The Doppler spectrum for accelerating objects - IEEE Xplore

The Doppler Spectrum for Accelerating Objects

Eq. (lb) is the one that is widely used by radar engineers. The model of the perfectly reflecting mirror is the basis for understanding the Doppler effect in radar, hence generalizations of this model to include non-inertial motion should provide additional information about the Doppler spectrum. There are two methods for treating the non-inertial movement of boundaries correctly. One method is to transform the fields in such a way that the boundary "appearst1to be at rest to the fields. The boundary conditions are applied in the rest frame. The scattered fields are then retransformed back to the original reference frame. The second method is to transform the boundary conditions to a moving reference frame and solve the scattering problem with moving boundary conditions. Each method has advantages and disadvantages. The second method is easier to apply for one-dimensional fields, but becomes computationally difficult in three dimensions. The first method works well in three dimensions when there is a rotational axis of symmetry for the scattering body (Refs. 5-7). Renewed interest in this area was sparked by Van Bladel in (Ref. 9 ) and MO in (Ref. 8 ) . A comprehensive treatment is contained in Van Bladel (Ref. 5) which considers the entire subject of relativity. Censor, Cooper, and De Smedt (Refs. 101 2 and 1 3 , pp. 633, as Well as Ref. 5, pp. 309-315) have independently arrived at .a general method for treating the arbitrary motion of mirrors that makes them amenable to examination in the Fourier domain. Their method uses the transformed boundary conditions instead of the transformed fields. The formulation that they have developed was not extended to the point that it could be applied to determine the Fourier spectrum of various types of waveforms. In a recent paper (Ref. 14), this formalism was extended to determine the Fourier spectrum for arbitrary motion of a moving mirror with an emphasis on continuous wave (CW) waveforms. The relationship between the incident and scattered field is now derived. In Figure I, let the incident (I) electric and magnetic fields be represented by

John E. Gray Naval Surface Warfare Center Dahlgren, Va. 2 2 4 4 8 (703-663-1110)

Abstract In this paper, the formalism for computing the Doppler spectrum for perfectly reflecting mirrors undergoing various types of accelerations is reviewed. This method is an amplification of work done by Censor and Cooper for one dimensional waves. For sinusoidal waves, the formalism provides a computationally easy algorithm that enables determination of the Doppler spectrum. This method is exact and does not ignore the effects of motion on the amplitude a s is normally done. The mirror is an alternative means of determining the Doppler spectrum of noint particles. From the exact result, an approximation method is derived that is of use to radar engineers. Extending these, results to other commonly used radar nraveforms is then considered. Part I Introduction The Doppler effect (Refs. 1 - 2 ) is a result of the relative motion of a signal source and manifests itself as a frequency shift of the wave. This covers a wide variety of phenomena that is presented in Gill (Ref. 2 ) . The frequency shift of a wave provides valuable information in some applications, such as a Doppler radar, while in other applications the shift is an unwanted effect that is designed around, such as in a TrackWhile-Scan (TWS) radar. The problem with the treatment of the Doppler effect in radar books (Ref. 3) is that the method of approximation cannot be generalized to obtain information about the effects of non-uniform motion on the spectrum of a return signal. The Doppler effect i s a purely relativistic effect that is derived by transforming a waveform to an object's moving frame, having it undergo a reflection, and then a translation back to the rest frame of the transmitted waveform. An exact derivation of the Doppler effect starts with either a point particle or an infinite, perfectly reflecting mirror. Dynamically, the assumption of a point particle is really the same as a mirror. The perfectly reflecting mirror was first solved by Einstein in his famous paper on relativity (Ref. 4). The formula for the shifted frequency, wR, is ( p = vR/c)

- p)/('

WR = w[(l

+

PI1

(la)

which can be approximated by wR = w(1

-

2p),

for p