Continuous-time signals

Discrete-time and continuous-time signals
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Contents:

  • Explore our Catalog
  • Organisation
  • Connexions

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    Access provided by: anon Sign Out. Note that in discrete time we almost always work with frequency in units of radians. This reveals the golden relationship between continuous-time and discrete-time frequency :.

    Continuous-Time Signals

    Let's run through a few examples. Suppose you have a continuous-time sinusoid with frequency Hz and sample it with a sampling frequency of Hz so seconds. The frequency of the discrete-time signal is thus radians. Of course we can also use this relationship to convert from discrete-time to continuous-time frequency given the sampling interval. If the sampling frequency is Hz seconds , then a discrete-time sinusoid with frequency radians corresponds to a continuous-time sinusoid of frequency Hz.

    Discrete-time sinusoids with different frequencies are only unique for frequencies within a interval. Indeed, the future dynamics of the data could be forecasted based on that of previous patterns similar to the considered one Farmer and Sidorowich, In this way, it approximates the estimation of entropy, which is the rate of information production Eckmann and Ruelle, Its definition requires the steps detailed below, which can be interpreted within the framework of time series embedding and prediction Kantz and Schreiber, Notice that the proposed interpretations hold for a sampled continuous time signal.

    Integration of Continuous-Time Signals

    However, ApEn found many important biomedical applications to such data, like in the fields of neuronal spiking activity Yang et al. The scalar time series is embedded into a phase space of vectors also called runs or templates of delayed coordinates or phases. The correlation integral indicates the probability that the embedded vector X i is similar to other vectors within a tolerance r.

    The vectors X n , where n indicates the arbitrary sample running on time, describe a trajectory in the phase space. Thus, the definition of correlation integral C i m r requires to count the number of recurrences N i r of the trajectory to points close to X i and to divide by the number of possible pairs thus, estimating the percentage of neighboring points of X i or the probability that the trajectory has recurrences close to it.

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    The study of recurrences also called neighboring points or matches is very important in the non-linear analysis of time series and lead to the definition of many indexes of complexity and non-linearity Webber and Zbilut, ; Kantz and Schreiber, Finally, ApEn is defined based on the correlation integral, computed for two embedding dimensions. The number of recurrences is higher in the lower dimension. Thus, the reduction of the number of recurrences is related to the divergence of trajectories that were close in dimension m , but not when adding an additional sample.

    Such a divergence is a marker of complexity and an indication of low predictability of the time series measured also by other non-linear indexes, like Lyapunov exponent, Kantz, , or determinism in recurrence quantification analysis, Webber and Zbilut, Notice that the study of the divergence of the trajectory in the phase space is feasible only if the embedding dimension is large enough to rule out the false near neighbors Kantz and Schreiber, Indeed, when m is small, the trajectory may show intersections or recurrences due to its projection in a low dimensional space.

    The dynamics of the system around false neighbors i. Thus, the inclusion of false recurrences can bias the estimation of ApEn. The following considerations suggest that parameters should be chosen carefully in order that the information extracted by ApEn is reliable and that possible problems tested in the following are avoided. As the delay between phases is fixed to be 1, over-sampling a signal corresponds to a linearization, which is expected to reduce ApEn.

    Indeed, it is simpler to predict the subsequent sample of a time series if data are over-sampled, as the new sample is close to the previous ones. Thus, the number of recurrences remains about constant in different embedding dimensions the trajectory has not enough time to diverge and the time series appears to be more predictable and hence less complex.

    Notice also that the maximum value of ApEn is ln N - m which is enlarging as the sampling frequency increases as N becomes larger if the epoch duration is fixed ; thus, the estimation of ApEn relative to its maximum possible value is further decreased by over-sampling the data. In the applicative papers quoted in the Introduction, a filter usually selected a specific bandwidth of EEG. On the other hand, if only an anti-aliasing filter is used, the over-sampled data include high frequency noise.

    In such a case, the divergence of a recurrent point is related to the noise, more than to the deterministic dynamics of the trajectory in the phase space. Thus, in such a condition, ApEn is expected to provide an estimation of the complexity of the noise, failing to decode the determinism of the system.

    Some considerations can be given to justify the relation between the embedding dimension and the number of samples needed to study reliably the time series. If the time series is embedded in a space of dimension m , the number of boxes in which to find recurrences is about As each recurrent point should have many neighbors, e. The previous example should be considered as a worst case, as deterministic trajectories are usually attracted on a portion of the phase space with dimension that is less than m Kantz and Schreiber, In such a case, the measure of complexity or predictability related to the percentage of neighbors that remain close to each other when extending the dimension is not statistically feasible.

    Using short epochs, the estimation of ApEn could become quite unstable, as it depends on the behavior of rare events that are poorly represented. Indeed, similar patterns of activity are not identified as recurrences if they are biased by a trend that translates them around different mean values with difference larger than r. Thus, the use of a high-pass filter with cutoff related to the duration of the epoch is suggested to remove low frequency trends. However, more recurrences should be studied to assess statistically the predictability of low frequency oscillations. However, notice that the removal of trends could be admissible only under some assumptions on the data.

    For example, it is reasonable if the signal is self-similar Mandelbrot, , so that the same behavior can be found at different scales, as in the case of fractal dynamics or if different frequency components are independent e. On the other hand, there could be conditions in which low frequency trends affect the behavior of other components; indeed, they could be related to a slow change in the state of the system generating the time series, which could affect the patterns of interest.

    In such cases, low frequency trends cannot be removed and could be studied only if the investigated epoch is extended. This could be admissible only if the data are stationary within it, so that a trade-off arises between the needs of studying reliably low frequency trends requiring a long epoch and considering stationary the data imposing a short epoch.

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    This allows to remove possible singularities, i. However, this introduces a bias toward low values of ApEn, as N - m -1 self-recurrences are always found in both embedding spaces Pincus and Goldberger, As an alternative, a simpler way to overcome the problem could be the introduction of a Theiler window Kantz and Schreiber, , i. Based on the previous observations, the modified ApEn is proposed.

    The following variations are included. This idea was also considered in the minimum numerator count method Lake and Moorman, This allowed to get stable estimations of quadratic sample entropy SampEn modified to make it stable to a variation of r with very short RR series.

    Organisation

    Consider also that the mathematical meaning of ApEn as an index of complexity could even fail when parameters are out of specific ranges. A sequence can be defined analytically in closed form, as for example:. In such a case, the divergence of a recurrent point is related to the noise, more than to the deterministic dynamics of the trajectory in the phase space. As we can infer from this simple analysis, IIR filters are much more delicate entities than FIR filters; in the next Chapters we will also discover that their design is also much less straightforward and offers less flexibility. Which operator do you think it is associated to? There is the suspect that even when it is not clearly stated parameters have been chosen adapting to the data in order to get specific outcomes, e.

    The literature proposed also other alternatives, as that of choosing the value of r corresponding to a maximal ApEn or an approximation of it, in order to reduce the computational burden, Lu et al. This method showed good results in the literature Lu et al. Further advanced techniques have been proposed to select an optimal value of the tolerance based on the asymptotic theory of bandwidth selection for kernel density estimators Lake, ; Darmon, or building efficiently an ApEn profile using different values of tolerances Udhayakumar et al.

    In this way, the index is always non-negative. ApEn and the modified index were tested on simulations Figures 1 — 5 and experiments Figure 6. Representative examples are considered for which the reliability of ApEn is pushed to the limit. Estimation of complexity indexes of signals with different sampling frequencies. Estimation of complexity indexes considering different embedding dimensions m. The same signals as in Figure 1 are considered, sampled at three times the bandwidth of the signals.

    Estimation of complexity indexes from epochs of different durations. The same signals as in Figure 1 are considered, sampled at three times their bandwidth. Points without matches and their impact on complexity estimation. A The same regular signal considered in Figure 1 called signal 1 and sampled at 3 times the bandwidth is processed, considering different embedding dimensions ED and numbers of samples N a single epoch was considered for each test.

    Connexions

    B Same as A , but for the chaotic signal considered in Figure 1 signal 2. C Percentage difference of the ApEn of the chaotic and regular signals positive number expected. D Same as C , but considering the modified ApEn. Estimation of complexity indexes considering fractional Brownian motions fBm with different Hurst exponents.

    A Epochs of different signals they were assumed to be sampled at Hz. B ApEn median, quartiles, range, and outliers shown individually estimated for 25 epochs of fBm with Hurst exponent ranging from 0. The mean values of the estimates versus Hurst exponents were interpolated by a line and the percentage root mean squared error with respect to the overall mean is indicated. C Same as B , but considering de-trended data.

    E Modified ApEn applied to de-trended data. Example of experimental application. Two EEGs were recorded in rest conditions from a healthy control and a patient who entered a vegetative state with closed eyes duration of the recording s, sampling frequency Hz; before processing, data were resampled at Hz after low-pass filtering with cutoff 32 Hz using a Chebyshev Type 2 filter of order Data from channel F7-RF are considered.

    A Example of a portion of data from the two subjects. D Same as C , but considering a high-pass filter with cutoff 1 Hz Chebyshev Type 2 filter of order 6.