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Using the procedure stated in section 3.4, we studied the performance of the two-sensor tandem network in the good signal region of the first quadrant for the following cases:

(i) Weaker sensor as the FC (ii) Better sensor as the FC

The performance of tandem network for cases (i) and (ii) listed above, and the perfor-mance of central LRT are listed in Tables 3.1-3.4. We can notice from Tables 3.1-3.4 that case (ii) outperforms case (i) when we compare the optimized (i.e minimum) probability of error achieved by choosing the corresponding optimum t₁ values in each case. However, the difference between minimum errors in (i) and (ii) is minimal.

Table 3.1. Performance comparison of tandem network configuration and central LRT for correlation coefficient ρ = 0.3, π₀ = 0.5

s1 s2

Table 3.2. Performance comparison of tandem network configuration with central LRT for correlation coefficient ρ = 0.5, π₀ = 0.5

s1 s2

Table 3.3. Performance comparison of tandem network configuration with central LRT for correlation coefficient ρ = 0.3, π0 = 0.3

s₁ s₂ Tandem Pe

Table 3.4. Performance comparison of tandem network configuration with central LRT for correlation coefficient ρ = 0.5, π₀ = 0.3

s₁ s₂ Tandem P_e

Table 3.4 continued from previous page

In Figs. 3.3-3.8 we show the variation of probability of error C(W ), as a function of threshold of sensor 1, t₁, for the cases (i) and (ii). We can observe from Figs. 3.3-3.8 that the minimum probability of error is sensitive to threshold t₁ in case (i), whereas when the better sensor is at FC, the minimum probability of error is only very slightly dependent on t₁, i.e. it is not sensitive to threshold t₁ of first sensor. Also, in the Figs 3.9-3.11, we can see the error performances of single sensor, better sensor as FC, weaker sensor as FC and central LRT, as a function of ρ. We can observe that, at each ρ, C(W ) is minimum for central LRT followed by the case of better sensor as FC. At ρ = 0, that is when the observations of the sensors are statistically independent, we observe a significant difference between the probabilities of error. The probabilities of error increase gradually with increasing ρ. In Fig 3.10, we observe that, when the difference between s₁and s₂is quite large, the performance of tandem network when the weaker sensor is placed at the bottom is same as the standalone better sensor performance. As observed in Figs 3.9 -3.12, better sensor as the FC is the preferred choice for signals in the good region. This conclusion is arrived based on extensive numerical study, for various correlation coefficients, signal levels, and prior probabilities, even though only representative results are shown here. In Figs. 3.13 -3.15, we observe that, when the better sensor is placed at the bottom, the error becomes maximum at the break-point ρ^∗ = ^s_s¹

2 and then starts to decrease gradually afterwards, as ρ increases further.

Again, the trend observed in Figs. 3.9-3.15 seem to be valid for many signal levels and prior probabilities. In Figs. 3.13-3.15, curves for ρ > ρ^∗ are obtained for the sub-optimal case of

sensor 1 employing a single threshold LRT with the second sensor quantization optimized accordingly to the results in Appendix B.

Figure 3.3. Probability of error versus threshold t₁, better sensor as fusion center, s₁ = 2, s2 = 3

Figure 3.4. Probability of error versus threshold t₁, weaker sensor as fusion center, s₁ = 3, s₂ = 2

Figure 3.5. Probability of error versus threshold t₁, better sensor as fusion center, s₁ = 2, s₂ = 4

Figure 3.6. Probability of error versus threshold t₁, weaker sensor as fusion center, s₁ = 4, s₂ = 2

Figure 3.7. Probability of error versus threshold t₁, better sensor as fusion center, s₁ = 1, s₂ = 3

Figure 3.8. Probability of error versus threshold t₁, weaker sensor as fusion center, s₁ = 3, s₂ = 1

Figure 3.9. Probability of error versus correlation coefficient ρ for good region, received signals, min(S) = 1.5, max(S) = 2 prior probability π₀ = 0.5

Figure 3.10. Probability of error versus correlation coefficient ρ for good region, received signals,min(S) = 1, max(S) = 4, prior probability π₀ = 0.5

Figure 3.11. Probability of error versus correlation coefficient ρ for good region,received signals, min(S) = 3, max(S) = 4, prior probability π₀ = 0.9

Figure 3.12. Probability of error versus correlation coefficient ρ for good region, received signals, min(S) = 1.5, max(S) = 3, prior probability π₀ = 0.3

Figure 3.13. Probability of error versus correlation coefficient ρ, when better sensor is at fusion center

Note: ρ > ^s_s¹

2 corresponds to sub-optimal single threshold LRT at sensor 1

Figure 3.14. Probability of error versus correlation coefficient ρ, when better sensor is at fusion center

Note: ρ > ^s_s¹

2 corresponds to sub-optimal single threshold LRT at sensor 1

Figure 3.15. Probability of error versus correlation coefficient ρ

Note: ρ > ^s_s¹

2 corresponds to sub-optimal single threshold LRT at sensor 1

CHAPTER 4

CONCLUSION AND FUTURE RESEARCH

In this thesis, we studied error performance of a two-sensor decentralized detection system that detects the presence of signals in Gaussian noise. We first present an analysis of detection of Gaussian signals in Gaussian noise using parallel fusion algorithm employing an AND or OR fusion rule at the FC. Results show :

1. For equal prior probabilities, the AND rule performs at-least as well as OR rule.

2. For equal prior probabilities and AND rule, the probability of error decreases gradually as ρ → 1. However, under same conditions, the performance of OR rule is about the same as the standalone performance of better sensor as single sensor.

We also studied Bayes error performance of two sensor tandem network for detection of deterministic signals in correlated Gaussian noise. Specifically, we identified the good region for signal points, where the optimum tests for both sensors are single threshold tests based on the observations at the sensors. Therefore, in good region, simple numerical computations can be used to find the optimum decision threshold levels at each sensor. Numerical results show that placing the sensor with better signal quality at the bottom gives a lower error performance as compared to the reverse configuration, for all signal points located in the good region. Notably, when the better sensor is at the top, the optimum performance is sensitive to the threshold of the top sensor, but not when the weaker signal sensor is at the top. We also notice that in the bad region, increasing noise correlation is expected to lead to decreasing error. Hence, it is important to study this region.

For the tandem network with the GBU model, we have to resort to a sub-optimal search algorithm, based on a genetic algorithm, in order to obtain quantization intervals in the bad

signal region. We defer this to a future study. Also, changing the optimization criterion to Neyman-Pearson may lead to different results than those obtained using the Bayes criterion.

For example, for a certain range of probability of false alarm at FC, would weaker sensor as the FC might lead to higher probability of detection?

BIBLIOGRAPHY

BIBLIOGRAPHY

[1] R. Viswanathan and P. K. Varshney, “Distributed detection with multiple sensors part i. fundamentals,” Proceedings of the IEEE, vol. 85, no. 1, pp. 54–63, Jan 1997.

[2] S. Haykin, “Cognitive radio: brain-empowered wireless communications,” IEEE Journal on Selected Areas in Communications, vol. 23, no. 2, pp. 201–220, Feb 2005.

[3] H. V. Trees, Detection, estimation and modulation theory. John Wiley and Sons, Inc, 2004.

[4] H. Poor, An introduction to signal detection and estimation. Springer Science and Business Media, 2013.

[5] P. Willett, P. F. Swaszek, and R. S. Blum, “The good, bad and ugly: distributed detection of a known signal in dependent Gaussian noise,” IEEE Transactions on Signal Processing, vol. 48, no. 12, pp. 3266–3279, Dec 2000.

[6] H. Kasasbeh, L. Cao, and R. Viswanathan, “Hard decision based distributed detection in multi-sensor system over noise correlated sensing channels,” in 2016 Annual Conference on Information Science and Systems (CISS), March 2016, pp. 280–285.

[7] L. Khalid and A. Anpalagan, “Cooperative sensing with correlated local decisions in cognitive radio networks,” IEEE Transactions on Vehicular Technology, vol. 61, no. 2, pp. 843–849, Feb 2012.

[8] H. Kasasbeh, R. Viswanathan, and L. Cao, “Noise correlation effect on detection: Sig-nals in equicorrelated or autoregressive (1) Gaussian,” IEEE Signal Processing Letters, vol. 24, no. 7, pp. 1078–1082, July 2017.

[9] H. Kasasbeh, L. Cao, and R. Viswanathan, “Soft-decision-based distributed detection with correlated sensing channels,” IEEE Transactions on Aerospace and Electronic Sys-tems, vol. 55, no. 3, pp. 1435–1449, June 2019.

[10] R. Viswanathan, S. C. A. Thomopoulos, and R. Tumuluri, “Optimal serial distributed decision fusion,” IEEE Transactions on Aerospace and Electronic Systems, vol. 24, no. 4, pp. 366–376, July 1988.

[11] J. D. Papastavrou and M. Athans, “On optimal distributed decision architectures in a hypothesis testing environment,” IEEE Transactions on Automatic Control, vol. 37, no. 8, pp. 1154–1169, Aug 1992.

[12] E. Akofor and B. Chen, “On optimal fusion architecture for a two-sensor tandem dis-tributed detection system,” in 2013 IEEE Global Conference on Signal and Information Processing, Dec 2013, pp. 129–132.

[13] E. Akofor and B. Chen, “Interactive fusion in distributed detection: Architecture and performance analysis,” in 2013 IEEE International Conference on Acoustics, Speech and Signal Processing, May 2013, pp. 4261–4265.

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6473, Oct 2014.

APPENDICES

APPENDIX A

DERIVATION OF DECISION REGION RULES FOR EACH SENSOR IN TW0-SENSOR TANDEM NETWORK

In this appendix we show the derivations that lead to equations First, let us minimize C(W ) by finding the decision region R_W at sensor 2 given the decision region at sensor 1 is already fixed. Let the whole sample space be denoted by Ω . Using the Bayes rules, we can simplify the risk as follows:

The above equation can be further simplified as :

C(W ) = π₁+

To minimize C(W ), we need to minimize each integral in equation (A.2). First integral can be minimized by assigning all those points x₂ that make the term present inside the square bracket non-positive to the region R_{W |V =1}. Writing this out explicitly yields the collection of all x₂ points that satisfy

p(x₂|V = 1, H₁)p_d p(x₂|V = 1, H₀)p_f ≥ π₀

π₁

We can further simplify the numerator conditional density term on the left of the inequality

Similarly, further simplification of the conditional probability density function in the denom-inator to the left of the inequality leads to equation (3.4) given earlier. Similarly, we can minimize the second integral by assigning all those points x₂ that make the term present in-side the square bracket non-positive to the region R_{W |V =0}. Writing this out explicitly yields the collection of all x₂ points that satisfy

p(x₂|V = 0, H₁)(1 − p_d) p(x₂|V = 0, H₀)(1 − p_f) ≥ π₀

π₁

We can further simplify the numerator conditional density term on the left of the inequality as shown below:

Similarly, further simplification of the conditional probability density function in the denom-inator to the left of the inequality leads to equation (3.5) given earlier.

Next, let us minimize C(W ) by finding the decision region RV at sensor 1, given that the

decision rules at sensor 2 is already fixed. We can write the overall cost as follows:

C(W ) = π₀[P (W = 1|V = 0, H₀)(1 − p_f) + P (W = 1|V = 1, H₀)p_f] + π1[P (W = 0|V = 0, H1)(1 − pd) + P (W = 0|V = 1, H1)pd].

(A.3)

The above equation can be simplified as

C(W ) = π₀P (W = 1|V = 0, H₀) + π₁P (W = 0|V = 0, H₁) + p_fπ₀[P (W = 1|V = 1, H₀) − P (W = 1|V = 0, H₀)]

+ p_dπ₁[P (W = 0|V = 1, H₁) − P (W = 0|V = 0, H₁)].

(A.4)

The terms involving p_f and p_d, are the only terms that depend on the determination of R_V regions . Hence, the sum of these two terms in the above equation needs to be minimized by appropriately choosing the decision region RV. This sum can be considered as the expected cost, given that the decision regions R_{W |V =1} and R_{W |V =0} are already determined. Hence, this sum is minimized by assigning all those points x₁ to R_V that will make the integrand below non-positive. (Given decision regions of sensor 2 for V = 1 and V = 0 are fixed, the probabilities involving W are constants and hence can be taken inside the integral shown below)

= Z

RV

[π0(P (W = 1|V = 1, H0) − P (W = 1|V = 0, H0))p(x1|H0)

− π₁(−P (W = 0|V = 1, H₁) + P (W = 0|V = 0, H₁))p(x₁|H₁)]dx₁

(A.5)

The above integrand can be further simplified using the results P (W = 0|V = 0, H₁) = 1 − P (W = 1|V = 0, H₁), P (W = 0|V = 1, H₁) = 1 − P (W = 1|V = 1, H₁). Writing this

out explicitly yields the collection of all x₁ points belonging to R_V region as

π₀(P (W = 1|V = 1, H₀) − P (W = 1|V = 0, H₀))p(x₁|H₀)

< π1(P (W = 1|V = 1, H1) − P (W = 1|V = 0, H1))p(x1|H1)

Writing out the above conditional probabilities in terms of integrals of conditional densi-ties lead to the equation (3.6) given earlier. Designing sensors’ decision rules as given by (3.5), (3.4) and (3.6) is a requirement for person-by-person optimal solution for minimiz-ing C(W ). As mentioned in [5], this is a requirement for globally optimal solution as well.

Although similar derivation of decision rules at the two sensors is provided in [12] for the detection of random signal in independent noise, the derivation provided here is much more straightforward and simpler.

APPENDIX B

DERIVATION OF GOOD REGION FOR SENSOR 2 IN TANDEM NETWORK FOR THE SUB-OPTIMAL CASE OF SENSOR 1 HAVING A SINGLE THRESHOLD LRT

In this appendix we derive sufficiency condition of good region in signal plane (s₁, s₂) for the sub-optimal case when sensor 1 is forced to be a single threshold LRT. We assume decision regions of sensor 1 to be semi-infinite intervals. In (3.5), let µ1(x2) − µ0(x2) = δ ( for simplicity, dependence on x₂ is not shown explicitly) where µ₁(x₂) is conditional expectation of G(s₁− ρs₂+ ρx₂, 1 − ρ²) (under H₁) and µ₀(x₂) is conditional expectation of G(ρx₂, 1 − ρ²)

According to Lemma 1 of GBU [5], we have the following result

δ > 0 if µ > 0

δ = 0 if µ = 0

δ < 0 if µ < 0 (B.2)

We now study different cases of signal (s₁, s₂) regions.

1. s1 ≥ 0 s2 ≥ 0 (i.e first quadrant)

We can write the expression (3.10) when sensor 1 decision is 1 (V = 1) as

x2 : g1(x2) = e

Let µ^∗ = √^s1−ρs2 decreases monotonically if µ^∗ < 0. Also, ^dy_dx²

2 = √^−ρ

The above condition is satisfied when µ^∗ < 0. Hence, R_{W |V =1} is a single semi-infinite region of the form (t_w1, ∞) if µ^∗ < 0, or we can say when

For s₂ > 0, h₁(x₂) is monotonic increasing with x₂. g₀(x₂) will be monotonic increasing w.r.t x₂ when h₂(y₂) will be monotonically increasing with x₂. This is true when −µ^∗ > 0 or µ^∗ < 0. The condition is same as in V = 1 or s₂ > ^s_ρ¹. Now, let us look another way.

Rewriting equation (3.7), we have the following

∂L(x₂) (t_w0, ∞) and will be optimal. Hence, sufficient condition becomes

s₂ > ρs₁ and s₁ > ρs₂, or

ρs₁ < s₂ < s1

ρ (B.11)

Combining (B.11) and (B.7) we observe that the sufficient condition for the good region in first quadrant is simply multiple disjoint intervals. This can be observed numerically when s₂ < ρs₁.

2. s₁ < 0, s₂ > 0 (second quadrant)

For s1 < 0, the decision region RV is of the form (−∞, t1). So, when V = 0, we have

[since R_V = (−∞, t₁), V = 1 type region for s₁ > 0 case becomes V = 0 for s₁ < 0 case and decreasing w.r.t x₂, we require h₂(y₂) to be monotone decreasing w.r.t x₂. This happens when µ^∗ > 0. Thus, R_{W |V =0} will be of the form (−∞, t_w0), if

Alternatively, considering equation (B.9), for ^dL(x_δx²⁾

2 < 0, sufficiency condition becomes s₁ − ρs₂ < 0 and s₂− ρs₁ < 0, or

s₁

ρ < s2 < ρs1 (B.18)

Putting (B.18) and (B.17) together, sufficiency condition becomes

s₂ < ρs₁ (B.19) (−∞, t_w1), signals must satisfy the following condition

s₂ < s₁

ρ (B.21)

The above condition will always be satisfied in the fourth quadrant as s₁ > 0, s₂ < 0.

Similarly, when V = 0, we have

x₂ : g₀(x₂) = e

monotonically decreasing w.r.t x₂. If µ^∗ > 0, then this condition will be satisfied and R_{W |V =0} will be an optimal semi-infinite region of the form (−∞, t_w0). This is the same condition as what we get for R_{W |V =1} in fourth quadrant of signal plane. Hence, over the entire fourth quadrant, R_{W |V =1} : x₂  (−∞, t_w1) and R_{W |V =0} : x₂  (−∞, t_w0), will be optimal.

The result here shows that for (s₁ > 0, s₂ > 0) case, if the first sensor is forced to be single threshold LRT, then the good region for sensor 2 will be valid as long as s₂ > ρs₁. Hence, for a given (s1, s2) and ρ^∗ = _max(s^min(s1,s2)

1,s2), ρ > ρ^∗ will be in good region for sensor 2, for both cases of weaker sensor as the FC and the better sensor as the FC. However, notice that for globally optimum tandem network case, ρ > ρ^∗ puts the first sensor test in possibly bad region and hence, single threshold LRT at sensor 1 is not likely to be globally optimal for ρ > ρ^∗. In fact, for a globally optimal tandem network, P_e → 0 as ρ → 1 (see comments in section 3.6). Graphs in 3.13-3.15 show for the sub-optimal case considered here in Appendix B, P_e decreases with ρ beyond ρ^∗, but never approaches zero as ρ approaches one.

VITA

Born in Vadodara, India in 1994, Shailee Yagnik received the B.E degree in instrumen-tation and Control engineering from Gujarat Technological University, India in June 2017.

Starting August 2017, she is working towards the M.S degree in electrical engineering at The University of Mississippi, University, MS. During her M.S. program, she worked as a teaching assistant for Dr. Mustafa Matalgah and then as a grader for Dr. Ramanarayan Viswanathan.

In document EFECTOS Y CONSECUENCIAS DEL 11-S. UNA PERSPECTIVA ÉTICO-POLÍTICA Autor: Alejandro Vélez Salas Director: Francisco Fernández Buey TESI DOCTORAL UPF/2011 INSTITUT UNIVERSITARI DE CULTURA DEPARTAMENT d’HUMANITATS (página 76-94)