Prediction and Representation of Array Performance under Sensor - - PowerPoint PPT Presentation

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Prediction and Representation of Array Performance under Sensor Failure

Erdal MEHMETCIK,

Underwater Acoustic Systems Defence Systems Business Sector, ASELSAN Inc.

• Prof. Dr. Çağatay CANDAN,

Electrical and Electronics Engineering Department Middle East Technical University

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Outline

• Problem definition

?

• Performance bounds
• Bayesian bounds
• Deterministic bounds
• Signal models

Fading channels

• Results and conclusion

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A practical problem

• How much should the sonar operator trust the system

when some of the sensors fail (or disabled)?

? ?

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How to report?

• In general the user manual of a given system provides the

following information;

• The system is operational, up to N sensor failures.
• The system is partially operational, up to M sensor failures.
• The system is not operational, after K sensor failures.





≈ ×

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How to report?

• The drawback here is these bounds are loose.
• i.e. The system may tolerate the loss of sensors in certain

angular sector.

• The bounds have to be loose, as the number of possible

combinations become very large, as number of sensors increase.

2𝑂subsets

→ 𝑇𝑢𝑗𝑚𝑚 𝑣𝑡𝑏𝑐𝑚𝑓 𝑗𝑜 𝑢ℎ𝑗𝑡 𝑡𝑓𝑑𝑢𝑝𝑠



×

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Performance prediction

• System performance can be assessed using

performance lower bounds.

• Two classes of performance bounds exist
• Bayesian bounds

– The parameter to be estimated is a random variable with a known a-priori distribution.

• Deterministic bounds

– The parameter to be estimated is a non-random parameter

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Performance Prediction

 Deterministic CRLB can be evaluated.

 Bound is optimistic,  Under low SNR.  Under fading conditions.

 Deterministic version of the Ziv-Zakai

Bounds can be used.

 Tighter lower bound, even at low SNR

conditions!

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SNR Regions



CRLB can not model gross-error events, since it only considers the second derivative of the beampattern at the mainlobe.



Consequently the errors produced by side-lobe jumps (gross errors at low SNR values) can not be modeled.



CRLB is not a tight bound at low SNR values.



Performance bounds are generally divided into three regions w.r.t. SNR:

Apriori region: Region in which the estimate is uniformly distributed in the a priori domain of the unknown parameter (region of low SNRs). Threshold region: Region of transition between the apriori and asymptotic regions (region of medium SNRs). The mean squared error is dominated by gross error events. Asymptotic region: Region in which the CRLB is achieved (region of high SNRs). Gross error probability is negligibly small.

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Signal model

෤ 𝑧𝑜𝑙 = ǁ 𝑡𝑙𝜽𝒐𝑓𝑘𝜄𝑑 + ෩ 𝑶𝒐𝒍 𝜃𝑜~𝐷𝑂( ҧ 𝜃𝐽𝑜 + 𝑘 ҧ 𝜃𝑅𝑜, 2𝜏2 ) ෤ 𝑧𝑜𝑙 = ǁ 𝑡𝑙 ҧ 𝜃𝐽𝑜 + 𝑘 ҧ 𝜃𝑅𝑜 𝑓𝑘𝜄𝑑 + ǁ 𝑡𝑙 ҧ 𝜊𝐽𝑜 + 𝑘 ҧ 𝜊𝑅𝑜 𝑓𝑘𝜄𝑑 + ෩ 𝑂𝑠 Specular component Random component 𝐿 = 𝑡𝑞𝑓𝑑𝑣𝑚𝑏𝑠 𝑞𝑝𝑥𝑓𝑠 𝑠𝑏𝑜𝑒𝑝𝑛 𝑞𝑝𝑥𝑓𝑠 = ҧ 𝜃𝐽 2 + ҧ 𝜃𝑅

2

𝑤𝑏𝑠 𝜃𝐽 + 𝑤𝑏𝑠 𝜃𝑅 = 𝑛𝐽

2 + 𝑛𝑅 2

2𝜏2 Rician factor The signal model under Rician fading is as follows;

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Channel Fading

Near 9 dB loss between two consecutive pulses.

Receiver Projector

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Ziv-Zakai and Cramer-Rao Bounds

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Cramer-Rao Bound under Rician Fading

The Fisher Information Matrix for a complex Gaussian pdf is as follows;

FIM = 𝐉𝜊 𝑗𝑘 = 𝑢𝑠 𝐃𝑦

−1 𝜊 𝜖𝐃𝑦 𝜊

𝜖𝜊𝑗 𝐃𝑦

−1 𝜊 𝜖𝐃𝑦 𝜊

𝜖𝜊𝑘 + 2Re 𝜖𝝂𝑰 𝜊 𝜖𝜊𝑗 𝐃𝑦

−1 𝜊 𝜖𝝂 𝜊

𝜖𝜊𝑘 CRLB = FIM−1

𝑠

𝑜 = 𝜃𝑜𝐵𝑡 exp 𝑘𝜄𝑜 + 𝑜𝑜

𝒔 = 𝑠 𝑠

1

⋮ 𝑠

𝑂−1

, 𝒃𝜄 = 1 𝑓𝑘𝜄 ⋮ 𝑓𝑘𝜄(𝑂−1) 𝜽𝑜~𝐷𝑂(𝜈, 2𝜏𝜃

2),

𝑜𝑜~𝐷𝑂(0,2𝜏𝑜

2),

𝑂, 𝐵𝑡 ∈ ℝ 𝝂 = 𝐹 𝒔 = 𝜈𝐵𝑡𝒃𝜄 𝐃𝑦 = 𝐹 𝒔 − 𝝂 𝒔 − 𝝂 𝑰 = 2𝜏𝜃

2𝐵𝑡 2𝒃𝜄𝒃𝜄 𝑰 + 2𝜏𝑜 2𝑱𝑂

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Ziv-Zakai Bounds

𝑎𝑎𝐶 = න

∞

𝜉 𝐵 ℎ 𝑄

𝑓(ℎ) ℎ𝑒ℎ

𝐵 ℎ = න

−∞ ∞

min 𝑞𝑣 𝑣 ,𝑞𝑣 𝑣 + ℎ 𝑒𝑣 𝑣~𝑣𝑜𝑗𝑔 0,2𝜌 → 𝐵 ℎ = 2𝜌 − ℎ 2𝜌 𝑎𝑎𝐶 = න

2𝜌

𝜉 𝑄

𝑓 ℎ 2𝜌 − ℎ

2𝜌 ℎ𝑒ℎ

*Figure taken from: K.Bell, Y.Eprahim, H.L.Van Trees, “Explicit Ziv Zakai Lower Bound for Bearing Estimation”, IEEE Transactions

• n Signal Processing, Vol. 44, No:11, Nov. 1996.

*Valley filling function No angular dependence ! 𝑄

𝑓 ℎ ∶ Minimum probability of error

between deciding 𝐼0: 𝑣 and 𝐼1: 𝑣 + ℎ. Dividing the parameter space into smaller intervals does not solve the issue, as the bound ignores gross errors larger than the sub-interval size.

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Approximate Deterministic ZZB

Discretize the parameter space

This is an approximate lower bound for ML type estimators.

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Stein’s unified analysis of error probability

Summary of the results for FSK in Stein’s paper*:

𝑨 𝑗𝑔 = 𝑛𝑗𝑔 + 𝑘𝜈𝑗𝑔 = 𝑨 𝑗𝑔 𝑓𝑘𝜄𝑗𝑔 , 𝑗 = 1, 2 𝑇𝑗𝑔 = 1 2 𝑨 𝑗𝑔

2 𝑛𝑗𝑔

𝑂𝑗𝑔 = 1 2 𝑨𝑗𝑔 − 𝑨 𝑗𝑔

𝜍𝑔 𝑂

2 𝑨1𝑔 − 𝑨 1𝑔

𝜍𝑔 = 𝜍𝑑𝑔 + 𝑘𝜍𝑡𝑔 = 1 2 𝑂

𝑨1𝑔 − 𝑨 1𝑔

1 2 𝑨1𝑔 − 𝑨 1𝑔 𝑨2𝑔 − 𝑨 2𝑔 = 0 𝜚 = arg 𝜍𝑑𝑔 + 𝑘𝜍𝑡𝑔 𝑏 𝑐 = 1 2 𝑇1𝑔 + 𝑇2𝑔 + 2 𝑇1𝑔𝑇2𝑔 cos 𝜄1𝑔 − 𝜄2𝑔 + 𝜚 𝑂

+ 𝑇1𝑔 + 𝑇2𝑔 − 2 𝑇1𝑔𝑇2𝑔 cos 𝜄1𝑔 − 𝜄2𝑔 + 𝜚 𝑂

∓ 2 𝑇1𝑔 − 𝑇2𝑔 𝑂

𝐵 = 𝑂

𝑂

𝑄 = 1 2 1 − 𝑅1 𝑐, 𝑏 + 𝑅1 𝑐, 𝑏 − 𝐵 2 exp − 𝑏 + 𝑐 2 𝐽0 𝑏𝑐 .

𝑨1𝑔 = 𝑐𝑢

𝑔 and 𝑨2𝑔 = 𝑐𝑚 𝑔

*S. Stein, “Unified analysis of certain coherent and non-coherent binary communication systems,” IEEE Trans. Inf. Theory, vol. IT-10, January 1964, pp. 43–51.

Where 𝑅1 is the Marcum-Q function and 𝐽0 is the modified Bessel function of the first kind.

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Online Performance Prediction

• Case study:

Circular array with directional sensors Sensor patterns: 𝐶𝑜 𝜚 =

1 2 cos 𝜚 − 𝜚𝑜 + 1 2 2

𝜚𝑜 = 𝑜−1 𝜌

6

, 𝑜 = 1, 2, … , 12.

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Online Performance Prediction

K = 10

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Online Performance Prediction

K = 10

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Summary

• Deterministic lower bounds can be utilized for on-

line performance estimation of the system.

• Extension of Ziv-Zakai bounds for an approximate

deterministic bound is used in this work.

• Closed form expressions are available.
• Actual systems can be equipped with a system

performance prediction tab, to provide the user with current system capabilities.