Figure 2

(a–e) Use of the FFT with time-domain windowing to determine the phase difference between two mono-sinusoidal signals. (a) Simulated 12-bit reference (red) and object (blue) signals, with a phase difference of \(\,{\phi }_{{\rm{obj}}}-{\phi }_{{\rm{ref}}}=-\,\pi /3\). (b) Input signals after time-domain windowing using the Hann function. (c) DFT magnitude spectra of windowed input signals, calculated using the FFT. The vertical line at \(k=9.83038\) denotes the mean signal frequency extracted from the DFT using Eq. (8). (d) DFT phase-spectra of windowed input signals, calculated using the FFT. (e) Differential phase-spectrum \(\Delta \phi ={\phi }_{{\rm{obj}}}-{\phi }_{{\rm{ref}}}\), wrapped to lie in the range \(-\pi < \Delta \phi \le \pi \). The phase-difference \(\Delta \phi =-\,1.04711\) is obtained by linear interpolation of the differential phase-spectrum to \(k=9.83038\). (f–k) Use of the sDFT with frequency-domain windowing to determine the phase difference between two mono-sinusoidal signals. (f) Simulated 12-bit reference (red) and object (blue) signals, with a phase difference of \(-\pi /3\). (g) DFT magnitude spectra of input signals, calculated using the sDFT. (h) DFT phase-spectra of input signals, calculated using the sDFT. (i) DFT magnitude spectra of input signals after frequency-domain windowing. The vertical line at \(k=9.83038\) denotes the signal frequency extracted from the windowed DFT using Eq. (8). (j) DFT phase-spectra of input signals after frequency-domain windowing. (k) Differential phase-spectrum \(\Delta \phi ={\phi }_{{\rm{obj}}}-{\phi }_{{\rm{ref}}}\), wrapped to lie in the range \(-\pi < \Delta \phi \le \pi \). The phase difference \(\Delta \phi =-\,1.04711\) is obtained by linear interpolation of the differential phase-spectrum to \(k=9.83038\).