4.3 Antenna arrays

Fundamentals of antenna arrays

An antenna array is a set of multiple antenna elements arranged in a specific geometry so that their individual radiation patterns combine to produce a composite pattern with improved characteristics. By controlling the number of elements, their spacing, excitation amplitudes, and relative phases, you can shape the overall radiation pattern to achieve higher gain, narrower beams, and electronic beam steering that a single element simply can't provide.

Arrays show up everywhere: wireless communications, radar, satellite links, radio astronomy. The reason is straightforward. More elements give you more degrees of freedom to control where your radiated power goes.

Antenna array definition

Each element in an array can be identical (the most common case) or different, depending on the application. The overall radiation pattern is determined by three main design parameters:

• Element spacing $d$: the physical distance between adjacent elements
• Excitation amplitudes $I_n$: how much current or voltage drives each element
• Excitation phases $\beta_n$: the relative phase offset applied to each element

The total radiated field is the vector sum of the fields from all elements. This superposition is what gives arrays their pattern-shaping power.

Advantages over single antennas

Compared to a single antenna, arrays provide:

• Higher gain and directivity, which translates to increased range and better signal-to-noise ratio
• Electronic beam steering, meaning you can redirect the main beam by adjusting element phases rather than physically rotating the antenna
• Null placement in the direction of interfering signals, improving signal quality
• Pattern flexibility, allowing you to shape the beam for specific coverage requirements (e.g., sectorized cells in wireless networks)

Types of antenna arrays

• Linear arrays: Elements arranged along a straight line. Common examples include dipole arrays and patch arrays.
• Planar arrays: Elements arranged on a two-dimensional grid, such as rectangular microstrip patch arrays. These give you control in both azimuth and elevation.
• Conformal arrays: Elements mounted on a curved surface (cylindrical, spherical, or arbitrary shapes), useful when the array must conform to a vehicle or aircraft body.
• Phased arrays: Any array geometry where adjustable phase shifters enable electronic beam steering. The term emphasizes the steering capability rather than the geometry.

Linear antenna arrays

Linear arrays are the simplest array geometry: all elements sit along a single line. Despite this simplicity, they form the foundation for understanding more complex configurations. The radiation pattern of a linear array separates neatly into two factors: the individual element pattern and the array factor.

Uniform linear arrays

A uniform linear array (ULA) has equally spaced elements with identical excitation amplitudes. Its array factor is:

$AF = \frac{\sin(N\psi/2)}{N\sin(\psi/2)}$

where $N$ is the number of elements and $\psi = kd\cos\theta + \beta$. Here $k = 2\pi/\lambda$ is the wavenumber, $d$ is the element spacing, $\theta$ is the angle measured from the array axis, and $\beta$ is the progressive phase shift between adjacent elements.

ULAs are easy to analyze and build, but they offer limited control over sidelobe levels and null positions since the only free parameters are $N$, $d$, and $\beta$.

Non-uniform linear arrays

When you allow unequal excitation amplitudes (and sometimes unequal spacing), you gain additional control over the radiation pattern. Three classical non-uniform designs are:

• Binomial arrays: Excitation amplitudes follow binomial coefficients. These eliminate all sidelobes but produce a wider main beam.
• Dolph-Chebyshev arrays: Excitation amplitudes are chosen so that all sidelobes have equal height at a specified level below the main beam. This gives the narrowest possible main beam for a given sidelobe constraint.
• Taylor arrays: A practical compromise that tapers sidelobe levels smoothly, reducing the highest sidelobes near the main beam while keeping the far-out sidelobes at a lower level.

Array factor for linear arrays

The array factor (AF) captures the contribution of the array geometry and excitations to the total pattern, independent of the individual element pattern. For $N$ elements along a line:

$AF = \sum_{n=1}^{N} I_n e^{j(n-1)\psi}$

where $I_n$ is the complex excitation of the $n$th element and $\psi = kd\cos\theta + \beta$.

The total radiation pattern is the product of the AF and the single-element pattern. This is the pattern multiplication principle, and it's one of the most useful results in array theory. It means you can design the element and the array factor separately, then multiply them to get the full pattern.

The AF alone determines the main lobe width, sidelobe levels, and null positions.

Directivity and gain

Directivity measures how well an array concentrates radiation in a particular direction relative to an isotropic radiator. For a broadside ULA, the directivity scales roughly as:

$D \approx 2Nd/\lambda$

when the spacing is around $\lambda/2$. More elements and wider apertures yield higher directivity.

Gain equals directivity multiplied by the radiation efficiency, which accounts for ohmic losses, mismatch losses, and feed network losses. You can increase gain by adding more elements, optimizing element spacing, or reducing losses in the feed network.

Planar antenna arrays

Planar arrays extend the linear concept into two dimensions, placing elements on a flat grid. This gives you independent control over the radiation pattern in both the elevation ($\theta$) and azimuth ($\phi$) planes, producing a pencil-shaped main beam that you can steer in any direction above the array.

Uniform planar arrays

A uniform planar array arranges elements on a rectangular grid with spacing $d_x$ and $d_y$ along the two axes. A key simplification: the array factor of a rectangular planar array is separable, meaning it equals the product of the array factors of the two constituent linear arrays along the $x$ and $y$ axes.

This separability makes analysis and synthesis much more tractable. The resulting main beam is pencil-shaped, with sidelobes appearing in both principal planes.

Non-uniform planar arrays

Non-uniform planar arrays break the regular rectangular grid by using unequal spacing, non-identical excitations, or alternative geometries:

• Circular arrays: Elements placed on a ring, useful for omnidirectional coverage in azimuth
• Concentric ring arrays: Multiple rings of different radii, providing more pattern control
• Aperiodic (sparse) arrays: Irregular element placement, often optimized numerically to suppress grating lobes with fewer elements

These configurations provide greater flexibility in shaping the pattern and controlling sidelobes at the cost of more complex analysis.

Array factor for planar arrays

For an $M \times N$ rectangular planar array, the array factor is:

$AF = \sum_{m=1}^{M} \sum_{n=1}^{N} I_{mn} \, e^{j(m-1)\psi_x} \, e^{j(n-1)\psi_y}$

where:

• $\psi_x = kd_x\sin\theta\cos\phi + \beta_x$
• $\psi_y = kd_y\sin\theta\sin\phi + \beta_y$

$I_{mn}$ is the complex excitation of the element at position $(m, n)$, and $\beta_x$, $\beta_y$ are the progressive phase shifts along each axis. These phase shifts control the beam pointing direction.

Grating lobes in planar arrays

Grating lobes are spurious high-intensity lobes that appear when the element spacing is too large. They occur because the phase difference between adjacent elements reaches an integer multiple of $2\pi$, creating additional directions of constructive interference.

To prevent grating lobes for all scan angles, the element spacing must satisfy:

$d < \frac{\lambda}{1 + |\sin\theta_{\text{max}}|}$

For a broadside array ($\theta_{\text{max}} = 0$), this reduces to $d < \lambda$. For full hemispherical scanning, the requirement tightens to $d < \lambda/2$. This half-wavelength rule is one of the most important practical constraints in array design.

Phased antenna arrays

Phased arrays add electronically controllable phase shifters to each element, enabling rapid beam steering without any mechanical motion. This makes them essential for applications that require fast scanning, such as radar tracking and 5G beamforming.

Principles of phased arrays

To steer the main beam to an angle $\theta_0$ from broadside, you apply a progressive phase shift across the array:

$\beta = -kd\cos\theta_0$

This creates a linear phase gradient that shifts the direction of constructive interference. The beam can be steered to any angle within the scan volume by updating the phase shifts, typically in microseconds.

Phase shifters in antenna arrays

Phase shifters introduce a controllable phase delay in each element's signal path. The two main categories are:

• Analog phase shifters: Ferrite-based or PIN diode-based devices that provide continuous phase control
• Digital phase shifters: Switched delay lines or vector modulators that provide discrete phase steps (e.g., 3-bit gives 8 states at 45° resolution, 5-bit gives 32 states at 11.25° resolution)

The phase shifter's resolution directly affects beam pointing accuracy and quantization sidelobes. Coarser resolution means higher quantization lobes.

Beam steering and scanning

Beam steering is the act of pointing the main beam in a specific direction. Scanning is the process of sweeping the beam across a range of angles, either continuously or in discrete steps.

• 1D scanning: The beam is steered in one plane only (e.g., a linear phased array scanning in elevation)
• 2D scanning: The beam is steered in both azimuth and elevation (requires a planar array with independent phase control along both axes)

A key limitation: as the beam scans away from broadside, the projected aperture shrinks, causing the beam to broaden and the gain to drop. This is sometimes called the scan loss, and it follows roughly a $\cos\theta$ dependence.

Applications of phased arrays

• Radar systems: Rapid electronic scanning enables simultaneous tracking of multiple targets and fast search patterns
• Satellite communications: Dynamic beam pointing allows a single array to serve multiple ground stations and adapt to changing link conditions
• 5G wireless networks: Massive MIMO base stations use large phased arrays for beamforming and spatial multiplexing, serving many users simultaneously
• Radio astronomy: Phased array feeds and aperture arrays are used in interferometric systems for high-resolution sky imaging

Array synthesis techniques

Array synthesis is the inverse problem: given a desired radiation pattern, determine the element excitations (and sometimes positions) that produce it. The goal is typically to achieve a specified main beam shape while keeping sidelobes below a required level.

Schelkunoff polynomial method

The Schelkunoff method represents the array factor as a polynomial in the variable $z = e^{j\psi}$:

$AF(z) = I_1 + I_2 z + I_3 z^2 + \cdots + I_N z^{N-1}$

This is a polynomial of degree $N-1$, so it has $N-1$ zeros. Each zero corresponds to a null in the radiation pattern. By placing these zeros at specific locations on the unit circle in the complex $z$-plane, you control where the nulls appear in the angular pattern. This method is intuitive for small arrays but becomes unwieldy for large ones.

Fourier transform method

This method exploits the Fourier relationship between the element excitations and the array factor. The steps are:

1. Specify the desired radiation pattern $AF_d(\psi)$ in the angular domain
2. Compute the inverse discrete Fourier transform to obtain the element excitations $I_n$
3. Truncate or window the excitations to fit the finite array aperture

The Fourier method works well for specifying a main beam shape and sidelobe envelope, but the truncation to a finite number of elements introduces Gibbs-phenomenon ripples. Windowing (e.g., Hamming, Kaiser) can reduce these ripples at the cost of a wider main beam.

Woodward-Lawson method

The Woodward-Lawson method decomposes the desired pattern into a sum of orthogonal pencil beams (sinc-like basis functions), each pointing in a different direction. The coefficient of each basis function is simply the value of the desired pattern at that beam's pointing direction.

This approach is straightforward to implement and naturally produces realizable excitations. It works well for shaped beams (e.g., cosecant-squared patterns for radar) but doesn't directly optimize sidelobe levels.

Optimization methods for array synthesis

When classical methods can't handle the design constraints, numerical optimization takes over. The synthesis problem is formulated as: minimize the error between the desired and achieved patterns, subject to constraints on sidelobe levels, null positions, excitation dynamic range, etc.

Common optimization algorithms include:

• Genetic algorithms: Population-based search inspired by natural selection; good at avoiding local minima
• Particle swarm optimization: Swarm-based search; often converges faster than genetic algorithms for array problems
• Convex optimization: When the problem can be cast in convex form, it guarantees a global optimum; particularly useful for sidelobe-constrained designs

These methods can also account for practical effects like mutual coupling, element failures, and excitation quantization.

Mutual coupling in antenna arrays

Mutual coupling is the electromagnetic interaction between array elements. When one element radiates, some of that energy is received by neighboring elements, altering their currents and impedances. Ignoring mutual coupling leads to inaccurate predictions of array performance.

Effects of mutual coupling

• Pattern distortion: The actual radiation pattern deviates from the ideal (no-coupling) pattern, with increased sidelobe levels and possible beam pointing errors
• Impedance changes: Each element's input impedance shifts from its isolated value, causing mismatch losses and reduced radiation efficiency
• Scan blindness: At certain scan angles, mutual coupling can cause a dramatic impedance mismatch, effectively making the array "blind" in that direction

The severity of coupling depends on element spacing (closer elements couple more strongly), element type, and polarization.

Compensation techniques for mutual coupling

Several strategies exist to mitigate coupling effects:

• Excitation correction: Modify the applied excitations based on the measured or computed coupling matrix to recover the desired pattern
• Element pattern compensation: Use the actual active element patterns (which include coupling) rather than the isolated element pattern when computing excitations
• Impedance matching networks: Design matching circuits that account for the coupling-modified impedances, not just the isolated impedance

Active element pattern

The active element pattern (AEP) is the radiation pattern of a single element measured while all other elements in the array are present and terminated in their nominal impedances. It differs from the isolated element pattern because it includes the effects of mutual coupling, scattering from neighboring elements, and edge effects.

Accurate AEPs are essential for realistic array analysis. You can obtain them through full-wave electromagnetic simulation or by measuring each element individually while the rest of the array is terminated.

Scanning impedance and matching

The scanning impedance $Z_s(\theta_0)$ is the input impedance of an array element as a function of the scan angle $\theta_0$. Mutual coupling causes $Z_s$ to vary significantly with scan angle.

At certain angles, $Z_s$ can become highly reactive, leading to severe mismatch and the phenomenon of scan blindness. This typically occurs when a surface wave or grating lobe condition is excited.

Mitigation approaches include:

• Wideband matching networks designed for the expected scan range
• Adaptive impedance tuning circuits that adjust in real time
• Element designs that inherently reduce surface wave excitation (e.g., cavity-backed patches)

Wideband antenna arrays

Wideband arrays must maintain stable radiation characteristics across a broad frequency range. This is challenging because the electrical spacing between elements (measured in wavelengths) changes with frequency, and mutual coupling is frequency-dependent.

Challenges in wideband arrays

The core difficulty is that element spacing $d$ is fixed physically, but its electrical size $d/\lambda$ increases with frequency. This means:

• At low frequencies, the array may be undersampled (small aperture in wavelengths, low directivity)
• At high frequencies, $d/\lambda$ can exceed 0.5, causing grating lobes to appear
• Mutual coupling strength and character change across the band, affecting impedance matching and pattern shape

Frequency-independent antenna arrays

Frequency-independent arrays use elements whose radiation properties are inherently broadband:

• Log-periodic dipole arrays (LPDAs): A series of dipoles with lengths and spacings that scale logarithmically, maintaining a self-similar active region across frequency
• Spiral antennas: Planar spirals radiate from a region whose size scales with wavelength

These designs achieve bandwidths of 10:1 or more, but they tend to be physically large and may have limited scan capability.

Timed arrays and true-time delay

Phase shifters introduce a frequency-independent phase shift, which means the beam pointing direction changes with frequency. This effect is called beam squint, and it becomes significant for wideband signals.

True-time delay (TTD) systems solve this by introducing a frequency-dependent time delay $\tau$ rather than a fixed phase shift. Since a time delay $\tau$ corresponds to a phase shift of $\omega\tau$ (which scales linearly with frequency), the beam points in the same direction at all frequencies.

TTD can be implemented with switched delay lines, fiber-optic delay lines, or photonic beamforming networks. The tradeoff is increased complexity and cost compared to simple phase shifters.

Wideband array feeding networks

The feed network distributes power to all elements and must maintain consistent amplitude and phase relationships across the operating bandwidth. Common architectures:

• Corporate (parallel) feeds: Binary power divider trees; good bandwidth but path lengths must be carefully equalized
• Series feeds: Elements tapped sequentially from a transmission line; compact but inherently dispersive (beam squint)
• Traveling-wave feeds: The transmission line is terminated in a matched load; low reflections but some power is wasted in the load

Key feed network requirements are low insertion loss, good impedance matching, and minimal dispersion across the band.

Smart antenna systems

Smart antennas combine antenna arrays with digital signal processing to adapt the radiation pattern in real time. Instead of fixed beams, the array weights are continuously updated based on the received signals, allowing the system to track desired users and suppress interference automatically.

Adaptive antenna arrays

Adaptive arrays adjust their complex element weights $w_n$ (amplitude and phase) based on statistical properties of the received signals. The optimization criterion is typically to maximize the signal-to-interference-plus-noise ratio (SINR) or minimize the mean squared error (MSE) between the array output and a reference signal.

Common adaptive algorithms:

• Least Mean Squares (LMS): Simple gradient-descent update; low computational cost but slow convergence
• Recursive Least Squares (RLS): Faster convergence than LMS but higher computational cost (scales as $N^2$ per snapshot)
• Direct Matrix Inversion (DMI): Computes the optimal weights directly from the sample covariance matrix; fast but requires matrix inversion at each update

Beamforming algorithms

Beamforming computes element weights to form a beam toward the desired signal while placing nulls toward interferers.

• Delay-and-sum (conventional) beamforming: Steers the beam by applying phase shifts corresponding to the desired look direction. Simple but has limited interference rejection.
• Capon (MVDR) beamforming: Minimizes output power subject to a unity-gain constraint in the look direction. Produces narrower beams and deeper nulls than conventional beamforming, but requires knowledge of the signal direction and the noise covariance matrix.
• Blind beamforming: Techniques like the constant modulus algorithm (CMA) exploit signal properties (e.g., constant envelope) rather than direction information. Useful when the signal direction is unknown.

Direction-of-arrival estimation

Direction-of-arrival (DOA) estimation determines the angular directions of incoming signals from the array's received data. It's a prerequisite for many beamforming and localization tasks.

Key algorithms:

• MUSIC (Multiple Signal Classification): Exploits the eigenstructure of the covariance matrix to identify signal subspace and noise subspace. Provides high-resolution DOA estimates but requires knowledge of the number of sources.
• ESPRIT (Estimation of Signal Parameters via Rotational Invariance Techniques): Uses the shift-invariance structure of the array to estimate DOAs without a spectral search. Computationally cheaper than MUSIC.
• Maximum likelihood methods: Statistically optimal but computationally expensive, especially for multiple sources.

Both MUSIC and ESPRIT can resolve sources that are closer together than the classical Rayleigh resolution limit of the array, which is why they're called super-resolution methods.

MIMO antenna arrays

MIMO (Multiple-Input Multiple-Output) systems use multiple antennas at both transmitter and receiver. Unlike phased arrays that form a single coherent beam, MIMO arrays can create multiple independent spatial channels simultaneously.

The channel capacity of a MIMO system scales as $\min(M, N)$ times the single-antenna capacity (in the high-SNR, rich-scattering regime), where $M$ and $N$ are the number of transmit and receive antennas.

MIMO arrays use precoding at the transmitter and combining at the receiver, both based on channel state information (CSI). In 5G massive MIMO, base stations with 64 or more antenna elements serve dozens of users simultaneously through spatial multiplexing.

Measurement techniques for antenna arrays

Validating an array design requires measuring its radiation pattern, gain, impedance, and polarization. The choice of measurement technique depends on the array size, operating frequency, and required accuracy.

Far-field measurement methods

Far-field measurements are taken at distances beyond $R > 2D^2/\lambda$, where $D$ is the largest array dimension. At this distance, the wavefront is approximately planar and the pattern has reached its final shape.

• Outdoor far-field ranges: The array under test (AUT) is mounted on a positioner and illuminated by a source antenna at the required distance. Simple but susceptible to ground reflections and weather.
• Compact ranges: A shaped reflector (or lens) converts a spherical wave from a nearby feed into a plane wave, creating a far-field-equivalent environment in a much shorter distance. These are typically housed indoors in anechoic chambers.

Near-field measurement methods

For large arrays, the far-field distance can be impractically long. Near-field measurements solve this by scanning a probe close to the array aperture and then mathematically transforming the data to obtain the far-field pattern.

Three scanning geometries are used:

1. Planar scanning: The probe moves over a flat plane in front of the array. Best for high-gain arrays with narrow beams.
2. Cylindrical scanning: The probe moves along a cylinder surrounding the array. Captures the pattern over a full 360° in azimuth.
3. Spherical scanning: The probe moves over a sphere enclosing the array. Provides the complete 3D pattern but is the most complex and time-consuming.

The near-field to far-field transformation relies on the equivalence principle and typically uses FFT-based algorithms.

Array calibration techniques

Calibration corrects for amplitude, phase, and position errors across the array elements. Without calibration, beamforming accuracy and null depth degrade significantly.

• External reference calibration: A known source at a known location transmits a signal; each element's response is measured and compared to the ideal, yielding correction factors
• Mutual coupling calibration: Uses the known mutual coupling between elements as a built-in reference to extract amplitude and phase errors
• Built-in test (BIT) calibration: Internal couplers or sensors inject and measure calibration signals without an external source, enabling in-field calibration

Diagnostic techniques for array performance

When an array isn't performing as expected, diagnostic techniques help identify the problem:

• Power monitoring and VSWR measurements: Detect failed or degraded elements by checking the power delivered to and reflected from each element
• Near-field holography: Measures the aperture field distribution and back-propagates it to the array surface, revealing amplitude and phase errors at each element location
• Far-field pattern diagnosis: Compares the measured far-field pattern to the expected pattern; deviations indicate element failures or excitation errors. Inverse techniques can localize the faulty elements from the far-field data alone