Joint time–frequency offset detection using the fractional Fourier transform

In this paper, a parametric correlation measure is introduced based on the fractional Fourier transform. Based on fundamental properties of this transform, maximum correlation is identified with a line structure in the time–frequency plane. Correlating measured and model data for different values of the parameter yields intersecting lines in the time–frequency plane, whose intersection is identified as the joint time–frequency offset of the measured data, as compared to the model data. Both theoretical and practical aspects of the method are discussed, while examples illustrate both the performance of the method and the geometric concept behind. The approach is particularly designed for analysing measured signals with different onset time (travel time) and frequency shift (Doppler), as compared to a given signal model.