Presentation Title Pierre-Richard Cornely, Ph.D., June 24, 2018 A Complete Amplitude and Phase Characterization of Ionospheric Scintillation Pierre-Richard Cornely, Ph.D., This document contains data whose export/transfer/disclosure is restricted by U.S. law. Dissemination to non-U.S. persons whether in the United States or abroad requires an export license or other authorization. Speaker Name

Background - Motivation Presentation Title June 24, 2018 Background - Motivation Patchy Clouds Foliage Target & Target Complexes Rain - Hail Coastal Input Waveform Environment System System Under Inquiry Inquiring System Take Away: The radar signals amplitude and phase are affected by the propagation environment Speaker Name

Background - Motivation Presentation Title June 24, 2018 Background - Motivation At present, we model radar channel fluctuations as follows: Single Pulse, Constant Amplitude, Known Phase (Rayleigh PDF) Single Pulse, Constant Amplitude, Uniform Phase (Ricean PDF) Multiple Pulse,Constant Amplitude, Uniform Phase (Chi-Square PDF with 4 degrees of Freedom) Multiple Pulse, Swerling Target, Uniform Phase (Chi-Square PDF with 4 degrees of Freedom) All of the above models do not include severe fluctuations due to the environment In particular, these models do not include target fluctuations due to Ionospheric Scintillations. Ionospheric Scintillation is the Rapid Fluctuation in Signal Amplitude of the Radar Echo from a Target due to Interaction with Free Electrons in the Ionosphere Measurements of Radar Cross Section are Directly Proportional to Signal Amplitude In Highly Scintillated Ionosphere, even Spherical Targets (e.g., satellite 5398 which normally have constant RCS) exhibit large Fluctuations in RCS Amplitude Target fluctuations due to ionospheric scintillations are Nakagami-m. All of the above models approximate the Nakagami-m but for special cases only The original Nakagami-m amplitude model assumes no correlation within a pulse In addition, no implicit Phase information was included in the model Take Away: There is a need to model target fluctuations due to ionospheric scintillations Speaker Name

Background - Motivation Presentation Title June 24, 2018 Background - Motivation Swerling-0 Target 1-m2 RCS, 0-dBsm ~20 dB RCS (dBsm) Fluctuations up to 20 dB due to scintillations The S4 index is the ratio of standard deviation and mean RCS S4 Index Take Away: The environment can both benefit and affect radar detection, tracking and discrimination Speaker Name

Background - Previous Work Presentation Title June 24, 2018 Background - Previous Work Uncorrelated Amplitude Nakagami-m Available in the literature and straight forward to implement [1] Both Phase and higher order statistics are not well defined in original work Correlated Amplitude Nakagami-m Available in the literature but more difficult to implement numerically [2] Correlation does not satisfy any Predefined Correlation Model Correlated Amplitude and Phase Nakagami-m Some work available on Correlated Nakagami-m with correct Phase Property [3],[4] Joint Nakagami-m PDF being used is only valid for integer values of m [4] Correlated Amplitude and Phase Nakagami-m /Predefined Autocorrelation Some work available on Correlated Nakagami-m with correct Phase Property and Predefined Autocorrelation [3], [4] The joint Nakagami-m PDF being used is only valid for integer values of m [4] Take Away: Correlated Nakagami-m with the Correct Phase satisfying Predefined Correlation Model are Open Problems Speaker Name

Presentation Title June 24, 2018 Proposed Approach The class of signals of interest are usually complex (I & Q) A complete model of the signal fluctuations must consider both I & Q x(n) = I(n) + jQ(n) To include both the real and imaginary parts of the signal x(n) in the fluctuation model We must be able to describe either the second order statistics of (I,Q), i.e., the joint PDF or the autocorrelation function of x(n) We must also be able to describe the statistics of the phase of x(n) Our approach proposes the following: 1. Creation of independent mappings of the real and imaginary parts of x(n) to Nagakami-m sequences, starting with Gaussian sequences 2. Simulation of complex Nakagami-m sequences which satisfy a pre-specified Autocorrelation Model 3. Simulation of complex Nakagami-m sequences which satisfy the theoretical Nakagami-m Phase Take Away: A Nakagami-m model and simulator that includes both amplitude and phase fluctuations Speaker Name

Background – The Nakagami-m Model Presentation Title June 24, 2018 Background – The Nakagami-m Model Beacon Satellite Or Radio Star Target Noise from the environment One-Way Scintillation Amplitude follows the Nakagami-m Model 1-WAY SCINTILLATION Amplitude Intensity 2-WAY SCINTILLATION Amplitude Intensity Receiver Radar Take Away: Ionospheric Scintillations modify the received signal Intensity, Phase and Polarization Speaker Name

Background – The Nakagami-m Model Presentation Title June 24, 2018 Background – The Nakagami-m Model The Nakagami-m distribution for one-way scintillation amplitude m: Shape Parameter, Ω: Scaling Parameter (Ω =1 in Fig) Scintillation Increases as m decreases Relationship between one way intensity Sm and two way intensity S4 Take Away: Amplitude Scintillations are well modeled by Nakagami-m Amplitude PDF Speaker Name

The Phase & Autocorrelation Models Presentation Title June 24, 2018 The Phase & Autocorrelation Models The theoretical Nakagami-m Phase The Nakagami-m phase is a function of m as well as the angle of arrival Caveat: The Nakagami-m phase is not uniform for m1 The Theoretical Autocorrelation model (“Jake’s Model”) Chosen correlation model is a function of signal bandwidth Take Away: Accepted Theoretical Nakagami-m phase and Autocorrelation can be used to anchor our results Speaker Name

Proposed Methodology Block Diagram Walk Through Presentation Title June 24, 2018 Proposed Methodology Block Diagram Walk Through Start with Gaussian sequences G(0,1) and create [G(0,1)]2 Map [G(0,1)]2 to Chi-Square (2) sequences with N degrees of freedom Map each (2) sequence to a Gamma sequence with parameter m/2 Map the Gamma sequences to Nakagami-m sequences Speaker Name

Simulator Input/Output Realizations Presentation Title June 24, 2018 Simulator Input/Output Realizations Take Away: Gaussian to Nakagami-m Mapping works even for the worse case m=0.5 Speaker Name

Simulator PDF vs. Theory Presentation Title June 24, 2018 Simulator PDF vs. Theory Take Away: Both the real part and the magnitude of the simulated sequence are consistent with the Nakagami-m model Speaker Name

Simulator Autocorrelation vs. Theory Presentation Title June 24, 2018 Simulator Autocorrelation vs. Theory 10 20 30 40 50 60 70 80 90 100 110 120 -0.4 -0.2 0.2 0.4 0.6 0.8 1 Theoretical, Jakes model Simulated m = 0.5 y Normalized correlation R Lags Take Away: The correlation of the Nakagami-m sequence agrees with the Predefined Correlation Model Speaker Name

Simulator Input/Output Realizations Presentation Title June 24, 2018 Simulator Input/Output Realizations Take Away: Rayleigh to Nakagami-m Mapping works even for the worse case m=0.5 Speaker Name

Simulator Autocorrelation vs. Theory Presentation Title June 24, 2018 Simulator Autocorrelation vs. Theory Take Away: The correlation of the Nakagami-m sequence from the Rayleigh model agrees with the Predefined Correlation Model Speaker Name

Simulator Phase vs. Theory Presentation Title June 24, 2018 Simulator Phase vs. Theory Take Away: The simulated phase agrees with theory for m=1.0,2.0,3.0,4.0 For m=1.0, it is uniform, as predicted by research in the available literature Speaker Name

Presentation Title June 24, 2018 Summary An Accurate and systematic radar channel simulation technique is critical for performance evaluation of our radar systems. Our current models do not include severe target fluctuations due to environment Amplitude Target fluctuations due to ionospheric scintillations are Nakagami-m. No Phase or correlation information was included in the original model In this presentation, we introduced a Nakagami-m model with the following properties: It satisfies an arbitrary pre-specified Autocorrelation function (open problem) It satisfies a Nakagami-m Phase consistent with the pre-specified Autocorrelation function (open problem) It allows means for mapping a Rayleigh model to the Nakagami-m model Our results show that the proposed Nakagami-m model satisfies: The theoretical properties of Nakagami-m amplitude The theoretical properties of Nakagami-m phase An arbitrary pre-specified Autocorrelation model Speaker Name

Literature Cited and References Presentation Title June 24, 2018 Literature Cited and References M. Nakagami, “The m-distribution, a general formula of intensity distribution of rapid fading,” in Statistical Methods in Radio Wave Propagation, W. G. Hoffman, Ed., Oxford, U.K., Pergamon, 1960. A. Papoulis, “Probability, Random Variables, and Stochastic Processes,” 3rd ed. New York: McGraw-Hill, 1995. N. Beaulieu, C. Cheng, “Efficient Nakagami-m Fading Channel Simulation”, IEEE Transactions on Vehicular Technology, Vol. 54, No2, March 2005. M. D. Yacoub, G. Fraidenraich and J.C.S, Filho, “Nakagami-m Phase-envelope joint distribution”, Electronics Letters 3rd ed, Vol. 41, No. 5, March 2005 Speaker Name

QUESTIONS?? Final Comments Presentation Title June 24, 2018 Speaker Name