Summary

We arrive at the same conclusion as (3.13), which suggests the simplified ML estimate is actually nothing but a modification from the natural voting scheme.

3.6 Summary

An accurate approximation formula (3.11) to the true ML solution is derived and it has a natural interpretation related with the straightforward voting scheme. This formula is more tractable in that it is an affine function in the received signal r ’s. To _i accomplish the GLRT test, the FC then adopts this simplified ML estimate in the LRT statistic and compares it with a predetermined threshold. However even with the simplified formula, the achievable detection performance of GLRT, in particular the impact from channel impairments, remains quite difficult to characterize especially when the number of sensors is finite. It motivates us to propose an alternative detection rule which can exploit the affine nature of π and result in analytic study of the link ˆ₁ error effects.

Chapter 4

Proposed Detection Method

4.1 Proposed Simple Detection Rule

It is shown that GLRT is merely a heuristic approach, nor does it involve any optimality criteria in deriving this rule. Even if the proposed ML approximation is simple, it helps little after being adopted into the GLRT statistic, which motivates another simpler fusion rule that can benefit from the ML approximation. Note that at receiving r ’s, the FC is actually applying the ML estimate ˆ_i π for Pr{s_i =1}. Because Pr{s_i =1 |H₁}=π₁ and Pr

{

s_i =1 |H₀

}

=π₀ , ideally, πˆ≈ π₁ when

H occurs and 1 πˆ≈π₀ when H occurs. A simpler and more natural alternative is ₀ to obtain ˆπ first and then compare it with the known p_f =π₀. More specifically, the FC can be designed to make the following decision

0 0

where γ was the predetermined threshold.

The main advantage of the proposed decision rule (4.1) is that, unlike the GLRT in (3.4), the test statistic in (4.1) is affine in the estimate ˆπ , and hence is affine in the received data r ’s. The proposed method also directly utilizes the parameter that really _i reflects the different hypotheses. Based on these attractive features, performance can be characterized analytically as shown below.

4.2 Performance Analysis

and T is substantially the equivalent test statistic. Since r_i ∈{ }0,1 , T assumes a finite number of alphabets, which are to be specified. First, for each 0≤ ≤k N , define

I I I IC to be the collection of all distinct k-element subsets of {1, , }N , where C_k^N N!/[ !(k n−k)!] and I⁽⁰⁾ ={ }φ . Each element in I^{( )k}

maps to a possible value of T , thus for each 0≤ ≤k N , let’s define S be the set ^{( )k}

consisting of all possible values of T when k sensors are active, that is,

{ } {

^{( ) ( ) ( )}

}

To facilitate further investigation, assume without loss of generality that, for each 1≤ ≤k N , the elements in S are arranged so that ^{( )k} ₁^{( )} ₂^{( ) ( )}_N

( ) ( 1)

By definition of the detection probability _{0 1}

1

Similarly, for the false-alarm probability _{0 0}

1 associated lower bound can be obtained as

( )L

Observe that the performance bounds in (4.8) and (4.10) depended on the link error probability ε ’s. This allows for further discussions on the impact of the channel _i effect on the detection performance, as in the next sections.

The performance a thirty-sensor WSN is simulated as the blue ROC curve in Figure 2.1. The channel model is the same as that in Chapter 4.4 and the average cross-over probability is 0.023, which is small enough to validate the bound derivation.

The local detection and false-alarm probability are 0.6 and 0.4, respectively. The proposed bound is plotted as the red ROC curve. It can be seen that the proposed bound is tight enough to evaluate the performance of the proposed fusion rule.

0 0.2 0.4 0.6 0.8 1

Figure 4.1: Accuracy of the proposed performance bound

4.3 Impact due to Channel Effects

The performance formulas in (4.8) and (4.10) remain non-linear functions of ε ’s. _i It is still difficult to assess the effects of non-ideal communication channels.

Remember that the proposed ML solution is derived under the high-SNR assumption.

By further exploiting the assumption that ε ’s are small, (4.8) and (4.10) can be _i simplified considerably, as in the next lemma.

Lemma 4.1: For small ε ’s, _i

where

Based on (4.8) and (4.10), (4.11) and (4.13) are obtained by neglecting the high-order terms of ε ’s and then some manipulations. The binomial coefficient comes from the _i

summation of all possible combinations given k sensors are active. □ While the bounds (4.8) and (4.10) are quite complicated functions of ε ’s, in the _i high-SNR regime, the detection performance is closely related to the summed cross-error probabilities, namely

∑

= does not explicitly indicate how the variation of

1 performance. However, under some reasonable assumptions on p and _d p , this work _f proves that minimizing

∑

= guarantees a better detector performance in terms of the ROC curve, as precisely stated in the following theorem.

Theorem 4.2: Assume π₀ <0.5<π₁. Given a fixed false-alarm probability P , let _f

( )L

Pd and P ′_d^{( )L} be two detection probability lower bounds associated with two different summed link errors

1

Proof: See Appendix A

Theorem 4.2 suggests that, when the condition π₀ <0.5<π₁ is fulfilled, the global detection performance improves if the summed link error rate can be made small. The assumption π₀ <0.5<π₁ is actually not too demanding for any reasonable sensors. Inspired by Theorem 4.2, a sensor power allocation scheme for enhancing the global detection performance is developed next.

4.4 Proposed Power Allocation Strategy

Recall that the ith sensor transmits s = when it claims _i 1 H and transmits ₁ nothing when it claims H . Namely, the sensors report their one-bit decisions using ₀ on-off keying to conserve energy. After incorporating the power allocation strategy, the ith sensor actually transmits s ∈_i { }0,1 multiplied with an amplitude factor a , _i

which is to be designed later on, and the corresponding power allocated to this sensor is p_i =a_i². Assume the communication channel between the ith sensor and the FC is flat and Rayleigh distributed with the current channel coefficient h , with the average _i power normalized to 1. Knowing these current channel coefficients, the FC can then apply the coherent detection and the received signal y from the ith sensor could be _i described by a commonly used discrete-time baseband model

, 1

i i i i i

y =h a s +n ≤ ≤i N (4.15)

where n is the zero-mean Gaussian noise of variance _i N₀ 2. The corresponding

cross-over probability of the ith link is then

2

Under a total transmit power budget P , the optimization problem can be formally stated as

The optimization problem of the form (4.16) has been addressed in the context of MIMO wireless communications [19, 20]. Note that the cost function and the inequality constraint are convex and the equality constraint is linear. The optimization problem is thus convex and the Kuhn-Tucker conditions are necessary and sufficient conditions for finding { }p_i^{∗ N}_i=1.

Define the Lagrangian function as:

2

(3.) p_i^∗ ≥ , 0 1≤ ≤i N (4.) u ≥ , _i 0 1≤ ≤i N (5.) u p_{i i}^∗ = , 0 1≤ ≤i N Condition (1.) turns to

, 1

To clarify further, demonstrated below is the flow chart of the power allocation procedure:

Figure 4.2: Flow chart of power allocation strategy

Start

( )0 0, λ2 λ=λ < Δ =λ

Calculate the Transmit Power for Each Sensor

The Current Power Used

1

4.5 Summary

To solve the problems that GLRT is not optimal and the GLRT statistic is too complicated, a simpler and more straightforward fusion rule is proposed. The proposed fusion rule conserves the affine properties of the approximated ML estimate and is in turn affine in the received signals, which enables the performance analysis. A tight bounds of P and _d P are proposed and then simplified under the assumption that _f the SNR is moderately high. These bounds are derived not only to evaluate the performance, but also to facilitate investigation of the channel effects, which is accomplished by locating the simple channel-related term

∑

^N_i=1ε_i ^{in the} approximation formulas. However, it remains unclear how to improve the ROC curve because both of these approximations remain complicated functions of

∑

^N_i=1ε_i ^. Despite of the fact, it is proved that under some more reasonable assumptions, minimizing

∑

_i^N₌₁ε_i guarantees a better performance. At the end, a power allocation strategy aiming at minimizing

∑

^N_i=1ε_i is proposed. Simulations of this work are demonstrated in the next chapter.

Chapter 5

Computer Simulations and Discussions

5.1 Computer Simulations

The performances are simulated for the proposed detection rule with and without transmit power allocation, and then are compared with those for GLRT. The simulations of the LRT performance are also provided, which serve as the upper bound of any possible detector designs. In all simulations, the channel coefficients are assumed flat and Rayleigh distributed with the average power normalized to 1.

Figure 5.1 shows the ROC curve for an WSN of twenty sensors with uniform local detection probability p =_d 0.6 and local false-alarm probability p =_f 0.4. The noise power is N =₀ 0.05. The blue line is the performance of the LRT detector with power allocation proposed in this work. The black solid and the red dash curves are the ROC curves of the proposed fusion rule with and without power allocation, respectively. The black solid and the red dash curves with circles are the ROC curves of GLRT with and without power allocation designed for the proposed fusion rule, respectively.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.3

0.4 0.5 0.6 0.7 0.8 0.9 1

Pf

Pd

ROC curve

LRT

Proposed scheme with power loading Proposed scheme without power loading GLRT with power loading

GLRT without power loading

Figure 5.1: ROC curve of twenty-sensor network

From the above figure, the proposed fusion rule without power allocation (red dash line) outperforms GLRT in P by 10 to 15 percent given a value of _d P small _f enough to have practical interests. After power allocation, the performance of the proposed fusion rule even approaches the optimal LRT bound (blue solid line).

Figure 5.2 and Figure 5.3 are based on similar environment settings as Figure 5.1, but the number of sensors changes to 30 and 50, respectively. As can be seen, similar results appear but the extent of increase in P becomes smaller. The phenomenon _d will be described shortly in this section.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Proposed scheme with power loading Proposed scheme without power loading GLRT with power loading

GLRT without power loading

Figure 5.2: ROC curve of thirty-sensor network

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Proposed scheme with power loading Proposed scheme without power loading GLRT with power loading

GLRT without power loading

Figure 5.4 illustrates the influence of the number of sensors on the relative increase in P of the proposed fusion rule with power allocation over GLRT, given the global _d

f 0.1

P = . As the number of sensors increases, the detector is expected to perform better because it has more information-bearing reports to make a correct decision.

Asymptotically, when the number of sensors grows to infinity, all detector designs that use the information in the received signals have so much information available that their ROC curves approach to the left-upper corner, same for GLRT. That is why the improvement of the proposed fusion rule over GLRT diminishes as the number of sensors increases, and the proposed fusion rule improves P significantly for smaller _d number of sensors.

20 30 40 50 60 70 80

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16

Number of sensors

Improvement in Pd

Figure 5.4: Relative increases in P over GLRT vs. number of sensors _d

Figure 5.5 demonstrates the relations of the relative increase in P of the _d proposed rule with power allocation over that without power allocation versus the average transmit power per sensor. The P is fixed to 0.1, and other environment _f settings remain the same, i.e., p =_d 0.6, p =_f 0.4 and N =₀ 0.05. Interestingly, the improvement in P is smaller when the average transmit power is too high or too _d low, and there is a peak improvement when the average transmit power is equal to about 0.8.

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 0.03

0.032 0.034 0.036 0.038 0.04

Average power per sensor

Improvement in Pd

Figure 5.5: Relative increase in P from power allocation _d vs. average power for N =₀ 0.05

2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 0.03

0.032 0.034 0.036 0.038 0.04

Average power per sensor

Improvement in Pd

Figure 5.6: Relative increase in P from power allocation _d vs. average power for N =₀ 0.15

Figure 5.6 also demonstrates the relations of the improvement in P from power _d allocation versus the average transmit power per sensor, but this time the noise variance increases by three times, i.e., N =₀ 0.15. Note that the peak of the improvement in P now moves to the average power of about 2.4, which is three _d times larger than the peak average power in Figure 5.5. The reason for the relation between Figure 5.5 and Figure 5.6 is explained as follows. In Equations (4.11) and (4.13), the channel effects only go into the value of ε ’s. Given a realization of _i h ’s, _i

(

^{2 2 0}

)

i Q h pi i N

ε = only depends on p and _i N . If the average power ₀ p and _i

N keeps the same ratio, they would produce the same value of 0 ε ’s. The cross-error _i probabilities before and after power allocation will be the same for all the same ratio of

p to i N . The effects of fixing ₀ p and changing _i N match that of fixing ₀ N and ₀ changing p . Specifically, the improvement in _i P at average power 0.8 in Figure 5.5 _d matches that at average power 2.4 in Figure 5.6.

To illustrate why too high or too low average power leads to smaller improvement in P , two ROC curves are illustrated for these two extreme scenarios. Figure 5.7 _d shows the case where the average transmit power is 0.1, which is extremely small, and

Figure 5.8 shows the case where the average transmit power is 3, which is comparatively high.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Pf

Pd

ROC curve

LRT

Proposed scheme with power loading Proposed scheme without power loading

Figure 5.7: ROC curve of thirty-sensor network with average power 0.1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Proposed scheme with power loading Proposed scheme without power loading

Figure 5.8: ROC curve of thirty-sensor network with average power 3

As expected, when there is no power allocation, the performance of the proposed fusion rule with higher transmit power is already better than that with lower transmit power. Also note that in the case of higher average power, the gap between the proposed fusion rule without power allocation and LRT (with power allocation) is already smaller than that in the low-average-power case. It is reasonable because when the average transmit power is high, even if there are no power allocations, the ε ’s are _i already quite small. Power allocation cannot make the performance of the proposed detector better than that of LRT and thus the increase in P is small. It can also be _d explained as follows: the improvement in P becomes small because the power _d allocation changes ε ’s from already small ones to smaller ones. _i

When the average transmit power is small, it is seen from Figure 5.7 that the proposed fusion rule without power allocation performs much poorly than LRT with

power allocation. In contrast to high-average-power case where the improvement in P is somewhat “upper bounded” by LRT, the low average transmit power deprives d

the FC of the margin to tackle with the channel impairments. Thus, the ε ’s decrease _i insignificantly after power allocation, which make the improvement in P small. _d

5.2 Discussion on Proposed Method

Simulations indicate that the proposed detection rule outperforms GLRT significantly, in terms of ROC curve. After the transmit power allocation, the ROC curve of the proposed rule moves to the left-upper corner and even approaches the LRT bound. It means that the proposed method with power allocation is nearly optimal.

Also, we can see that for any given value of P , the improvement in _f P has a peak _d at a certain average transmit power, termed p_peak . Average transmit power lower than

ppeak is too small to have large improvement in P , and the ROC curve of the _d proposed method after power allocation still has gap from that of LRT. Average transmit power greater than p_peak is large enough to raise the ROC curve of the proposed fusion rule to the LRT’s ROC curve. But as the performance gap between LRT and the proposed fusion rule without power allocation becomes less, the extent of improvement is thus less. The ROC curve of the proposed rule with power allocation is somewhat “upper bounded” by that of the optimal LRT. Although p_peak generates the peak improvement, it is not really the so-called optimal transmit power from the global perspective. Indeed, this work suggests using the average power greater than p_peak because the system performance can approach that of the optimal LRT, but note that

corner. Since the performance of our work with power allocation approaches that of LRT, the resulting P is still increasing with the total power, although the extent of _d improvement in P is decreasing. After all, designers can choose a certain transmit _d power larger than p_peak to obtain the desired optimal ROC curve.

To find the reason GLRT performs poorly, we first notice that in LRT all the parameters in p r H

(

; 1

)

and p r H

(

; 0

)

are assumed known by FC, which can directly adopt these parameters in calculating the likelihood ratio. Most importantly, these values of parameters do not depend on whether the underlying situation is H or ₀ H . ₁ However in GLRT, we only obtain the formula for estimating these parameters and these estimates are different in H and ₀ H . Take the system in this work for example, ₁ although GLRT asks for Pr

{

s =_i 1

}

given that H is true, the derived ML estimate ₁ approximates π only when ₁ H is actually happening. In other words, it is not ₁ possible to obtain the estimate of π if it is ₁ H that is happening. In such case, the ₀ FC can only obtain Pr

{

s =_i 1 |H , which is close to ₀

}

π . Consequently, although ₀ GLRT takes a similar form as LRT, it does not actually behave the same. Even in asymptotic case, i.e. through extensive computer simulations, there is still a gap between the ROC performance of LRT and GLRT.

5.3 Summary

Simulations indicate that the proposed simple fusion rule with power allocations outperforms GLRT rule significantly, especially when the number of sensors is small.

With average transmit power moderately large, the performance of the proposed fusion rule with power allocations can even approach the ROC curve of the optimal clairvoyant LRT detector. When the transmit power is too low, power allocation does

not improve much and there is still a gap from the optimal ROC curve of LRT. When the transmit power is larger than a threshold, power allocations can raise the ROC curve of the proposed fusion rule to that of LRT. However, the relative increase in P _d diminishes because the performance with power allocation is upper bounded by the ROC curve of LRT.

Chapter 6

Conclusions and Future Works

In the beginning, this thesis traces some important developments of distributed detection systems, and a popular system model called canonical distributed detection system is introduced, where sensors transmit their reports reliably and directly to FC through parallel channels. Two problems in canonical distributed detection systems are to be solved: the fusion rule design and the signal processing algorithm at local sensors, and our work focuses on the first one. In the WSN cases where channels cannot be assumed reliable anymore, following the important concept that fusion rule design and the channel effects should be considered jointly, this thesis surveys many works of channel-aware fusion rule design and finds out that most of these works do not address the problems where sensor performances are not known to FC. If the sensor performances are necessary in fusion rule design, estimations must be conducted at FC.

Although GLRT can be applied to tackle these problems, there are rooms for improvement because firstly, the GLRT statistic is too complicated and secondly, GLRT does not guarantee the optimal performance.

In Chapter 3, the system model in our work is described in detail. The probability distributions of the received signal at FC are derived, which are indispensable in many fusion rule design, including GLRT. The formula of GLRT statistic is then derived for our system, and as is mentioned above, it is too complicated to analyze; moreover, the ML solution in the GLRT statistic is also complicated. This thesis then proposes an accurate approximation of the ML solution in high SNR, which is an affine function in the received signals, and can be reasonably interpreted as a modification of the voting

scheme. However, even after replacing the simplified ML solution in the GLRT

scheme. However, even after replacing the simplified ML solution in the GLRT

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