RF / antennas / phased arrays
In developmentAetherArray
An electronically steered array, and what it takes to make one point where it is told to
Objective
- Question
- A phased array only points where its calibration says it points. How many physical measurements does it take to know that?
- Scope
- Four channels, phase-only control, and the calibration treated as an inverse problem rather than as a trim step
- Counted resource
- The number of physical measurements, not compute time
- Evidence
- github.com/DanielMBouyou/AetherArray, the public lab notebook
State of the project
- Stage
- Architecture research and feasibility study
- Rev A schematic
- Captured in KiCad, electrical rule check clean, ready for review
- Layout
- Not started
- Hardware
- Nothing fabricated. No array exists.
- Measurement
- No measurement, no calibration run, no pattern.
- Frequency
- Still open, blocked on the instrument and environment audit
What exists, and what does not
Nothing is built. No array, no fabricated board, no bench, no calibration. What exists is the public lab notebook (mathematics, architecture decisions, benchmark specification) and a Rev A beamformer schematic, captured and ERC clean. Every figure on this page is a target or a definition.
Array factor and beam steering
A set of elements fed separately. Each contribution reaches the observer with a geometric phase shift \(n k d \sin\theta\); the applied phase \(\phi_n\) is the control knob. Cancel the geometric term in advance and the beam points at \(\theta_0\), with no moving part. That is the whole idea, and it is the last part of the project that is simple.
Orders of magnitude, four elements at half-wavelength spacing
- Beam width
- About 25 degrees. Four elements do not make a narrow beam.
- Side-lobe level
- About -13 dB for uniform weighting.
- Array gain
- About 6 dB over a single element. Doubling the element count adds 3 dB.
- Source
- Theory, for a perfect array. None of these three is a measurement.
Hardware non-idealities
One calculation justifies the entire project. In ordinary coaxial cable the wave travels at about 66 percent of the speed of light, so at 2.4 GHz the wavelength inside the cable is around 82 mm and one millimetre of length is about 4.4 degrees of phase. A centimetre of length difference between two cables is roughly 44 degrees of phase error. Cutting cables by hand destroys the pattern. That figure is worked for 2.4 GHz as an illustration; the operating frequency of this array is not yet settled.
| Source | Origin | Order of magnitude | Correctable |
|---|---|---|---|
| Cable length | Fabrication | Tens of degrees | Yes, by calibration |
| Component tolerance | Manufacturing spread | A few degrees to a few dB | Yes |
| Mutual coupling | The elements see each other | Depends on spacing, often significant | Partly |
| Thermal drift | Temperature change | A few degrees | Yes, if you recalibrate |
| Connectors | Tightening, wear | A few degrees | Yes, but variable |
| Environment | Reflections off nearby objects | Highly variable | No: the site has to be controlled |
For random phase errors of standard deviation \(\sigma\) in radians. At \(\sigma = 30^{\circ}\) that is around 1.2 dB of gain lost, and, more importantly, a rise in the side lobes, which is usually the more annoying of the two. An uncalibrated array works, but badly and unpredictably.
The y = Hx calibration model
Rather than treating each defect separately, gather them into one complex matrix. \(\mathbf{x} \in \mathbb{C}^{N}\) is the vector of commands applied, one complex value per channel; \(\mathbf{y} \in \mathbb{C}^{N}\) is what actually leaves each element; \(\mathbf{H} \in \mathbb{C}^{N \times N}\) holds per-channel gain and phase error on the diagonal and coupling off it. If \(\mathbf{H}\) were the identity the array would be perfect. Calibrating means measuring \(\mathbf{H}\) and pre-compensating with it, and that is where the project becomes applied mathematics: each measurement costs time, and the inversion is unstable when the matrix is poorly conditioned.
Mathematical toolbox
The five questions the project exists to answer
- How many measurements does calibration take?An N-channel array has at least N complex unknowns, and many more with coupling. Each measurement is a mechanical move, a settling time and a noisy acquisition.
- Can you calibrate without measuring phase?Many cheap setups measure power only. Recovering phase from power alone is a classical and non-trivial problem.
- Is a classical method enough?Least squares, regularisation and direct inversion are old, proven and cheap. How far they go has to be established before anything else is proposed.
- Can a learned method cut the measurement count?Not for a first calibration. At four elements the classical baselines already sit at the information bound for power-only measurement, so there is nothing left to save. The surviving question is narrower: once an array has been calibrated before, can a prior learned from its own drift history recalibrate it in fewer measurements than starting again?
- How long does a calibration stay valid?Rarely addressed, easy to measure, directly useful, and the question the learning track now depends on.
The original learning claim did not survive a counting argument; question four is the narrower claim that replaced it.
Calibration methods under comparison
Every configuration is judged against the same metrics and against the information bound.
| Configuration | What it represents |
|---|---|
| Ideal simulation | What theory predicts |
| Uncalibrated | What you get for free, and the reference case |
| Classical calibration | Direct inversion, least squares, regularisation |
| Rotating element field vector | The power-only baseline, at its own minimum rather than padded |
| Orthogonal coding | All elements measured at once, the strongest count baseline |
| Mutual coupling | Calibration with no external probe, if the board allows it |
| Adaptive measurement selection | Choosing each measurement for information gain |
| Learned drift prior | Recalibrating from history rather than from nothing |
Metrics
Definitions fixed before any run. None has a value yet.
Pattern metrics
- Pointing error
- Requested direction minus observed maximum, in degrees
- Gain
- Received power in the wanted direction relative to one element, in dB
- Side-lobe level
- Highest side lobe relative to the main lobe, in dB
- Half-power beam width
- Angle between the -3 dB points
- Pattern deviation
- Overall difference between achieved and wanted pattern
Cost and robustness
- Physical measurements
- The counted resource of the project
- Measurements to recover
- Measurements needed to bring pointing error back under target after drift, starting from the previous calibration. The primary metric of the learning track.
- Distance to the information bound
- Measurement count divided by 4N-4
- Stability over time
- Pattern degradation after hours without recalibration
- Thermal sensitivity
- Variation with temperature
- Reconnection sensitivity
- Effect of disconnecting and reconnecting cables
Rev A hardware, as decided and captured
Rev A is decided and its beamformer schematic captured. Layout has not started; nothing is fabricated or ordered.
- TopologyTwo boards, phase-only, switched-line phase shifting at three bits, four channels. Antennas and connectors on one board, switches and combiner on the other, joined by jumpers.
- Why two boardsPer-element access exists by construction, which keeps the mutual-coupling calibration route and the cable-error experiment available. An integrated splitter with no per-element connector would close both permanently.
- Why three bits and not a digital phase shifterAn integrated per-channel digital phase shifter would cost the entire budget and is specified outside the band that would be used. A varactor reflection-type shifter is cheap but temperature dependent, and temperature-dependent phase drift is the thing this project exists to measure.
- ChannelAn enable switch that either passes the signal or terminates the channel in 50 ohm, then three cascaded switched-line bits of 45, 90 and 180 degrees. Seven switches per channel.
- Common nodeA four-way Wilkinson divider, and a path-select switch that gives the common node either to the analyser or to an on-board logarithmic detector.
- Identical by constructionOne hierarchical channel sheet instantiated four times, with the net topology signature of the four channels compared after generation. There is no second copy to drift.
- Generated, not drawnThe schematic is produced by a generator script, which is the source of truth. The electrical rule check is clean.
In the repositoryDecision 0003: Rev A architectureRev A schematic and ERC
The difficulty not to underestimate
Characterising a radiation pattern requires enough distance for the wave to be planar, \(R > 2D^{2}/\lambda\). For four elements at 2.4 GHz spaced 6.25 cm that is about 0.58 m, which fits on a table. The distance is not the problem. In an ordinary room the signal reflects off walls, floor, furniture and the operator, and those echoes are the same order of magnitude as what is being measured.
- Careful free-space measurementAbsorbers, distance and differential measurement. Realistic, but of uncertain quality, and absorbers cost money.
- Conducted, channel by channelVery repeatable, but does not measure radiated coupling.
- Near fieldAccurate, but needs precise mechanical movement.
- Time-domain gatingVery effective at separating the direct path from later echoes, and free if the instrument supports it.
- Acoustic cross-validationAt 40 kHz in air the wavelength is around 8.6 mm, transducers cost a few euros, and the array theory applies unchanged. Not a fallback: it separates the algorithmic risk from the RF measurement risk instead of adding them together.
Not decided yet. The instrument and environment audit that settles it, along with the operating frequency and the board dimensions, has not been run.
Limitations
No S-parameter, coupling matrix, pattern, gain or side-lobe figure exists. The array has not been built.
The operating frequency, and even the kind of wave used, are still open. They depend on what the measurement environment allows, and that audit has not been run.
Four elements bound what can be shaped, and nothing will be extrapolated to larger arrays from this platform.
The original learning claim, that a learned estimator would cut the measurement count of a first calibration, was withdrawn: at four elements the classical baselines already sit at the information bound for power-only measurement. It is recorded in the repository rather than removed from it.