The weight-subtraction rule of Lesson 24 holds the beam at broadside and nulls an interferer at \(\theta_1 = +22.5^\circ\):

The table converts the weights to what the GUI accepts: element gain as a percentage of the largest, \(100\vert w_n\vert/\max\vert w\vert\), and phase offset as \(\angle w_n\), entered with its sign. It is worked at 10.443 GHz, the frequency Find HB100 reported for phaser-00, one of the course kits. Part 1 of the packet has you compute the same table at your own kit’s frequency; at the nominal 10.525 GHz the outer gains read 75 % and the phases of elements 2 and 7 read ±3.1°, so the table moves only slightly from kit to kit.

The predictions assume the Element Gains slider is linear in voltage, which it is on current Phaser software. On phaser-00 the main lobe dropped 2.5 dB against the predicted 2.0.

The gains are symmetric about the array center and the phases are antisymmetric, the conjugate mirror pairs of Lesson 24’s weight circle. Neither column nulls anything on its own; only the eight weights together do.

Lesson 24 estimated the depth that small, independent channel errors allow, about 48 dB below the uniform peak, and the 2026-10-23 dry run on phaser-00 reached 46 to 50 dB. Rounding the weights to the ADAR1000’s 2.8125° and 1 % steps does not set that limit: the rounded table keeps its mirror symmetry, so its null survives and moves only a fraction of a degree. On a freshly calibrated kit the notch landed at +21.7°. In a cramped space on 2026-10-25 the same table bottomed out 22 to 24 dB down, and so did the difference null of the next frame: the HB100’s own tone was still there in the null, so a reflection arriving from another direction, about 22 dB below the direct path, was filling both.

When a notch reads much shallower than 40 dB, look at the bench before the arithmetic. A wall or a person beside the kit fills it, a stale calibration fills it, and a sign error on the phase column moves it to −22.5°, where it is easy to miss.

There is a second null on this board that needs no computation at all. Each ADAR1000 sums its four elements into one RF channel, so the PHASER hands the Pluto two digital subarray channels, one for elements 1–4 and one for elements 5–8. Adding them is the ordinary sum beam. Subtracting them, by setting Beam 1 Phase to 180°, cancels the two halves wherever they arrive in phase, which is boresight. The result is a null there with twin peaks near ±11°: the monopulse difference beam, which Procedure D and Module 4 use for angle measurement. On phaser-00 it measured −41.5 dB on an open bench and −22 dB in a cramped one, centered on boresight, with the peaks at ±12.0°. It needs no arithmetic, so its depth is a direct reading of what the room lets through.

The +22.5° notch exists because we solved for eight weights, and it moves only when we solve again. The boresight null exists because the two subarrays are identical, and it stays at boresight no matter what the signal does.

Procedures A and B both needed something known in advance: the interferer’s angle in A, the array’s symmetry in B. The MVDR (minimum-variance distortionless-response) beamformer needs neither. It estimates the covariance of what the two channels receive over \(K\) snapshots,

and solves for the weights that minimize total output power while holding unit gain in the look direction \(\mathbf{s}\). Whatever is loud and is not in the look direction is suppressed, and the beamformer never has to be told where it is.

Unit gain means the two MVDR weights sum to 1 in the look direction, where Manual’s two weights of 1 sum to 2. The Phaser multiplies the MVDR output by 2, the number of channels, so a target in the look direction reads the same level under either mode and the two are compared on one scale.

Both limits belong to the architecture rather than to the algorithm. A four-element subarray at \(d = 14\) mm has its first null where \(\sin\theta = \lambda/4d\), about 31°, so an interferer there barely reaches either channel and there is little left for the digital layer to remove. The two channels sit four elements apart, so their phase difference, \(4kd\sin\theta\), runs through a full cycle across the subarray beam: a source near +30° presents very nearly the same channel phase difference as one on boresight, and the two channels cannot tell them apart. The digital layer can only work on what the analog layer passes it, and only on directions the two channels can distinguish.

At +30° both limits act at once, and the bench shows what follows. MVDR still spends its one null on the interferer, but to the two channels the interferer looks like the look direction, and the HB100 sits a fraction of a degree off it, so the null lands on the target too. On phaser-00 the target dropped about 10 dB, and MVDR did worse against the interferer than Manual. Diagonal loading is the guard: the Phaser adds \(\delta\ \text{tr}(\hat{\mathbf{R}})/2\) to the diagonal of \(\hat{\mathbf{R}}\) before inverting it, which caps how far the weights can move from Manual’s. At \(\delta = 0.1\) the target loss at +30° fell to about 2 dB, and at +20° MVDR still suppressed the interferer by about 17 dB, so Procedure C uses \(\delta = 0.1\). At \(\delta = 1\) the weights stay so close to Manual’s that the suppression falls to about 5 dB.

Procedure C also uses \(K = 1024\) snapshots rather than 128. At the kit’s 3 MSPS, 128 samples span only four cycles of the 100 kHz offset between target and interferer, so the two are partly correlated in each estimate of \(\hat{\mathbf{R}}\) and the null wanders from one update to the next. 1024 samples span 34 cycles; on phaser-00 the spread of ten suppression readings fell from about 25 dB to under 10.

Manual null steering has eight degrees of freedom and can place several nulls at once, but every one of them is our arithmetic, computed for a geometry we assumed. MVDR has two degrees of freedom on this board, one constraint and one null, and it finds that null itself, from data, several times a second. On an array with a digitizer behind every element, adaptive beamforming has both advantages at once. On the PHASER we can see what the hybrid architecture costs: the analog sums discard the per-element information before the algorithm sees it.

Every measurement so far has come from a sweep: the array steps its commanded angle across the field of view and records power at each step. A sweep takes time, and a tracking radar does not have it. Monopulse forms both beams at once from the two subarray channels, adding them for \(\Sigma\) and subtracting them for \(\Delta\). The sum beam is flat at its peak, so its level barely changes as the target drifts a degree off axis. The delta beam is zero on boresight and climbs steeply out of that zero, so its level changes a great deal over the same degree. Dividing one by the other turns that small angle into a large, readable number.

The two subarrays sit four elements apart, so \(\Delta\) arrives in quadrature with \(\Sigma\), and the useful quantity is the imaginary part of their product. The error function is zero on boresight, positive on one side and negative on the other, and straight within about ±5°, well inside the array’s 13° beamwidth. A target one degree off axis reads about 0.11, and a tracker that drives that reading back to zero holds the beam on the target far more finely than the beamwidth alone would allow, from one look and with no sweep.

The GUI plots a normalized form of the same comparison, taking its sign from the phase difference between the two channels and its magnitude from the two channel levels in dB,

so the trace stays bounded and on screen however strong the target is.

The rules come from the Module 3 dry runs. A peak near −10 dBFS keeps the receiver linear with room above the floor; at Rx Gain 10 dB the FFT peak read +2 to +3 dBFS at these distances and the receiver clipped. Every prediction uses the frequency Find HB100 reports for your kit, not the nominal 10.525 GHz; the kits sit anywhere from 10.41 to 10.46 GHz. Level comparisons need the source at 1.2 m or more, because closer in the residual curvature across the aperture puts errors of a few tenths of a decibel on every peak.

In the phaser-00 dry run, a wall beside the kit produced a lobe at −37° to −41° only 9 dB below the peak, and a person seated beside the kit left a −17 dBc shelf on that side. Either one fills a notch. A stale calibration made the uniform trace’s two first sidelobes read unequal, and recalibrating fixed it. Signal BW returned to zero after each of Calibrate, Find HB100, and the Rectangular preset, so check it before every sweep you record.

Lesson 25 derives the radar range equation and asks the question this module never had to: how much power comes back from a target that transmits nothing of its own. Everything Module 3 built, beamwidth, sidelobes, scan loss, and steering, becomes the antenna terms in that equation, and the array you have been sweeping becomes the front end of a radar. The monopulse pair of Part D measures a target’s angle from a single look, which is what a radar needs to track something that moves; the range equation supplies the signal that feeds it. Keep your Table D numbers: the slope of the error function near boresight sets how finely a tracking radar can measure angle, and Module 4 returns to it.