Mathematical questions raised by the non-uniform Doppler effect John E. Gray Electromagnetic and Sensor Systems Department, Naval Surface Warfare Center Dahlgren, 18444 FRONTAGE ROAD SUITE 328, DAHLGREN VA 22448-5161 ICERM Mathematical and Computational Aspects of Radar Imaging (October 16-20, 2017) Description: This workshop will bring together mathematicians and radar practitioners to address a variety of issues at the forefront of mathematical and computational research in radar imaging. Some of the topics planned include shadow analysis and exploitation, interferometry, polarimetry, micro-Doppler analysis, through-the-wall imaging, noise radar, compressive sensing, inverse synthetic-aperture radar, moving target identification, quantum radar, multi-sensor radar systems, waveform design, synthetic-aperture radiometry, passive sensing, tracking, automatic target recognition, over-the-horizon radar, ground-penetrating radar, and Fourier integral operators in radar imaging. Abstract The non-uniform motion Doppler effect in radar occurs when an object being tracked by a radar is undergoing any type of motion that is other than constant speed. Examples include: accelerations, jerk motion, exponential slowdown, and periodic motion such as rotation, vibration, or what is now termed micro-Doppler. I review a physics model based on a perfectly reflecting mirror which captures the essential features of the physics of non-uniform laws of motion. Then, we discuss the frequency spectrum of these types of motion as well as raise some questions for signal analysis for this type of physics. Finally, we pose some interesting questions that might be of interest to mathematicians concerning direct and inverse problems associated with observing non-uniform motion data.
INTRODUCTION: NON-UNIFORM MOTION DOPPLER AND ONE DIMENSIONAL SIGNALS Since the inception of coherent waveforms, it has been realized that the effect of the motion of a non-point like object can induce structure in the return spectrum of the waveform, see for example in the electromagnetic literature. The explosion of interest since Chen's seminal papers has been centered on micro-Doppler, which is based on periodic motion about the central Doppler line, which have proven to have enormous applications (see Chen's recent books). While not as extensive in terms of potential applications, the non-uniform Doppler has applications to automobile radars, astrophysics, and moving sources, as well as other applications. We review the non-uniform Doppler and demonstrate some useful information that can be found in the spreading of the Doppler spectrum for the motion models: acceleration, jerk, quadric, and exponential slowdown as examples well as a characteristic of periodic motion. 2
INTRODUCTION: NON-UNIFORM MOTION DOPPLER AND ONE DIMENSIONAL SIGNALS The laws of motion for non-uniform behavior can be represented functionally by equations that are non-periodic that can be represented by Taylor series 𝑛 (𝛽𝑢) 𝑗+1 𝑠 𝑜 (𝑢) = ±𝑤 𝑆 𝑢 + ∑ . 𝑗=1 The laws of motion for non-uniform behavior can be represented functionally by the equation that are 𝑛 𝑞 (𝑢) = ±𝑤 𝑆 𝑢 + ∑ cos(𝜕 𝑗 𝑢) 𝑠 𝑐 𝑗 𝑗=1 for periodic motion. The periodic motion produces a modulation term in the FM portion of the signal. Assume that 𝑠(𝑢) represents the law of motion of a perfectly reflecting electromagnetic boundary, what is a referred to as a mirror in the optical domain; and let the return signal be (𝜐) which has interacted with a moving boundary. 3
INTRODUCTION: NON-UNIFORM MOTION DOPPLER AND ONE DIMENSIONAL SIGNALS • If we assume that r(t) represents the law of motion of a perfectly reflecting electromagnetic boundary, what is a referred to as a mirror in the optical domain; then the return signal, (𝜐) , from the boundary can be shown to be (𝜐) = − 𝑒𝐺(2ℎ(𝜐) − 𝜐) , 𝑒𝜐 where ℎ(𝜐) is the solution to the functional equation: ℎ(𝜐) + 𝑠(ℎ(𝜐)) = 𝜐. 𝑑 𝜐 𝑔(𝑦)𝑒𝑦. • Note the broadcast waveform's functional form is 𝑔(𝜐) and 𝐺(𝜐) = ∫ • Note, the functional equation for ℎ(𝜐) can be solved sometimes for some types of boundaries undergoing non-uniform motion, but for most radar applications it is sufficient to use the radar 𝑠(𝜐) approximation: ℎ 1 (𝜐) = 𝜐 − 𝑑 , so the return signal can be represented as 1 (𝜐) = 2𝑠(𝜐) 𝑒 − 𝑒𝜐 𝐺 1 (𝜐 − 𝑑 ). 4
INTRODUCTION: NON-UNIFORM MOTION DOPPLER AND ONE DIMENSIONAL SIGNALS • Note, the functional equation for ℎ(𝜐) can be solved sometimes for some types of boundaries undergoing non-uniform motion, but for most radar applications it is sufficient to use the radar approximation: ℎ 1 (𝜐) = 𝜐 − 𝑠(𝜐) , 𝑑 • The return signal can be represented as 1 (𝜐) = − 𝑒 1 (𝜐 − 2𝑠(𝜐) 𝑒𝜐 𝐺 ). 𝑑 • For all our subsequent work, this is sufficient for analyzing the return signal's spectrum. 5
Observables in the Doppler Spectrum for CW Waveform Recall the CW spectrum for a law of motion 𝑠(𝜐) is 1 (𝜐) = 𝐵 exp(𝑗𝜕 0 𝜐 − 𝑗𝑙𝑠(𝜐)) [Θ(𝑢) − Θ(𝑢 − 𝑈)] 2𝜕 0 where 𝑙 = 𝑑 . From the definition of instantaneous frequency 𝜒 𝑗 (𝑢) is for this spectral function 𝜒 𝑗 (𝜐) = 𝜕 0 + 𝑙𝑠′(𝜐) , so 〈𝜕〉 , 〈𝜕 2 〉 , 𝜏 𝜕 2 and therefore 2𝑠′(𝜐) ∞ T 〈𝜕〉 = ∫ ∗ = 𝐵 2 𝜕 0 ∫ (1 + 1 (𝜐)𝜒 𝑗 (𝜐) 1 (𝜐) ) 𝑒𝜐. −∞ 0 𝑑 The standard deviation in the frequency is 2 T 2 = ∫ (𝐵 2 𝜕 0 (1 + 2𝑠′(𝜐) ) − 〈𝜕〉) 𝜏 𝜕 𝑒𝜐 , 𝑑 0 because 𝐵 2 (𝜐) = 𝐵 2 . The examples of different motion models in the Doppler Section can be easily computed in the 𝜐 domain. 6
Observables in the Doppler Spectrum for CW Waveform All of the subsequent integrals 𝐻 1 𝑠(𝜐) (𝜕) we consider can be written in the form 𝑈 𝑓 −𝑗𝜕 ′ 𝜐 exp(−𝑗𝑙𝑠(𝜐))𝑒𝜐 = 𝐻 𝑠(𝜐) (𝜕) ∗ 𝑄 𝑈 (𝜕 ′ ) 𝑠(𝜐) (𝜕) = A′ ∫ 𝐻 1 0 where ∞ 𝑓 −𝑗𝜕 ′ 𝜐 exp(−𝑗𝑙𝑠(𝜐)) 𝑒𝜐, 𝐻 𝑠(𝜐) (𝜕) = 𝐵′ ∫ −∞ and 𝑈 ∞ [Θ(𝑢) − Θ(𝑢 − 𝑈)] 𝑓 −𝑗𝜕 ′ 𝜐 𝑒𝜐 = exp(−𝑗𝜕 ′ 𝜐) = 𝑈 2 𝑓𝑦𝑞 [ 𝑗𝑈𝜕′ 2 ] 𝑡𝑗𝑜𝑑 [ 𝑈𝜕′ 𝑄 𝑈 (𝜕 ′ ) = ∫ | 2 ]. 𝑗𝜕′ −∞ 0 Thus, for some of the integrals we evaluate the integral explicitly, while other times we determine 𝐻 𝑠(𝜐) (𝜕) only. Note all of the boundaries that we present in subsequent examples do not include a velocity term. If one wants to include the velocity all that is needed in the subsequent examples is to reinterpret 𝜕′ as ±𝑙𝑤 𝑝 − 𝜕 ′ = ω′ in subsequent equations. 7
Examples (Constant Acceleration) For a constant acceleration or CA-boundary, the law of motion is 𝑠(𝜐) = − 1 2 𝑏 0 𝜐 2 , where 𝑏 0 is the acceleration of the boundary. 𝑏 0 𝜕 0 Then, the Doppler spectrum is 𝑙 = 𝑑 2 2 𝑈 𝑈 𝑓 −𝑗𝜕 ′ 𝜐 exp(𝑗𝑙 ′ 𝜐 2 ) 𝑒𝜐 = 𝐵′𝑓𝑦𝑞 (−𝑗 { 𝜕′ exp (𝑗 [√𝑙 ′ 𝜐 − 𝜕′ 𝐻 𝐷𝐵 1 (𝜕) = 𝐵′ ∫ 2√𝑙 ′ } ) ∫ 2√𝑙 ′ ] ) 𝑒𝜐, 0 0 2 𝐵′𝑓𝑦𝑞(−𝑗{ 𝜕′ 2√𝑙′ } ) 𝜕 ′ Let √𝑙 ′ 𝜐 − 2√𝑙 ′ = 𝑧 , 𝑒𝑧 = √𝑙 ′ 𝜐, 𝐵′′ = , then 𝑙 √𝑙 ′ 𝑈− 𝜕′ exp(𝑗𝑧 2 ) 𝑒𝑧 = 𝐵′′ [𝐿 ( 𝜕 ′ 2√𝑙 ′ ) − 𝐿 (√𝑙 ′ 𝑈 − 𝜕 ′ 2√𝑙′ 𝐻 𝐷𝐵 1 (𝜕) = 𝐵′′ ∫ 2√𝑙 ′ )] − 𝜕′ 2√𝑙′ where 𝐿 is the Fresnel integral. T 〈𝜕〉 𝑑 = 𝐵 2 𝜕 0 ∫ (± 2𝑏 0 𝜐 ) 𝑒𝜐 = ±𝐵 2 𝜕 0 𝑈 2 (𝑏 0 𝑑 ) 𝑑 0 2 2 𝑈 3 2 2 2𝐵 2 𝜕 0 𝑏 0 2 = (𝐵 2 𝜕 0 ) 2 ( 2𝑏 0 T 𝑈 and 𝜏 𝜕 𝑑 ) ∫ (𝜐 − ) 𝑒𝜐 ≈ ( ) 3 . 0 2 𝑑 8
Examples (Constant Jerk) For a law of motion 𝑠(𝜐) = 1 6 𝑘 0 𝜐 3 , where 𝑘 0 is the "constant jerk" of the boundary. Then, the spectrum given by the expression 1 ( 𝑙𝑘0𝑈3 3 1 𝑈 ) 3 𝑧 exp (− 𝑗𝑧 3 𝑓 −𝑗𝜕 ′ 𝜐 exp (−𝑗 𝑙𝑘 0 𝐵′ 𝑓 −𝑗𝜕 ′ ( 𝑙𝑘0 2 2 ) 𝐻 𝐷𝐾 6 𝜐 3 ) 𝑒𝜐 1 (𝜕) = 𝐵′ ∫ ∫ 3 ) 𝑒𝑧. 1 𝑙𝑘 0 0 3 0 ( 2 ) 𝑧 3 Now with the definition of the Airy function : 𝐵𝑗(𝑦) = ∫ ∞ exp (−𝑗 ( 3 + 𝑦𝑧)) 𝑒𝑧 , with, 𝑦 = −∞ 1 1 1 𝑙𝑘 0 𝑈 3 𝜕 ′ ( 𝑙𝑘 0 3 𝑙𝑘 0 𝑈 3 , we have ( ( 3 2 ) 16 ) ⁄ ( 2 ) ) = 2 1 1 1 ) ∗ 𝑓𝑦𝑞 [𝑗 (𝑙𝑘 0 𝑈 3 𝜕 ′ ] 𝑡𝑗𝑜𝑑 [𝑗 (𝑙𝑘 0 𝑈 3 1 (𝜕) = − 𝐵′𝑈 2 𝐵𝑗 (𝜕 ′ (𝑙𝑘 0 3 3 3 𝐻 𝐷𝐾 𝜕 ′ ] 2 ) 16 ) 16 ) Similar expressions occur in diffraction theory for circular apertures. 9
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