Fourier transform approximations for sweeps and phase‐encoded sweeps

Abstract

The Fourier transform of a real sweep is often approximated by a complex exponential. This approximation fails under certain conditions related to taper length and bandwidth. Our intuitive graphical explanation using the Cornu spiral is supported by rigorous proof that the radius of the spiral decreases monotonically to zero. This proof is also useful elsewhere in diffraction study and other Fourier analyses of sweeps. With this approach, we found that the Fourier phase function of an approximate formulation of a phase‐encoded sweep is affected in the same manner by the sweep parameters and the length of the taper. If the mean phase value is computed over the bandwidth, this mean phase agrees better with the ideal encoded phase. This simple phase‐encoded sweep can also be improved computationally so that its individual phase component approximates more closely with the ideal. We conclude that, other than some time‐compressed phase‐encoded wavelets known to be useful in direction determination, our approximate and improved phase‐encoded sweeps are equally applicable as well.