Scale-TimeDynamics
The wheel is not turning backward
Helicopter blades stand still under a strobe. Wagon wheels roll forward in a film while their spokes drift backward. Neither effect belongs to the motion. Both belong to the readout: a discrete observation preserves the phase advance only up to a whole number of turns, and the turns are what went missing.
Engineering Note EN-001 takes that ordinary nuisance and asks what can honestly be done about it. Its answer starts with a refusal. Alias correction is routinely treated as an inversion problem, as though the right algorithm could undo the wrapping. It cannot, and the note spends its first pages proving so before it proposes anything.
One number instead of two
The formulation rests on a single change of variable. Rather than tracking a rotation rate and a sampling rate separately, the note works in the normalized phase advance, measured in cycles per sample. One dimensionless ratio absorbs both the physical speed and the observation cadence, and the paradoxes stop being paradoxes: they are statements about where that ratio sits relative to the sampling lattice.
In those terms the wrapped observation is whatever remains after the whole cycles are discarded, and it always lands in the half-open band between minus one half and one half. Every candidate differing by a whole number of cycles per sample produces exactly the same record.
Aliasing is a loss of injectivity
That last sentence is the note's central claim, and it is stated as a proposition with a proof rather than as a caution. The admissible set behind a single wrapped observation is countably infinite and evenly spaced, and the likelihood is invariant under integer shifts, so the maximum-likelihood estimate is set-valued rather than unique. A constant motion at 2.7 cycles per sample and one at 0.7 or at minus 1.3 are not hard to tell apart. They are indistinguishable.
So aliasing is not noise and not distortion. It is a loss of injectivity, and no processing of the wrapped observations alone can recover the missing integer branch. Anything that appears to invert the wrapping is importing an assumption from somewhere else, usually without saying so.
Four ways to buy the information back
The rest of the note is an inventory of where that assumption can legitimately come from. Four classes are studied, each with a reference algorithm and each stated as an assumption that can be checked and can fail.
- Known-start continuity. The motion is known to begin inside the principal band and to evolve smoothly, so each step can be attached to the branch nearest a continuity prediction. Classical phase unwrapping. It fails the moment the true ratio jumps by more than half a cycle between samples, and every later estimate inherits the error.
- Bounded two-rate congruence. The same process is observed at two sampling rates, and the true frequency is known to lie in a bounded interval. Given rates without a large common factor and a bound tight enough, the two ambiguous observations agree on a single candidate inside it.
- Fiducial timing. A physical mark passes once per revolution, and the interval between detections gives the rate directly. This route bypasses the wrapped phase stream rather than repairing it.
- External coarse prior. A second instrument supplies an estimate good enough to pick the branch, which needs an accuracy of only half a wrap.
Six demonstrations, beginning with impossibility
Each method is exercised on synthetic data, and the first demonstration is deliberately the impossibility baseline: a constant signal, no assumption supplied, no recovery attempted. The remaining five supply what their recovery method needs, and each recovers the motion: four demonstrate a single assumption class, and one seeds a coarse prior then hands over to continuity tracking. Residual errors run to a few thousandths of a cycle per sample in the two noisy tracking examples, while the idealized noiseless cases recover exactly or effectively exactly.
The demonstrations are illustrative rather than competitive. Their point is not that these algorithms are fast or best, but that recovery succeeds precisely when its assumption holds and fails when it is absent. The combination that looks most useful in practice is the least glamorous one: a coarse prior to seed the branch, then continuity tracking to hold it.
The design question changes
The engineering consequence is a different question at the start of a project. Not which de-aliasing algorithm is best, but which disambiguating information source is physically available in this sensor system. Each of the four assumption classes points at a different answer: build the acquisition so the motion starts in band, run two clocks, put a mark on the shaft, or fuse a second modality. Rotor diagnostics, encoders, wheel-speed estimation, machine vision and stroboscopic imaging all sit somewhere in that table.
The note is explicit about its own narrowness. It offers no universal de-aliasing method, benchmarks nothing against standard industrial practice, and its demonstrations are synthetic and idealized. A companion stress test adds dropout, jitter and abrupt jumps, and a later appendix works on surrogates built to match the published metadata of a public rotor dataset, but those remain synthetic too. Validation on the real recordings is named as future work rather than claimed. What the note does deliver is a piece of bookkeeping: the hidden assumption in every alias recovery, written down where it can be argued with.