Distributed Diversity in Ultrawide Bandwidth Wireless Sensor Networks Tony Q.S. Quek and Moe Z. Win
Marco Chiani
Laboratory for Information & Decision Systems (LIDS) Massachusetts Institute of Technology Cambridge, MA 02139 Email : {qsquek,moewin}@mit.edu
DEIS, University of Bologna Bologna, Italy Email:
[email protected]
Abstract— In this paper we consider a decentralized detection problem in ultrawide bandwidth (UWB) wireless sensor network (WSN). The system model assumes that all sensor nodes have established an ad-hoc network for sharing the information before sending the common message to the fusion center. The architecture is asynchronous and communication channels between the local sensors to the fusion center are subjected to fading and noise. Our communication strategy employs a UWB transmittedreference (TR) signaling scheme. We first analyze the asymptotic performance of our communication strategy via the Chernoff bound and show that the upper bound on the probability of decision error at the fusion center decays exponentially with increasing number of nodes. Simple link budget analysis is given to demonstrate the need for distributed diversity in energy constrained large scale WSN. We also derive the required average transmission power per bit of each node to achieve a target BER at the fusion center and quantify the energy efficiency of our communication strategy.
I. I NTRODUCTION With the development of low-cost, low-power transceivers, sensors and embedded processors, there has been a recent interest in wireless sensor networks (WSN) for a wide variety of applications from surveillance and security to biological applications involving data collection [1]. Such WSN differ from traditional ad-hoc networks in many ways: distinct features of WSN include high density of nodes, severe power constraint, low mobility, large number of low-cost nodes, and high correlation of data among the nodes. An example of a large scale WSN is the reachback problem where no sensor node in the network is powerful enough to communicate reliably with the fusion center [2]. In such WSN, the primary issues are intelligent integration of multi-sensor data, and the design of an efficient and scalable communication strategy [3]. A typical decentralized binary detection problem involves geographically dispersed sensor nodes receiving information about a phenomenon; each sensor node is required to transmit a summary of its own observation to a fusion center [4]. Based on the received data, the fusion center produces an estimate of the phenomenon. The problem of decentralized detection has been extensively studied in the literature [5]–[8]. However, most studies made idealistic assumptions like dedicated and noise-free communication channels between local sensors and fusion center, and transmission from local sensors are completely synchronized. These assumptions are impractical
0-7803-8887-9/05/$20.00 (c)2005 IEEE
particularly for large scale WSN with a transmission power constraint. Moreover, multiple access (MAC) layer design for large scale WSN to coordinate such decentralized transmission can be a difficult task. Recently, there has been some work that aims to tackle such issues [2]. In this paper, we focus on a large scale WSN without any MAC protocol [2], where the underlying UWB physical layer is used to integrate the multi-sensor data [9]. We provide a justification why distributed diversity is necessary for reliable communication in large scale WSN from a simple link budget analysis. Unlike [2], we do not assume that sensor nodes can synchronize themselves such that coherent combining can be achieved at the fusion center. However, we assume that an ad-hoc network protocol has been established and information to be sent has been shared via efficient flooding. Our communication strategy employs a UWB transmitted-reference (TR) signaling scheme that allows a simple autocorrelation receiver at the fusion center [10]–[12]. TR signaling does not require any channel estimation or network synchronization. TR signaling with distributed diversity is capable of exploiting the spatial distribution of the sensor nodes to achieve robust communication even when the sensors are operating in low SNR environments. We assume that the communication channels are subjected to both frequency selective Rayleigh fading and noise. The remainder of the paper is organized as follows. In Section II, we develop the signal and channel models for UWB TR signaling. In Section III, we introduce the receiver structure and derive the asymptotic upper bound on the probability of decision error at the fusion center in large scale WSN. In Section IV, a simple link budget analysis of the reachback problem is given and the energy efficiency of our communication strategy is quantified. Some numerical results are presented in Section V. Finally, we give our conclusions in the last section.
II. S IGNAL AND C HANNEL M ODELS We consider a large scale WSN with K identical sensors deployed over a wide area. We consider the binary hypothesis
written as
H0 H1
y1
yK
y2
Sensor 1
Sensor 2
br (t)
Sensor K
u Modulator 1
u Modulator K
s1(t)
s2(t)
s K(t)
Fading channel h 2 (t)
=
=
Fading channel h K (t)
=
System model
phenomenon of interest absent phenomenon of interest present,
g(t) = where all K local sensors collect observations generated under the above hypothesis. We want to decide whether the phenomenon of interest is present in the area at the fusion center. All the local sensors are capable of establishing an ad-hoc network so that information can be disseminated via efficient flooding (sensor gossip). As shown in Fig. 1, all the sensors will agree upon the common message u (H0 or H1 ) to be sent to the fusion center. However, we do not assume that the sensors can synchronize themselves to coherently combine at the fusion center. The common transmitted signal from each sensor for the i-th message is given by v−1 (t), v+1 (t),
when u = 0 when u = 1,
(1)
where 1 ≤ k ≤ K and the UWB TR waveform [11], [12] for the i-th message is given by vdi (t)
K L
αk,l sk (t − τk,l ) + n(t),
(4)
where αk,l and τk,l denote respectively the attenuation and delay of the l-th path of the k-th sensor. The parameter L is the number of multipath components and is assumed to be the same for all sensors. For simplicity, we also assume that the power delay profile is uniform and the fading distribution is Rayleigh. For a slow fading channel, we can model the processes αk,l and τk,l as constant over several symbol duration. Due to the linear time-invariant property (LTI) of the channel, we can represent the composite channel impulse responses as
testing problem with the following hypotheses:
sk (t) =
(3)
k=1 l=1
Fusion Center
Ep p(t − j2Tf − Tr ),
where p(t) is the signal +∞ pulse with duration Tp and it is normalized so that −∞ |p(t)|2 dt = 1. The energy of the transmitted pulse is then Ep = Es /Ns and symbol energy is Es . The inter-pulse delay between each data-modulated pulse and its corresponding reference pulse is given by Tr . The composite received signal at the fusion center for the i-th message can then be written as
n(t)
: :
−1 j=0
r(t)
H0 H1
Ep p(t − j2Tf )
Ns 2
∑
Fig. 1.
−1 j=0
bd (t)
Sensor Gossip u Modulator 2
Fading channel h 1 (t)
Ns 2
= br (t − iNs Tf ) + di bd (t − iNs Tf ),
(2)
where Tf is the average repetition period, di = ±1 is the data symbol, and each block has symbol interval Ns Tf . Within each block, there are Ns /2 transmitted signal pulses and it can be
K L
αk,l δ(t − τk,l ),
(5)
k=1 l=1
where τk,l is assumed to be uniformly distributed over [0, Td ) and Td is the maximum excess delay. Without loss of generality, we assume that τ1,1 is the minimum delay and the receiver at the fusion center is synchronized to this delay. Unlike in conventional channel models, g(t) consists of both overlapped and non-overlapped clusters of multipaths induced by both the spatial location of the sensor nodes and the scattering of the environment. In order to preclude intra-symbol interference and inter-symbol interference (ISI), Tf and Tr are chosen to satisfy Tg ≤ Tr < 2Tf − Tg , where Tg = Tp + Td . As a result, the data modulated pulse and reference pulse still maintain orthogonality. III. A SYMPTOTIC PERFORMANCE ANALYSIS The autocorrelation receiver at the fusion center first passes the composite received signal through a bandpass zonal filter (BPZF) with bandwidth W and center frequency fc to eliminate the out-of-band noise [11], [12]. If W is wide enough, then the signal spectrum will pass undistorted and, consequently, ISI and intra-symbol interference caused by filtering will be negligible. The autocorrelation receiver then correlates the composite received modulated pulses with their corresponding reference pulses received Tr seconds earlier over the interval T (0 < T ≤ Tg ), and combines all the Ns /2
Ns 2 −1 2W T 1 ( √ wj,m − β2,j,m )2 E exp −r W
correlated values to form a global decision statistic for the i-th message as −1
Ns 2
Z
=
=
Ns 2
r(t) r(t − Tr )dt
=
j2Tf +τ1,1 +Tr
j=0 Ns 2
j=0 m=1
j2Tf +τ1,1 +Tr +T
min r≥0
−1
Ui,j ,
−1 2W T
2 E exp rβ1,j,m
j=0 m=1
1 E exp −r( √ wj,m − β2,j,m )2 , W (11)
(6)
j=0
where r(t) is the filtered composite received signal. Theorem 1: Consider a composite multipath fading channel as defined in (5). Let bm be the variance of the m-th multipath component. By employing UWB TR signaling with Ns pulses and energy per pulse Ep , a distributed transmission from K local sensor nodes to the fusion center can be achieved. The probability of decision error at the fusion center of such a communication strategy satisfies
2 WT N s Ep 2 bm , (7) Pe ≤ exp − 2 N0 m=1 where W is the bandwidth of receiver BPZF and T is the integration interval of the correlator at the receiver. Proof: Following the sampling expansion approach in [11], [12], we can represent Ui,j in (6) by 2W T samples as follows Ui,j
=
2W T 1 2 (di wj,m + wj,m η2,j,m W m=1
+ di wj,m η1,j,m + η1,j,m η2,j,m ),
(8)
where wj,m , η1,j,m , and η2,j,m are the m-th sample of wj (t), η1,j (t) and η2,j (t) respectively defined in [11], [12]. Conditioned on di , we can express (8) in the form of a summation of squares Ui,j|di =+1
=
2W T
1 2 [( √ wj,m + β1,j,m )2 − β2,j,m ] (9) W m=1
2W T
1 2 [−( √ wj,m − β2,j,m )2 + β1,j,m ], Ui,j|di =−1 = W m=1 (10) √ where β1,j,m = (η √2,j,m + η1,j,m )/2 W and β2,j,m = (η2,j,m − η1,j,m )/2 W . Note that β1,j,m and β2,j,m are uncorrelated Gaussian r.v.’s. By involving the Chernoff bound, we can derive the upper bound of BER at the fusion center as follows Pe
w
E b
where √j,m ∼ N (0, p2 m ) and bm is the variance of the m-th W multipath component. We can then express (11) as N2s 2W T 1 1 Pe ≤ min r≥0 N0 r N0 1 − r 2W 1 + W ( 2 + Ep bm ) m=1 (12) since bm = bm+W T for m = 1, . . . , W T , we have Pe
≤ min r≥0
W T
"
m=1
1 # $# N0 1 − r 2W 1+
% N2s
1
r W
$ ( N20 + Ep bm ) (13)
To continue the proof of Theorem 1, we will use Lemma 1. (Proof is omitted due to space &Wconstraint) T Lemma 1: The function m=1 f (r) in (13) is convex and has constrained minimum. From Lemma 1, we optimize r in (13) to obtain rmin = Ep bm W/(N0 ( N20 + Ep bm )). By substituting rmin back into (13), we have N2s Ep W T 1 + 2 N bm 0 Pe ≤ ( )2 E m=1 1 + Np0 bm =
W T
(4pm (1 − pm ))
Ns 2
,
(14)
m=1 E
E
where we define pm 1/2(1 + Np0 bm ). For Np0 bm 11 , then E pm ≈ 1/2 − bm 2Np0 by first order Taylor series approximation and the upper bound in (14) becomes " 2 % N2s W T Ep Pe 1 − bm N0 m=1
2 WT Ns E p 2 ≈ exp − bm , (15) 2 N0 m=1
= P {Zi > 0|di = −1} where the last line invokes the first order Taylor series approx Ns 2 −1 2W T 1 imation of exp function. 2 [−( √ wj,m − β2,j,m )2 + β1,j,m ]>0 = P Corollary 1 (Non-overlapped clusters): When the clusters W j=0 m=1 in (5) are non-overlapped, we can further decompose the Ns 2 −1 2W T 2 1 This assumption is reasonable since Ep is generally very small when N ≤ min E exp r β1,j,m s N0 r≥0 j=0 m=1
is large in UWB systems.
summation in the error exponent of (15) into
2 K L Ns E p b2k,l Pe ≤ exp − 2 N0 k=1 l=1 2
K Ns Ep = exp − , (16) 2 N0 L + , 2 where bk,l = E αk,l = 1/L and T = Tg . From (16), we can obtain several interesting conclusions regarding the asymptotic performance of our communication strategy. Firstly, for a fixed Ep /N0 , the BER is decreasing exponentially with increasing Ns [11]. Moreover, the BER is decreasing exponentially with increasing K, thereby, showing the asymptotic performance of distributed diversity in large WSN. More importantly, we can observe that even with overspreading or large L [13], the gain in distributed diversity still dominates over poor performance in non-coherent combining by having a large K. IV. ENERGY EFFICIENCY ANALYSIS TABLE I R ECEIVED SNR AT FUSION CENTER (fc = 100MH Z , W = 100MH Z ) d 1 km 10 km
Pt = 1 W 48.78 dB 28.78 dB
Pt = 1 mW 18.78 dB -1.22 dB
Pt = 1 µW -11.22 dB -31.22 dB
in Table I and II.3 Given the small size and low energy consumption constraint of each node [15], Table I and II demonstrates that each sensor node may not have sufficient transmitted power to ensure reliable communication at the fusion center. Below we demonstrate how distributed diversity can alleviate such a problem. B. Average Transmission Power Theorem 2: Consider a composite multipath fading channel as defined in (5). Let bm is the variance of the m-th multipath component. By employing the UWB TR signaling with Ns pulses and energy per pulse Ep , a distributed transmission from K local sensor nodes to the fusion center can be achieved. The average transmission power per bit of each node for a given target BER at the fusion center is approximated by -" % . 2 1 2N (4πd) (Kb W ) . s / 0 ln , (18) Pt ≈ WT 2 Gt Gr λ20 La P e m=1 bm where W is the bandwidth of the receiver BPZF and T is the integration interval of the correlator at the receiver. Proof: From (17), we can obtain the transmitted power per sensor node as 2
Pt TABLE II R ECEIVED SNR AT FUSION CENTER (fc = 1GH Z , W = 500MH Z ) d 1 km 10 km
Pt = 1 W 41.79 dB 21.79 dB
Pt = 1 mW 11.79 dB -8.21 dB
Pt = 1 µW -18.21 dB -38.21 dB
A. Link Budget We consider a sensor field with K nodes having the same processing power and energy constraint. The fusion center is d meters away from the vicinity of the sensor field. In the following, we investigate the link budget2 between a single sensor node and the fusion center. From [14], the received SNR at the fusion center can be obtained as follows 2 λ0 Gr Pr = Pt Gt La , (17) N 4πd Kb W where Pr is the received signal power at the fusion center, N is the noise power, Pt is the transmitted power from a single sensor node, Gt is the transmitter antenna gain, λ0 is the carrier wavelength, La is the atmospheric loss, Gr is the receiver antenna gain at the fusion center, W is the transmission bandwidth, Kb = k T where k is the Boltzman’s constant and T is the noise temperature in degrees kelvin. From (17), we can then calculate the received SNR in free space propagation for different d, Pt , fc and W as shown 2 Note that the application of this link budget to analyze UWB systems is primarily restricted to a rule of thumb since it is more suitable for narrowband systems.
=
(4πd) (Kb W ) Eb . Gt Gr λ20 La N0
(19)
From (7), we can write the upper bound for the required average energy consumption per bit for each sensor node while meeting a target BER at the fusion center as -" % . . 1 2N s Eb ≤ N0 / 0W T ln . (20) 2 P e m=1 bm By approximating the bound in (20) as an equality, we can approximate the average required transmission power per bit by using (19) and (20) and we have -" % . 2 1 2N (4πd) (Kb W ) . s / 0 ln Pt ≈ . (21) WT 2 Gt Gr λ20 La P e m=1 bm Corollary 2 (Non-overlapped clusters): When the clusters in (5) are non-overlapped, we can rewrite (21) as 1 2 1 (4πd) (Kb W ) 2LNs ln (22) Pt ≈ Gt Gr λ20 La K Pe , + 2 = 1/L and T = Tg . From (22), it can where bk,l = E αk,l √ be seen that Pt ∝ L. Thus, if L is too large, we need to increase Pt and, intuitively, this is reasonable since the average 3 The results are based on the assumptions that T = 300K, G ≈ 1, t La = 1, Gr is calculated according to Gr = 05( λπ )2 , where parabolic 0 antenna of diameter is 1m, and illumination efficiency factor is 0.5. Note that these calculations simply serve as a guideline since it is more suited for narrowband signals.
−3
0
10
10
−4
−1
10
10
−5
−2
10
Pt (W)
BER
10
−3
10
K = 10 K = 100
−6
10
K = 1000 −4
10
Ep/N0 = −5 dB
−7
10
L = 20
−5
10
Ns = 2 N =4 s N =6 s Ns = 8 N = 10
−8
10
L = 20 N =2
s
s
−9
−6
10
0
10
100
Fig. 2.
200
300
400 500 600 700 Number of sensor nodes K
800
900
1000
Effect of Ns on the decision error bound in (16).
energy of each multipath component becomes so small that it the combining ability of the noncoherent receiver is degraded √ [13]. Observe also that Pt ∝ 1/ K. Thus, we can reduce Pt by increasing K or exploiting greater distributed diversity. V. NUMERICAL ANALYSIS In this section, some numerical results are given to illustrate the reliability and robustness of WSN by using distributed diversity. For simplicity, here we have restricted results to non-overlapping clusters of i.i.d. Rayleigh fading channels with L = 20. Fig. 2 shows the decision error bound of (16) with different Ns values. Two significant observations can be made. First, the decision error bound decreases with increasing K, thereby showing the effectiveness of distributed diversity in increasing the reliability of the communication. Next, the decision error bound decreases with increasing Ns . Thus, this provides us with a means to increase reliability and robustness of our WSN by either exploiting distributed diversity gain (increasing K) or power gain (increasing Ns ). In Fig. 3, the average transmission power per bit of a single sensor node is plotted against the transmission range for different K. The values of the parameters used follows that of Table I. A higher distributed diversity order (larger K) reduces the required average transmission power per bit to maintain the target BER. For example, for BER = 10−5 and d = 500m, Pt = 2µW for K = 1000 and Pt = 20µW for K = 10. Note that our analysis has not included the required energy to conduct sensor gossip in the WSN and this may be significant when K is very large. VI. CONCLUSIONS We analyze the problem from the link budget perspective to justify that distributed diversity is necessary for reliable communication between the sensors and the fusion center. Our communication strategy employs UWB TR signaling with a simple autocorrelation receiver at the fusion center. It exploits the spatial distribution of the sensors to achieve robust communication even when the sensors are operating
0
100
200
300
400
500 d (m)
600
700
800
900
1000
Fig. 3. Effect of K on Pt using (22). The solid and dotted lines denote the target BER of 10−2 and 10−5 , respectively.
in low SNR environment. We adopt the Chernoff bound to analyze the asymptotic performance analysis and quantify the energy efficiency of our communication strategy. Moreover, we quantify that by increasing the order of the distributed diversity, we are able to increase reliability and robustness of the WSN. R EFERENCES [1] C.-Y. Chong and S. P. Kumar, “Sensor networks: evolution, opportunities, and challenges,” Proc. IEEE, vol. 91, pp. 1247–1256, Aug. 2003. [2] S. Servetto, “Distributed signal processing algorithms for the sensor broadcast problem,” Proc. Conf. Information Sciences and Systems, Mar. 2003. [3] J.-F. Chamberland and V. V. Veeravalli, “Asymptotic results for decentralized detection in power constrained wireless sensor networks,” IEEE J. Select. Areas Commun., vol. 22, pp. 1007–1015, Aug. 2004. [4] R. Tenney and N. S. Jr., “Detection with distributed sensors,” IEEE Trans. Aerosp. Electron. Syst., vol. 17, pp. 501–510, Aug. 1981. [5] J. N. Tsitsiklis, “Decentralized detection,” in Adv. Statist. Signal Process., vol. 2, 1993, pp. 297–344. [6] P.-N. Chen and A. Papamarcou, “New asymptotic results in parallel distributed detection,” IEEE Trans. Inform. Theory, vol. 39, pp. 1847– 1863, Nov. 1993. [7] R. Viswanathan and P. K. Varshney, “Distributed detection with multiple sensors: Part ifundamentals,” Proc. IEEE, vol. 85, pp. 54–63, Jan. 1997. [8] R. S. Blum, S. A. Kassam, and H. V. Poor, “Distributed detection with multiple sensorspart ii: Advanced topics,” Proc. IEEE, vol. 85, pp. 64– 79, Jan. 1997. [9] T. Q. S. Quek, “Distributed diversity for large scale wireless sensor networks,” 6.263 Project Report, Nov. 2003. [10] J. D. Choi and W. E. Stark, “Performance of ultra-wideband communications with suboptimal receivers in multipath channels,” IEEE J. Select. Areas Commun., vol. 20, no. 9, pp. 1754 –1766, Dec. 2002. [11] T. Q. S. Quek and M. Z. Win, “Performance analysis of ultrawide bandwidth transmitted-reference communications,” Proc. IEEE Semiannual Veh. Technol. Conf., May 2004. [12] ——, “Ultrawide bandwidth transmitted-reference signaling,” Proc. IEEE Int. Conf. on Commun., pp. 3409 – 3413, June 2004. [13] I. E. Telatar and D. N. Tse, “Capacity and mutual information of wideband multipath fading channels,” IEEE Trans. Inform. Theory, vol. 46, pp. 1384 – 1440, July 2000. [14] J. G. Proakis, Digital Communications, 4th ed. New York, NY, 10020: McGraw-Hill, Inc., 2001. [15] A. Goldsmith and S. Wicker, “Design challenges for energy-constrained ad hoc wireless networks,” IEEE Commun. Mag., pp. 8–27, Aug. 2002.
