiver’s antenna
Eq. (1) dB W 0 dBi Fig. 1 2,500 MHz calculated M 0 dB 0 dB 0 dB 1,000,000 Hz 500 Degrees Kelvin 300 Degrees Kelvin Tr + Te 0 dB Eq. (2) dB W
SNIR (SNIR = S/N ). Both the total received signal power level and the total noise level are random variables. Hence the quality of service delivered across a radio channel is random as well. We utilize the following propagation model to evaluate the communications quality of service. The total received power level S depends on many elements including transmitted power, antenna gain, path loss, shadow fading and noise [8], S = Pt + G t + G r − Ct − Cr − Pl
(1)
Here Pt (dBi)1 is the transmitter’s power level, G t (dBi) is the transmitter’s antenna gain in the direction towards the receiver, G r (dBi) is the receiver’s antenna gain in the direction of the transmitter, Ct is the transmitter’s cable attenuation, Cr is the receiver’s cable attenuation and Pl is transmission path loss, which we will discuss carefully in Sect. 2.2. Ct and Cr are assumed to be zero here. At the receiving unit the total noise level is N = k + dB(Tr + Te ) + dB(BW) + I
(2)
where k is Boltzmann constant equal to −228.6 dB(Watts/(Hertz × Degree Kelvin)). Tr is noise temperature at the receiver’s antenna and Te is environment noise temperature at the receiver’s antenna. The receiving bandwidth is of course matched to communication signal’s bandwidth BW. The final term I is the total interference power level. This interference can be a combination of network self interference generated by other transmitters within the mobile ad hoc network, active hostile jamming, and other radio signals generated by both man-made sources and natural sources. The impact of interference is assumed to be zero in our study. The meaning of elements of Eqs. (1) and (2) are listed in Table 1. The values of these variables are given by experience. In radio communication systems, Effective Isotropic Radiated Power (EIRP) is the gain of a transmitting antenna multiplied by the net power accepted by the antenna from the connected transmitter in a given direction [10]. As Pt and G t are measured in dB, EIRP can be calculated as EIRP = Pt + G t . 1 dB = 10log (x) 10
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X. Lu et al. Table 2 Terrain type parameters
Terrain type
Average distance to horizon (m)
Range of values of n
Rural-open Rural-trees Suburban Urban
1,000 300 200 100
2–2.5 2–4.0 2–5.0 2–6.0
2.2 Adaptive Path Loss Formula Path loss is the attenuation undergone by an electromagnetic wave in transit between a transmitter and a receiver. Path loss may be due to many effects, such as free-space loss, refraction, reflection, and absorption [11]. Path loss is also influenced by terrain, environment (urban or rural, vegetation and foliage), propagation medium (dry or moist air), the distance between the transmitter and the receiver, and the height and location of antennas [11]. Free-space path loss (FSPL) is the loss in signal strength of an electromagnetic wave that would result from a line of sight path through free space, with no obstacles nearby to cause reflection or diffraction [11]. This loss is calculated using the following formula: pl(d, f ) = c + 20log10 (d) + 20log10 ( f )
(3)
where d is the distance from the transmitter to receiver, the radio frequency is f , and c is a constant that depends of the units of measure for d and f . With the units of measure for d and f listed in Table 1, c = −27.55. Past line of sight, communications is still possible, but there is additional attenuation due to shadowing. Additionally it is well know that the average receive power level, measured in dBW, around a circle at a constant distance from the transmitter and beyond the line of sight is a lognormally distributed random variable. Let pl(d, f, n) be the path loss when the distance from the receiver to the transmitter is larger than the line of sight distance. We modify the FSPL formula and propose an adaptive path loss formula equation (4). pl(d, f, n) = −27.55 + n10log10 (d) + 20log10 ( f )) + L s
(4)
where n is determined by the terrain type and L s is shadow fading. In our analysis, the coefficient n is a random variable that depends of the type of terrain, i.e. how rugged the terrain is to radio frequency waves. Typical terrain types include open rural, rural trees and rolling hills, suburban, and urban. For each of the terrain types there is an average distance to the edge of the unobstructed line of sight given. Beyond this limit, the value of n is drawn uniformly random between the values listed in Table 2 with the possibility that there are locations that have direct line of sight beyond this average. Shadow fading is a phenomenon that occurs when a mobile moves behind an obstruction and experiences a significant reduction in signal power [12]. From various experimental results, the value of shadow fading Ls can be characterized by a Normal distribution in the logarithmic scale with zero mean and certain standard deviation that is also determined by the terrain type in the magnitude of 8–10 dB [13]. 2.3 Antenna Gain Antenna gain refers to the antenna’s ability to direct its radiated power in a desired direction, or to receive energy preferentially from a desired direction [3]. It is defined as the
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Fig. 1 A case of directional antenna gain curve
Fig. 2 Illustration of d and θ . d is referred to the distance from the transmitter to the detection system, and θ is referred to the angle between the direction of the detection system and the direction of the transmitter’s main lobe
ratio of the radiation intensity of an antenna in a given direction to the intensity of the same antenna as it radiates in all directions (isotropically). Antenna gain is expressed in dBi. For an omni-directional antenna, because the ratio of the radiation intensity is 1, the antenna gain is 10log10 (1) = 0. For a directional antenna, gains from different directions are different, and because the radiation intensity of the main lobe direction is greatest, antenna gain in this direction is largest. At the same time, no directional antenna is able to radiate all of its energy in one preferred direction. Some is inevitably radiated in other directions. These smaller peaks are referred to as side lobes, commonly specified in dBi down from the main lobe. Figure 1 shows a case directional antenna gain in main lobe, side lobes and back lobe.
3 Detection Probability Model 3.1 Detection Probability at Arbitrary Location Now we study the issue of the probability that a hostile node detects a transmitter. We refer to such a hostile node as a detection system in this paper. Let the direction of the main lobe’s peak radiation intensity lie on the positive x axis and the star node to be a hostile node in Fig. 2. The distance from the transmitter to the detection system is d and the angle between the direction of the detection system and the direction of the transmitter’s main lob is θ . We will use d and θ in the following sections of this paper with the same meanings defined here. If the power received by a detection system is strong enough, the detection system can distinguish the transmission signals from electromagnetic noise. Hence, the detection event occurs if and only if the SNIR is larger than a threshold λ at the detection system. Assume
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detection systems do not know the transmitter’s radio frequency and location, and their antennas work in omni-directional mode. Pr(Detection) = Pr(SNIR > λ) SNIR = S − N
(5)
Substitute Eqs. (1), (2) and (4) into SNIR. SNIR = Pt + G t (θ ) + 27.55 − n10log10 (d) − 20log10 ( f ) −k − d B(Tr + Te ) − d B(BW) + G r
(6)
where G t (θ ) is the transmitter’s antenna gain function as Fig. 1 shows, and G r = 0. As Pt , 20log10 f , d B(Tr + Te ) and d B(BW) are constants, let K = Pt + 256.15 − 20log10 f − d B(Tr + Te ) + d B(BW)
(7)
Substitute (7) into the definition of the SNIR, and the probability of the detection event can be written as K + G t (θ ) − λ Pr(SNIR > λ) = Pr(K + G t (θ ) − n10log10 d > λ) = Pr >n (8) 10log10 d Now we discuss the value of n, for each of the terrain types listed in Table 1, there is an average distance to the edge of the unobstructed line of sight given, which we defined as d0 . When the distance from the hostile to the transmitter is smaller than d0 , we set n equal to 2. If the distance to the transmitter is greater than d0 , the value of n is a random variable between the values listed in Table 1. ⎧ K + G t (θ ) − λ ⎪ ⎪ > 2, d ≤ d0 ⎨ 10log d 10 (9) Pr(SNIR > λ) = K + G t (θ ) − λ ⎪ ⎪ ⎩ > n, d > d0 10log10 d where K is given by Eq. (7). 3.2 Detection Probability in an Area Using Eq. (9), we can calculate the probability of detecting a transmitter at any location. We assume that the operational area is a finite area L × L and that an adversary could be located anywhere within this region with equal probability. According to the total probability theorem, the probability of detecting a transmitter in an area, DP, is DP = l x px (10) x∈L×L
where l x is the probability of a detection system being at location x and px is the probability of detecting the transmitter at location x.
4 Detection Probability Estimation We model adversaries as passive. Adversaries in this model are assumed to be able to receive any transmitter’s signals but are not able to modify these signals.
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Detection Probability Estimation Fig. 3 DP function of L with both the directional and omni-directional antenna employing the same transmit power
0.14 Omni−directional antenna Directional antenna
0.12 0.1
DP
0.08 0.06 0.04 0.02 0 1
1.5
2
2.5
3
3.5
4
4.5
5
L (100 Km)
If a set of adversaries detect a transmitter in a synchronous manner, they may be able to compute the transmitter’s position with localization algorithms. It is dangerous to reveal the position information to adversaries, because adversaries may find the transmitter and catch it according to its position. Now we calculate the probability of detecting a transmitter when the transmitter sends data using either an omni-directional or a directional antenna. Let the terrain is rural-open. 4.1 Using the Same Transmit Power First we assume that both the omni-directional and directional antennas employ the same transmit power Pt is 1 W. We place the transmitter at the center of the operational area. Figure 3 shows the detection probability function of L. From this figure, we know that if the operational area is small, an adversary has a larger probability to detect an omni-directional antenna than to detect a directional antenna. With the increase of the operational area, the probability of detecting a directional antenna becomes larger than detecting an omni-directional antenna. It is reasonable that both the detection probabilities decrease with the increase of the operational area, because with the increase of L, the power received by a detection system that is far away from the transmitter is less and less and the probability of detecting the transmitter is lower and lower. Thus the DP decreases with the increase of L. Figure 4 shows the detection probabilities at detection systems that are in the directional antenna’s main lobe direction. The x axis in this figure is the distance from the direction system to the transmitter. This figure reveals that if the transmitter uses a directional antenna to send data, the probabilities of detecting it in locations that are in its main lobe direction are much higher than the probabilities of detecting this transmitter if it uses omni-directional antenna. We conclude that directional antennas can not decrease the detection probability if both the omni-directional and directional antennas employ the same transmit power. In some scenarios, directional antennas can bring higher detection probability if some hostile detection systems happen to lie in the directional antenna’s main lobe direction. 4.2 Using the Same EIRP As a directional antenna exhibits more antenna gain in the main lobe direction than an omnidirectional antenna does, it can transmit signals to the receiver using much less transmit power
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Directional antenna Omni−directional antenna
0.9 0.8
Detection Probability
Fig. 4 Probabilities of detecting a transmitter at detection systems that are in the directional antenna’s main lobe direction with both the directional and omni-directional antenna employing the same transmit power
0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
5
10
15
20
25
Distance (10Km) Fig. 5 DP function of L with both the directional and omni-directional antennas employing the same EIRP
0.14
Omni−directional antenna Directional antenna
0.12 0.1
DP
0.08 0.06 0.04 0.02 0
1
1.5
2
2.5
3
3.5
4
4.5
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L (100Km)
than that employed by an omni-directional antenna. We assume the omni-directional antenna and directional antenna both provide the same EIRP in the direction of the receiver. Assume an omni-directional antenna’s transmit power is 1 W and the gain function of the directional antenna is as Fig. 2 shows, the transmit power for the directional antenna is 0.01 W. In Fig. 5, the probability of detecting a directional antenna is lower than that of detecting an omni-directional antenna at each value of L. In this case, the average DP of an omnidirectional antenna is 0.0378 and the average DP of a directional antenna is 0.00376, which is only 10% of the DP of the omni-directional antenna. Now we compare the detection probabilities at detection systems that lie in the direction of the receiver when the transmitter sends data using a directional antenna and omni-directional antenna. Figure 6 shows that the detection probabilities at these detection systems are almost the same no matter whether the transmitter sends data using a directional antenna or not. This figure also indicates the receiver that the transmitter sends data to can receive the same signal power because the signal power received by the receiver depends on the transmitter’s EIRP. Figures 5 and 6 reveal that a directional antenna can reduce the probability of being detected by detection systems by over 90% if the EIRP that two antennas provide in the receiver’s direction are the same while the transmit power the directional antenna employs to send data is less than that employed by the omni-directional antenna.
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Omni−directional antenna Directional antenna
0.9
Detection Probability
Fig. 6 Probabilities of detecting a transmitter at detection systems that are in the directional antenna’s main lobe direction with both the directional and omni-directional antenna employing the same EIRP
0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 5
10
15
20
25
Distance (10Km)
Fig. 7 Detection probability versus power
0.45 Omni−directional antenna Directional antenna
0.4 0.35
DP
0.3 0.25 0.2 0.15 0.1 0.05 0 1
1.5
2
2.5
3
3.5
4
4.5
5
Transmit power (Watt)
5 Parameters Analysis 5.1 Transmit Power It is clear that with the increase of transmit power, both the DP of omni-directional antennas and directional antennas rise accordingly. Figure 7 shows the DP function of the transmit power, where L is 100 km. This figure shows that with the increase of transmit power, the probability of detecting an omni-directional antenna increases clearly, however the probability of detecting a directional antenna does not increase markedly. Hence, we can make a conclusion that transmission through a directional antenna can bring benefit in security compared with transmission through an omni-directional antenna if the directional antenna employs less transmit power than an omni-directional antenna but provides the same EIRP in the direction of the receiver as the omni-directional antenna.
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(a) 100
0.8
90 0.7
Distance (1000m)
80 0.6
70 60
0.5
50
0.4
40
0.3
30 0.2 20 0.1
10 30
60
90
120
150
180
0
Angle (degree)
(b) 100
0.8
90 0.7
Distance (1000m)
80 0.6
70 60
0.5
50
0.4
40
0.3
30 0.2 20 0.1
10
0 30
60
90
120
150
180
Angle (degree) Fig. 8 Distribution of probability of detection with omni-directional antenna and directional antenna. a Omnidirectional antenna. b Directional antenna
5.2 Distance d and Angle θ Figure 8 shows the detection probability distribution with different distance d and angle θ . Y axis of each figures is the distance d from the hostile detection system to the transmitter, and x axis is θ . In Fig. 8, different colors mean different detection probabilities. As omni-directional antennas radiate signals in all directions equally, θ has no impact on the detection probability. However, the detection probability becomes lower and lower with the increase of distance as Fig. 8a shows. Figure 8b shows that the detection probability distribution of directional antenna is strongly related to θ . In this figure, there is a threshold of θ denoted as θ0 . In this case, the threshold θ0 is 60o that is related to the antenna gain function (Fig. 2). When θ < 60o , the antenna gain decreases with the increase of angle θ , so the detection probability decreases rapidly with the increase of θ when d > 10 km. When
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θ > 60o , the antenna gain is a constant as Fig. 1 shows, so the detection probability does not vary with the increase or decrease of θ at a given distance. The probability of detecting directional antennas becomes lower and lower with the increase of distance d at a given angle, which is the same as omni-directional antennas. Comparing these two figures, we can find that the area where the detection probability being zero in Fig. 8b is much larger than that in Fig. 8a. This can explain why a directional antenna has lower detection probability than an omni-directional antenna if they employ the same EIRP). 6 Conclusions In a network that is untrustworthy, it is very important for the transmitter to avoid being detected by adversaries. In the paper, we explore the problem of detection probability estimation of omni-directional antennas and directional antennas. We propose a detection probability model to calculate the probability of detecting a transmitter at any location. Our study shows that with the same transmit power, directional antennas can not reduce the transmitter’s probability of being detected by detection systems. If there are hostile detection systems that happen to lie in the directional antenna’s main lobe direction, the probability of detecting a directional antenna is even higher than that of detecting an omni-directional antenna. However, if the directional antenna employs less transmit power but provides the same EIRP in the direction of the receiver as that provided by an omni-directional antenna, the directional antenna can reduce the detection probability by over 90%. This is the benefit in anonymity and security provided by directional antennas. Based on this advantage, we can employ directional antennas as relays to bypass hostile detection systems to transmit data. In the future, we would study how to make use of directional antennas to build a secure routing path which has the lowest detection probability. Acknowledgments We would like to gratefully acknowledge ITA Project. Our research was sponsored by the U.S. Army Research Laboratory and the U.K. Ministry of Defence. This work is also partly supported by National Natural Science Foundation of China (60803120) and the Graduate Innovation and Practice Foundation of Beihang University.
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Author Biographies Xiaofeng Lu is currently a Ph.D. candidate in the school of Computer Science of Beijing University of Aeronautics and Astronautics. He received his B.S. and M.S. degrees in computer science from North China University of Technology, in 1999 and 2005, respectively. During 2007 to 2008, He visited the Computer Laboratory of University of Cambridge. His research interests include wireless communication, network security, and anonymous communication, etc.
Fletcher D. Wicker received his B.S., M.A., and Ph.D. degrees in mathematics all from the Pennsylvania State University in 1968, 1969, and 1975 respectively. From 1974 to 1977 he worked as an operations research analyst for the Naval Personnel Research and Development Center. In 1977 he joined the Aerospace Corporation in El Segundo, CA. During the academic year of 2007–2008 he was a visiting scholar at both the Statistical Laboratory and the Computer Laboratory at the University of Cambridge, UK. His current research interests include wireless network performance evaluation and control.
Don Towsley received the B.A. degree in physics and the Ph.D. degree in computer science, both from University of Texas University, in 1971 and 1975 respectively. He is currently a Distinguished University Professor in the Department of Computer Science at the University of Massachusetts — Amherst, where he co-directs the Networking Research Laboratory. He has been a Visiting Scientist at AT&T Labs - Research, IBM Research, INRIA , Microsoft Research Cambridge, and the University of Paris. He currently serves as Editor-in-Chief of the IEEE/ACM Transactions on Networking and on the editorial boards of Journal of the ACM and IEEE Journal of Selected Areas in Communications. He has twice received IBM Faculty Fellowship Awards, and is a Fellow of the IEEE and the ACM. Dr. Towsley’s research interests include network measurement, modeling, and analysis.
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Detection Probability Estimation Zhang Xiong is currently a Professor in the school of Computer Science of Beijing University of Aeronautics and Astronautics. He received his M.S. degree in computer science from Beijing University of Aeronautics and Astronautics in 1984, and visited the Michigan State University from 1989 to 1992. Professor Xiong was awarded China National Golden Medal for Progress in Science and Technology in 1994. His current research interests include wireless sensor networks, multimedia, RFID, etc.
Pietro Lio’ is Senior Lecturer at the University of Cambridge, Computer Laboratory in England and Fellow and Director of Studies at Fitzwilliam College of University of Cambridge. He is currently modeling biological processes on networks; modeling stem cells; developing transcription and phylogenetic applications on a grid environment. He is also interested in bio-inspired design of wireless networks; epidemiological networks.
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