Interestingly we find that neither the. PAC (parallel-access channel) nor the MAC (multiple-access channel) based transmission scheme is always optimal. Our.
DECENTRALIZED DETECTION USING NONCOHERENT MODULATION: IS MAC THAT GOOD? Christian R. Berger, Marco Guerriero, Shengli Zhou, and Peter Willett University of Connecticut ECE Department Storrs, CT 06269 USA ABSTRACT In this paper we study the decentralized detection problem in wireless sensor networks when the sensors’ observations have to be communicated to a fusion center through a fading channel subject to an energy constraint. We analyze the noncoherent case based on a simple on/off scheme combined with optimal sensor “censoring”. Interestingly we find that neither the PAC (parallel-access channel) nor the MAC (multiple-access channel) based transmission scheme is always optimal. Our results show that in large sensor networks, for reliable observations at the sensors an optimal limited number of sensors should be queried using PAC, while for weak sensors’ observations the MAC scheme is preferrable. 1. INTRODUCTION In a wireless sensor network (WSN), observations often are transmitted to a fusion center (FC) for central processing. For simplicity the usual architecture assumes that each sensor transmits through a parallel access channel (PAC). Recently, it has been suggested to exploit the wireless medium directly as a multiple-access channel (MAC) for decentralized detection, as the FC will observe a superposition of the signals sent by the individual sensors and hence perform fusion automatically. In this context most previous work, has assumed that the FC observes the coherent sum of the signals sent by the sensors [1, 2]. Under this channel model, the received signal-to-noise ratio (SNR) at the FC grows with the number of sensors, due to an array-processing gain, making the noise irrelevant as the number of sensors tends towards infinity. This is clearly not realistic since it assumes phase synchronization between all sensors and the FC. It was actually proven, in [3], for type-based multiple access and later in [4] for decentralized inference in more generality, that if the channel is noncoherent, i.e., the random channel coefficients are zero-mean, the performance is in fact limited by the channel noise, even for an infinite number of sensors, as there is no array-processing gain. As the performance of a traditional PAC based scheme is also noise limited, it is not This work was supported by the Office of Naval Research under contract N00014-07-1-0429 and N00014-07-1-0055.
clear how these different approaches will compare in the noncoherent case. Therefore we are motivated to study noncoherent decentralized detection. We consider the case where distributed sensors make conditionally independent observations. The sensors have to communicate their information to a FC across a fast varying channel, without any knowledge of the channel state, subject to an average energy constraint. We want to analyze and compare the detection performance using parallel or multiple channel access, studying varying channel models, using the simple additive white Gaussian noise (AWGN) channel, and the noncoherent case, for both a fast fading Rayleigh channel and a random phase channel. The rest of the paper is organized as follows: In Section 2 we formalize our problem, in Section 3 we characterize the PAC and MAC performance for noncoherent decentralized detection focussing on Rayleigh fading, in Section 4 we provide simulation results and in Section 5 we conclude our work. 2. SCENARIO DESCRIPTION We are interested in a decentralized detection problem using a wireless sensor network (WSN). In the WSN, N sensors make statistically independent observations conditioned on the hypothesis, denoted as {H0 , H1 }, see Fig. 1. The sensors process their observations to a sufficient statistic yn , n = 1, . . . , N , e.g., the log likelihood ratio (LLR) or a monotonic transform thereof. The yn are distributed as f0 (yn ) and f1 (yn ) conditioned on the two hypotheses respectively. Then, the sensors forward a function U (yn ) of the sufficient statistic to a fusion center (FC), which combines them to reach a decision. In decentralized detection, the added challenge is caused by an imperfect communication channel between the sensors and the FC, as the FC observes the channel output zn and not directly the yn , see Fig. 1. We therefore introduce the following: • Communication Bound: The performance of any decentralized detection setup is always bounded by an equivalent communication problem. This is defined as the case, where all sensors have perfect obervations and the only uncertainty is introduced by the communica-
U
y1 H0 / H1
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(a) PAC
(b) MAC
Fig. 1. Diagram of the general decentralized detection framework for parallel (PAC) and multiple (MAC) channel access. tion channel. • Observation Bound: The detection performance is always limited by the quality and number of available observations. In decentralized detection this bound is achieved when the communication channel is noiseless. 3. DECENTRALIZED DETECTION USING NONCOHERENT MODULATION For “type-based” estimation [3] and later for decentralized inference in general [4], it has been shown that if the channel is noncoherent and the total power is fixed, then the performance does not improve with the number of sensors beyond a certain point, in contrast to the coherent case. Instead of the rate constraint, in practical wireless communication it makes more sense to accurately model the channel’s effects [5,6]. In a PAC, the FC receives noisy versions of the sensors’ transmissions, zn , e.g., an additive white Gaussian noise (AWGN) channel: zn = hn U (yn ) + wn (1) or a noisy superposition in the case of a MAC, z=
N X
hn U (yn ) + w.
(2)
n=1
In this setup, it is most realistic to enforce an average energy usage constraint on the function U (yn ), of the general form1 : (1 − η)E |U (yn )|2 | H0 + ηE |U (yn )|2 | H1 ≤ Eη /N (3) 1 This constraint can cover several scenarios, both Bayesian and NeymanPearson, where η = 0 only constrains the energy usage under H0 , namely P0 , and for a Bayesian formulation the average energy usage, where η = π1 and 1 − η = π0 are the priors of the hypotheses.
The case where hn is distributed zero-mean and the channel state is unknown at the sensor nodes is denoted as noncoherent. Examples are the Rayleigh fading channel, with hn ∝ CN (0, 1), or the unknown phase channel, hn = ejφn , φ ∈ [0, 2π]. Motivated by earlier works [7] we adopt a simple on/off scheme and the FC will base its decision on the variance of the channel, indicating the sum of sensors that detected a target. This on-off keying (OOK) scheme is the obvious noncoherent binary modulation and also links to the “censoring” concept [8,9], as the “off” symbol doesn’t expend any energy resources. Accordingly, we define the function U (yn ) as a simple step function (see Fig. 2), based on an LR test, A, yn ≥ τ U (yn ) = (4) 0, yn < τ. The local detection threshold τ can be chosen according to the censoring concept to minimize utilized energy. At the FC, we assume an energy detector, H1 , ξ > Γ (5) d(ξ) = H0 , otherwise where ξ is the total energy received across all parallel channels. As for the noncoherent case with many sensors an optimal forwarding function U (yn ) is not known, the censoring concept allows us to improve performance to some degree. Due to the censoring approach, we will have to: • jointly choose the optimal τ , A and Γ, in the sense of maximizing detection performance, while adhering to the average energy usage constraint in (3); and • compare to the communication and observation bounds, as the optimal forwarding function is unknown.
U(y)
A0
A τ τ0
y
Fig. 2. A binary function similar to on-off-keying (OOK) performs close to optimal; a threshold test on the likelihood ratio is used as in the “censoring” approach. Note that the censoring level and output amplitude can be traded-off: (τ, A) and (τ 0 , A0 ) may require the same average energy.
We now compare the PAC approach with the MAC scheme. Since the joint optimization of τ , A and Γ is not available in closed form, we will develop expressions for the performance given a certain parameterization. This way the threedimensional parameter space can be efficiently evaluated, and the optimal parameterization can be found numerically.
3.1. PAC Detection Performance In the PAC, each sensor is received separately see Fig. 1(a), we focus on the Rayleigh fading model: zn = hn U (yn ) + wn ,
(6)
with hn ∝ CN (0, 1) and wn ∝ CN (0, N0 ) We compare the total received energy to a threshold; this is not the optimal decision statistic in this case, but has been shown to be close to optimal for low SNR, c.f. [10], ξP AC =
N X
|zn |2 .
(7)
n=1
It can be shown that the probability of detection and probability of false alarm at the FC are given by: N X N k Pd = p (1 − pd )N −k FΞ (Γ|k) k d k=0 N X N k Pf a = p (1 − pf a )N −k FΞ (Γ|k) k fa
(8)
(9)
k=0
R +∞ R +∞ where we define pf a = τ f0 (y)dy and pd = τ f1 (y)dy. The expressions for the conditioned complementary cumulative probability densities FΞ (Γ|k) = Pr(ξ > Γ|k), for k = 1, . . . , N − 1 can be derived based on (7) but they are omitted for conciseness.
We also reproduce the optimal decision statistic, developed in [10], which is the LLR: ! 2 2 N X p1 β2 e−β2 |zn | + (1 − p1 )β1 e−β1 |zn | Λ= log p0 β2 e−β2 |zn |2 + (1 − p0 )β1 e−β1 |zn |2 n=1 (10) 1 1 and β = . In [10], it was shown with β1 = A2 +N 2 N0 0 that for vanishing SNR on the sensor-FC link, A2 /N0 → 0 (β1 → β2 ), this combining rule is equivalent to (7). Additionally it is easy to see that for increasingly reliable observations at the sensors, pd → 1, pf a → 0, the same is true, as (7) is the optimal decision statistic under a regular OOK communication regime. To calculate the communication bound for the PAC scheme, we note that N parallel channels are used, the same transmitted on each them. This can be linked to noncoherent transmit diversity, but with the less common asymmetric OOK modulation. Still, as can be found in standard textbooks [11], noncoherent parallel channels always have a “combining loss”, i.e., after combining the equivalent SNR is less than if all energy had been sent through a single channel. The output of a noncoherent fading channel under OOK modulation is simply χ22N distributed, N −1 X
1 n!
Γ Pd = E /(N η) + N0 η n=0 n N −1 X 1 Γ Pf a = e−Γ/N0 n! N 0 n=0
n
e
− Eη /(NΓη)+N
0
(11)
(12)
3.2. MAC Detection Performance As in Fig. 1(b), the FC observes the superposition of the waveforms: N X z= hn U (yn ) + w (13) n=1
where we assume the same complex Gaussian fading coefficients and noise of variance N0 . Because of the random phase, clearly the received power, ξMAC = |z|2 ,
(14)
is a sufficient statistic and the threshold test in (5) is the optimal decision rule in this case. It can be easily shown that the detection and false alarm probabilities at the FC are given by a mixture of exponentials: N X N k Γ N −k Pd = p (1 − pd ) exp − 2 (15) kA + N0 k d k=0 N X Γ N k Pf a = pf a (1 − pf a )N −k exp − 2 kA + N0 k k=0
(16)
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(a) N = 5, Eη /N0 = 13 dB
(b) γobs = 3 dB, Eη /N0 = 10 dB
Fig. 3. Comparison of receiver operating characteristics (ROC) of MAC and PAC based noncoherent decentralized detection; in case of few sensors with reliable observations, PAC is preferable, while for many sensors with unreliable observations MAC is superior; LR denotes the likelihood ratio based PAC combining rule in (10); η = 0.5. The communication bound in the MAC case is simpler, as the output of the channel will always be complex Gaussian, with variance either N A2 + N0 or N0 . Since in this case A2 = Eη /(N η), the N cancels and we note that the channel output is actually independent of N . This is because the signals emitted by each sensor add up noncoherently, the observed energy is not affected by its split. Therefore the probability of detection and false alarm for the communication bound is identical to the PAC communication bound for N = 1. 4. SIMULATIONS As an example we consider exponential observations at the sensors. With that, given a local SNR γobs at the sensors, yn are exponentially distributed with mean λ0 = 1 and λ1 = − τ (γobs + 1) leading to pf a = e−τ and pd = e γobs +1 . For the observation bound, if the observations were directly available at the FC, the optimal decision statistic (both for PAC and MAC schemes) would be the sum, distributed as χ22N . The probability of detection and false alarm are accordingly, Pd =
N −1 X n=0
Pf a =
N −1 X n=0
1 n!
Γ γobs + 1
1 (Γ)n e−Γ , n!
n
e
Γ obs +1
−γ
,
(17)
(18)
The joint optimization problem, which involves (τ, A, Γ), can be implemented as a simple one-dimensional search for the most efficient censoring rule: for any given local detection threshold τ , an amplitude A can be found to satisfy the energy
constraint, N A2 [(1 − η)pf a + ηpd ] ≤ Eη
(19)
Then varying Γ a receiver operating characterstic (ROC) is generated. The optimal ROC is the convex closure of all ROCs corresponding to a specific local threshold τ . To get some intuition of the performance behavior, we plot the optimal receiver operating characteristic (ROC) for several configurations of N , Eη and γobs in Fig. 3. We include both closed form results based on the energy detector (ED) and Monte-Carlo simulation, where using simulation we can also evaluate the optimal PAC combining rule based on the likelihood ratio (LR). Interestingly we find that neither the PAC nor the MAC based transmission scheme is always optimal. For certain constellations the PAC outperforms the MAC, e.g., in Fig. 3(a) where there are few sensors with high quality observations. On the other hand, for less reliable observations, see Fig. 3(b), but a large number of sensors, the bandwidth-efficient MAC transmission scheme has superior performance. The optimal combining rule based on the likelihood ratio (LR) increases the PAC performance, but shows generally the same trends. To verify these trends, it is necessary to compare the performance of the MAC and PAC schemes over a large range of N and γobs . Therefore we next plot the probability of error. Fig. 4 shows the values for three fixed values of γobs for increasing number of sensors N . Surprisingly MAC and PAC have different trends as N increases: the MAC performance approaches a constant determined by the communication bound, constant in N and γobs ; the PAC performance decreases after exceeding a certain N , which varies with γobs . This shows the varying tradeoff between diversity gain and
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Fig. 4. Comparing the probability of error for the noncoherent Rayleigh fading case, we see that while the MAC performance is ultimately bounded by the communication bound, independently of N and γobs , the PAC achieves its optimal performance for a finite N that varies with γobs ; Eη /N0 = 10 dB and η = 0.5. combining loss, where the effective SNR on the channel is affected by the local pd and pf a . In comparison to the coherent case, where the PAC peformance was constant for large N and improving for MAC, now the peformance of PAC is affected negatively by N over a certain threshold, while the MAC performance is constant for large N . This points to the fact that in large sensor networks, for reliable observations at the sensors a limited number of sensors should be queried using PAC, while for weak observations all sensors should be combined using the MAC approach. 5. CONCLUSION We analyzed the performance of decentralized detection using parallel or multiple channel access for the noncoherent case. We found that for noncoherent modulation, there is no processing gain using the MAC for decentralized detection, but compared to parallel-access channels (PACs) the MAC avoids the noncoherent combining loss. Still the performance of the MAC approach is only of diversity one, as the output of the MAC is approximately a zero-mean complex Gaussian random variable for a large number of sensor. 6. REFERENCES [1] K. Liu and A. M. Sayeed, “Type-based decentralized detection in wireless sensor networks,” IEEE Trans. Signal Processing, vol. 55, no. 5, pp. 1899–1910, May 2007. [2] K. S. Gomadam and S. A. Jafar, “Optimal relay functionality for SNR maximization in memoryless relay networks,” IEEE
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