Fig. 1. (a) Support of the continuous signal f(x) on x. (b) Support of the continuous IAF of f(x) on x and τ.

Fig. 2. (a) Naive sampling method. (b) CM sampling method.

Fig. 3. (a) PSD of the continuous test signal f(τ) based on its LCT width and (b) based on its Fourier bandwidth.

Fig. 4. (a) PSD of f(τ2) based on f(τ)’s LCT width. (b) PSD of f(τ2) based on f(τ)’s Fourier bandwidth. (c) PSD of f*(−τ2) based on f(τ)’s Fourier bandwidth.

Fig. 5. (a) PSD of f(x+τ2) based on f(τ)’s LCT width and (b) PSD of f*(x−τ2) based on f(τ)’s Fourier bandwidth.

Fig. 6. (a) and (b) Two extreme conditions of the shift process in the convolution.

Fig. 7. Conditions (a) and (b) define the vertical lines’ width of Rf(x,τ)’s PSD. (c) The PSD of the IAF, Rf(x,τ).

Fig. 8. (a) PSD of Rf(nT,2mT). PSD of Rf(nT,2mT) (b) after the first chirp multiplication, (c) after the discrete-space Fourier transform, (d) after the scaling operation, and (e) after the second chirp multiplication.

Fig. 9. PSD of the IAF (a) after multiplying by a chirp and (b) after periodizing.

Fig. 10. (a) PSD of the chirp-periodic IAF. Its PSD (b) after the sampling process, (c) after the first chirp multiplication, (d) after the DFT, (e) after the scaling operation, and (f) after the second chirp multiplication.

Fig. 11. (a) Spectrum of the Gaussian test signal. (b) The LCT of the Gaussian test signal when a=4, b=3, and d=1.

Fig. 12. (a) Simulated discrete BDF when a=1, b=3, d=1, and fs=13Hz. (b) Analytical discrete BDF. (c) Error between the simulation and analytical solution.

Fig. 13. (a) Simulated discrete BDF when a=1, b=3, d=1, and fs=6.5Hz. (b) Analytical discrete BDF. (c) Error between the simulation and analytical solution.

Fig. 14. Error curve of the discrete BDF for different sampling frequencies and different b values.

Fig. 15. Error curve of the discrete BDF for different sampling frequencies and different a values.