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 (Gray and Addison, Gray et-al). There are many ways to think about this, but a newer one is to think of the law of motion induced onto the scattered waveform as being "generated" by an operator to produce the law of motion (Gray and Parks). One can use this observation to design a receiver's matched filter response specific to these type of operators that generate a law of motion. We use Poisson bracket method for generating a Taylor series solution to a functional equation to illustrate how to do this. In addition, one can also do the same for three dimensional periodic motion using the rotation matrices. We illustrate this by developing the matched filter response for a rotating object in terms of the rotation matrix.
John E. Gray, Allen D. Parks, and Jeremiah J. Hansen, "The operator approach to the non-uniform Doppler to radar: implications for signal processing," Proc. SPIE 10188, Radar Sensor Technology XXI, 101880R (Presented at SPIE Defense + Security: April 11, 2017; Published: 1 May 2017); https://doi.org/10.1117/12.2262126.
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