DFT of a Pure Real Tone: A Geometrical Perspective

This paper gives a new exact formula for the Discrete Fourier Transform (DFT) of a pure real tone and compares it to several existing approaches. The Goertzel method is used as a tool for the theoretical demonstration. A major benefit of our approach is that the phase of the sine signal is now disentangled from the different parameters of interest. This new expression leads us to some insightful observations in the context of phase tracking. Indeed our new formulation reveals that the DFT bin is moving along an elliptical path if only the phase of the tone is changing and the frequency is fixed. A simulation study as well as an FMCW radar experiment demonstrates the validity of the approach by reformulating phase tracking as the locus estimation of a point on an ellipse.