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AC07 · L2

Count the cycles between exposures

Sandbox Physics ·

A sampling alias exists when two different continuous signals give the same measurements. To find it, write the signal at the actual exposure times. The lost information becomes visible in the algebra before we compute a periodogram.

A frequency counts cycles per unit time

Let time be measured in days relative to a chosen origin. A frequency counts cycles per day; period is days per cycle. The angular argument of a sine uses radians, so multiply frequency by time and by a full turn in radians. An initial phase specifies where the star begins in its cycle.

The browser source has mean normalized intensity one, semi-amplitude 0.23 and period 0.635 days. A semi-amplitude is the distance from the mean to a maximum. This is a synthetic sinusoid; real RR Lyrae stars have a more asymmetric pulse shape. The source keeps this period when the schedule changes.

f=1P,F(t)=F0+Asin⁡(2πft+ϕ),ftrue=10.635 d=1.57480 d−1f=\frac1P,\qquad F(t)=F_0+A\sin(2\pi f t+\phi),\qquad f_{\rm true}=\frac1{0.635\,\mathrm d}=1.57480\,\mathrm d^{-1}
F is linear intensity in the browser. The observed OGLE tutorial later fits magnitudes as a different numerical observable.

Reading: VanderPlas, Understanding the Lomb–Scargle Periodogram, arXiv v1 · OGLE-IV LMC RR Lyrae official data README

Regular sampling cannot count an extra whole turn

Suppose each visit occurs after the same interval, with an initial time offset. Replace the signal frequency by the original frequency plus an integer multiple of the visit frequency. At visit number n, the new angular argument adds an integer number of full turns plus a constant. Adjusting the initial phase removes the constant; every sampled sine value is identical.

A negative frequency can be represented by a positive one with a reflected phase. The family of aliases therefore uses an absolute value. This derivation assumes strictly regular timestamps and an unknown sinusoidal phase and amplitude. Adding more visits at those same phases does not distinguish the family.

For a sufficiently band-limited signal sampled regularly, the Nyquist theorem selects a unique representation below half the visit frequency. That restriction is an assumption about which continuous signals are allowed. A star with an unknown faster rhythm can still leave an indistinguishable slower sequence.

tn=t0+nΔt,fs=1Δt,f′=f+kfs,ϕ′=ϕ−2πkfst0t_n=t_0+n\Delta t,\quad f_s=\frac1{\Delta t},\quad f'=f+kf_s,\quad \phi'=\phi-2\pi kf_s t_0
Substitution adds a whole number of turns at every sampled point.
falias=∣f+kfs∣,k∈Z,fNyquist=fs2f_{\rm alias}=|f+kf_s|,\quad k\in\mathbb Z,\qquad f_{\rm Nyquist}=\frac{f_s}{2}
The Nyquist relation is for a regular sampling lattice. It does not define a universal cutoff from the average spacing of irregular visits.

Reading: VanderPlas, Understanding the Lomb–Scargle Periodogram, arXiv v1 · Astropy 8.0.2 Lomb–Scargle documentation

Worked example: a fifteen-hour star looks slower

Observe the injected 0.635-day star once each day. Its frequency is about 1.57480 cycles per day. Subtracting twice the daily visit frequency and reflecting gives 0.42520 cycles per day, a period of about 2.352 days. Subtracting only one gives another candidate near 1.740 days. All are daily aliases of the same samples.

The browser searches a fixed frequency grid, so its reported values are discretized. With seed 707, the once-nightly run selects a 2.35294-day peak with power 0.9352. Spreading three visits across each nightly window selects 0.634921 days. The star was not slowed by the first camera schedule or accelerated by the second.

These are representative synthetic detector results, generated by acquireCadence and saved with the fixed source, twenty-four nominal nights and five-minute exposures. Detector noise changes with the exported seed. The worked alias relation itself is exact for instantaneous regular samples; a known exposure response changes amplitudes, discussed below.

∣1.57480−2∣=0.42520 d−1,Palias=10.42520 d−1≃2.352 d|1.57480-2|=0.42520\,\mathrm d^{-1},\qquad P_{\rm alias}=\frac1{0.42520\,\mathrm d^{-1}}\simeq2.352\,\mathrm d
A positive slow frequency can hide a negative-frequency branch of a faster source.

Reading: VanderPlas, Understanding the Lomb–Scargle Periodogram, arXiv v1

Irregular visits leave a characteristic window

For irregular times there is no single visit frequency to put into the exact alias equation. Instead add a unit arrow for each observing time at every trial frequency. The arrow angle is that frequency times the observing time, converted to radians. A complex exponential is a short way to store its two coordinates, cosine and sine.

When the arrows line up, their sum is large. A regular daily schedule makes them align at one cycle per day and its multiples. Jitter rotates individual arrows away from that alignment. Squaring the length and dividing by the squared number of visits defines a window power equal to one at zero frequency.

This window depends only on timestamps. It predicts where the sampling can generate competing structure; it is not itself a measured stellar spectrum. A floating-mean fit to a constant signal would return no variability. We calculate the window directly from the timestamp arrows, including its zero-frequency component.

W(f)=1N2[(∑jcos⁡2πftj)2+(∑jsin⁡2πftj)2]W(f)=\frac1{N^2}\left[\left(\sum_j\cos 2\pi ft_j\right)^2+\left(\sum_j\sin 2\pi ft_j\right)^2\right]
The dimensionless direct transform uses equal timestamp weights here. A likelihood-weighted window is another declared choice.

Reading: VanderPlas, Understanding the Lomb–Scargle Periodogram, arXiv v1 · Astropy 8.0.2 Lomb–Scargle documentation

The shutter averages during an exposure

A camera does not take an infinitely short point sample. A rectangular exposure averages intensity over a duration centered on its timestamp. Integrating the sine gives the same midpoint phase multiplied by a sinc factor. Slow variations survive nearly unchanged; a rapid oscillation can partly cancel within one exposure.

For the five-minute exposure and 0.635-day period, the attenuation is about 0.99995. The point-sample approximation is excellent for that source. At much higher frequencies the exposure response becomes important even if irregular visit times could otherwise distinguish the phase pattern. Timestamp precision and integration duration are different limitations.

F‾(t)=1Te∫t−Te/2t+Te/2F(u)du=F0+Asin⁡(πfTe)πfTesin⁡(2πft+ϕ)\overline F(t)=\frac1{T_e}\int_{t-T_e/2}^{t+T_e/2}F(u)du=F_0+A\frac{\sin(\pi f T_e)}{\pi f T_e}\sin(2\pi ft+\phi)
The sinc factor equals one in the zero-duration limit. The measured timestamp is the exposure midpoint; an alert can only be produced after its end.

Reading: VanderPlas, Understanding the Lomb–Scargle Periodogram, arXiv v1 · Patterson et al., ZTF Alert Distribution System, arXiv v1

A short flash can fit entirely in the gap

Consider ideal instantaneous visits separated by a gap and a rectangular flash of duration D, with a uniformly random start phase. If the flash is shorter than the gap, the fraction of phases that put a visit inside it is D divided by the gap. This simple completeness calculation is about overlap, before noise or a threshold.

Finite exposures enlarge the set of overlapping phases, but partial overlap dilutes the measured intensity and may fail the detection threshold. The browser therefore averages its Gaussian flash through each actual exposure before adding noise and checking the trigger. Processing time affects the alert’s arrival after detection, not which incident photons were collected.

poverlap=min⁡ ⁣(1,DΔt)p_{\rm overlap}=\min\!\left(1,\frac{D}{\Delta t}\right)
This probability assumes a stationary regular train of instantaneous visits and a uniformly random flash phase. Finite exposures and detection thresholds require the more detailed calculation described above.

Reading: VanderPlas, Understanding the Lomb–Scargle Periodogram, arXiv v1 · Patterson et al., ZTF Alert Distribution System, arXiv v1

Exercises and explanations

  1. You collect twice as many once-daily samples at exactly the same clock hour. What improves, and what exact ambiguity remains?

    Read the explanation

    More samples reduce random errors and sharpen finite-baseline structure. They still cannot distinguish frequencies differing by integer cycles per day under the same sinusoidal model and freely fitted phase/amplitude. Move some observations to other phases to address that ambiguity.

  2. An irregular sequence has a mean gap of one day and a few five-minute pairs. Should you reject every trial frequency above half a cycle per day?

    Read the explanation

    No. The mean gap is not a universal Nyquist lattice. Inspect the actual timestamps, time precision and exposure response, choose a physically motivated frequency range, and calibrate the complete chosen search.

  3. An instantaneous camera visits every two hours and a flash lasts fifteen minutes at a random phase. What is the ideal overlap probability?

    Read the explanation

    Fifteen divided by 120 gives one eighth, or 12.5 percent. Real finite exposures and a required signal threshold need an exposure-integrated detection calculation.

Try the prediction

Predict the aliases before adding more exposures

  1. Calculate the daily aliases of the injected frequency and locate them in the once-nightly periodogram.
  2. Increase only the window duration until several exposures fit inside it. Compare the one-cycle-per-day window peak and the competing stellar frequencies.
  3. Keep the schedule fixed and add processing time. Explain why the saved light curve stays the same while the alert arrives later.

Sources and calculation scope

  1. VanderPlas, Understanding the Lomb–Scargle Periodogram, arXiv v1

    28 March 2017; Sections 3.2/4.2 and Figures 7–14 for windows and aliases; Sections 6.1–6.2 for weighted floating mean; Section 7.1 and 7.4.2/Figure 26 for grid and FAP. Published journal counterpart: ApJS 236,16 (2018). This guide pins the arXiv version.

  2. Astropy 8.0.2 Lomb–Scargle documentation

    Consulted 28 October 2026; page updated 15 October 2026. Additional Arguments, standard normalization, peak significance and RR Lyrae example. Default floating mean, same-times/same-grid FAP, and no general average-Nyquist cutoff for irregular data.

  3. OGLE-IV LMC RR Lyrae official data README

    Column definitions for HJD minus 2450000, I-band magnitude and uncertainty; public-data citation requirement. Frozen per-object file OGLE-LMC-RRLYR-00001.dat and catalog row are retained with SHA-256 and HTTP metadata.

  4. Patterson et al., ZTF Alert Distribution System, arXiv v1

    6 February 2019; III.3,IV,V, Table 1 and Figure 2. Packet/filter/distribution stages and explicitly dated performance, with different latency reference points. These numbers are not presented as current infrastructure measurements.

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