Energy-Efficient Noncoherent Signal Detection for Networked Sensors Using Ordered Transmissions

Transcription

1 Energy-Efficient Noncoherent Signal Detection for Networked Sensors Using Ordered Transmissions Ziad N. Rawas, Student Member, IEEE, Qian He, Member, IEEE, and Rick S. Blum, Fellow, IEEE Abstract Energy efficiency has been a topic of recent interest in networks employing wireless sensors used for signal detection. Energy efficient signal detection is of particular interest in systems where such sensors carry their own energy sources with a limited capacity. It has been demonstrated that ordering the sensor transmissions and halting them upon accumulating enough evidence can save transmissions (energy) without degrading the system s detection performance. We extend this concept to a system employing noncoherent signal detection where the hypothesis testing problem involves a log-likelihood ratio which can only take on nonnegative values. We propose a new algorithm for ordering the transmissions for such systems and show that the new algorithm tends to provide more significant savings than the existing methods, requiring as little as one sensor transmission in certain cases. We utilize MIMO radar as an example in our paper, and provide results demonstrating the savings achieved. I. INTRODUCTION Wireless sensor networks have gained increased popularity in recent years in a wide variety of applications [1], [2]. Such networks frequently employ sensor-equipped nodes, which we refer to as sensors, with self-contained energy sources. Since the energy sources often have a limited lifespan while the sensors are expected to operate for an extensive period of time, conserving energy is of critical importance. Energyefficiency in sensor networks has been investigated in many recent studies. In most of the previous work [3]-[8], energy saving is achieved at the expense of detection performance. More recently, in [9], an approach for energy-efficient signal detection was developed in which sensor transmissions are ordered according to the informativeness of their observations. This work considered a hypothesis testing problem where the log-likelihood ratios could take on both nonnegative and negative values. Further, a Bayesian hypothesis testing approach Copyright 2011 IEEE. Published in the 45th Conference on Information Sciences and Systems (CISS) in Mar 2011 in Baltimore, MD, USA. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works, must be obtained from the IEEE. Contact: Manager, Copyrights and Permissions / IEEE Service Center / 445 Hoes Lane / P.O. Box 1331 / Piscataway, NJ , USA. Telephone: + Intl This material is based on research supported by the U.S. Army Research Office under grant No. W911NF , the Air Force Research Laboratory under agreement No. FA , the National Science Foundation under Grant No. CCR , and by the Fundamental Research Funds for the Central Universities under grant No. ZYGX2009J019. The authors are with the ECE Dept, Lehigh University, Bethlehem, PA USA. Q. He is also with the EE Dept, University of Electronic Science and Technology of China, Chengdu, Sichuan China. s: {znr209; qih207; rblum@lehigh.edu}. was considered that assumed prior probabilities to be known. In this paper, we describe an ordering approach for a class of noncoherent signal detection problems where the log-likelihood ratios at each sensor can take on nonnegative values. There are many practical binary hypothesis where this is the case. An example of an important practical problems of this type is presented in a later section. Note that unlike the previous approaches which always lead to some loss in performance, the ordering approach presented in this paper reduces the average number of transmissions while achieving exactly the same performance as if all data has been transmitted to the fusion center. II. ORDERED TRANSMISSION APPROACH Consider a system with N networked sensors, each of which takes data for the purpose of binary hypothesis testing between hypotheses H 0 and H 1. Let Y k denote a random variable for the observation at sensor k and f Yk (y k H j ) the probability density function 1 (pdf) of Y k conditioned on H j. Denote the log-likelihood ratio (LLR) of the k th sensor s observation as L k = ln (f Yk (y k H 1 )/f Yk (y k H 0 )). If all sensors transmit their data to a central location, the fusion center, then the optimum decision is made at the fusion center by comparing N the overall LLR, L k, to a fixed threshold τ and a decision for H 1 is made if the overall LLR is larger than τ, otherwise a decision for H 0 is made 2. For the Neyman-Pearson criterion, τ is chosen to fix the false alarm probability. For the Bayesian criterion, where the prior probabilities P rob(h 0 ) and P rob(h 1 ) are known, if uniform cost is assumed then τ = P rob(h 0 )/P rob(h 1 ). For the minimax criterion, we set τ as the Bayesian optimum threshold for the least favorable priors [12]. The following assumptions are made throughout the paper: Assumption 1: Assume all sensors receive independent and identically distributed (iid) observations, conditioned on the binary hypothesis. For simplicity, assume each sensor uses an orthogonal noise-free communication channel to transmit its data to a fusion center. Assumption 2: The LLR at every sensor is strictly positive such that 0 L k, k. Algorithm 1: Consider an approach where the sensors order their transmissions to the fusion center so that the sensor with 1 If Y k is a discrete random variable, then f Yk (y k H j ) will denote the probability mass function of Y k conditioned on H j. 2 The results in this paper are valid for randomized tests.

2 the largest LLR transmits its data to the fusion center first. Denote the ordered LLRs as L [1] > L [2] > > L [N]. Thus the first sensor to transmit will transmit L [1] = max(l 1,..., L N ) and the next L [2], the next largest in the set {L 1,..., L N }, and so on. In fact, the sensors can decide when to transmit in a completely distributed manner. The k th sensor could transmit 3 after a time equal to C/L k for some common constant C. After each transmission, the fusion center accumulates the sum of all sensor LLRs that have been transmitted so far and compares this sum to the two thresholds τ and t L. If it is assumed at a given time that all but n UT sensors have transmitted and that the last sensor transmission was L [UT ], then, we set t L = τ n UT L [UT], where τ is the previously defined fixed threshold for the optimum N-sensor test, the test where all N sensors transmit to the fusion center. If the accumulated sum of the LLRs exceeds the fixed threshold τ, L [k] > τ, (1) we halt transmissions and decide for H 1. Alternatively, if the stopping condition L [k] < t L (2) is met, we halt transmissions and decide for H 0. The following Theorem describes the capability of Algorithm 1 to save transmissions. Theorem 1: Consider a sensor network with N sensors which is employing Algorithm 1 to solve a binary hypothesis testing problem. Under Assumptions 1 and 2, the approach described in Algorithm 1 will always make the same decision as the optimum approach where all N sensors transmit while generally using a smaller average number of sensor transmissions. Proof: First, let us focus on the case where the fixed threshold τ is exceeded. According to Assumption 2, L k > 0 k. Thus we can bound any partial sum of the LLRs using L [k] L k (3) for any n UT > 0. Therefore, if the left side of (3) exceeds the threshold τ, then the overall sum of the LLRs from all N sensors will also be greater than τ and we can halt transmissions and make the same decision as the test which uses all the sensor data. Note that there is generally a nonzero probability that we stop transmissions before all N sensors transmit. Thus, we will reduce the average number of sensor transmissions. Now, consider the case where the sum of the LLRs transmitted is smaller than the lower threshold t L so that t L > L [k] (4a) 3 While we use this approach in our numerical results, other approaches are also possible as long as the ordering is maintained. Using t L = τ n UT L [UT ], (4a) becomes τ n UT L [UT] > L [k] (4b) Since the transmissions have been ordered, the LLRs that have not yet transmitted will be strictly smaller than the last one transmitted L [UT ]. Then (4b) implies that τ > L [k] + n UT L [UT] L k (5) so if (4a) occurs then we can stop transmissions and make a decision for H 0. Based on (5), we know we will make the same decision as the optimum test which uses the data from all N sensors. Note that there is generally a nonzero probability that we stop transmissions before all N sensors transmit. Thus, we will reduce the average number of sensor transmissions. Note that if all the sensors hear all the transmissions, then distributed decision making could be implemented and each sensor can perform the computations to know when to stop the transmissions. Otherwise, a single stop signal can be sent to all sensors from the fusion center to halt the transmissions. While we have shown we can save some transmissions without loss in performance, the magnitude of the savings have not been quantified. Next, large savings are shown to be achieved in some cases of great interest. III. SAVINGS UNDER DIFFERENT CONDITIONS The following assumption is useful for some quantification of the possible savings in some specific cases for general binary hypothesis testing problems. Assumption 3: For the binary hypothesis testing problems considered, we assume the existence of a distance measure s, which describes the distance between f Yk (y k H 1 ) and f Yk (y k H 0 ), such that Pr(L k > τ H 1 ) 1 as s for any finite τ. Intuitively, Assumption 3 implies that as s gets larger it becomes easier to tell the difference between observations under H 0 and H 1. Such distance measures can be found for many hypothesis testing problems of practical interest. For example, in many cases where we are testing noise only, H 0, versus signal-plus-noise, H 1, the signal-to-noise-ratio (SNR) plays the role of s. Later in Section IV, we discuss an example hypothesis testing problem involving a MIMO radar system which is such a case where SNR is a distance measure s as described in Assumption 3. Define N s as the number of transmissions saved and N t as the number of transmissions after which a decision can be made using Algorithm 1, then N s = N N t. The following theorems present savings that can be achieved for Algorithm 1. Theorem 2: Under Assumptions 1-3 for the case where H 1 is true, the probability that only a single transmission is needed when Algorithm 1 is employed with a finite τ approaches 1 as s. It follows that E{N t H 1 } = 1 as s.

3 Proof: Given Assumption 3, we have Pr(N t = 1 H 1 ) 1 and Pr(N t > 1 H 1 ) 0 as s. (6) Thus, E{N t H 1 } = k Pr(N t = k H 1 ) 1 as s. (7) The rest of this section focuses on the Neyman-Pearson hypothesis test, where the threshold τ is set by the probability of false alarm, P F A. We make the following assumption relating τ and P F A. Assumption 4: For the binary hypothesis testing problem considered, P F A 1 as τ 0, and P F A 0 as τ. Further 0 < L k < k with probability one under H 0 or H 1. Such an assumption implies L k has no point masses of probability at zero and infinity and inherently implies finite SNR. The following Theorems describe the impact of P F A on the transmission savings. Theorem 3: Assume the Neyman-Pearson criterion is employed and Assumptions 1-4 hold. If Algorithm 1 is employed, the probability that only a single transmission is needed approaches 1 as P F A 0, under the case where H 0 or H 1 is true. It follows that Pr(N s = N 1 H j ) 1 as P F A 0 for j = 0, 1. Proof: As P F A 0, by referring to Assumption 4 we have τ. Recalling that L k < with probability one from Assumption 4, Pr(L [1] < t L H j ) = Pr ( L [1] < τ (N 1)L [1] H j ) Pr(L [1] < H j ) = 1, as P F A 0, for j = 0, 1. (8) Thus Pr(N t = 1 H j ) 1. Therefore, Pr(N s = N 1 H j ) 1 as P F A 0 for j = 0, 1. Theorem 4: Assume the Neyman-Pearson criterion is employed and Assumptions 1-4 hold. If Algorithm 1 is employed, the probability that only a single transmission is needed approaches 1 as P F A 1, under the case where H 0 or H 1 is true. It follows that Pr(N s = N 1 H j ) 1 as P F A 1 for j = 0, 1. Proof: As P F A 1, by referring to Assumption 4 we have τ 0. Recalling that L k > 0 with probability one from Assumption 4, Pr(L [1] > τ H j ) Pr(L [1] > 0 H j ) = 1 as P F A 1. (9) Thus, Pr(N t = 1 H j ) 1. Therefore, Pr(N s = N 1 H j ) 1 as P F A 1 for j = 0, 1. In the next section, we describe an example hypothesis testing problem to concretely illustrate the utility of the ideas we have presented thus far. IV. MIMO RADAR AS AN EXAMPLE APPLICATION Consider a MIMO radar system that has M transmit and N receiver antennas. The transmit antennas are placed at the known positions (x t l, yt l ), l = 1,, M and the receive antennas are placed at the known positions (x r k, yr k ), k = 1,, N in a two-dimensional Cartesian coordinate system. The low-pass equivalent of the signal transmitted from the l-th transmitter is Es l (t), where E denotes the transmitted energy per transmit antenna, and the waveform is normalized so that s l(t) 2 dt = 1. Assume the hypothesis test will be for H 0 : no target (Gaussian noise only) versus H 1 : a target moving with velocity (v x, v y ) that is located at (x, y) observed in Gaussian noise. The time delay τ kl and Doppler shift f kl involved in the path from transmitter l to receiver k, via the target reflection, are τ kl = (d t l + dr k )/c and f kl = (v x (x t l x) + v y(yl t y))/(λdt l ) + (v x (x r k x) + v y(yk r y))/(λdr k ), where c denotes the speed of light, d t l = (x t l x)2 + (yl t y)2 the distance between the target and the l-th transmitter, d r k = (x r k x) 2 + (yk r y)2 the distance between the target and the k-th receiver, and λ the wavelength of the carrier. In the noncoherent processing approach, we assume all transmitter and receiver nodes have oscillators which are locked in frequency, possibly due to the use of a beacon. We further assume the transmitted signals are approximately orthogonal and maintain approximate orthogonality after reception for time delays and Doppler shifts of interest [10]. The noise corresponding to the kl-th path w kl (t) is a temporally white, zero-mean complex Gaussian random process with E{w kl (t)wkl (u)} = σ2 wδ(t u), where σ w is a constant, and δ(t) is a unit impulse function. The noise components are spatially white, such that E{w kl (t)wk l (u)} = 0 if l l or k k. Since scaling the observations changes nothing, we set σw 2 = 1 without loss of generality. When the antennas are separated widely enough such that they are in different target beamwidths, the reflection coefficients for different transmit-receive paths are independent. Denote the complex reflection coefficient for the kl-th path by ζ kl, which is assumed to be a Gaussian random variable with variance σkl 2 which remains constant over the observation interval. We assume the values of σkl 2 for various l, k are known and finite, possibly calculated from the known position probed for a target. Thus, under the target present (H 1 ) hypothesis, the received signal at receive antenna k due to the transmission from transmit antenna l is modeled as [10] H 1 : r kl (t) = Eζ kl s l (t τ kl )e j2πf klt + w kl (t). (10) Under the target absent (H 0 ) hypothesis, only noise is received so that H 0 : r kl (t) = w kl (t). (11) The Neyman-Pearson optimum hypothesis test for H 0 (no target) versus H 1 (target with position and velocity [x, y, v x, v y ])

4 is to compare the overall LLR [10] A. Effects of Varying Signal-to-Noise Ratio (SNR) M σkl 2 L k = E 2 While Theorem 2 implies only a single transmission is σ 2 l=1 kl E + 1 r kl (t)s l (t τ kl )e j2πfklt dt needed at infinite SNR, the results in Table 1 demonstrate this occurs at finite SNR. Table I presents the minimum SNR, (12) denoted SNR min, for which only one transmission is needed to a threshold set to fix the false alarm probability, where r kl (t) to halt transmissions, using Algorithm 1, for various number represents the actual observed received signal. Note that the of transmitters (M) and receivers (N). Table 1 shows that L k > 0 in (12) so it satisfies the assumptions made previously. increasing M generally reduces SNR min. This reduction is The 2 terms take on a chi-square distribution as described substantial when N is small: For N = 2, SNR min is almost in [11], and we set the threshold according to the following halved when M is increased from 2 to 6. On the other hand, equation: increasing N initially decreases SNR min but later increases Eσkl 2 SNR min. This appears to result from the initial increase in τ = 2(Eσkl 2 + (1 P 1)F 1 χ 2 F A ) (13) performance due to the increase in diversity. However this 2MN benefit quickly decays with N and hence only overcomes the increase in τ with N (see (13)) for small N. where F 1 denotes the inverse cumulative distribution χ 2 2MN function of a chi-square distribution with 2M N degrees of freedom, and P F A denotes the desired probability of false alarm. Next we show that for the MIMO radar problem considered, SNR is a distance measure s of the type described in Assumption 3, where we define SNR= σkl 2 E/σ2 w. Due to the normalization of the noise variance, SNR is reduced to σkl 2 E. Under H 1, if we plug (10) into (12), we can rewrite L k as L k = M l=1 σ 2 kl E σ 2 kl E + 1 Eζkl + z kl 2, (14) where z kl = w kl(t)s l (t τ kl)e j2πfklt dt denotes the output of the matched filter with noise as an input. Since we assume σkl 2 are known and finite, for any nonzero realization of ζ kl and z kl we have L k as SNR. Hence, for any finite τ, lim Pr(L k > τ H 1 ) = Pr( > τ H 1 ) = 1. (15) SNR That is, Pr(L k < τ H 1 ) 1 as SNR. This justifies Assumption 3 for the proposed MIMO radar system where SNR is considered a distance measure. In this system, we order the transmissions of the N receivers to the fusion center according to the approach described in Algorithm 1. We begin summing the LLRs at these receivers in order until either the stopping rule (1) or (2) is satisfied, then we decide for H 1 or H 0 respectively. V. SIMULATION RESULTS In this section, we present some simulation results to show savings achieved for the example system described in Section IV. In our simulations, we set σkl 2 =1 for all k, l for simplicity. Assume the target is located at the coordinates (150m, 127.5m) and is moving under the velocity vector (50m/s, 30m/s). Antennas are located along the circumference of a circle of radius 7000m centered at the origin. The transmit antennas are equidistant from each other, and so are the receive antennas. The carrier frequency is set to 1 GHz. With these assumptions, we discuss the effects of changing parameters in the MIMO radar system on performance and savings. TABLE I MINIMUM SNR UNDER ALGORITHM 1 WHERE ONLY ONE TRANSMISSION IS REQUIRED TO TRIGGER ONE OF THE TWO STOPPING CONDITIONS WHEN P FA = SNR min (db) N = 2 N = 10 N = 20 N = 30 N = 40 N = 50 M = M = M = M = M = B. Effects of Varying Probability of False Alarm (P FA ) Let us now consider the number of transmissions saved for different probabilities of false alarm, P FA. Assume H 0 is true. The number of receivers is N = 10 and the number of transmitters is M = 2, 4, or 8. We employ Algorithm 1 to halt transmissions and make detection decisions. In Fig. 1, each of the graphs represents the resulting average percentage savings, Ns /N 100%, as a function of P FA, where N s denotes the average of N s computed by a Monte Carlo simulation. We see that for P FA = 1, Ns /N = 90%. That is to say, Ns = 9 = N 1 transmissions are saved, which agrees with Theorem 4. For 0 < P F A < 1, Ns is generally smaller than N 1 and the exact value of Ns depends on P FA. Let P FAC (M, N) denote the minimum of the corresponding curve for a plot similar to that in Fig. 1 when we employ M transmit antennas and N receive antennas. The difference in the behavior for P FA above and below the minimum P FAC (M, N) is explained by looking at the proofs of Theorems 3 and 4. Note that in Theorem 3, for small P FA, the stopping condition in (2) applies. On the other hand, in Theorem 4, for large P FA, the stopping condition in (1) applies. Thus P FAC (M, N) denotes the point where we switch from one stopping condition dominating to the other dominating. The results in Fig. 1 also show that Ns approaches N 1 as P FA approaches zero, which agrees with Theorem 3. A similar analysis under H 1 is shown in Fig. 2, assuming SNR= 10 db and here P FAC (M, N) takes on a much smaller value. For convenience we do not show these small values of P FA.

5 Average Transmissions Saved (%) H0, M=2 H0, M=4 H0, M= P FA Fig. 1. Average transmissions saved as a function of P FA for different numbers of transmitters in the absence of a signal (H 0 ). In the above setup, N = 10. Average Transmissions Saved (%) H1, M=2 H1, M=4 H1, M= P FA Fig. 2. Average transmissions saved as a function of P FA for different numbers of transmitters in the presence of a signal (H 1 ). In the above setup, N = 10 and SNR= 10 db. C. Effects of Varying Number of Transmitters (M) and/or Receivers(N) As shown in Fig. 1, where H 0 is assumed to be true, the average number of transmissions saved decreases with M for any fixed P FA. In this case, increasing M leads to an increase in τ for any given P FA as per (13), while no diversity can be exploited in this noise-only scenario. We also notice that P FAC (M, N) increases as M increases. On the other hand, as shown in Fig. 2, when H 1 is assumed to be true, the average number of transmissions saved increases with M for any fixed P FA. This is attributed to the extra diversity gain obtained by increasing M. Similar results were obtained for cases with a different N, with the same conclusions. collected and transmissions can be halted, resulting in savings. We demonstrated that great savings in transmissions can be achieved while the obtained detection performance remains the same as if all sensors have transmitted. Under the Neyman- Pearson criterion, we showed that large savings (only a single transmission is needed) can be achieved when the false alarm probability approaches zero or one. We applied these results to the noncoherent MIMO radar system and obtained numerical results which are explained by our theoretical analysis. We also analyzed the effects of varying SNR and the number of transmit and receive antennas on the average percentage savings. REFERENCES [1] I. F. Akyildiz, W. Su, Y. Sankarasubramaniam, and E. Cayirci, A survey on sensor networks, IEEE Commun. Mag., vol. 40, no. 8, pp , Aug [2] B. M. Sadler, Fundamentals of energy-constrained sensor network systems, IEEE Aerosp. Electron. Syst., vol. 20, pp. 17-5, Aug [3] C. Rago, P. Willett, and Y. Bar-Shalom, Censoring sensors: A lowcommunication-rate scheme for distributed detection, IEEE Trans. Aerosp. Electron. Syst., vol. 32, pp , Apr [4] S. Appadwedula, V. V. Veeravalli, and D. L. Jones, Energy-efficient detection in sensor networks, IEEE J. Sel. Areas Commun., vol. 23, pp. 693C702, Apr [5] N. Patwari and A. O. Hero, and B. M. Sadler, Hierarchical censoring for distributed detection in wireless sensor networks, in IEEE International Conference on Acoustics, Speech, and Signal Processing, (ICASSP), pp. 848C851, [6] S. Marano, V. Matta, P. Willett, and L. Tong, Cross-layer design of sequential detectors in sensor networks, IEEE Trans. Signal Process., vol. 54, no. 11, pp. 4105C4117, Nov [7] J. Lunden, V. Koivunen, A. Huttunen, and H. V. Poor, Censoring for collaborative spectrum sensing in cognitive radios, in Proc. 41st Asilomar Conf. Signals, Syst., Comput., Pacific Grove, CA, Nov. 4-7, [8] R. Jiang, Y. Lin, B. Chen, and B. Suter. Distributed sensor censoring for detection in sensor networks under communication constraints, in Proc. Asilomar Conference on Signals, Systems, and Computers, Pacific Grove, CA, Nov [9] R.S. Blum and B.M. Sadler, Energy Efficient Signal Detection in Sensor Networks Using Ordered Transmissions, IEEE Trans. on Sig. Proc., vol. 56, no. 7, pp , Jul [10] Q. He, R. S. Blum, and A. M. Haimovich, Non-coherent MIMO Radar for location and velocity estimation: More antennas means better performance, IEEE Transactions on Signal Processing, vol. 58, no. 7, pp , Jul [11] E. Fishler, A. Haimovich, R. S. Blum, L. J. Cimini, D. Chizhik, and R. A. Valenzuela, Spatial Diversity in Radar - Models and Detection Performance, IEEE Trans. on Sig. Proc., vol. 54, no. 3, pp , Mar [12] H.V. Poor, An Introduction to Signal Detection and Estimation, 2nd ed. New York: Springer-Verlag, VI. CONCLUSION In this paper, we consider the detection of a target of interest using ordered transmissions in the case of noncoherent signal detection where the log-likelihood ratio is nonnegative. We presented an algorithm where sensors can operate in an autonomous way and determine whether or not to send their data to the fusion center until enough data has been