The Parallel Sensor System with an Additional Broadcast Sensor

Now, we temporarily turn our attention to a system with a different configuration from the parallel system in the previous sections, that is, a parallel sensor system with n−1 “ordinary”

sensors and an additional special sensor. For convenience, we will index the special sensor as the n-th sensor. In operation, the broadcast sensor will broadcast its local decision to all other sensors and the fusion center before each of them makes its own decision. More precisely, the broadcast sensor makes its decision u_n based on its own observation X_n first, and then sends u_n to the fusion center and the remaining n − 1 ordinary sensors. The i-th (ordinary) sensor afterwards makes its decision u_i based on its own observation X_i and the received u_n, and then conveys u_i to the fusion center individually. Once all of {u_i}ⁿ_i=1 are received, the fusion center performs a likelihood ratio test of (u₁, . . . , u_n−1, u_n), and decide whether the hypothesis H1 or the hypothesis H0 is true. In subsequent discussions, we restrict ourselves to the special case that the n − 1 “ordinary” sensors are all identical, and only the broadcast sensor can have a different local decision rule.

It can be shown that the likelihood ratio test of received (u1, . . . , un−1, un) at the fusion center still results in a majority voting fusion rule, i.e., a k-out-of-(n − 1 + m) fusion rule with the broadcast sensor has m ballots, while each of other “ordinary” sensors has only one ballot. For conciseness, we will refer the conventional parallel n-sensor system as system Γ_n and the system described above (with the identical ordinary sensors) as system Ξ_n hereafter.

Before we research on the general Ξ_n system, let us take a look at the simplest kind of it, i.e., system Ξ₂. It turns out that system Ξ₂ is equivalent to the decentralized 2-sensor tandem (serial) system in literature. Since the only non-broadcast sensor in system Ξ₂ has acquired all necessary information in making its own decision, we can just let the first sensor in Ξ₂ be integrated with the fusion center, and take u₀ = u₁.

In the following, we shows that for the classification of exponential sources problems, the optimal serial two-sensor strategy is to adopt identical local decision rules for both sensors, and an OR fusion rule at the first sensor.

4.4.1 The Serial Two-sensor System

A serial two-sensor system operates equivalently as system Ξ2. The second sensor makes its decision u₂ according to its observation X₂, and then conveys u₂ to the first sensor. The first sensor then makes the overall decision based on the received u₂and its own observation X₁. It is known that for the tandem configuration of two-sensor network, the optimal local decision rules are [32] that for the first sensor, two local likelihood ratio thresholds are required (one for u₂ = 0 and the other for u₂ = 1), while for the second sensor, only one the local likelihood ratio threshold is sufficient.

Denote the LLRT threshold of the second sensor by η. Let the LLRT threshold of the first sensor for u₂ = 0 as θ₀, and that for u₂ = 1 as θ₁. Then, the probability of error can be

Equating the above derivatives with zero, we obtain the necessary conditions for the optimal error probability as

ηP_D(θ₁) − P_D(θ₀)

P_F(θ₁) − P_F(θ₀) = 1 − r

r , (4.3)

θ₁P_D(η)

PF(η) = 1 − r

r , (4.4)

and

θ0

1 − P_D(η)

1 − P_F(η) = 1 − r

r . (4.5)

Since for the ROC curve, ^P_P^D(η)

F(η) > η > ^1−P_1−P^D(η)

F(η), we have θ₁ < ^P_P^{D(θ1)−PD(θ0)}

F(θ1)−PF(θ0) < θ₀.

Let us take a look at two extreme cases, namely, θ₀ = ∞ and θ₁ = 0. It turns out that¹ the cases of θ₀ = ∞ and θ₁ = 0 are equivalent to that the first sensor makes a local decision u₁ according to its own observation X₁ only, and then applies the AND and OR fusion rules, respectively, to decide the overall output u₀.

Lemma 4.2. For the serial two-sensor system with θ₀ = ∞, the optimal strategy is to let the first and the second sensors make their local decisions u₁ and u₂ according to the LLRTs of their own observations X₁ and X₂, respectively, and then apply the AND fusion rule at the output of the first sensor, i.e., u₀ = u₁⊗ u₂.

Proof.

P_e(η, θ₁, ∞) = rP_D(η)(1 − P_D(θ₁)) + (1 − r)P_F(η)P_F(θ₁)

+ r(1 − PD(η))(1 − PD(∞)) + (1 − r)(1 − PF(η))PF(∞)

= rP_D(η)(1 − P_D(θ₁)) + (1 − r)P_F(η)P_F(θ₁) + r(1 − P_D(η))

= r(1 − P_D(η)P_D(θ₁)) + (1 − r)P_F(η)P_F(θ₁), where we have used a property of the ROC: PD(∞) = PF(∞) = 0.

Lemma 4.3. For the serial two-sensor system with θ₁ = 0, the optimal strategy is to let the first and the second sensors make their local decisions u₁ and u₂ according to the LLRTs of their own observations X₁ and X₂, respectively, and then apply the OR fusion rule at the output of the first sensor, i.e., u₀ = u₁⊕ u₂.

1Here, with a slight abuse of notations, we let the intermediate product, i.e., the result of the LLRT at the first sensor, be denoted by u1, and let the ultimate output of the first sensor be denoted by u0).

Proof.

P_e(η, 0, θ₀) = rP_D(η)(1 − P_D(0)) + (1 − r)P_F(η)P_F(0)

+ r(1 − P_D(η))(1 − P_D(θ₀)) + (1 − r)(1 − P_F(η))P_F(θ₀)

= (1 − r)PF(η) + r(1 − PD(η))(1 − PD(θ0)) + (1 − r)(1 − PF(η))PF(θ0)

= r(1 − P_D(η))(1 − P_D(θ₀)) + (1 − r)(1 − (1 − P_F(η))(1 − P_F(θ₀))), where we have used a property of the ROC: P_D(0) = P_F(0) = 1.

In both of the above cases, the serial two-sensor systems function exactly like the parallel two-sensor system with corresponding fusion rules. In general, the optimal serial two-sensor system uses two finite and nonzero LLRT thresholds at the first sensor, and therefore does not necessarily reduce to some equivalent parallel two-sensor system. In the next theorem, we show that for the considered classification problem of exponential sources, the optimal serial two-sensor system is one of the above extreme cases. More precisely, the optimal serial two-sensor system is equivalent to the optimal parallel two-sensor system.

Theorem 4.2. For the classification problem of exponential sources, the optimal strategy for the serial two-sensor system is to let the first and the second sensors make their local decisions u₁ and u₂ according to the LLRTs of their own observations X₁ and X₂ with the two equal thresholds θ₀ and η, respectively, and then apply either AND or OR fusion rules at the output of the first sensor.

Proof. Firstly, define a function B(λ) = _PD(λ)^λ

PF (λ)

. Then, (4.3) becomes

B(η)PD(η) P_F(η)

PD(θ1) − PD(θ0)

P_F(θ₁) − P_F(θ₀) = 1 − r r . Combining the above equation with the (4.4), we establish

θ₁ = B(η)P_D(θ₁) − P_D(θ₀) P_F(θ₁) − P_F(θ₀).

By using the identity θ1 = B(θ1)^P_P^D_F(θ^(θ₁¹⁾₎, the above equation can be written as

F(λ) increases monotonically with respect to λ, we have θ₁ ≥ θ₀,

which contradicts the aforementioned proposition: θ0 ≥ θ1. Hence, _B(θ^B(η)₁₎ < 1. Yet, for the classification of exponential sources problem, we have B(η) = B(θ₁) = ξ; thus, the optimal (η, θ₁, θ₀) must lie on the boundary, i.e., the two extreme cases.

Remark 4.1. For the classification problem of the additive Gaussian sources, B(λ) is a monotonically increasing function of λ; thus, η < θ1 < θ0.

Corollary 4.1. For the classification problem of exponential sources, the optimal perfor-mance of the serial two-sensor system is equal to the optimal perforperfor-mance of the parallel two-sensor system.