Applications to Radar and Scattering
In this chapter we propose an application of electromagnetic wavelets to radar signal analysis and electromagnetic scattering. The goal in radar, as well as in sonar and other remote sensing, is to obtain information about objects (e.g., the location and velocity of an airplane) by analyzing waves (electromagnetic or acoustic) reflected from these objects, much as visual information is obtained by analyzing reflected electromagnetic waves in the visible spectrum. The location of an object can be obtained by measuring the time delay ז between an outgoing signal and its echo. Furthermore, the motion of the object produces a Doppler effect in the echo amounting to a time scaling, where the scale factor s is in one-to-one correspondence with the object's velocity. The wideband ambiguity function is the correlation between the echo and an arbitrarily time-delayed and scaled version of the outgoing signal. It is a maximum when the time delay and scaling factor best match those of the echo. This allows a determination of s and ז, which then give the approximate location and velocity of the object. The wideband ambiguity function is, in fact, nothing but the continuous wavelet transform of the echo, with the outgoing signal as a mother wavelet! When the outgoing signal occupies a narrow frequency band around a high carrier frequency, the Doppler effect can be approximated by a uniform frequency shift. The wideband ambiguity function then reduces to the narrow band ambiguity function, which depends on the time delay and the frequency shift. This is essentially the windowed Fourier transform of the echo, with the outgoing signal as a basic window. The ideas of Chapters 2 and 3 therefore apply naturally to the analysis of scalar-valued radar and sonar signals in one (time) dimension. But radar and sonar signals are actually waves in space as well as time. Therefore they are subject to the physical laws of propagation and scattering, unlike the unconstrained time signals analyzed in Chapters 2 and 3. Since the analytic signals of such waves are their wavelet transforms, it is natural to interpret the analytic signals as generalized, multidimensional wideband ambiguity functions. This idea forms the basis of Section 10.2.
KeywordsElectromagnetic Wave Conformal Transformation Lorentz Transformation Mother Wavelet Ambiguity Function
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