Curved structures in recurrence plots: The role of the sampling time

We give a theoretical explanation of the formation of the curved macropatterns in the recurrence plots of sinusoidal signals, with nonstationarity in the phase or in the frequency. We show that the large time scales observed and the curved structures are the artificial product of the discretization of the signal. Recurrence plots are highly sensitive to the phase error introduced by the sampling, and we show that this characteristic can be used to detect very small $(\sim 0.5\%)$ phase or frequency shifts of the carrier frequency.

Figure 1

The RP of the signal $\sin \left(2\pi 1000t\right)$, with $f_s=11025\mathrm{Hz}$. Here two time scales are clearly visible. The smaller is related to the period of the signal, the larger is due to the interference effect.

Figure 2

(a) Recurrence plot of the signal $\sin (2\pi 1000t+2\pi 50t^2)$. (b) $\sin (2\pi 1000t+2\pi 15t^2)$, $f_s=11025\mathrm{Hz}$.

Figure 3

(a) Recurrence plot of the signal $\sin [2\pi 1000t+2\pi \sin (2\pi 10t\left)\right]$. (b) Recurrence plot of the signal $\sin [2\pi 1000t+2\pi \sin (2\pi 10t\left)t\right]$. $f_s=11025\mathrm{Hz}$.

Figure 4

The plot of the signal $\sin \left(2\pi f_{c}t\right)$, with $f_c=1000\mathrm{Hz}$ and $f_s=11025\mathrm{Hz}$, every 11 samples. This shows how the position of the zeros changes with time. The other samples follow the same sinusoidal path.

Figure 5

(a) Motion of the state vector ${\mathbf{x}}_k\left(0\right)$ in the phase space [$(x,y)$ plane] and in time ($z$ axis). Instead of moving on a straight line, the path is a helix. (b) The path of the sample $s_k\left(0\right)$ in the time. The circles indicate the time at which a new correct recurrence happens.

Figure 6

(Color online). Recurrences of the state vector $\mathbf{x}\left(0\right)$ with ${\mathbf{x}}_{i{k}^{\prime }}\left(i\right)$ $(i=1,\dots ,\alpha )$. Because of the motion of the vectors, $\mathbf{x}\left(0\right)$ recurs with itself and with the other $\alpha$ vectors.

Figure 7

The MF in function of $f_c$ for a nonmodulated sinusoid. Despite the simplicity of the signal, the MF shows with the increase of the frequency a nontrivial behavior. The frequencies corresponding to $\beta =0$ correspond theoretically to $f_M=0$.

Figure 8

(Color online). The orders of the MF $\left(f_{M_n}\right)$ for the unmodulated signals.

Figure 9

The MF in function of the time, $f_c=1000$, $\gamma =15$.

Figure 10

The MF in function of the time, $f_c=1000$, $f_m=10$.

Figure 11

The MF in function of the time, $f_c=1000$, $f_m=10$.