Arrays, Diversity
& Beamforming

Failure: the second antenna sees the same mode

Both ports look matched. Why does the same node orientation still fail?

The fictional condition-monitoring node still operates near 2.45 GHz. Its gateway team proposes two antennas to cover changing orientations, then a four-element board to “get more range.” In one possible failure, the two feeds excite nearly the same chassis-current mode. Turning the node into that mode’s polarization mismatch or angular null reduces both branch signals together. The second connector has not created a useful alternate response.

Even a small reflected fraction at both real 50 Ω R2 ports cannot settle this. Use the signs carefully: small reflection means a very negative signed S11 and a large positive return loss. Neither describes the joint received-channel distribution. No measured match or pattern is supplied for this gateway.

Keep the node’s earlier N1 requirement intact. The new G4-ULA and G2-PAIR examples are local synthetic variants with separate fixture identities; no scalar pattern from 06.1–06.3 has been promoted to a complex embedded field. We first isolate the spatial mechanism that a coherent array can control.

Spatial phase comes from path difference

Which feed must lag when the beam points toward +x?

Place element n at (nd,0,0), n=0…N−1. Broadside is +z. Our local angle α is signed from +z toward +x in the xz plane, so u=(sinα,0,cosα). Positive α puts the positive-x elements closer to a distant observation point. Their paths are shorter by nd sinα.

Remove the common propagation factor e^−jkr. The relative outward phase is positive. To make all contributions align at positive α₀, put a negative progressive phase in the feeds. Do this once: putting steering into both the weights and a second compensating exponent would steer the wrong pattern.

ADI’s array-pattern derivation, Part 1 supplies accessible path-difference context. The signs and axes above are derived for this lesson. Having aligned the phasors, we must still ask what field each element contributes.

Element pattern times array factor—under assumptions

Does a perfect feed phase guarantee radiation in that direction?

An array factor accounts for positions and excitations. It cannot supply a field where an individual element has none. If the elements have identical aligned patterns and polarization, see the same environment, are negligibly coupled, and operate in the far field with a narrowband excitation, their common element field can factor out of the sum. These are model assumptions, not attributes of every four-port board.

Our element choices are Epow=1, an isotropic teaching element, and Epow=cos²α, a broadside scalar power envelope in this cut. Neither is a full physical vector antenna. An isolated pattern belongs to one antenna in its stated surroundings. An embedded pattern belongs to a driven element with the other elements present and their ports terminated as declared. Embedded patterns need not be identical; then sum their complex vector fields instead of multiplying by one scalar pattern.

The cosine envelope reduces power at the commanded 20° by 0.540284 dB. Its combined cut maximum shifts to about 18.292°, because the envelope favors broadside. This is distinct from frequency squint. Total real-array scan loss would also need active match, element efficiency, feed loss, polarization and actual embedded patterns.

ADI Part 1’s pattern-multiplication discussion motivates this separation; Balanis, fourth edition, Chapters 6 and 8 is the book route for arrays and coupling. Only its publisher contents were accessible here. Next, fix the input energy so the comparison means something.

Spacing, steering, beamwidth, and sidelobes

When four fields add, what happened to the total feed power?

For fixed-phase hardware at f₀, start with nonnegative taper aₙ and apply gain and phase offsets. Uniform taper uses aₙ=1; binomial uses C(N−1,n), giving [1,3,3,1] for N=4. Normalize the complete impaired vector, so changing taper or deterministic imbalance does not secretly increase the total feed input.

These are independent, uncoupled, matched R2 feed waves. A coherent uniform sum then has field √N and power N. For N=4, |wₙ|=.5, AF(20°)=2+j0, and |AF|²=4. The 6.020599913 dB coherence factor compares this direction to one unit-normalized isotropic element; it does not calibrate absolute antenna gain. An unnormalized sum of four unit fields gives power 16 while using total feed energy 4.

More aperture generally narrows a main lobe, but also changes which replicas fit in the visible domain. Steering toward endfire broadens angular width because a fixed interval in sinα covers more degrees near |α|=90°. Binomial weighting trades coherent peak and beamwidth for suppression of intervening sidelobes in the usual half-wave broadside example. It cannot remove geometric spatial aliases.

HPBW uses the two crossings at half the selected lobe’s power, −3.010299957 dB. Our 0.1° plot grid locates brackets; bounded refinement resolves reported extrema and crossings to better than 0.01°. The sidelobe metric searches outside both first-null boundaries and includes replicas. Without two separating nulls and an external maximum it is “not defined in this cut.” A two-element broadside response must not invent a sidelobe from its missing region.

1. Predict the negative feed-phase sign, then read the default weight table.
2. Change only s from .5 to .75 and update. Inspect physical spacing and all replica flags.
3. Change only α₀ from 20° to 30°; find m=−1. Compare the grating preset.
4. Use the squint preset: hold d fixed at f=1.1f₀. Switch hardware to true delay and update.
5. Use the default, then add only δ=30°. Compare the common-reference ideal overlay and the actual alternating weights.
6. Switch to combining. Vary only ρ through 0, .7 and 1, retaining the eight samples and both SNR settings.

ADI Part 3 discusses taper tradeoffs. The finite four-element sidelobe value here comes from the disclosed local model, not a universal −13 dB approximation. Now distinguish an ordinary sidelobe from a fully coherent replica.

Grating lobes, scan range, and beam squint

Could a second direction satisfy exactly the same phase progression?

Neighboring phase differences that differ by an integer 2π are indistinguishable to the ideal regularly spaced array. Solve for every integer m, not merely the highest sampled peak. m=0 is the intended coherent branch; all nonzero m within the visible domain are replicas, including an equality at endfire.

At the operating frequency, the strict condition for no visible replica over symmetric scan |α₀|≤αmax with steering referenced to that frequency is d/λ<1/(1+sinαmax). Equality touches a boundary. For an ideal half-wave array at that frequency, all interior steering angles are free of these replicas; the exact endfire limit has an opposite-endfire ambiguity. This is a precise qualification, not a claim that half-wave spacing arbitrarily fails at its design frequency.

A phase shifter retains −nk₀d sinα₀ as f changes. A true delay τₙ=nd sinα₀/c instead supplies phase −2πfτₙ. Relative delays can include an arbitrary common delay so every physical path is causal, even for a negative scan command. Real delay devices still have loss, range, resolution and calibration limits.

ADI Part 2’s grating-lobe and beam-squint sections provide the engineering context. The inclusive endpoints, fixed physical spacing and local α mapping above govern this explorer. Real ports introduce another boundary condition: each element can alter the others.

Mutual coupling and embedded element patterns

What does an “inactive” antenna do when its neighbor is driven?

Its conductor remains in the field and can carry induced current. Changing its termination changes that response. A nearby element can redistribute currents, reflected waves, dissipation and radiation, so a good isolated match need not survive a scanned excitation. The relevant product includes the loads, feed network, enclosure and cables.

To collect an embedded pattern, drive one declared port, leave all other physical elements in place and terminate every inactive port in a named load. Record complex Fθ and Fφ on a common angular grid and phase origin, the driven incident-wave normalization, reference impedances and source conditions. Repeat for each port. Moving the reference origin without the associated phase transformation corrupts a coherent field sum.

The active reflection ratio depends on the whole excitation vector. Under mutually driven ports its magnitude can exceed one because other sources deliver coupled power; it is not an isolated passive one-port efficiency test. Total incident and reflected multiport power must be accounted for together. The lesson supplies no complex S, so active reflection and coupled accepted power are unknown, not calculated zeros.

Honma and Murata (2020), §§2.5–2.6, explicitly connect embedded patterns, common reference points and port conditions. The unknown evidence row below asks for those data before attempting correlation.

Pattern, polarization, and spatial diversity

Different antenna shapes are visible. Are different useful channels established?

Pattern diversity seeks complementary angular responses; polarization diversity seeks different transverse couplings; spatial diversity seeks different path phases and amplitudes at distinct positions. Each depends on which waves arrive. Two orthogonal free-space polarizations may be useful in one scattering environment and leave a dominant single-polarized path weak on one branch. Separation in wavelengths alone does not assign a universal correlation.

Before using an envelope-correlation coefficient, define a complex field overlap. Let Fᵢ(Ω)=[Fθ,ᵢ,Fφ,ᵢ]ᵀ be embedded fields in a common eθ/eφ basis with one phase origin, compatible loads and consistent incident-power normalization. Let W(Ω) be a Hermitian positive-semidefinite 2×2 angular/polarization power weighting in that same basis. It describes the chosen incoming-wave population; it is not supplied by the antenna alone.

Interpreting this overlap as channel correlation assumes an appropriate incoherent angular-scattering model and the declared polarization statistics. Direction-to-direction coherent multipath may require a fuller correlation kernel. A normalized complex channel covariance, a correlation of powers |h|², and an envelope statistic of |h| are different definitions. Under certain jointly circular Gaussian channel assumptions, power correlation relates to |r|²; an “ECC” formula must name its particular convention and assumptions.

Honma and Murata’s §§2.2–2.6 distinguish correlation definitions and the lossless limitation of the port shortcut. Our weighted expression extends the disclosed basis to a named channel population; no product W is assumed. Tse and Viswanath, Chapters 2–3, supply channel/diversity context. Next use a completely specified population to make the combining arithmetic testable.

Selection, MRC, and beamforming families

Are we selecting an observation, aligning signals, or steering a field?

For direct-sum coefficients cᵢ, the desired field is Σcᵢhᵢs. Independent noise powers contribute Σ|cᵢ|²σᵢ². With perfect instantaneous channel knowledge and no interference, maximizing output SNR gives the conjugate channel scaled by noise variance. The alternative Hermitian-vector notation already contains the conjugation.

For h=[1,j], use direct-sum c=[1,−j], giving two aligned signal terms. Using c=[1,j] instead cancels this example. Colored receiver noise or interference requires a covariance-aware solution; the explorer neither estimates covariance nor applies this sum rule to that unsupported case. Tse and Viswanath, Chapter 3, §3.2.1, equation (3.33), and §3.3.1, develop matched-filter combining and receive diversity under their channel assumptions.

Use real ρ∈{0,.7,1}, γ̄=10 linear (10 dB) per branch, and a 10-linear requirement. Both branches retain unit mean channel power and complex correlation ρ. γ̄ already represents the branch signal/noise balance after the declared ideal receive transformations; do not subtract the same feed or receiver loss again. No antenna impedance, absolute received power or angular population is assigned to these channel numbers.

For ρ=0, selection is always 12.041199827 dB and MRC always 13.010299957 dB. For ρ=1, MRC is exactly 3.010299957 dB above each branch in every sample, yet four rows remain below the requirement. Common fading and independent receiver noise are separate facts.

Eight ordered samples can verify arithmetic and expose a mistaken claim. They cannot identify asymptotic diversity order, channel capacity, a real-site outage probability or a certification result. The missing link between a useful coherent sum and physical hardware is calibration.

RF-chain coherence and calibration drift

The geometry stayed fixed. Why did yesterday’s weights stop aligning today?

The propagation phase is only one contribution. Cables, filters, amplifiers, mixers, converters and clocks add branch-dependent transfer functions. Their amplitude and delay can vary across frequency and temperature. A phase value calibrated at the center frequency does not remove a differential group delay across the band.

The explorer’s δ=30° case is deliberately deterministic: even elements get +15°, odd elements −15°. At the original command with uniform N=4, zero gain error and isotropic elements, Fcut=4cos²15°=3.732050808, −0.301124437 dB on the common reference. The selected maximum can shift. Do not label δ a 30° RMS random error or infer a thermal coefficient from it.

ADI’s Calibration Implementation Techniques for Multichannel Phased Array Subsystems discusses synchronization and calibration across frequency and temperature. Its example hardware results are not assigned to this gateway. With the ideal mechanisms and open hardware evidence separated, make a conditional design choice.

Choose the gateway objective and evidence plan

Which proposal answers the actual installation problem with defensible evidence?

Begin with orientation coverage for the monitoring node, unless the installation has a known sector or a named fading-outage target. Provisionally prototype G2-COVERAGE: two efficient, distinct installed modes with selection, recording switching loss and quality-estimation delay. This is an experiment choice. It is not evidence that two antennas have already met coverage.

Freeze frequency edges, element positions and orientations, reference planes, loads, polarization basis, gain/phase normalization, mechanics and specimen IDs before collecting data. Record total efficiency as well as relative pattern shape. Pair the installed node and gateway with a named population of locations, orientations, times and interference conditions. Choose the coverage/outage statistic and uncertainty decision rule before looking for a favorable result.

The complete local p06-evidence-map-v1 snapshot below preserves M01-INSTALL-UNKNOWN and records new module-owned rows. Requirements are proposed thresholds; illustrative rows are equations; unknown rows have no observation. None is relabelled as measurement. R2-TX/R2-RX map to R2, S0 names space, and R3-C is the combined detector/decision boundary. Incident, accepted, available receive, delivered receive, EIRP and TRP remain different quantities.

The next decisions are channel distributions in 06.5 and measurement reconciliation in 06.6. Check the path overview for current availability. Carry the open population, uncertainty and calibration questions forward; the present lesson establishes reasoning and reproducible model arithmetic.