This interactive tutorial demonstrates how linear antenna arrays work through a real-time visualization. A linear antenna array is a collection of antenna elements arranged in a line, where each element can have different positions, amplitudes, and phases. By controlling these parameters, you can shape the radiation pattern (array factor) to steer the main beam, control sidelobes, and optimize antenna performance.

The visualization consists of two main panels: (1) Array Editor (Spatial Domain) (top) - a horizontal line showing the antenna array axis with draggable circular nodes representing antenna elements, and (2) Radiation Pattern (Angle Domain) (bottom) - a polar plot showing the normalized Array Factor as a function of angle. The visualization uses a dark theme (black background) with bright colors for optimal visibility. You can drag antenna elements left/right to change their spacing, click elements to adjust their amplitude and phase, and see the radiation pattern update in real-time.

The tutorial uses the generalized summation formula for arbitrary spacing, allowing you to create non-uniform arrays with any number of elements. The Array Factor is calculated using Euler's identity to handle complex exponentials, summing real and imaginary components separately before computing the magnitude. The pattern is normalized by the sum of amplitudes to ensure the maximum value is always 1.0 (0 dB). The visualization also includes a grating lobe warning system that highlights element pairs with spacing ≥ λ/2 in red, indicating potential unwanted radiation lobes.

NOTE : The tutorial uses the generalized Array Factor formula for arbitrary linear arrays: AF(θ) = Σ_n=0^N-1 A_n × e^{j(k×x_n×sin(θ) + φ_n)}, where A_n is the amplitude of the n-th element, x_n is its position, φ_n is its phase, k = 2π/λ is the wavenumber, and θ is the scan angle. The formula uses Euler's identity (e^jθ = cos(θ) + j×sin(θ)) to compute the complex exponential, summing real and imaginary parts separately before calculating the magnitude. The wavelength λ is set to 100 pixels for visualization purposes.

Mathematical Model

The Array Factor (AF) for a linear antenna array with arbitrary spacing is given by:

Array Factor Formula:

AF(θ) = Σ_n=0^N-1 A_n × e^{j(k×x_n×sin(θ) + φ_n)}

where:

• A_n: Amplitude of the n-th element (0.0 to 1.0)
• x_n: Position of the n-th element in pixels (relative to center)
• φ_n: Phase shift of the n-th element in degrees (-180° to 180°)
• k: Wavenumber = 2π/λ (where λ is the wavelength in pixels)
• θ: Scan angle from -90° (end-fire) to +90° (end-fire), with 0° being broadside
• N: Number of antenna elements

Calculation Method: Using Euler's identity, the complex exponential is decomposed into real and imaginary parts: e^jθ = cos(θ) + j×sin(θ). For each element, we compute the phase term k×x_n×sin(θ) + φ_n, then calculate A_n×cos(phaseTerm) for the real part and A_n×sin(phaseTerm) for the imaginary part. We sum all real parts and all imaginary parts separately, then compute the magnitude as |AF| = √(realSum² + imagSum²). Finally, we normalize by dividing by the sum of all amplitudes to ensure the maximum value is 1.0.

Grating Lobes: When the spacing between adjacent elements is greater than or equal to λ/2 (half wavelength), unwanted radiation lobes called "grating lobes" can appear in the pattern. These are additional main beams that can cause interference and reduce antenna performance. The visualization highlights element pairs with spacing ≥ λ/2 in red to warn about potential grating lobes. For optimal performance, element spacing should be kept below λ/2.

 

Usage Example

Follow these steps to explore the Linear Antenna Array tutorial:

1. Initial State: When you first load the simulation, you'll see two main panels: Array Editor (top) showing a horizontal line with 8 antenna elements (blue circles) arranged with uniform spacing, and Radiation Pattern (bottom) showing a polar plot of the array factor. The default array has 8 elements with uniform spacing of 50 pixels, all with amplitude 1.0 and phase 0°. The radiation pattern shows a main beam at broadside (0°) with sidelobes on either side.
2. Observe the Array Editor: The top panel shows the antenna array as a horizontal line with circular nodes (blue circles) representing antenna elements. The center of the line (marked with "0") represents the reference point. Elements are positioned relative to this center. The line connecting elements is green when spacing is safe (< λ/2) and red when spacing is ≥ λ/2 (grating lobe warning). Each element shows its position label below it.
3. Observe the Radiation Pattern: The bottom panel shows a polar plot of the normalized Array Factor as a semicircle on the right side. The center represents the array location, and the distance from center represents the magnitude (normalized to 0-1). The plot shows angles from -90° (bottom) to +90° (top), with 0° pointing to the right (horizontal). Grid arcs show magnitude levels (0.25, 0.5, 0.75, 1.0), and radial lines show angles every 15° from -90° to +90°. The green curve shows the radiation pattern, and a yellow dashed line indicates the main beam direction.
4. Drag Elements to Change Spacing: Click and drag any antenna element left or right to change its position. As you drag, watch the radiation pattern update in real-time. Notice how changing the spacing affects the pattern - closer spacing creates a wider main beam, while wider spacing creates a narrower beam but may introduce grating lobes (shown as red connections).
5. Select and Adjust Element Properties: Click on any antenna element to select it (it turns yellow). The "Selected Element Controls" panel will appear, allowing you to adjust the amplitude (0.0 to 1.0) and phase (-180° to 180°) using sliders. Watch how changing amplitude affects the pattern shape, and how changing phase steers the main beam direction.
6. Add Elements: Click the "Add Element" button to add a new antenna element. The element is automatically positioned according to the currently selected spacing preset, and its amplitude is set according to the currently selected amplitude preset. You can then drag it to any desired location. Double-clicking on the array canvas also adds an element at the clicked position, which is then repositioned according to the spacing preset. Experiment with different numbers of elements to see how array size affects the radiation pattern.
7. Remove Elements: Select an element by clicking on it, then click the "Remove Selected" button to delete it. You must have at least one element remaining. After removal, the remaining elements are automatically repositioned according to the currently selected spacing preset, and amplitudes are recalculated according to the currently selected amplitude preset. Notice how removing elements changes the pattern - fewer elements typically result in a wider main beam and higher sidelobes.
8. Understand Grating Lobes: When two adjacent elements have spacing ≥ λ/2 (half wavelength), the connection line turns red. This indicates potential grating lobes - unwanted radiation beams that can cause interference. Try dragging elements far apart to see the red warning appear, and observe how the radiation pattern develops multiple main beams.
9. Steer the Beam with Phase: Select an element and adjust its phase slider. Positive phase values steer the beam toward positive angles (right), while negative phase values steer it toward negative angles (left). Try applying progressive phase shifts across multiple elements to create a phased array that steers the beam electronically.
10. Use Steering Angle Control: Use the "Steering Angle" slider to set a desired beam direction from -90° to +90°. Check the "Enable Steering" checkbox to automatically calculate and apply the required phase shifts to all elements. The simulation uses the formula φₙ = -k × xₙ × sin(θ₀) to compute the phase for each element based on its position and the desired steering angle. This demonstrates electronic beam steering - you can change the beam direction without physically moving the antenna. When steering is enabled, manually adjusting individual element phases will disable steering mode.
11. Control Sidelobes with Amplitude Tapering: Reduce the amplitude of edge elements (tapering) to lower sidelobe levels. For example, set the center elements to amplitude 1.0 and the edge elements to 0.5. This creates an amplitude taper that reduces sidelobes at the cost of a slightly wider main beam.
12. Use Engineering Presets: The tutorial provides two separate preset dropdowns for independent control:
    • Amplitude Presets: Control the amplitude distribution across elements:
      • Uniform (default): All elements have equal amplitude - maximum gain, higher sidelobes
      • Binomial: Amplitudes follow binomial coefficients - lowest sidelobes, widest main beam
      • Cosine Tapered: Cosine squared amplitude distribution - good sidelobe reduction
      • Hamming Window: Hamming window function - balanced sidelobe/beamwidth tradeoff
      • Edge Tapered: Linear reduction from center to edges - simple tapering approach
    • Spacing Presets: Control the geometric arrangement (positions) of elements:
      • Uniform (λ/2) (default): Standard spacing of 50 pixels - optimal for avoiding grating lobes
      • Uniform (λ/4): Tighter spacing of 25 pixels - narrower beam, no grating lobes
      • Uniform (λ): Wide spacing of 100 pixels - demonstrates grating lobes (red warning lines)
      • Log-Periodic: Progressive spacing that increases geometrically - broadband behavior
      • Random (Perturbed): Uniform base with random jitter - suppresses periodic grating lobes
    Each preset dropdown has an "Apply" button to reapply the currently selected preset. Selecting a preset automatically applies the corresponding distribution (amplitude or geometry) to all elements. When you add or remove elements, both presets are automatically reapplied to maintain the selected configuration. The amplitude presets demonstrate different tradeoffs between main beam width, sidelobe levels, and gain. The spacing presets demonstrate how element positions affect beamwidth, grating lobes, and array performance.
13. Reset and Experiment: Click "Reset Array" to restore the default 8-element uniform array. Try creating different array configurations: uniform spacing, non-uniform spacing, amplitude tapering, phase steering, and combinations of these techniques. Observe how each parameter affects the radiation pattern.

Tip: The key to understanding antenna arrays is recognizing that the radiation pattern is determined by the interference of waves from all elements. When elements are in phase and properly spaced, they constructively interfere in the desired direction (main beam) and destructively interfere in other directions (sidelobes). Phase shifts create path length differences that steer the beam, amplitude tapering controls sidelobe levels, and spacing affects both beamwidth and the potential for grating lobes. Start with a simple uniform array, then experiment with one parameter at a time to understand its effect. The real-time visualization makes it easy to see cause-and-effect relationships.

Parameters

Followings are short descriptions on each parameter

• Antenna Element Position (x_n): The horizontal position of each antenna element in pixels, measured relative to the center of the array (marked as "0"). Positive values are to the right, negative values are to the left. Elements can be positioned at any location, allowing for both uniform and non-uniform spacing. The position directly affects the phase relationship between elements and thus the radiation pattern shape and beam direction.
• Amplitude (A_n): The excitation amplitude of each antenna element, ranging from 0.0 to 1.0. An amplitude of 1.0 means the element radiates at full power, while 0.0 means it doesn't radiate. Different amplitude distributions (e.g., uniform, tapered) affect the radiation pattern - uniform amplitudes maximize gain but may have higher sidelobes, while amplitude tapering reduces sidelobes at the cost of slightly wider main beam and lower gain.
• Phase (φ_n): The phase shift of each antenna element in degrees, ranging from -180° to +180°. Phase controls the relative timing of radiation from each element, which determines the beam direction. Progressive phase shifts across elements create beam steering - positive phase shifts steer the beam toward positive angles (right), negative phase shifts steer toward negative angles (left). Uniform phase (all 0°) creates a broadside beam (perpendicular to the array axis).
• Wavelength (λ): The wavelength of the electromagnetic wave, set to 100 pixels for visualization purposes. The wavelength determines the wavenumber k = 2π/λ, which is used in the Array Factor calculation. The ratio of element spacing to wavelength (d/λ) is critical - spacing greater than λ/2 can cause grating lobes (unwanted radiation beams).
• Wavenumber (k): The spatial frequency of the wave, calculated as k = 2π/λ. The wavenumber relates the physical spacing of elements to the phase differences in the radiated waves. It appears in the Array Factor formula as k×x_n×sin(θ), representing the phase contribution from element position and scan angle.
• Scan Angle (θ): The angle from broadside (perpendicular to the array axis) at which the Array Factor is calculated. The angle ranges from -90° (end-fire left) to +90° (end-fire right), with 0° being broadside. The Array Factor is calculated for all angles to generate the complete radiation pattern shown in the polar plot.
• Array Factor (AF): The normalized radiation pattern of the array, calculated using the generalized summation formula: AF(θ) = Σ_n=0^N-1 A_n × e^{j(k×x_n×sin(θ) + φ_n)}. The Array Factor represents how the array radiates power as a function of angle. It is normalized by the sum of amplitudes so the maximum value is always 1.0 (0 dB), making it easy to compare different array configurations.
• Grating Lobe Warning: A visual indicator (red connection lines) that appears when the spacing between adjacent elements is ≥ λ/2 (half wavelength). Grating lobes are unwanted radiation beams that appear when element spacing is too large. They can cause interference and reduce antenna performance. For optimal performance, element spacing should be kept below λ/2, which is indicated by green connection lines.
• Steering Angle (θ₀): The desired beam direction in degrees, ranging from -90° (end-fire left) to +90° (end-fire right), with 0° being broadside. When steering is enabled, the simulation automatically calculates the required phase shift for each element using the formula φₙ = -k × xₙ × sin(θ₀), where k is the wavenumber, xₙ is the element position, and θ₀ is the steering angle. This creates a progressive phase shift across the array that steers the main beam to the specified angle. Steering can be enabled/disabled with a checkbox, and when enabled, all element phases are automatically updated when the steering angle changes.

Controls and Visualizations

Followings are short descriptions on each control

• Add Element Button: Adds a new antenna element to the array. The new element is automatically positioned according to the currently selected spacing preset, and its amplitude is set according to the currently selected amplitude preset. The element has phase 0° by default. You can then drag it to any desired position and adjust its properties. Double-clicking on the array canvas also adds an element at the clicked position, which is then repositioned according to the spacing preset. Adding elements allows you to experiment with different array sizes and configurations. Both amplitude and spacing presets are automatically reapplied when elements are added.
• Remove Selected Button: Removes the currently selected antenna element from the array. You must first click on an element to select it (it turns yellow), then click this button. The array must have at least one element remaining. After removal, the remaining elements are automatically repositioned according to the currently selected spacing preset, and amplitudes are recalculated according to the currently selected amplitude preset. Removing elements allows you to simplify the array and observe how fewer elements affect the radiation pattern.
• Reset Array Button: Resets the array to the default configuration: 8 elements with uniform spacing of 50 pixels (λ/2), all with amplitude 1.0 and phase 0°. Both amplitude and spacing presets are reset to "Uniform". The steering angle is reset to 0° and steering is disabled. This provides a clean starting point for new experiments. The radiation pattern is recalculated immediately after reset.
• Amplitude Slider: Adjusts the amplitude of the currently selected antenna element. The slider ranges from 0 to 100 (representing 0.0 to 1.0). The current value is displayed next to the slider. Changing amplitude affects the radiation pattern - lower amplitudes reduce the element's contribution, which can be used for amplitude tapering to control sidelobes. The slider only appears when an element is selected.
• Phase Slider: Adjusts the phase of the currently selected antenna element. The slider ranges from -180° to +180°. The current value is displayed next to the slider in degrees. Changing phase steers the main beam direction - positive phase shifts steer right, negative phase shifts steer left. Progressive phase shifts across multiple elements create electronic beam steering. The slider only appears when an element is selected.
• Array Editor Canvas (Top Panel): Displays the antenna array as a horizontal line with circular nodes representing elements. The center is marked with "0" and a vertical line. Elements are shown as blue circles (yellow when selected), with position labels below. Connection lines between elements are green for safe spacing (< λ/2) and red for grating lobe warning (≥ λ/2). You can click elements to select them, drag them to change position, and double-click to add new elements. The canvas updates in real-time as you interact with elements.
• Radiation Pattern Canvas (Bottom Panel): Displays a polar plot of the normalized Array Factor as a semicircle on the right side. The center represents the array location, and distance from center represents magnitude (0 to 1.0). The plot shows angles from -90° (bottom) to +90° (top), with 0° pointing to the right (horizontal). Grid arcs show magnitude levels (0.25, 0.5, 0.75, 1.0), and radial lines show angles every 15° from -90° to +90°. The green curve shows the radiation pattern, and a yellow dashed line indicates the main beam direction. The pattern updates in real-time as you modify the array configuration.
• Selected Element Controls Panel: Appears when an element is selected, providing sliders to adjust amplitude and phase. The panel shows the current values and updates the radiation pattern immediately as you adjust the sliders. This allows precise control over individual element properties to shape the radiation pattern. Note: Manually adjusting an element's phase will disable steering mode. Manually adjusting an element's amplitude will change the element from the selected amplitude preset distribution, but you can reapply the preset using the "Apply" button.
• Amplitude Preset Dropdown: Selects a well-known engineering amplitude distribution for the array. Available presets include Uniform (equal amplitudes, maximum gain, higher sidelobes), Binomial (lowest sidelobes, widest main beam), Cosine Tapered (cosine squared distribution, good sidelobe reduction), Hamming Window (balanced tradeoff between sidelobes and beamwidth), and Edge Tapered (linear reduction from center to edges, simple tapering). Selecting a preset automatically applies the corresponding amplitude distribution to all elements and resets element positions to uniform λ/2 spacing. Each preset demonstrates different tradeoffs between main beam width, sidelobe levels, and array gain. The presets are calculated based on the current number of elements in the array. An "Apply" button next to the dropdown allows you to reapply the currently selected preset.
• Spacing Preset Dropdown: Selects a geometric arrangement (spacing pattern) for the array elements. Available presets include Uniform (λ/2) - standard spacing of 50 pixels optimal for avoiding grating lobes, Uniform (λ/4) - tighter spacing of 25 pixels for narrower beam, Uniform (λ) - wide spacing of 100 pixels that demonstrates grating lobes (red warning lines appear), Log-Periodic - progressive spacing that increases geometrically for broadband behavior, and Random (Perturbed) - uniform base with random jitter to suppress periodic grating lobes. Selecting a preset automatically applies the corresponding position distribution to all elements and resets amplitudes to uniform. The spacing presets demonstrate how element positions affect beamwidth, grating lobes, and array performance. An "Apply" button next to the dropdown allows you to reapply the currently selected preset.
• Steering Angle Slider: Controls the desired beam direction from -90° to +90°. The current angle is displayed next to the slider. When steering is enabled, changing this slider automatically calculates and applies the required phase shifts to all elements to steer the beam to the specified angle. This demonstrates electronic beam steering - the ability to change beam direction without physically moving the antenna.
• Enable Steering Checkbox: Activates or deactivates automatic beam steering. When checked, the simulation calculates phase shifts for all elements based on the steering angle. When unchecked, element phases remain at their current values (or can be manually adjusted). Enabling steering after manually adjusting phases will overwrite the manual phase settings with calculated values based on the steering angle. The checkbox state is preserved when applying amplitude or spacing presets - the Apply buttons do not change the steering checkbox state.

Key Concepts

• Array Factor: The Array Factor (AF) is the radiation pattern of an antenna array, representing how the array radiates power as a function of angle. It is calculated by summing the contributions from all elements, taking into account their positions, amplitudes, and phases. The Array Factor is normalized so the maximum value is always 1.0 (0 dB), making it easy to compare different array configurations. The Array Factor does not include the individual element pattern - it assumes all elements are isotropic radiators.
• Constructive and Destructive Interference: The radiation pattern is created by the interference of waves from all array elements. When waves from different elements are in phase (constructive interference), they add together to create a main beam. When waves are out of phase (destructive interference), they cancel each other out, creating nulls in the pattern. The Array Factor formula calculates this interference mathematically by summing complex exponentials representing the phase relationships.
• Beam Steering: Beam steering is the ability to change the direction of the main radiation beam without physically moving the antenna. This is achieved by applying progressive phase shifts across the array elements. A linear phase progression creates a path length difference that steers the beam in the direction of increasing phase. Electronic beam steering is a key advantage of phased arrays, allowing rapid beam pointing without mechanical movement. The required phase shift for element n at position xₙ to steer the beam to angle θ₀ is: φₙ = -k × xₙ × sin(θ₀). The tutorial includes a steering angle control that automatically calculates and applies these phase shifts when enabled.
• Grating Lobes: Grating lobes are unwanted radiation beams that appear when element spacing is too large (≥ λ/2). They occur because the array pattern repeats periodically, and when spacing exceeds half a wavelength, additional main beams appear at angles where the phase relationships match. Grating lobes can cause interference and reduce antenna performance. The visualization warns about potential grating lobes by showing red connection lines when spacing ≥ λ/2.
• Amplitude Tapering: Amplitude tapering is the technique of using different excitation amplitudes for different elements, typically reducing amplitudes toward the edges of the array. This reduces sidelobe levels at the cost of a slightly wider main beam and lower gain. Common taper functions include uniform (all 1.0), binomial, cosine, and Hamming window distributions. The tutorial provides amplitude presets (Uniform, Binomial, Cosine Tapered, Hamming Window, Edge Tapered) that automatically apply well-known amplitude distributions. You can also manually adjust individual element amplitudes to experiment with custom tapering.
• Geometry and Spacing Presets: The tutorial includes spacing presets that control the physical arrangement of elements. Uniform spacing (λ/2, λ/4, or λ) provides predictable patterns, with λ/2 being optimal for avoiding grating lobes. Log-Periodic spacing uses progressive geometric spacing for broadband behavior. Random spacing adds jitter to suppress periodic grating lobes. When a spacing preset is selected, element positions are automatically reset to the specified pattern, and amplitudes are reset to uniform. This allows you to experiment with how spacing affects the radiation pattern independently from amplitude tapering.
• Uniform vs. Non-Uniform Spacing: Uniform spacing (equal distances between all adjacent elements) is the simplest configuration and is commonly used. However, non-uniform spacing can provide advantages such as reduced sidelobes, wider bandwidth, or fewer elements for a given aperture size. The generalized Array Factor formula supports arbitrary spacing, allowing you to experiment with non-uniform arrays.
• Broadside vs. End-Fire: Broadside radiation (θ = 0°) occurs when the main beam is perpendicular to the array axis. This is achieved when all elements have the same phase. End-fire radiation (θ = ±90°) occurs when the main beam is along the array axis. This requires specific phase relationships and is typically used for different applications than broadside arrays.
• Euler's Identity in Array Calculations: The Array Factor formula uses complex exponentials e^jθ, which are computed using Euler's identity: e^jθ = cos(θ) + j×sin(θ). The tutorial implements this by calculating real and imaginary parts separately for each element, then summing them before computing the magnitude. This approach is necessary because standard programming languages don't have native complex exponential functions.
• Normalization: The Array Factor is normalized by dividing by the sum of all amplitudes. This ensures the maximum value is always 1.0 (0 dB), regardless of how many elements are in the array or what their amplitudes are. Without normalization, adding more elements or increasing amplitudes would make the pattern grow continuously, making it difficult to compare different configurations.
• Real-Time Visualization: The radiation pattern updates in real-time as you modify the array configuration. This immediate feedback makes it easy to understand cause-and-effect relationships - you can see how dragging an element, changing its amplitude, or adjusting its phase affects the pattern instantly. This interactive approach helps build intuition about antenna array behavior.
• Applications: Linear antenna arrays are fundamental to many applications:
  • Radar Systems: Phased array radars use electronic beam steering for rapid target tracking
  • Wireless Communications: Base station antennas use arrays for beamforming and spatial multiplexing
  • Satellite Communications: Ground station antennas use arrays for tracking and high gain
  • Radio Astronomy: Large arrays combine signals from many elements for high resolution
  • 5G/6G Networks: Massive MIMO systems use large arrays for capacity and coverage