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

Prediction and Representation of Array Performance under Sensor Failure Erdal MEHMETCIK, Prof. Dr. Çağatay CANDAN , Underwater Acoustic Systems Electrical and Electronics Engineering Department Defence Systems Business Sector, ASELSAN Inc. Middle East Technical University #UDT2019

Outline ? • Problem definition • Performance bounds - Bayesian bounds - Deterministic bounds • Signal models Fading channels • Results and conclusion #UDT2019

A practical problem • How much should the sonar operator trust the system when some of the sensors fail (or disabled)? ? ? #UDT2019

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.   ≈ × #UDT2019

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.  × → 𝑇𝑢𝑗𝑚𝑚 𝑣𝑡𝑏𝑐𝑚𝑓 𝑗𝑜 𝑢ℎ𝑗𝑡 𝑡𝑓𝑑𝑢𝑝𝑠 - The bounds have to be loose, as the number of possible combinations become very large, as number of sensors increase. 2 𝑂 subsets #UDT2019

Performance prediction • System performance can be assessed using performance lower bounds. • Two classes of performance bounds exist o Bayesian bounds – The parameter to be estimated is a random variable with a known a-priori distribution. o Deterministic bounds – The parameter to be estimated is a non-random parameter #UDT2019

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! #UDT2019

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. #UDT2019

ҧ ҧ ҧ ҧ Signal model The signal model under Rician fading is as follows; 𝑡 𝑙 𝜽 𝒐 𝑓 𝑘𝜄 𝑑 + ෩ 𝑧 𝑜𝑙 = ǁ ෤ 𝑶 𝒐𝒍 𝜃 𝑅 𝑜 , 2𝜏 2 ) 𝜃 𝑜 ~𝐷𝑂( ҧ 𝜃 𝐽 𝑜 + 𝑘 ҧ 𝜃 𝑅 𝑜 𝑓 𝑘𝜄 𝑑 + ǁ 𝜊 𝐽 𝑜 + 𝑘 ҧ 𝜊 𝑅 𝑜 𝑓 𝑘𝜄 𝑑 + ෩ 𝑧 𝑜𝑙 = ǁ ෤ 𝑡 𝑙 𝜃 𝐽 𝑜 + 𝑘 ҧ 𝑡 𝑙 𝑂 𝑠 Specular component Random component 2 2 + 𝑛 𝑅 𝜃 𝐽 2 + 2 𝐿 = 𝑡𝑞𝑓𝑑𝑣𝑚𝑏𝑠 𝑞𝑝𝑥𝑓𝑠 𝜃 𝑅 = 𝑛 𝐽 𝑠𝑏𝑜𝑒𝑝𝑛 𝑞𝑝𝑥𝑓𝑠 = 2𝜏 2 𝑤𝑏𝑠 𝜃 𝐽 + 𝑤𝑏𝑠 𝜃 𝑅 Rician factor #UDT2019

Channel Fading Near 9 dB loss between two consecutive pulses. Receiver Projector #UDT2019

Ziv-Zakai and Cramer-Rao Bounds #UDT2019

Cramer-Rao Bound under Rician Fading The Fisher Information Matrix for a complex Gaussian pdf is as follows; 𝑠 1 0 𝑠 𝑓 𝑘𝜄 1 𝑠 𝑜 = 𝜃 𝑜 𝐵 𝑡 exp 𝑘𝜄𝑜 + 𝑜 𝑜 𝒔 = , 𝒃 𝜄 = ⋮ ⋮ 𝑠 𝑓 𝑘𝜄(𝑂−1) 𝑂−1 2 ) , 2 ) , 𝜽 𝑜 ~𝐷𝑂(𝜈, 2𝜏 𝜃 𝑜 𝑜 ~𝐷𝑂(0,2𝜏 𝑜 𝑂, 𝐵 𝑡 ∈ ℝ 𝝂 = 𝐹 𝒔 = 𝜈𝐵 𝑡 𝒃 𝜄 𝑰 + 2𝜏 𝑜 𝒔 − 𝝂 𝑰 = 2𝜏 𝜃 2 𝐵 𝑡 2 𝒃 𝜄 𝒃 𝜄 2 𝑱 𝑂 𝐃 𝑦 = 𝐹 𝒔 − 𝝂 + 2Re 𝜖𝝂 𝑰 𝜊 −1 𝜊 𝜖𝐃 𝑦 𝜊 −1 𝜊 𝜖𝐃 𝑦 𝜊 −1 𝜊 𝜖𝝂 𝜊 FIM = 𝐉 𝜊 𝑗𝑘 = 𝑢𝑠 𝐃 𝑦 𝐃 𝑦 𝐃 𝑦 𝜖𝜊 𝑗 𝜖𝜊 𝑘 𝜖𝜊 𝑗 𝜖𝜊 𝑘 CRLB = FIM −1 #UDT2019

Ziv-Zakai Bounds *Valley filling function ∞ 𝑎𝑎𝐶 = න 𝜉 𝐵 ℎ 𝑄 𝑓 (ℎ) ℎ𝑒ℎ 0 ∞ 𝐵 ℎ = න min 𝑞 𝑣 𝑣 ,𝑞 𝑣 𝑣 + ℎ 𝑒𝑣 −∞ 𝑣~𝑣𝑜𝑗𝑔 0,2𝜌 → 𝐵 ℎ = 2𝜌 − ℎ 2𝜌 𝑄 𝑓 ℎ ∶ Minimum probability of error between deciding 𝐼 0 : 𝑣 and 𝐼 1 : 𝑣 + ℎ. 2𝜌 𝑓 ℎ 2𝜌 − ℎ 𝑎𝑎𝐶 = න 𝜉 𝑄 ℎ𝑒ℎ 2𝜌 Dividing the parameter space into 0 smaller intervals does not solve the issue, as the bound ignores gross No angular dependence ! errors larger than the sub-interval size. #UDT2019 *Figure taken from: K.Bell, Y.Eprahim, H.L.Van Trees, “ Explicit Ziv Zakai Lower Bound for Bearing Estimation ” , IEEE Transactions on Signal Processing, Vol. 44, No:11, Nov. 1996.

Approximate Deterministic ZZB Discretize the parameter space #UDT2019 This is an approximate lower bound for ML type estimators.

Stein ’ s unified analysis of error probability Summary of the results for FSK in Stein ’ s paper*: 𝑔 and 𝑨 2 𝑔 = 𝑐 𝑚 𝑔 𝑨 1 𝑔 = 𝑐 𝑢 𝑨 𝑗𝑔 = 𝑛 𝑗𝑔 + 𝑘𝜈 𝑗𝑔 = 𝑨 𝑗𝑔 𝑓 𝑘𝜄 𝑗𝑔 , 𝑗 = 1, 2 𝑇 1 𝑔 + 𝑇 2 𝑔 + 2 𝑇 1 𝑔 𝑇 2 𝑔 cos 𝜄 1 𝑔 − 𝜄 2 𝑔 + 𝜚 𝑇 1 𝑔 + 𝑇 2 𝑔 − 2 𝑇 1 𝑔 𝑇 2 𝑔 cos 𝜄 1 𝑔 − 𝜄 2 𝑔 + 𝜚 𝑏 𝑐 = 1 + 𝑇 𝑗𝑔 = 1 2 = 1 𝑂 𝑗𝑔 = 1 2 2 + 𝜈 𝑗𝑔 2 , 2 2 2 2 𝑨 𝑗𝑔 2 𝑛 𝑗𝑔 2 𝑨 𝑗𝑔 − 𝑨 𝑗𝑔 1 𝑔 + 𝑂 2 𝑔 + 2 𝑂 1 𝑔 + 𝑂 2 𝑔 − 2 𝑂 𝑂 1 𝑔 𝑂 2 𝑔 𝜍 𝑔 𝑂 1 𝑔 𝑂 2 𝑔 𝜍 𝑔 1 𝑔 𝑂 2 𝑔 = 1 ∗ 𝑨 2 𝑔 − 𝑨 2 𝑔 2 𝑇 1 𝑔 − 𝑇 2 𝑔 𝜍 𝑔 𝑂 2 𝑨 1 𝑔 − 𝑨 1 𝑔 ∓ 2 − 4 𝑂 2 𝑂 1 𝑔 + 𝑂 2 𝑔 1 𝑔 𝑂 2 𝑔 𝜍 𝑔 1 ∗ 𝑨 2 𝑔 − 𝑨 2 𝑔 𝜍 𝑔 = 𝜍 𝑑𝑔 + 𝑘𝜍 𝑡𝑔 = 𝑨 1 𝑔 − 𝑨 1 𝑔 2 𝑂 1 𝑔 𝑂 2 𝑔 1 = 0 𝑂 1 𝑔 − 𝑂 2 𝑔 2 𝑨 1 𝑔 − 𝑨 1 𝑔 𝑨 2 𝑔 − 𝑨 2 𝑔 𝐵 = 2 − 4 𝑂 2 𝑂 1 𝑔 + 𝑂 2 𝑔 1 𝑔 𝑂 2 𝑔 𝜍 𝑔 𝜚 = arg 𝜍 𝑑𝑔 + 𝑘𝜍 𝑡𝑔 𝑄 = 1 − 𝐵 2 exp − 𝑏 + 𝑐 2 1 − 𝑅 1 𝑐, 𝑏 + 𝑅 1 𝑐, 𝑏 𝐽 0 𝑏𝑐 . 2 Where 𝑅 1 is the Marcum-Q function and 𝐽 0 is the modified Bessel function of the first kind. #UDT2019 *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.

Online Performance Prediction - Case study: Circular array with directional sensors Sensor patterns: 2 1 2 cos 𝜚 − 𝜚 𝑜 + 1 𝐶 𝑜 𝜚 = 2 𝜚 𝑜 = 𝑜−1 𝜌 , 𝑜 = 1, 2, … , 12 . 6 #UDT2019

Online Performance Prediction K = 10 K → ∞ #UDT2019

Online Performance Prediction K = 10 K → ∞ #UDT2019

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. #UDT2019