Lesson 16 showed that the array factor repeats, and that a repeat reaching visible space is a grating lobe as tall as the main beam. This lesson turns that result into a design rule. Every array pattern in this module has been a function of one variable, Lesson 16’s inter-element phase with Lesson 18’s steering ramp subtracted,

and the uniform array factor built on it,

Replace \(\psi\) by \(\psi + 2\pi\) and both sines flip sign together, so \(\vert AF_N \vert\) is unchanged. The array factor is periodic in \(\psi\) with period \(2\pi\). The main lobe is the peak at \(\psi = 0\), and the same peak reappears at \(\psi = \pm 2\pi, \pm 4\pi, \ldots\) — every one of them a full-height beam.

Whether we ever see those repeats depends on how much of the \(\psi\) axis the real angles \(-90^\circ \le \theta \le +90^\circ\) reach. We call that span visible space, and the spacing sets its width: as \(\sin\theta\) runs from −1 to +1, \(\psi\) runs over a window of total width \(2kd = 4\pi d/\lambda\). Widening \(d\) brings more periods of the array factor into visible space. A repeat that lands inside visible space is a grating lobe. The figure draws both cases on the same axis: at 14 mm the window stops at ±173°, short of the repeats at ±360°, and at 42 mm it reaches ±519° and takes in both.

Set \(\psi = 2\pi m\) and solve:

for integer \(m = 1, 2, \ldots\) Any solution with \(\vert \sin\theta_g \vert \le 1\) is a real angle, and the array radiates a second beam there. The thinning example’s figure shows why from the elements’ side: at 43.9° the fed elements of the 42 mm row sit exactly one wavelength of path apart, so their phases read 0°, 360° and 720° and add exactly as they do at broadside.

Thinning has two effects. The three fed elements still span an 84 mm aperture, so the main beam stays narrow, but with three elements instead of eight the array collects less signal and the pattern between the beams is poorly controlled. Each fed element adds the same voltage at the peak, so the peak falls by the fraction of elements fed, every third element and then every fourth:

The phaser-00 test run measured −7.8 and −10.9 dB, about 1 dB short of both. The same shortfall appears when the full array is compared with its single elements, and its cause is still open. The narrow beam of a large aperture with the ambiguity of a sparse one is the classic thinned-array trade.

We want no grating lobe anywhere in visible space, for every angle we intend to scan to. The dangerous solution is the one closest to the horizon on the far side of the beam. For a beam steered to positive \(\theta_0\), that is \(m = 1\) with the minus sign, and pushing it past the horizon requires

The wider we intend to scan, the closer together the elements must be. Broadside operation tolerates \(d < \lambda\). Scanning to ±45° demands \(d < 0.586\lambda\), and ±60° demands \(d < 0.536\lambda\). Take the criterion to its limit, \(\theta_0 \to 90^\circ\), and you get \(d < \lambda/2\) — which is why half-wavelength spacing is the default in every scanning array. It is grating-lobe-free at any scan angle we can command. The figure, from Lesson 16, draws the criterion as a boundary: any spacing and scan angle below the curve is free of grating lobes, and the PHASER’s \(0.481\lambda\) sits below it everywhere.

The beam-steering computer sets each phase shifter once, in degrees. The angle that setting steers to is not fixed, because the phase a given path length produces depends on frequency. The figure shows the result across the aperture for the worked example below: the shifters apply 125.1° per element, set for 45° at 10.525 GHz, and a 10.025 GHz signal needs 119.2° per element to add in phase at 45°. The difference, 5.9° per element, is itself a straight ramp, and a straight ramp is a steer: the beam moves to a new angle and stays intact. Lesson 18 named the result, beam squint, and the L18 Lab showed it once on the kit; here we compute it.

Suppose the beam-steering computer sets element \(n\) to a lag of \(n\ k_0 d \sin\theta_0\), computed at the design frequency \(f_0\) where \(k_0 = 2\pi/\lambda_0\). Now a signal arrives at some other frequency \(f\) in the band. The elements add in phase toward whatever angle \(\theta\) makes the arrival phase cancel the applied lag:

so the beam points at

away from the commanded angle.

This is beam squint. It does not depend on \(N\) or on \(d\) — the phase ramp and the path lengths scale together, and only the ratio \(f_0/f\) survives. It vanishes at broadside, where the ramp is zero and there is nothing to scale. For small fractional offsets \(\delta = (f - f_0)/f_0\), expanding the arcsine gives the small-offset form:

The beam moves toward the horizon as the frequency drops, and it moves fastest when the beam is already far off broadside.

The figure plots both forms for signals 500 and 1000 MHz below 10.525 GHz. Squint is under a degree near broadside, 2.9° and 6.4° at 45°, and 5.4° and 13.1° at 60°. The small-offset form tracks the exact one at small angles and falls short at wide scan, by 3.7° at 60° for the 1000 MHz offset, so a budget at wide scan should use the exact form.

Squint affects two kinds of system differently. A wideband radar transmits a pulse whose spectrum is the bandwidth; each frequency component in that pulse points somewhere slightly different, so the pulse’s energy is spread in angle and the effective beam is broadened and depointed. A communications link running a modulated carrier loses gain at the band edges relative to the center, which shows up as amplitude tilt across the channel rather than as an angle error.

The cure is to stop steering with phase. If instead of a phase lag you insert a true time delay of \(\tau_n = n\ d \sin\theta_0 / c\) in each element, the delay compensates the geometric path difference itself, which is a length and not a phase. Every frequency in the band then arrives in phase at the same angle, and the squint is gone at all offsets. True time delay is expensive in analog hardware, which is why large wideband arrays usually compromise: time delay at the level of the subarray (a group of elements combined before the rest, as on the PHASER’s two four-element halves in Lesson 17) to control the squint across the aperture, phase shifters inside each subarray where the residual path differences are small. The figure shows why for one element of a 45° steer: a 33.0 ps delay produces a phase that grows in step with frequency, while a phase shifter holds its 125.1° at every frequency, so the two agree only at 10.525 GHz and differ by 5.9° at 10.025 GHz.

The ADAR1000 does not accept an arbitrary phase. It has a \(B\)-bit phase shifter, and \(B\) bits divide the circle into \(2^B\) steps. The smallest step, one least significant bit (LSB), is the grid Lesson 18 rounded its 30° phase table onto:

The ADAR1000 uses \(B = 7\), giving 128 states and an LSB of 2.8125°. The ADAR1000 rounds every commanded phase to that grid, which has two consequences.

The inter-element phase for a beam at \(\theta_0\) is \(\Delta\phi = 360^\circ (d/\lambda) \sin\theta_0\). If the controller can only command multiples of the LSB, only a discrete set of beam angles exists:

so the step near broadside is

For the PHASER at 10.525 GHz, \(d/\lambda = 0.491\), so the step is \(2.8125/(360 \times 0.491) = 0.0159\ \text{rad} = 0.91^\circ\) at broadside. Compared with the 13.2° beamwidth that is about one-fourteenth of a beam, which is fine. Off broadside the same step in \(\sin\theta_0\) spans a wider angle, because \(\Delta(\sin\theta_0) = \cos\theta_0\ \Delta\theta_0\). The step therefore grows as \(1/\cos\theta_0\), reaching 1.8° at 60°. Drop to \(B = 2\) and the same calculation gives 29° per step: a 2-bit array can only point at broadside and at about ±31°.

The figure shows the angles each bit count can reach on the course array. With 3 bits they are 0°, 14.7°, 30.6° and 49.7°, equally spaced in \(\sin\theta_0\) and so spreading apart in angle as the scan widens. A real controller rounds each element on its own, which fills in the angles between these and brings in the quantization lobes below.

A real controller does better than a uniform quantized increment. It computes the exact ramp for every element, then rounds each element independently, which puts the beam much closer to the commanded angle. What is left behind is a staircase: the difference between the exact ramp and the rounded one is a sawtooth error, bounded by half an LSB, that repeats across the aperture. A periodic phase error radiates, and where it radiates is a quantization sidelobe. The figure shows the staircase for a beam at 28° with 3 bits: the exact ramp climbs 83.1° per element, the shifters can only set multiples of 45°, and the error left at each element rises and falls within ±22.5°.

Its level follows from the size of the error. Rounding leaves at most half an LSB at each element. For a small phase error \(e^{j\phi} \approx 1 + j\phi\), so the error adds a small field to each element in proportion to \(\phi\), and the lobe that field radiates scales with the sawtooth’s peak:

Each extra bit halves the error, which halves the lobe’s field and lowers the quantization sidelobe level (QSLL) by 6 dB. That gives the rule of thumb, so 2 bits puts a lobe near −12 dB, 3 bits near −18 dB, and the ADAR1000’s 7 bits near −42 dB. Treat this as an RMS estimate for a large array. On an aperture as short as eight elements there are only a couple of sawtooth periods, so individual lobes scatter several dB either side of the rule. Taken at face value, the rule says that from 3 bits up the quantization lobes drop below the array’s own −13 dB sidelobes and no longer set the sidelobe level.

The hardware disagrees. In the phaser-00 test run, with the source fixed, the kit read the strongest lobe beyond ±35° at −8.5 dBc with 2 bits, −12.0 dBc with 3, −15.5 dBc with 4, and −17.4 dBc, the floor of that sweep, with 7. Every reading sits above the rule, by 3.5 dB at 2 bits and by 6 to 8.5 dB at 3 and 4 bits, so on this kit 3 bits is not enough to hide the quantization lobes under the −13 dB sidelobes. The Angle readout moved too: it read +8.4° at 4, 3, and 2 bits against +1.8° at 7, several times the pointing step derived above. The cause of the extra error has not been confirmed. At low bit counts, read the lobe levels on the sweep rather than the Angle readout.

The figure adds a third curve: the worst quantization lobe the course array can produce at any steer angle up to 50°, computed from the field the rounding error radiates on its own. It sits about 4 dB above the rule at every bit count, because the rule is an average for a large array and eight elements leave room for a worst case. That accounts for the 2-bit reading, −8.3 dB simulated against −8.5 dBc measured. The 3- and 4-bit readings still sit 2.5 and 4.7 dB above even the worst case, so rounding the steering ramp does not explain them by itself.

More bits improve two things: finer pointing, which scales as the LSB, and a lower sidelobe floor, which scales as 6 dB per bit. Seven bits is the standard choice because it puts the quantization floor far below every other error in the array.

Dither lowers the peak quantization lobe without new hardware. If the rounding error is the same sawtooth on every beam position, it radiates coherently into one lobe. Adding a small known phase offset per element before rounding breaks the periodicity, which spreads the same total error energy over many angles instead of concentrating it in one.

The effect needs a longer aperture than eight elements to see, so the figure uses 32. With plain rounding the error drifts about 4° per element and repeats as a sawtooth, and the field it radiates peaks at −16.9 dB. With a fixed, known offset built into each element and the command rounded around it, the error stays within the same ±22.5° but no longer repeats, and for a typical set of offsets its peak falls to −21.8 dB.

One thing quantization does not set is how deep a null can go. It is tempting to assume that a cancellation is only as good as the phase resolution behind it, but the LSB does not limit the null weights: rounding them to a 2.8125° grid with 1% gain steps leaves the pattern 56 dB down at the angle it was designed for, and the pattern still passes through zero a quarter of a degree away, at +22.3°. Rounding moves the notch; it does not fill it. Even independent errors of one step in every channel, which Lesson 24 works through, still allow a null about 48 dB down.

Our first estimate of the measured limit came from the receiver. Earlier sweeps on the PHASER showed a noise floor about 23 dB below the uniform-taper peak, and the weights that create a null reduce the main-lobe gain by about 2 dB, so we expected a sweep to show nothing deeper than about 21 dB below the new peak.

The phaser-00 test run did not bear that out. In the L24 Lab the notch reached −46 to −50 dB below the uniform peak, the difference null −41.5 dB, and the delta null −39.7 dBc. The deepest of these sits close to the −48 dB that one-step channel errors allow, so on that kit the sweep followed the pattern most of the way down. The same session’s L22–L23 Lab sweeps, however, read floors between −17 and −31 dBc, and a wall or a person beside the kit raised lobes as high as −9 dBc. The floor a sweep shows is therefore not one fixed number: it changed from lab to lab on the same kit, and the bench around the kit changed it further.

The three failures look different on a sweep and a designer fixes each one in a different place.

Part 2 of the L22–L23 Lab measures all three of today’s departures on your own kit. Lesson 24 then extends pattern control from the whole beam to one chosen angle. So far you have shaped the beam and the sidelobes as a whole; next you will place a null at a chosen angle on purpose, which is what an array does when a jammer or an interfering transmitter sits somewhere you know about. The weights come from subtracting one steering vector from another, and the arithmetic is short.

Both of today’s floors appear again in L24. Quantization moves the notch slightly off its commanded angle without filling it, and the floor of the sweep decides how much of the notch you can see. We predicted a measured depth near 21 dB; phaser-00 reached −40 to −50 dB, so in the L24 Lab compare your own notch against both numbers. Before Lesson 24, be able to state the LSB of a \(B\)-bit shifter and the pointing step it implies on the course array, and review the complex element weights \(w_n\) from Lesson 22 — the null-steering result is written entirely in that notation.