An introduction to the angular Fourier transform

The author introduces the angular Fourier transform (AFT), a generalization of the classical Fourier transform. The AFT can be interpreted as a rotation on the time-frequency plane. An AFT with an angle of alpha = pi /2 corresponds to the classical Fourier transform, and an AFT with alpha =0 corresponds to the identity operator. The angles of successively performed AFTs simply add up, as do the angles of successive rotations. A number of properties of the AFT are given. Most important among these are the AFT´s relationships with time-frequency representations such as the Wigner distribution, the ambiguity function, the short-time Fourier transform, and the spectrogram. These relationships have a very simple and natural form, which further enhances the AFT´s interpretation as a rotation operator. An example of the application of the AFT to the study of swept-frequency filters is given.<>