Antenna Arrays, Beamforming & Beam Scanning — ULA, URA, Phased Arrays

// The Motivation

Why Use an Array?

A single patch antenna at 2.4 GHz has a gain of about 5–8 dBi and a beamwidth of roughly 80°. That's fine for a WiFi router covering a room, but terrible for a 5G base station trying to serve 100 users simultaneously, or for a radar that needs to detect a drone 5 km away.

An array of antennas solves this by doing something physically beautiful: using the constructive and destructive interference of radio waves to concentrate energy in one direction and cancel it in others. No mechanical moving parts, no exotic materials — just careful control of the signal fed to each element.

The four things arrays give you that a single element cannot: (1) Higher gain — the beam concentrates power into a smaller solid angle. (2) Narrower beamwidth — so you can aim at a specific user or target. (3) Electronic beam steering — you can point the beam anywhere in microseconds by changing phase shifts, with no mechanical movement. (4) Multiple simultaneous beams — with digital beamforming, the same array can serve many users at the same time.

// The Physical Principle

How Arrays Work — Constructive & Destructive Interference

The secret of every array is the same physics that makes water waves add up or cancel each other. Each antenna element transmits (or receives) a wave. In certain directions, the waves from all elements arrive in phase — they add up constructively and the signal is strong. In other directions, the waves arrive out of phase — they cancel destructively and the signal is weak or zero.

The array "points" its beam toward the direction where all element contributions add up perfectly. The element spacing d and the wavelength λ determine at what angles this constructive interference occurs. The number of elements N determines how sharp the beam is — more elements means the constructive addition happens over a narrower angular range, so the beam is narrower.

// The Mathematics

The Array Factor — What Every Array Pattern Calculation Starts From

The total radiated pattern of an array = Array Factor (AF) × Element Pattern (EP). The element pattern is the pattern of a single antenna element (patch, dipole, etc.). The array factor is the pattern created purely by the geometry and excitation of the elements — it's the same regardless of what type of element you use.

Uniform Linear Array (ULA)

A ULA has N identical elements equally spaced by d along a straight line. Each element is excited with the same amplitude but a progressively increasing phase shift β·n (where n is the element index 0 to N−1). The array factor is:

ULA Array Factor

AF(θ) = Σₙ₌₀^(N-1) wₙ · e^(j·n·ψ)

where ψ = 2π·(d/λ)·sinθ + β
β = progressive phase shift between elements (controls beam direction)
wₙ = amplitude weight of element n (controls sidelobe level)
d = element spacing, λ = wavelength, θ = angle from broadside

For a uniform (unweighted) array: AF(θ) = sin(N·ψ/2) / sin(ψ/2) — a sinc-like function
Peak at ψ = 0 → sinθ = −β/(2π·d/λ) → beam points to angle where phase difference exactly cancels path difference

The term ψ = 2π·(d/λ)·sinθ + β is called the phase progression. When ψ = 0, all N contributions are in phase and add up to N — the maximum possible value. The beam points in the direction where ψ = 0, which is controlled entirely by choosing β.

Uniform Rectangular Array (URA)

A URA extends the ULA to two dimensions — Nx elements along x and Ny elements along y. Because it's separable (the x and y tapering are independent), the 2D array factor is simply the product of two ULA array factors:

URA Array Factor (separable)

AF(θ,φ) = AFx(θ,φ) × AFy(θ,φ)

AFx = Σₘ wx[m] · e^(j·m·ψx) where ψx = 2π·(dx/λ)·sinθ·cosφ + βx
AFy = Σₙ wy[n] · e^(j·n·ψy) where ψy = 2π·(dy/λ)·sinθ·sinφ + βy

The 2D weight on element (m,n) = wx[m] × wy[n]
This is the "separable taper product" — visible in the URA element diagram in the calculator

The URA is the dominant topology in modern wireless — 5G base stations use 64T64R (64 transmit, 64 receive) URA panels, automotive radar uses small 4×4 or 8×8 URAs, and satellite phased arrays can be 1000+ elements. The ULA is simpler and still very common in radar (elevation scanning), sonar, and any application where you only need beam control in one plane.

// Pointing the Beam

Beam Steering — Pointing Anywhere Without Moving

The most powerful feature of a phased array is electronic beam steering: the ability to point the beam to any angle in microseconds, purely by changing the phase shift applied to each element. No motors, no mechanical pivots, no delay.

The Phase Shift Per Element — How to Aim the Beam

To steer the beam to angle θ₀, you want all elements to contribute in phase toward that direction. The wavefront arriving from direction θ₀ has a path length difference of d·sinθ₀ between adjacent elements. To pre-compensate for this, you apply a phase advance of β = −2π·(d/λ)·sinθ₀ per element step. Element n gets phase n·β.

Beam Steering Phase per Element

β = −2π · (d/λ) · sinθ₀ (phase increment per element step)
Phase on element n: φₙ = n · β = −n · 2π · (d/λ) · sinθ₀

Example — 16-element ULA, d=0.5λ, steer to θ₀=30°:
β = −2π × 0.5 × sin30° = −2π × 0.5 × 0.5 = −π/2 = −90° per element
Element 0: 0°, Element 1: −90°, Element 2: −180°, Element 3: −270° (=+90°), ...

To verify: the beam peaks where ψ = 2π·(d/λ)·sinθ + β = 0 → sinθ = sinθ₀ ✓

Scan Loss — Why Beams Get Weaker at Wide Angles

When you scan the beam far from broadside (θ₀ = 0°), two things happen: First, the element pattern (e.g. a patch's cosine pattern) decreases with angle, reducing gain. Second, the projected aperture area reduces by cosθ₀ — the array "looks smaller" from the far-off direction. Combined, scan loss ≈ 10·log₁₀(cosθ₀) dB per factor, so scanning to 60° causes about 3 dB of scan loss.

Scan Loss

Scan Loss (dB) ≈ 10·log₁₀(cos θ₀) (element-factor scan loss only)
At θ₀ = 30°: loss = 10·log₁₀(cos30°) = 10·log₁₀(0.866) = −0.6 dB (small)
At θ₀ = 45°: loss = 10·log₁₀(cos45°) = 10·log₁₀(0.707) = −1.5 dB (modest)
At θ₀ = 60°: loss = 10·log₁₀(cos60°) = 10·log₁₀(0.5) = −3.0 dB (significant)
At θ₀ = 70°: loss ≈ −4.7 dB · At θ₀=80°: ≈ −7.6 dB

This is why phased arrays are typically limited to ±60° scan range in practice.

// Key Performance Metrics

Beamwidth & Directivity

Two numbers summarise an array's beam quality: HPBW (half-power beamwidth — how wide the main beam is) and directivity (how much better the array concentrates power compared to an isotropic radiator).

Beamwidth and Directivity Formulas

HPBW ≈ 0.886 · λ / (N · d · cosθ₀) [radians]
In degrees: HPBW ≈ 50.8° / (N · d/λ · cosθ₀)

Examples (d=0.5λ, broadside):
N=4: HPBW ≈ 50.8/(4×0.5) = 25.4°
N=8: HPBW ≈ 50.8/(8×0.5) = 12.7°
N=16: HPBW ≈ 50.8/(16×0.5) = 6.4°
N=64: HPBW ≈ 50.8/(64×0.5) = 1.6°

ULA Directivity ≈ 10·log₁₀(2·N·d/λ) dBi (uniform taper, d=0.5λ → D=10·log₁₀(N) dBi)
URA Directivity ≈ 10·log₁₀(Nx·Ny·4·(dx/λ)·(dy/λ)) dBi

Rule of thumb: Every doubling of N halves the HPBW and adds 3 dB of directivity.

Physical meaning of HPBW: If you're at the beam peak and walk sideways until the signal power drops by half (−3 dB), you've crossed the half-power point. The angle between the two half-power points is the HPBW. A 5G massive MIMO array with 64 elements at 3.5 GHz (λ=8.6 cm) has HPBW ≈ 1.6° — narrow enough to serve one user without interfering with the neighbour sitting next to them.

// The Unwanted Beams

Sidelobes — The Energy You Didn't Want to Send

Every real array has a main beam (the peak you want) and sidelobes (smaller peaks in other directions). Sidelobes waste power, cause interference to other systems, and in a radar or communications receiver, pick up unwanted signals from the wrong direction.

For a uniform (unweighted) array with N elements and d=0.5λ, the first sidelobe is −13.2 dBc below the main beam — this is a fundamental property of the rectangular/sinc pattern, regardless of N. The sidelobe level improves (becomes lower) with amplitude tapering at the cost of a wider main beam.

Sidelobe Levels — Uniform Array

First sidelobe: −13.2 dBc (always, for any N with uniform amplitude)
Second sidelobe: ≈ −17.8 dBc
Third sidelobe: ≈ −20.8 dBc
Sidelobes decrease slowly with order — only amplitude tapering can suppress the first sidelobe

In the calculator: switch between "Uniform" and other tapers to see SLL drop dramatically

// Controlling Sidelobes

Amplitude Tapering — Trading Beamwidth for Sidelobe Suppression

Amplitude tapering means reducing the excitation level of the outer elements of the array while keeping the centre elements at full power. This smooths the transition from the array to free space at the edges — reducing the "sharp cutoff" that causes high sidelobes — at the cost of a slightly wider main beam.

Think of it like a camera lens with a smooth aperture stop (apodization) rather than a hard circular edge. The smooth stop blurs the diffraction rings but doesn't degrade the main image resolution much.

Taper Types Compared

Taper	First SLL	HPBW factor	Taper Eff.	Best for
Uniform	−13.2 dBc	1.00× (narrowest)	100%	Maximum resolution, tolerant to interference
Hamming	−42 dBc	1.47×	75%	General purpose, good SLL with moderate BW loss
Hanning	−31 dBc	1.50×	75%	Spectral analysis, smooth tapering
Blackman	−58 dBc	1.73×	67%	Very low sidelobes, tolerate wide BW loss
Chebyshev	Adjustable	Minimum for SLL	~70–85%	Optimal: narrowest beam for given SLL spec
Taylor	Adjustable	Near-Chebyshev	~72–85%	Monotonically decreasing sidelobes — radar standard
Triangular	−26 dBc	1.44×	75%	Simple, moderate improvement
Binomial	No sidelobes	~2.0×	~50%	Zero sidelobes (small N only), very wide beam

Taper Efficiency — The Cost of Tapering

When you reduce the outer element amplitudes, you're not using those elements at full power — so you lose some directivity. Taper efficiency η measures how much directivity you retain compared to a uniform array of the same size:

Taper Efficiency

η = |Σ wₙ|² / (N · Σ |wₙ|²)
Ranges from 1.0 (100%) for uniform to ~0.5 for binomial
Gain Loss = 10·log₁₀(η) dB

Hamming: η ≈ 0.75 → Gain Loss = −1.25 dB
Chebyshev −30 dBc: η ≈ 0.80 → Gain Loss = −1.0 dB
Chebyshev −50 dBc: η ≈ 0.70 → Gain Loss = −1.5 dB
Blackman: η ≈ 0.67 → Gain Loss = −1.7 dB

The taper efficiency and gain loss are shown live in the calculator's "Taper Stats" panel.

Common misunderstanding: Tapering does NOT change the number of elements or the aperture size. A 16-element Hamming-tapered array has the same physical aperture as a 16-element uniform array — it just has slightly lower effective gain (−1.25 dB) and much lower sidelobes (−42 dBc vs −13 dBc). This trade-off is almost always worth it for communications and radar systems.

// The Aliasing Problem of Arrays

Grating Lobes — When Spacing Is Too Large

If the element spacing d is more than λ/2, a second full-strength main beam (called a grating lobe) appears in the pattern at a different angle. This is the spatial equivalent of aliasing in digital signal processing — when you undersample, a high-frequency signal appears as a false low-frequency alias.

Grating Lobe Condition

Grating lobes appear in visible space when: d > λ / (1 + |sinθ₀|)

At broadside (θ₀=0°): grating lobes appear when d > λ
Scanning to θ₀=30°: grating lobes appear when d > λ/1.5 = 0.667λ
Scanning to θ₀=60°: grating lobes appear when d > λ/1.866 = 0.536λ
Scanning to θ₀=90°: grating lobes appear when d > λ/2 = 0.5λ

The universal rule: d = λ/2 guarantees no grating lobes anywhere in the visible space (±90°).
The calculator shows "⚠ Present" in the Grating Lobes metric whenever this condition is violated.

Why d = λ/2 is the standard spacing: It's the largest spacing that prevents grating lobes even when scanning to ±90°. Using d = λ/2 also maximises the gain for a given number of elements (the aperture is maximised without aliasing). In 5G at 28 GHz, λ/2 = 5.4 mm — which is why 64-element mmWave arrays can be smaller than a credit card.

// Pattern Multiplication

Element Pattern × Array Factor = Total Radiated Pattern

The array factor assumes each element is isotropic (radiates equally in all directions). Real elements — patches, dipoles, slots — have their own directional pattern. The total pattern is the product of the two, a principle called pattern multiplication.

Pattern Multiplication

Total Pattern(θ) = Element Pattern(θ) × Array Factor(θ)

Isotropic element: EP(θ) = 1 → Total = AF(θ) only
Patch element: EP(θ) = cosθ → suppresses endfire region, enhances broadside
Half-wave dipole (⊥ array): EP(θ) = cos(π/2·sinθ)/cosθ → slightly reinforces array pattern
cos²θ element: EP(θ) = cos²θ → aggressively suppresses wide-angle sidelobes

In the calculator: change the "Element type" dropdown to see how each element pattern modifies the total beam shape. Patch elements are the most common in real phased array hardware.

The element pattern acts as a natural high-angle sidelobe suppressor. A patch antenna already has near-zero radiation at endfire (θ = ±90°), so any array sidelobes near endfire get naturally attenuated by the element pattern — for free.

// Implementation

Beamforming — Analogue vs Digital vs Hybrid

Beamforming is the signal processing (or hardware) that applies the correct amplitude and phase weights to each element to create the desired beam pattern. There are three major architectures:

Architecture	How it works	Pros	Cons	Used in
Analogue beamforming	Phase shifters in the RF path before ADC/DAC. One beam at a time.	Low cost, low power, simple	Only one beam, limited amplitude control	Radar, mmWave phones (28 GHz), early 5G
Digital beamforming	One ADC/DAC per element. All beamforming in DSP after digitisation.	Multiple simultaneous beams, full flexibility, adaptive nulling	Huge cost and power at high frequency (one full RF chain per element)	Sub-6 GHz 5G massive MIMO, military radar
Hybrid beamforming	Analogue phase shifters per element grouped into subarrays. One ADC/DAC per subarray. DSP combines subarrays.	Multiple beams with manageable cost and power	Some flexibility trade-offs vs full digital	Most 5G mmWave base stations, 28/39 GHz

Adaptive beamforming goes further than fixed beam steering — algorithms like MVDR (Capon's method) and LCMV dynamically place nulls toward interferers while maintaining the gain toward the desired user. The pattern updates every millisecond based on the measured signal environment. This is what makes 5G massive MIMO "smart" — the array can suppress 60+ dB of co-channel interference from users just a few degrees away.

// The Real-World Application

5G Massive MIMO — Arrays in Your Everyday Life

5G base stations use antenna arrays with 64, 128, or even 256 elements arranged in a URA panel. "Massive MIMO" means having far more antennas than users being served simultaneously. With 64 antennas and 8 simultaneous users, the array has 64−8 = 56 degrees of freedom to shape beams — pointing at each user while nulling toward all others.

System	Array	Frequency	Beamwidth	Gain	Notes
5G Sub-6 (3.5 GHz)	64T64R (8×8 URA typical)	3.5 GHz	~7°×7°	~24 dBi	Digital beamforming, MU-MIMO
5G mmWave (28 GHz)	256 elements, hybrid BF	28 GHz	~3°×3°	~30 dBi	λ/2 = 5.4mm, fits on a small panel
Automotive Radar	4×4 or 8×8 MIMO URA	77 GHz	~15°×15°	~18 dBi	Virtual aperture via MIMO
WiFi 6E AP	4×4 or 8×8 MU-MIMO	6 GHz	~25°	~15 dBi	Digital BF, serves 4 clients simultaneously
Satellite LEO (e.g. Starlink)	~1280 elements phased array	Ku/Ka band	<1°	~38 dBi	Electronically scans to track satellite

The massive in Massive MIMO doesn't just mean more elements — it means the array has so many antennas that the beam becomes pencil-thin and the channel "hardens" (becomes deterministic and predictable). At 64 or more elements per user, the random multipath fading that plagues single-antenna systems averages out, and the system capacity approaches the theoretical Shannon limit more closely.

// Quick Reference

Antenna Array Thumb Rules Cheat Sheet

These rules give you the right answer fast — within 10–15% — for any real array design situation. Verify with the calculator.

📐 Geometry & Spacing

✦d = λ/2 is the default. Guarantees no grating lobes anywhere in ±90° scan range. Any wider and you'll get grating lobes when scanning.

✦Aperture = (N−1)·d ≈ N·d for large N. Larger aperture = narrower beam. Doubling aperture halves HPBW.

✦Grating lobe free scan limit: d ≤ λ/(1+|sinθ_max|). Scan to ±60°? Use d ≤ 0.536λ. Scan to ±45°? Use d ≤ 0.586λ.

📡 Beamwidth & Gain

✦HPBW ≈ 51°/(N·d/λ) for broadside, uniform taper. Halves every time N doubles.

✦Array gain over 1 element ≈ 10·log₁₀(N) dB. N=4 → +6 dB, N=16 → +12 dB, N=64 → +18 dB.

✦URA gain ≈ 10·log₁₀(N_total) + element gain. An 8×8=64 element array with 5 dBi patches gives ≈ 5+18 = 23 dBi.

✦Scan loss at θ₀: −10·log₁₀(cosθ₀) dB. At 60°: −3 dB. At 45°: −1.5 dB. At 30°: −0.6 dB.

📊 Sidelobes & Tapering

✦Uniform array always has −13.2 dBc first sidelobe, regardless of N. Only tapering reduces this.

✦Hamming taper: first SLL −42 dBc, HPBW×1.47, gain loss 1.25 dB. Good all-around choice.

✦Chebyshev is optimal: gives minimum HPBW for a specified SLL. All sidelobes equal — none higher than specified.

✦Taper gain loss is always small: −1 to −2 dB for practical tapers. The sidelobe benefit (20–40 dB) far outweighs the gain cost.

✦URA separability: apply the same 1D taper independently to both x and y. 2D weight = wx[m] × wy[n].

⚡ System Design

✦Every 3 dB of gain → 2× reduction in required transmit power (or 2× increase in detectable range for radar, or 41% more range for comms).

✦Phase shifter resolution: 4-bit (22.5°) phase shifter gives a quantisation sidelobe of about −24 dBc. 5-bit (11.25°) gives −30 dBc. 6-bit gives −36 dBc.

✦Practical scan limit: ±60° with good performance. Beyond this, scan loss and pattern degradation become severe.

✦Digital BF: one RF chain per element = full flexibility but high cost. Sub-6 GHz arrays (few GHz) can afford this. mmWave arrays (28+ GHz) almost always use hybrid BF.

// Hands-On Learning

Using the ULA/URA Array Pattern Calculator

The best way to understand array concepts is to see them change live as you adjust parameters. Our interactive calculator lets you control every parameter — N, spacing, scan angle, taper window, element type — and immediately see the beam pattern, element weights, taper bar chart, steering phases, and all metrics update in real time.

🔭 Interactive ULA / URA Array Pattern Visualiser

Real-time polar and rectangular beam patterns · Element weight table with amplitude and phase per element · Side-by-side wx and wy taper charts for URA · Grating lobe detection · HPBW, SLL, directivity, taper efficiency metrics · CSV export

Open Calculator →

What the Calculator Shows You

1
Orientation diagram (top): Shows the physical array layout. For ULA — elements as coloured dots along a horizontal line (colour = amplitude weight, size = amplitude weight). For URA — a 2D dot grid with each dot coloured by wx[m]×wy[n]. Broadside direction shown as an upward arrow. Instantly see that a tapered array has brighter centre elements and dimmer edge elements.
2
Main beam pattern: Polar (circular dB plot) or rectangular (θ on x-axis, dB on y-axis). The pale purple outline is the Array Factor alone. The bright cyan curve is the total pattern (AF × element pattern). The dB range slider controls the dynamic range displayed. Scan the beam by changing the scan angle slider and watch the beam sweep around the polar plot.
3
URA 2D UV pattern: (URA mode only) A colour heatmap of the 3D radiation pattern in sine-space (u,v coordinates). The main beam appears as a bright spot. Grating lobes appear as secondary bright spots outside the central beam. Scan the beam by changing θ₀ and φ₀ and watch the main beam move.
4
Element diagram with weights and phases: For ULA — each element shown as a circle with size proportional to amplitude weight and colour (dark red = low weight → bright green = high weight). Amplitude value shown below each circle. Steering phase (purple) shown above. For URA — a full Nx×Ny colour grid where each cell = wx[m]×wy[n]. See the 2D taper pattern clearly.
5
Taper bar chart: For ULA — a single bar chart showing the amplitude weight of every element from left to right. For URA — two side-by-side charts: wx (cyan) for the x-axis and wy (purple) for the y-axis. Edge taper value shown on each chart. Change the taper type and watch the bars transform from uniform (all flat) to bell-shaped (Hamming/Taylor).
6
Element weight table: Every element's exact numbers — index, position in wavelengths, normalised amplitude (4 decimal places), amplitude in dB, and steering phase in degrees. For URA — a tabbed interface to switch between the wx table and wy table. Export everything to CSV with one click.
7
Metrics bar: HPBW (measured directly from the pattern), First SLL (dBc), Directivity (dBi), Scan Angle, Grating Lobe status (✓ or ⚠), Aperture (λ), Taper Efficiency (%), and Gain Loss (dB).
8
Taper stats panel: Five statistics below the taper chart — Taper Efficiency %, Gain Loss dB, Edge Taper dB (ratio of centre to edge amplitude in dB), Mean Weight, RMS Weight. For URA — also shows wy efficiency and wy gain loss separately in purple.

// Control Panel Overview

Array typeULA (1D) / URA (2D)

Elements N (ULA) or Nx, Ny (URA)2 → 64 (ULA) · 2 → 32 per axis (URA)

Spacing d/λ0.1λ → 2.0λ · Try 0.5λ first

Scan angle θ₀−90° to +90° · 0° = broadside

Amplitude taperUniform / Hamming / Hanning / Blackman / Taylor / Chebyshev / Cosine / Triangular / Binomial

SLL target (Chebyshev / Taylor)−15 dB to −60 dB · Adjustable slider

Plot typePolar (dB) / Rectangular (dB)

dB range20 / 25 / 30 / ... / 80 dB dynamic range

Element patternIsotropic / Dipole (⊥ or ∥) / Patch (cosine) / cos²θ

Experiment 1 — See How N Affects Beamwidth

// Try This in the Calculator

Select ULA mode, set taper = Uniform, d = 0.5λ, scan = 0°
Start with N=4. Note the HPBW metric and the wide beam in the polar plot.
Increase N to 8. Watch the beam halve in width and the first sidelobe stay at −13.2 dBc.
Increase to N=16, then N=32, then N=64. Each doubling halves the beamwidth and adds 3 dB to directivity.
What to observe: HPBW ≈ 51°/(N×0.5). At N=64: HPBW ≈ 1.6°. The element diagram shows more and more equally-sized dots. The taper bar chart stays flat (uniform).

Experiment 2 — Steer the Beam Across the Sky

// Try This in the Calculator

Set N=16, d=0.5λ, Uniform taper. Switch to Polar plot.
Slowly drag the Scan Angle θ₀ slider from 0° toward +60°. Watch the main beam rotate around the polar plot.
Look at the Element Weight Table — watch the Phase column change progressively (e.g. at θ₀=30°, you'll see phases like 0°, −90°, −180°, +90°, 0°, −90°... repeating every 360°).
Notice the HPBW slightly increases as you scan further from broadside — this is the cosθ broadening effect.
Scan to 70°–80°. See how the beam degrades — it becomes asymmetric and the pattern quality deteriorates. This is why arrays have a practical ±60° limit.
Now increase d to 0.8λ and scan to 45°. Watch the Grating Lobe warning appear (⚠). A second beam appears in the polar plot — this is the grating lobe.

Experiment 3 — Taper and Sidelobe Trade-offs

// Try This in the Calculator

Set N=16, d=0.5λ, scan=0°. Start with Uniform taper. Note: First SLL = −13.2 dBc, HPBW ≈ 6.4°, Taper Eff = 100%.
Switch to Hamming. First SLL drops to ≈−42 dBc. HPBW widens to ≈9.4°. Taper Eff ≈75%, Gain Loss ≈1.25 dB. The element diagram now shows smaller outer elements and larger central elements — a bell shape.
Switch to Blackman. SLL drops to ≈−58 dBc. Beam widens more. See the taper bar chart become more aggressive — outer elements almost zero.
Switch to Chebyshev. Set SLL target to −30 dB. All sidelobes become equal at −30 dBc — this is the Chebyshev property. Slide the SLL target from −15 to −60 and watch the trade-off live.
Switch to Taylor. Note how the sidelobes decrease monotonically away from the main beam — unlike Chebyshev's equal sidelobes. Taylor is preferred in radar where close-in sidelobes matter most.
What to observe: The Taper Efficiency and Gain Loss stats update immediately for each choice. The element diagram visually shows the taper shape.

Experiment 4 — Observe Grating Lobes

// Try This in the Calculator

Set N=16, scan=0°, Uniform taper.
Start with d=0.5λ. Confirm: Grating Lobes = ✓ None.
Increase d to 1.0λ. Watch a full second main beam appear at θ=−90° and θ=+90°. Grating Lobes = ⚠ Present.
Increase d to 1.5λ. More grating lobes appear — now at ±42°.
Reset d to 0.5λ. Now scan to θ₀=30°. Grating lobes are still absent — because d=λ/2 is safe for all scan angles.
Now try d=0.7λ with θ₀=45°. The grating lobe condition 0.7 > λ/1.707 = 0.586 is violated — grating lobe appears.
Physical insight: The taper bar chart and element diagram still look the same — tapering cannot eliminate grating lobes, only reduces their peak slightly. The only cure is reducing the spacing.

Experiment 5 — URA 2D Beam Scanning

// Try This in the Calculator (URA Mode)

Switch to URA mode. Set Nx=8, Ny=8, dx=dy=0.5λ, Hamming taper.
Look at the Element Diagram — the full 8×8 grid with colours. The centre cells are bright green (highest 2D weight = wx[4]×wy[4] ≈ 1.0). The corner cells are dark red (wx[0]×wy[0] ≈ 0.005). This is the separable 2D Hamming taper — a dome shape over the array aperture.
Look at the Taper bar charts — wx on the left (cyan) and wy on the right (purple). Both are identical bell-shaped Hamming windows. Compare taper efficiency for both axes.
Look at the UV pattern heatmap — a bright central spot surrounded by dark fringes.
Now scan by changing φ₀ (azimuth) from 0° to 90°. Watch the beam rotate in the UV plane — the bright spot moves from the u-axis toward the v-axis.
Change Nx and Ny independently. Make Nx=16, Ny=4. The beam becomes narrow in azimuth (x-axis) but wide in elevation (y-axis) — like a fan beam. Check the taper bar charts: wx has 16 bars, wy has 4 bars.
Switch to Chebyshev −30 dB taper. The element grid corners become even darker. Both wx and wy tables show the Chebyshev weight distribution.

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