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Detection of Aliasing in Persistent Signals

We explain why aliasing can be detected in a generic temporally-sampled stationary signal process. We then define a concept of stationarity that makes sense for single waveforms. (This is done without assuming that the waveform is a sample path of some underlying stochastic process.) We show how to use this concept to detect aliasing in sampled waveforms. The constraint that must be satisfied to make detection of aliasing possible is shown to be fairly unrestrictive. We use simple harmonic signals to elucidate the method. We then demonstrate that the method works for continuous-spectrum signals---specifically, for time series from the Lorenz and Rossler systems. Finally we explain how the method might permit the recovery of additional information about Fourier components outside the Nyquist band.


Introduction
Detection of aliasing from temporal samples alone, with no restrictions on the original continuous-time source, is impossible because any set of samples may be reconstructed (using Shannon's sinc filter) to a properly sampled signal having the same samples. However, quite often additional information about the source is available. It is, of course, obvious that tight constraints on the source would permit perfect reconstructions of vastly undersampled signals. For example, the constraint that the data comes from a linear function of time makes any two samples sufficient. A less extreme example is a signal with a lower as well as an upper frequency cutoff (a bandpass signal). For bandpass signals, it is wellknown that one can sample at a rate below twice the highest frequency while still achieving perfect signal recovery (see [4, p.138, theorem 13.3]).
What are the weakest constraints that one can put on the signal and still get something-detection of aliasing, for example? Here, we examine constraints of stationarity. In 1988 Hinich and Wolinsky [3] suggested a bispectral test for detecting aliasing in temporally sampled stationary stochastic processes 4 . The test aroused some controversy [6,1] which is examined in [2,7]. In [7] we show, in detail, that the test does detect aliasing in some signal processes and that it is the constraint of stationarity that makes the detection of aliasing possible. Briefly, if we undersample a stationary process and then reconstruct a continuous-time signal from the samples using the Shannon sinc filter, the reconstructed process will not, in general, be stationary. In contrast, a proper sampling followed by reconstruction will not destroy stationarity because this procedure just reconstructs the original signal. Detecting nonstationarity in the reconstructed process thus suffices to establish the existence of aliasing in the time series, provided it can be assumed that the original signal was stationary. These results are reviewed in Section 3.
Applying these concepts requires either a random sample of the paths of the process or an assumption of ergodicity which makes it possible to extract statistics from a single sample path. In this paper we attempt to generalize the results for stationary processes to the more common case where we have only a single sample path and can make no assumption of ergodicity. In other words, we look for ways to discover undersampling in a time series drawn from a single waveform, which may or may not be a sample path of some underlying stochastic process. We define sampling stationarity, a form of stationarity that makes sense for single waveforms, and show that it can be used to detect aliasing in complex, continuous-spectrum signals. We present reasons to believe that sampling stationarity should be a generic 5 property of signals and that the destruction of sampling stationarity by the process of undersampling and reconstruction should occur quite generally. Finally, we explain how it might be possible to use the reconstructed sample statistics plots (RSS plots) that we use to detect aliasing to obtain additional information about individual Fourier components beyond the Nyquist frequency.
The remainder of this paper proceeds as follows. After illustrating the key idea of this paper with an example in Section 2, we proceed, in Section 3, to demonstrate how a constraint of stationarity permits the detection of undersampling in some signal processes. Then in Section 4 we define sampling stationarity. In Section 4.1 we use examples to show that the concept of sampling stationarity does, indeed, enable detection of aliasing for nontrivial signals. In Section 4.2 we consider the case of periodic signals. For this class of signals, we provide a complete explanation of how (and when) the method of high-frequency detection works. The possible extension of this explanation to nonperiodic signals is then discussed in Section 4.3. In Section 4.4, we present some reasons to believe that the plots that we have used to detect aliasing may also be used to recover some portion of the original signal's high-frequency content. This is followed by suggestions for further work (Section 5) and a conclusion that summarizes the work in this paper (Section 6). Two appendices contain computational and mathematical details.

Example
The key idea of our approach is captured by a very simple example. Suppose that we sample a square wave that takes the values −1 and 1. There is a unique properly bandlimited signal that has this time series as its samples. We can compute this signal by applying the Shannon sinc filter to our time series. We may regard this computation as an attempt to reconstruct the original continuous-time signal. If we can reject this reconstructed signal as the source of our samples, then we must conclude that the time series contains aliased components.
Note that our given time series consists only of −1's and 1's. The reconstructed signal, on the other hand, is necessarily a continuous function of time, taking on all values in the interval [−1, 1] (and, in fact, beyond). The only way that we could have obtained a sequence of −1's and 1's by sampling the reconstructed signal is if we had chosen a particular sampling rate (or one of its subharmonics) and a unique shift of the sampling comb. Any other combination of sampling rate and shift would have produced a series that takes a continuum of values. The probability of having chosen the special sample rate and shift that give a sequence of −1's and 1's is clearly zero, provided that our sampling rate was chosen independently of the source. With this proviso, then, we can reject (with probability 1) the hypothesis that our time series consisting of −1's and 1's came from sampling the reconstructed signal.
The assumption that the sampling rate was chosen independently of the source is justified in most (but not all) cases of practical importance because we can rule out any interdependence between the source and the sampling rate on physical grounds. For example, if a signal produced by a distant source is sampled at a predetermined rate, such a coupling is clearly out of the question-it would amount to believing that the process that produced the signal "knew" when we were going to sample at a distant location.
We can draw valid conclusions from a sampled signal about Fourier components beyond the Nyquist frequency only if we can put constraints on the original continuous-time source. How can we characterize the constraints that we are imposing in this case? Effectively, we are assuming that the sample times (which are determined by the sample rate and shift) do not play a distinguished role in the source. Showing that the sampling times are distinguished in the reconstructed signal then suffices to reject the reconstructed signal as the original source of the samples.
How, then, can we extend this analysis to more general classes of signals? In the case of a square wave (or any signal that takes on a finite number of values), the appearance of the time series produced by sampling the reconstructed signal at the given sampling times could not be more different from the appearance of a time series produced by sampling at any other shift of the sampling comb. Thus it is clear what we mean when we say that the sample times are distinguished in the reconstructed signal. For more general signals, however, it is not so clear exactly what it means for the sample times to be distinguished.
There is one obvious case in which we can be assured that the sample times are not distinguished in the original signal and in which we can detect the distinguished character of the sample times in the reconstructed signal. If the original signal is, in fact, a stationary signal process, then, by definition, no time is distinguished. The appearance of nonstationarity in the reconstructed signal would then indicate the presence of aliasing in the time series. The detection of aliasing in time series from stationary signal processes is the subject of the next section. Following that, we use our example of sampling from a square wave and insights from the case of stationary processes to develop a method for detection of aliasing in single waveforms.

Detection of Aliasing in Stationary Processes
Consider the case of detection of aliasing in stationary signal processes. We start with the simplest stationary processes imaginable-randomly shifted periodic signals. If we have a waveform, x(t), with period T , then we can produce a stationary process by adding to t a random time shift, θ, that is evenly distributed on [0, T ). A sample path of our process then has the form x(t + θ) for a particular choice of θ.
Consider then the effect of undersampling and reconstruction on a simple sine process, where θ is evenly distributed on [0, f −1 ). If we undersample with a sampling interval ∆t, corresponding to the Nyquist band [−(2∆t) −1 , (2∆t) −1 ), and then reconstruct via convolution with the sinc filter, we get the sine process given by Here,f is the aliased frequency, given byf = f + k f /∆t where k f is the unique integer that placesf in the Nyquist band. The key point is that the phase of the reconstructed signal is the same as the phase of the source even though the frequency has changed to the aliased valuef . For a process with a single harmonic, the reconstructed signal remains stationary because the phase term, 2πf θ, is evenly distributed on 2π. Consider then a second signal process, where β is an integer multiple of α and θ is chosen randomly from the interval [0, α −1 ). Since the time shift, θ, is the same for both components, this is, for the various values of θ, just a shifted waveform of a given shape. Since θ is evenly distributed over the period, α −1 , the process is stationary. If we sample this process at a rate low enough for both components to be aliased and then reconstruct using the sinc filter, we get whereα andβ are the aliased frequencies. Although the phase terms, 2παθ and 2πβθ, are still evenly distributed over 2π, they now correspond to different time shifts for the two components. Thus, we no longer have a single shifted waveform and we can expect, in general, that stationarity will have been lost.
We illustrate this loss of stationarity on an example by setting α = 1.0 and β = 3.0 in Equation 3 and choosing a sample time, ∆t, equal to e. We may detect the loss of stationarity by examining the envelope of the reconstructed process. We define the envelope of a process, X t , as the support of the probability density of X t as a function of t. For a process produced by randomly shifting a periodic waveform, the envelope may be conveniently displayed by plotting the sample paths corresponding to a representative collection of time shifts as in Figure 1. Clearly, a stationary process must have a constant envelope. If we compute the envelope for the process defined in Equation 4 with the parameter values that we have specified, we get an oscillating figure (see Figure 1). This implies that the signal is nonstationary. In fact, it is cyclostationary with period equal to the sampling interval.
As explained above, we do not lose stationarity when our original signal is a single sine wave. Nor do we lose stationarity when ratios between frequencies are preserved under the aliasing. For example, if ∆t = 1.0 and the original frequencies are (10/9, 20/9, 30/9), they would alias to (1/9, 2/9, 3/9) and we would obtain another stationary process. But this situation is very special (non-generic). In general, more than one Fourier component is present and we do not have the special relationships between the sampling rate and the component frequencies that preserve ratios between frequencies when undersampling. Thus, we expect that, generically, a stationary process formed by randomly shifting a periodic waveform will lose stationarity upon undersampling and reconstruction.
Within the context of single shifted waveforms, the destruction of stationarity can occur in some remarkable situations. Consider that undersampling and reconstruction can break stationarity even when only one of the components (β, say) is aliased and when β aliases to α or −α. In other words, stationarity can be broken even when the two components, after undersampling, lie right on top of each other. We can see this by choosing α = 0.25, β = 0.75, and ∆t = 1.0 in Equation 3. The envelope for the reconstructed process is shown in Figure 2 where the nonstationarity is apparent. (Of course, we cannot possibly detect this loss of stationarity by examining only a single sample path, since the sample path will never be more than a single sine wave of some amplitude and phase.) Not all stationary processes are randomly shifted periodic waveforms. What can we say about more general stationary processes? It is clear that, if there exist no phase relationships between any of the Fourier components of the process, then undersampling and reconstruction will not destroy stationarity. For a generic stationary process, though, we would expect at least some sets of components to exhibit phase relations. In that case, we would expect stationarity to be destroyed because it is difficult to imagine how the destruction of stationarity associated with one set of components could somehow be canceled out by the presence of other incommensurate components.
This argument, together with the observation that there is simply no reason to believe that stationarity should be preserved under undersampling and reconstruction, suggests that the loss of stationarity should be a general feature of stationary processes.

Detection of Aliasing in Single Sample Paths
The method that we have just used to detect aliasing in a sampled stationary process requires complete knowledge of the discrete-time process obtained by sampling the original continuous-time source. Usually, however, we have available to us only a single sample path. Therefore, we require a method for detecting aliasing in a single waveform which may or may not be a sample path of a stochastic signal process.
We may develop such a method by reconsidering the example of sampling from a square wave discussed in Section 2 in light of our discussion of the effect of undersampling on stationary signal processes. Recall that the sampled time series from the square wave takes on the values −1 and 1. We may state this in statistical language by saying that the one-time probability density of the time series consists of two Dirac delta functions centered at −1 and 1, respectively. Now, we would have obtained the same one-time statistics if we had sampled the original square wave with any shift of the sampling comb. We will say that a waveform has sampling stationarity for a given sampling interval if the one-time sample statistics do not change as the position of the sample comb is shifted along the waveform.
Observing that the original square wave had sampling stationarity for the given sampling interval is essentially equivalent to saying that the sample times were not distinguished in the source. 8 Note that the signal obtained by applying the Shannon sinc filter to the time series does not have sampling stationarity-the one-time statistics of the reconstructed signal vary dramatically with shifts of the sampling comb (see Figure 3). This lack of sampling stationarity corresponds to the distinguished role of the sample times in the reconstructed signal. Of course, this distinguished role for the sample times is what allowed us to reject the reconstructed signal as a candidate for the original source of the samples and, thus, to conclude that the sampled series contained aliased components.
This discussion suggests the following test for aliasing in signals (no underlying stochastic process assumed). Collect the statistics on the recorded samples. Reconstruct the signal at various shifts of the sampling comb and collect the statistics at these reconstructed samples. Compare with the original statistics. (We use the term "statistics" loosely, without the assumption that the samples are independent samples of some underlying probability distribution.) If we find that the reconstruction has different statistics at some shift of the sampling comb, an assumption of sampling stationarity for the original signal implies that the reconstruction is not the original signal and therefore that the signal was undersampled.
For this test to be at all useful, two questions must be answered: 1. Are typical signals characterized by sampling stationarity?
2. Do typical undersamplings reconstruct to signals for which sampling stationarity is violated?
The answer to question 1 is clearly "yes" for sample paths of ergodic stationary processes and for signals from ergodic dynamical systems. It is also clear that there are other classes of signals which possess sampling stationarity. For example, general periodic signals (not just square waves) possess sampling stationarity if the sampling interval is incommensurate with the signal period (a generic condition). Below, we conjecture that sampling stationarity is a generic property of signals.
The examples that we consider next suggest that the answer to the second questions is also "yes".

Examples
In each of the examples listed below, the time series to which we apply our test for aliasing was split into two interleaving series, D 1 from the samples taken at [0,2∆t,4∆t, ...] and D 2 from the samples taken at [∆t, 3∆t, 5∆t, ...]. The sample statistics corresponding to D 1 and D 2 are plotted in blue and green, respectively. For original signals with sampling stationarity, these two histograms will coincide. We then produce a reconstruction from D 1 , computed at the times corresponding to D 2 . The sample statistics corresponding to this reconstructed series are shown in red. If the red histogram is significantly different from the blue, then the reconstructed signal does not have sampling stationarity.
Example: For a periodic signal, the generic condition of incommensurability of the sampling interval and the signal period implies that the signal has the property of sampling stationarity. But we also find that undersampling and reconstructing produces a signal that does NOT have sampling stationarity, as illustrated in Figure 4. The data for the plot were generated by sampling (∆t = e/16) a sum of sines with frequencies (0, 1, 2, ..., 10) and random amplitudes that ranged between .78 and 1.22.
Example: If our signal consists of a sum of sine waves with incommensurate frequencies, then we cannot detect aliasing by this method (see Figure 5). Although such a signal will have sampling stationarity, the sampling stationarity will not be broken by undersampling and reconstruction because it is impossible to have relationships between the phases of different Fourier components (see Section 4.2).
Example: The previous example might lead to the suspicion that this method works only for periodic signals (or step signals such as the square wave). However, the presence of incommensurate Fourier components does not necessarily destroy the ability to detect aliasing in a periodic waveform with more than one harmonic component. Figure 6 shows the result of combining a periodic waveform with incommensurate harmonics. The total power in the incommensurate harmonics is about 21% of the power in the periodic waveform. The sampling stationarity of the original signal and the breakdown of sampling stationarity with undersampling and reconstruction are apparent. This shows that, as long as some of our aliased Fourier components are commensurate with other components, the method can

work.
Example: So far, we have demonstrated that the method works for pure periodic signals and for periodic signals mixed with incommensurate harmonics. Figures 7, 8, and 9 show that the method works for much more complex signals with continuous spectra. The signals are taken from the Lorenz and Rössler systems (see Appendix A).
The success in detecting aliasing in time series from the Lorenz and Rössler systems suggests that the method may work for a very broad class of signals. Before attempting to determine how wide this class might actually be, we will look at the periodic case in order to begin to understand the precise mechanism of the method.

A Closer Look at the Periodic Case
If one samples a periodic signal incommensurately with the signal period, the samples end up mixing evenly around the waveform (see Appendix B). Thus, all shifts of a sampling comb with a sampling interval that is incommensurate with the period will produce the same statistics. This implies that: Theorem 1 A periodic signal will have sampling stationarity with respect to any sampling interval that is incommensurate with the period of the signal.
Conversely, sampling with an interval that is commensurate with the period will, in general, produce statistics that depend on the sampling shift. Theorem 1 implies

that:
Theorem 2 Every periodic signal has sampling stationarity with respect to all sampling intervals except for a set of intervals with (Lebesgue) measure zero.
Thus, the probability of choosing a sampling interval for which a given periodic signal does not have sampling stationarity is zero, provided the interval is chosen independently of the signal.
What, then, is the effect of undersampling and reconstruction on this sampling stationarity? The sample statistics are determined by the shape of the waveform (see Appendix B for the exact formula). It can be shown that the process of undersampling and reconstruction is equivalent to sampling the original waveform at the same rate with the individual Fourier components shifted with respect to each other. When different components experience different time shifts, the shape of the effective waveform changes. Consequently, the statistics change. We now explain this in detail.
When we sample a single harmonic with frequency f and phase ϕ every ∆t time units, we get the values We will temporarily suppress the phase and rewrite this expression as sin (2πf n∆t) = sin (2πf n∆t + 2πkn) = sin 2π f + k ∆t n∆t (6) for any integer k, so that the reconstruction of this component at points 1 + s, 2 + s, 3 + s, ... is given byŷ where the reconstruction chooses precisely one of the integral k's, which we will call k f , such that f +k f /∆t is in the interval [−1/2∆t, 1/2∆t). We can now rewrite the reconstructed harmonic aŝ where we drop 2πk f n since k f is an integer. Thus, the reconstructed signal has samples at a shift, s, as though we were sampling the original waveform, but with the phase of the individual Fourier component shifted by the amount 2πf s∆t + 2πk f s. The first term amounts to a time shift which is the same for all the components in the waveform. This implies that these first terms do not change the shape of the waveform and can be ignored. So we may consider the effective waveform (at a shift s) to be where we have reinserted the phase. The term 2πk f i s amounts to a time shift that is different for different f i . This difference in time shifts leads to a change in the shape of the effective waveform as s changes which in turn changes the sample statistics. For a given waveform, it is clear that almost any change in the shape of the waveform will change the sample statistics. (For example, a generic choice of s will change the heights of the extrema, changing the locations of the singularities in the histogram. See Appendix B.) Thus, we conclude that generically, periodic signals have sampling stationarity which is destroyed by undersampling and reconstruction.

Nonperiodic Signals
Now, we want to use the insight that we have gained for the case of periodic signals to get a better understanding of the answers to the two questions at the end of Section 4. We begin with some general questions about the kinds of signals to which our method might possibly apply.
Consider first the case of transient signals. In order to be able to talk about sampling stationarity at all, we have to be able to take as many samples as we want (at the given sampling rate) in order to be able to estimate the one-time probability distribution to arbitrary accuracy. This implies that we must think of our signals as functions of infinite time. In this context, any transient signal has trivial sampling stationarity-the probability distribution is a delta function at zero. By the same token, undersampling and reconstruction will not destroy this sampling stationarity. Thus, we need to restrict our attention to persistent (nontransient) signals.
Within the class of persistent signals, it is clear that we need the signals that we consider to have well-defined sample statistics for arbitrary sampling intervals and shifts. Given that we are discussing aliasing, our signals also need to have a Fourier transform (in some sense). The set of signals with well-defined power spectra (which will have, in general, singular components) will clearly meet these criteria, although the actual class to which our method applies may be larger. In the following, then, we may take the term persistent signal to refer to a signal with a well-defined, nonzero power spectrum.
Consider then the question of which signals have sampling stationarity for which sampling intervals. In the case of periodic signals, Theorems 1 and 2 provide what is essentially a complete answer-sampling stationarity holds for generic choices of signals and sampling intervals. At first glance one might try to generalize Theorem 1 to the following: Conjecture 1 Every signal has the property of sampling stationarity for every sampling interval ∆t that is not commensurate with the period of any singular component of its spectrum. (FALSE) Unfortunately this conjecture is false as may be seen from the following counterexample. If we undersample and reconstruct a signal with a purely continuous spectrum (such as our signal from the Lorenz system), we will introduce no new singular components. Thus, the reconstructed signal will have a purely continuous spectrum. If the conjecture were true, then, such a reconstructed signal would have sampling stationarity for all sampling intervals by virtue of having no singular components. Yet it is just the lack of sampling stationarity of this signal with respect to the given sampling interval that allows us to detect aliasing in this case. Thus, we know that there exist signals that lack sampling stationarity with respect to sampling intervals that are not commensurate with any singular component of their spectra and the conjecture is false. However, the reconstruction of an undersampled signal has a very special relationship to the interval with which the sampling was done. Thus, one expects that resampling the reconstruction with a new sampling interval not related to the original interval will yield statistics that are again stationary with respect to shifts in the sampling comb. Therefore, we arrive at the following conjecture: Conjecture 2 Every signal has the property of sampling stationarity for every sampling interval ∆t, except a set of ∆t's with (Lebesgue) measure zero.
This conjecture implies that, if one were to observe the reconstructed statistics varying with changes in the shift, this observation would be enough to conclude (with probability 1) that the samples came from an undersampled waveform. In other words, the truth of the conjecture would imply that the detection of undersampling by the proposed method is generically free of false positives Next, we want to know when undersampling and reconstruction of persistent nonperiodic signals will yield new signals which have lost the property of sampling stationarity. (In other words, we also want to know when we can get false negatives.) The analysis that we have presented for periodic signals suggests that undersampling and reconstruction should destroy sampling stationarity for general persistent signals in which at least some of the aliased Fourier components are commensurate with other components of the signal. The reasoning is that each individual component can be regarded as a part of a family of harmonics and that the effective shape of the waveform associated with this family is changing with shifts of the sampling comb. There does not appear to be any reason to believe that combining different periodic waveforms, each of which is changing its sample statistics with shifts of the sampling comb, would result in sample statistics that do not change. Note that the condition that the original signal must have at least some commensurability will be satisfied by any signal with a nonzero continuous part to its spectrum as well as by periodic signals. Therefore, it seems likely that, generically, persistent signals have sampling stationarity that is destroyed by undersampling and reconstruction. In order to turn this last statement into a well defined conjecture, it will be necessary to define precisely what is meant by "generically" in the case of persistent nonperiodic signals. The question of exactly how to define "persistent" must also be answered. Since the transformation that takes us from a waveform to sample statistics is extremely nonlinear, a proof is likely to be difficult.

Recovery of High-Frequency Information
The next question that presents itself is whether or not we can recover information about individual aliased Fourier components using the sampling-shift dependence of the reconstructed statistics. Ideally, we would like to know how much of the signal at an individual frequency, f , in the Nyquist band comes from each frequency that aliases to f . Consider the one-time probability density of the reconstructed signal, p(x), as a function of both x and shift s. We call this two-dimensional surface a Reconstructed Sample Statistics (RSS) plot. The RSS plot has dependencies on s tied directly to the quantities k f i . Each k f i , in turn, determines the particular copy of the Nyquist band in which its corresponding f i is located. This chain of dependencies suggests that the RSS plot contains the information necessary to determine the contribution of each band to the signal at a given frequency in the Nyquist band. The inverse problem is greatly complicated by the interaction of the Fourier components and the nonlinear "projection" that turns the waveform into statistics. This extremely nonlinear inverse problem is the main object of our current research.

Directions for Further Investigations
In addition to the work already alluded to on the inverse problem formed by the RSS plots, there are other issues to explore. Included among them are: • What are the effects of noise on this method for detection of aliasing?
• What is the effect of near commensurability of sample interval and signal period?
• What is the effect of finite time-series length?
• How does the departure of the statistics of the reconstructed signal from stationarity depend on the fraction of the total power that lies outside the Nyquist band?
These questions are important to the practical usefulness of the method of high-frequency detection/recovery.

Conclusion
Although the idea of detection of aliasing is typically dismissed with references to the Nyquist criterion and the Shannon reconstruction theorem, we have demonstrated that detection of aliasing is possible with what appear, at first glance, to be very weak prior assumptions. The key concept is that of sampling stationarity. We emphasize that this concept makes sense for single waveforms. Although this concept arose in the consideration of step signals like square waves, its usefulness extends far beyond these signals. In particular, our method enables the detection of aliasing in samples from nontrivial waveforms such as measurements from motion on the Lorenz or Rössler attractors. As indicated above, many questions remain. Some of these are important for the practical utility of the concept of sampling stationarity and the associated RSS plots.

A Computational Details
The calculations represented in the paper were done with Matlab. The Lorenz equations,ẋ =σ(y − x) were integrated with parameter values of σ = 10, R = 28, and b = 8/3 using Matlab's "ODE45" which is an adaptive step size routine. Relative tolerance was set to the default value of 1.0 × 10 −3 and absolute tolerance was set to the default 1.0 × 10 −6 . Initial conditions were set at x = y = z = 1. Values for the x, y, and z coordinates were saved every 0.5 time units. 200,001 samples were taken and split into two interleaving time series each 100,000 samples long. The first series was used to reconstruct a signal via convolution with a sinc filter of length 200,001. These very long series and filters were used to minimize the effects of truncating the convolution at the ends of the series. The histograms for the reconstructed signal were computed from the middle 50% of the reconstructed series.
The Rössler equations,ẋ = − z − ẏ y =x + aẏ z =b + z(x − c), were integrated in the same way with a sampling interval of 10 time units, and parameter values of a = 0.15, b = 0.2, and c = 10. In this way 200,001 samples were obtained and the splitting and reconstruction were done as described for the Lorenz equations. All the histograms presented here were originally calculated from much shorter time series (10, 000 samples). The features that allow us to conclude that aliased components are present were all clearly visible in the histograms made from shorter time series although, of course, the histograms were considerably rougher. We conclude that the results that we have presented are certainly not an artifact of finite-length time series.

B Sample Statistics for a Periodic Signal
If one samples a periodic signal, h(t), incommensurately with the signal period T , the samples end up mixing evenly around the waveform 9 . The resulting histogram is proportional to the reciprocal of the derivative of the waveform. This follows from the fact that the probability of getting any particular t (position along the waveform) is uniformly distributed over [0, T ) which in turn implies that the probability of the interval [y, y + dy) is the probability of the corresponding dt or (1/T )(dy/h ′ (t)). More precisely, the probability density for y is where Note that the density will have 1/ (y) singularities at the local maxima and minima of h(t). The form of the singularities follows from the fact that a generic waveform has maxima and minima with nonzero second derivative.