Figure 1

Aliasing illustrated in the time domain. Vertical bars indicate sampling times; the sampling interval $\Delta t$ is determined by the sampling frequency $f_s$. (a) The sampled data points (dots) are consistent with a sine wave of frequency $f_0=0.3f_s$, below the Nyquist frequency $f_N=0.5f_s$. (b) A sine wave of frequency $f=0.7f_s=f_s-f_0$, when sampled at the same times, will yield the same data points as in (a). For any frequency $f_0$ below the Nyquist frequency, there are an infinite number of frequencies $f=n{f}_s\pm f_0$ above the Nyquist frequency that yield identical data when sampled at the sampling frequency $f_s$; panel (c) shows another such alias, at a frequency of $f=1.3f_s=f_s+f_0$. (d) The solid black curve shows a waveform composed of the sine wave from (a), shown as a dotted line, summed with its two aliases from panels (b) and (c). When sampled at the sampling frequency $f_s$, this waveform yields data points that are consistent with a sine wave of frequency $f_0=0.3f_s$ (gray curve), but at an amplitude much greater than that frequency’s actual contribution to the waveform (dotted curve).Reuse & Permissions

Figure 2

Aliasing illustrated in the frequency domain. The true power spectrum for a continuous function is shown by the thin black line. Sampling this continuous function at a sampling frequency $f_s$, creates aliases of the true spectrum, shown by dotted lines, centered at $\pm f_s$, $\pm 2f_s$, $\pm 3f_s$, etc. Spectral power is typically calculated within the frequency range $0<f<f_N$ (where $f_N=0.5f_s$ is the Nyquist frequency), indicated by the broad shaded band. (a) If the true spectrum has no significant power at frequencies above $f_N$, measures of power at frequencies less than $f_N$ will reflect the true power spectrum. (b) If the true spectrum has significant power at frequencies up to $f_s$, frequencies in the range $f_N<f<f_s$ will be aliased into the range $0<f<f_N$. For noise spectra (in which the aliases and the true signal at any frequency will be randomly phase-shifted relative to one another), the measured spectral power at any frequency $f$ will be the sum of the true signal and the aliases at that frequency. This sum is shown by the heavy dashed curve and is highlighted by the thick black line over the range that the spectrum would normally be measured. (c) If the true spectrum has significant power at frequencies above $f_s$, multiple aliases will be superimposed on all frequencies $0<f<f_N$. The measured spectrum will be affected by aliasing over its entire frequency range, and aliased power will exceed the true power at the Nyquist frequency.Reuse & Permissions

Figure 3

Effects of undersampling on power spectra of synthetic power-law noises. Spectra of synthetic power-law noises (gray lines) deviate systematically from the theoretical spectra from which they were generated [straight black lines, $S_X\left(f\right)\sim 1∕f^{0.5}$] when time series are undersampled, i.e., the highest frequency in the sampled process $\left(f_{\max }\right)$ is greater than the Nyquist frequency of the sampling $(f_N=0.5f_s)$. Left-hand panels (a) and (c) show unsmoothed spectra of synthetic power-law noises composed of $2^{14}=16384$ points. Right-hand panels (b) and (d) show spectra after smoothing by taking weighted averages of spectral power over a Gaussian smoothing window with a scale factor of 0.05 log units. Scaling the smoothing window logarithmically means that at higher frequencies, more points are averaged together and thus the average spectrum is more precisely defined. If the noise process is undersampled by only twofold (top panels), each frequency $f$ has only one alias, at frequency $f_s-f$. In this case [see panel (a)] the spectral power forms an envelope between an upper limit of ${\mid X\left(f\right)\mid }^2+{\mid X(f_s-f)\mid }^2+2\mid X\left(f\right)\mid \mid X(f_s-f)\mid$ and a lower limit of ${\mid X\left(f\right)\mid }^2+{\mid X(f_s-f)\mid }^2-2\mid X\left(f\right)\mid \mid X(f_s-f)\mid$, depending on whether the alias from frequency $f_s-f$ is in phase or out of phase with the true signal at frequency $f$. Aliasing inflates the spectrum by an average amount $S_X(f_s-f)$ [see panel (b)], in agreement with Eq. (17). Under more severe undersampling (bottom panels), the random variation in the power spectrum is larger and more widespread [panel (c)]. The average spectral power is elevated above the true $1∕f^{0.5}$ spectrum across a wide range of frequencies [panel (d)], in agreement with Eq. (13).Reuse & Permissions

Figure 4

Effects of aliasing on power spectra of $1∕f^{\alpha }$ noises. (a) Apparent $\alpha$, as indicated by the log-log slopes of power spectra measured over a frequency range of three orders of magnitude ($0.001f_N$ to $f_N$). (b) Inflation of measured power at the Nyquist frequency $f_N$. (c) Variance due to aliases, as a fraction of the total variance in the measured time series (estimated as the integral of spectral power from $0.001f_N$ to $f_N$). Curves are shown for $1∕f^{\alpha }$ noises ranging from $\alpha =0.2$ to $\alpha =1.0$. Dotted lines show log-log slopes [panel (a)] and power levels [panel (b)] that would be measured in the absence of aliasing. At left edge of the plots, the maximum frequency in the noise equals the Nyquist frequency and aliasing does not occur. As the maximum frequency in the $1∕f^{\alpha }$ noise becomes a large multiple of the sampling frequency, the power levels (a) and log-log slopes (b) of the measured spectra become highly distorted, and aliases comprise a significant fraction of the total variance (c) in the sampled time series.Reuse & Permissions

Figure 5

Alias filtering applied to concentration fluctuations of chloride in streamflow at Plynlimon, Wales. Weekly measurements of chloride concentrations (a) appear to exhibit $1∕f^{0.5}$ scaling (c), whereas daily measurements (b) over the same time span exhibit clear $1∕f^{1.0}$ scaling (d). (e) Illustration of alias filtering (axes are expanded to show detail). A power-law model [Eq. (21)], including its aliases (dotted gray curve) is fitted to the smoothed spectrum of the weekly measurements (solid gray curve). The alias-filtered spectrum (solid black curve) is calculated from the measured spectrum by multiplying by the ratio of the power-law model with and without aliases (gray and black dashed lines). (f) Alias-filtered spectra for the weekly and daily data show consistent spectral scaling. In comparing panels (a) and (b), note that the data sets do not exactly coincide because the weekly samples are taken at a different time of day than the daily measurements; thus they are not an exact subset of the daily data.Reuse & Permissions

Figure 6

Alias filtering with a mis-specified model spectrum, illustrated with the weekly chloride spectrum of Fig. 5. (a) Specifying a model spectrum [Eq. (24)] with a scaling exponent of $\alpha =0.5$ rather than the best-fit value of $\alpha =1.01$ yields a modeled signal-plus-alias spectrum (dashed black line) that clearly deviates from the spectrum of the weekly chloride measurements (solid gray line); the best-fit signal-plus-alias model (dashed gray line) is shown for comparison. (b) Nevertheless, the mis-specified model yields an alias-filtered spectrum (solid black line) whose scaling behavior is almost exactly the same as that derived from the best-fit model (solid gray line).Reuse & Permissions

Figure 7

Alias filtering applied to stream discharge records at Plynlimon, Wales. Daily (a) and hourly (b) streamflow records are similar, with the hourly records showing more fine detail (records for 1985 are shown as an example). Daily instantaneous flows appear to exhibit $1∕f^{0.40}$ scaling (c) between frequencies of $3∕\text{year}$ and $183∕\text{year}(=0.5∕\text{day})$, whereas hourly measurements exhibit clear $1∕f^{0.67}$ scaling (d) over the same frequency range. Alias-filtering (e) is achieved by fitting a power-law model [Eq. (21)], including its aliases (dashed gray curve), to the smoothed spectrum of weekly measurements (solid gray curve). The alias-filtered spectrum (solid black curve) is derived by multiplying the measured spectral power (solid gray curve) by the alias filter, formed from the ratio of the modeled spectra with and without aliases (dashed gray line and dashed black line). After alias-filtering, the daily and hourly spectra show consistent spectral scaling (f), whereas the unfiltered daily spectrum diverges from the hourly spectrum.Reuse & Permissions

Figure 8

Alias filtering with a mis-specified model spectrum, illustrated with the daily streamflow spectrum of Fig. 7. (a) Specifying a model spectrum [Eq. (24)] with a scaling exponent of $\alpha =1$ rather than the best-fit value of $\alpha =0.59$ yields a modeled signal-plus-alias spectrum (dashed black line) that clearly deviates from the spectrum of the daily streamflow measurements (solid gray line); the best-fit signal-plus-alias model (dashed gray line) is shown for comparison. (b) Nevertheless, the mis-specified model yields an alias-filtered spectrum (solid black line) whose scaling behavior is almost exactly the same as that derived from the best-fit model (solid gray line).Reuse & Permissions

Figure 9

Reduction of aliasing by time-averaging before sampling. Thin lines indicate unfiltered signals. Dotted lines indicate moving averages over an interval of $\Delta t$; these moving averages lag the signal that they are averaging. Dots indicate averages sampled at the sampling interval $\Delta t$, and gray lines indicate the signal that would be inferred from the sampled averages. (a) The signal inferred from the sampled averages is slightly damped and phase-lagged relative to the true signal, at frequencies below the Nyquist frequency $f_N=0.5f_s$. (b), (c) Frequency components above the Nyquist frequency are more strongly damped by averaging, and thus their aliases, shown by the gray curves, are much smaller than they would be under point sampling (compare Fig. 1). (d) Thus averages of a complex waveform will primarily reflect the lower-frequency components, and will reduce aliasing of the frequency components that lie above the Nyquist frequency. The averages in panel (d) yield an inferred signal that resembles the true signal at that frequency [panel (a)], whereas point sampling [see Fig. 1d] does not suppress the aliases and leads to an inflated estimate of signal strength.Reuse & Permissions

Figure 10

Effects of time-averaged sampling on power spectra of synthetic power-law noises. The gray line is the spectrum of synthetic noise generated, as in Fig. 3, from an ideal $1∕f^{0.5}$ spectrum over a frequency range that extends to the sampling frequency $f_{\max }=f_s$, top panels) or 8 times the sampling frequency ($f_{\max }=8f_s$, bottom panels). The gray lines in the left-hand panels are unsmoothed spectra; those in the right-hand panels (with axes expanded to show detail) are smoothed by averaging over 10 adjacent frequencies. The dotted black lines show the ideal $1∕f^{0.5}$ spectrum with the damping of high-frequency fluctuations that would be expected from time-averaged sampling [Eq. (26)], in the absence of aliasing. The dashed black lines in the right-hand panels show the theoretical spectra that would result from the combined effects of time-averaged sampling and aliasing [Eq. (28)]; these correspond closely to the average spectral power of the synthetic noise (gray line). Note that, in comparison to Fig. 3, time-averaged sampling greatly reduces the scatter in the measured power spectrum (left-hand panels) that arises from aliasing of high-frequency noise components. As a result, aliasing has very little effect on the spectrum below roughly $f=0.1f_s$, even when the signal is severely undersampled [panel (c)]. Time averaging leads to damping of the expected spectrum (dashed black line) near the Nyquist frequency $f_N=0.5f_s$; this is partly offset by aliasing of higher-frequency components of the signal. As a result, the average spectral power of the synthetic noise (solid black line in left-hand panels, gray line in right-hand panels) closely approximates the ideal $1∕f^{0.5}$ spectrum.Reuse & Permissions

Figure 11

Effects of aliasing and time-averaging on power spectra of $1∕f^{\alpha }$ noises that are time averaged (over the sampling interval) before sampling (compare with Fig. 4). (a) Apparent $\alpha$, as indicated by the log-log slopes of power spectra measured over a frequency range of three orders of magnitude ($0.001f_N$ to $f_N$). (b) Reduction in measured power at the Nyquist frequency $f_N$; note that vertical axis is expanded 100-fold compared to Fig. 4b. Curves are shown for $1∕f^{\alpha }$ noises ranging from $\alpha =0.2$ to $\alpha =1.0$; labels on curves indicate true $\alpha$ values [also shown as dotted lines in panel (a)]. At left edge of the plots, the maximum frequency in the noise equals the Nyquist frequency and aliasing does not occur, but time-averaging reduces the high-frequency spectral power (to 40.5% of true power at $f_N$), and consequently steepens the measured log-log slope. As the maximum frequency in the $1∕f^{\alpha }$ noise becomes a large multiple of the sampling frequency, aliasing inflates the high-frequency power and thus partly offsets the effects of time averaging, and thus the log-log slopes of the measured spectra come closer to the true $\alpha$ values.Reuse & Permissions