Time-Domain Waveforms for Remote Materials Characterization

This report presents an alternative derivation of the solution given in 1 in terms of the general problem of determining the impulse response of a linear, time-invariant system. The integral equation with chirp excitation and matched filter operator arc shown to lead to a Fourier integral representation of the Dirac delta function, which immediately yields the scaling relation shown in 1. Important practical ramifications of this approach, including the noise properties of the matched filter and the effects of finite- duration chirps are explored. The continuous time integral equation is then discretized in a straight forward manner, and a magic number is presented that allows for the exact recovery of an arbitrart impulse response.