9.2 Beampattern and directivity

Beampattern Fundamentals

The beampattern describes how an array of sensors or antennas distributes energy across different spatial directions. It's the core tool you use to evaluate whether an array design actually does what you need: focusing sensitivity toward a target while rejecting signals from elsewhere. Everything in this unit builds on understanding what the beampattern tells you and how to shape it.

Definition of Beampattern

The beampattern is the complex amplitude of the far-field radiation pattern as a function of spatial angles (azimuth $\phi$ and elevation $\theta$). For a uniform linear array (ULA), it takes the form:

$B(\theta, \phi) = \sum_{n=1}^{N} w_n \, e^{j(n-1)kd\sin\theta\cos\phi}$

where $w_n$ are the complex element weights, $k = 2\pi/\lambda$ is the wavenumber, and $d$ is the inter-element spacing.

This function captures the directional sensitivity and spatial selectivity of the array. The weights $w_n$ are your primary design knobs for controlling the shape of this pattern.

Relationship to Array Geometry

The beampattern is directly tied to the physical layout of the array:

• Number of elements controls the angular resolution (more elements = narrower main lobe)
• Element positions and orientations determine the overall pattern shape; linear, planar, and circular arrays each produce fundamentally different beampatterns
• Inter-element spacing relative to wavelength governs where grating lobes appear and how wide the main beam is

Changing the operating frequency shifts the electrical spacing $d/\lambda$, so a fixed physical array produces different beampatterns at different frequencies.

Normalized Beampattern

To compare different array configurations on equal footing, you normalize the beampattern by dividing by its maximum value. This gives a unitless quantity bounded between 0 and 1.

In practice, you'll almost always view the normalized beampattern in decibels:

$B_{\text{dB}}(\theta, \phi) = 20\log_{10}\left|\frac{B(\theta, \phi)}{B_{\max}}\right|$

The dB scale makes it much easier to read sidelobe levels and null depths, which can span several orders of magnitude.

Directivity

Directivity quantifies how well an array concentrates energy in a particular direction compared to radiating uniformly in all directions. It directly affects SNR and interference rejection, so it's one of the first metrics you check when evaluating an array design.

Definition of Directivity

Directivity is the ratio of the peak radiation intensity to the average radiation intensity over the full sphere:

$D = \frac{4\pi \, |B(\theta_0, \phi_0)|^2}{\displaystyle\int_0^{2\pi}\int_0^{\pi}|B(\theta, \phi)|^2\sin\theta \, d\theta \, d\phi}$

Here $(\theta_0, \phi_0)$ is the direction of maximum radiation. A higher value of $D$ means the array is putting more of its energy into the main lobe and less into sidelobes and back lobes.

For a ULA with $N$ uniformly weighted elements spaced at $d = \lambda/2$, the directivity equals $N$ (or $10\log_{10}(N)$ in dB). This serves as a useful baseline.

Directivity vs. Beamwidth

Directivity and beamwidth are closely linked but not identical:

• Beamwidth is typically measured as the angular width between the half-power points (3 dB down from the peak) of the main lobe
• A narrower beamwidth generally means higher directivity, because the energy is more tightly concentrated
• However, the sidelobe structure also matters. Two arrays can have the same 3 dB beamwidth but different directivities if their sidelobe levels differ

For a ULA with uniform weights, the approximate 3 dB beamwidth is $\Delta\theta \approx 0.886\lambda/(Nd)$ radians at broadside.

Factors Affecting Directivity

• Number of elements: More elements increases the aperture and narrows the beam, raising directivity
• Array aperture: The total physical extent of the array is what fundamentally sets the achievable directivity at a given wavelength
• Element weights: Tapering the weights (e.g., for sidelobe control) typically reduces directivity compared to uniform weighting, because energy is redistributed away from the main lobe
• Operating frequency: Higher frequency means smaller wavelength, so a fixed aperture has higher directivity
• Element radiation pattern: Real elements aren't isotropic; their individual patterns multiply with the array factor and affect the composite directivity

Beampattern Synthesis

Beampattern synthesis is the inverse problem: given a set of desired pattern characteristics, find the array weights (and sometimes geometry) that produce them. This is where design meets optimization.

Desired Beampattern Characteristics

What you want from the beampattern depends on the application, but common goals include:

• Narrow main lobe for high angular resolution
• Low sidelobe levels to reject clutter and interference
• Specific null placement to cancel known interferers
• Main lobe shape: pencil beams for point targets, fan beams for search/surveillance
• Grating lobe suppression when element spacing exceeds $\lambda/2$

Synthesis Methods

Several families of methods exist:

1. Analytical (closed-form) methods produce weight vectors for specific criteria. Dolph-Chebyshev synthesis yields the narrowest main lobe for a given peak sidelobe level. Taylor synthesis provides a smooth transition between the main lobe and sidelobe region, which is more practical for implementation.

2. Optimization-based methods handle more flexible constraints. Convex optimization (e.g., semidefinite programming) can enforce sidelobe masks, null constraints, and main lobe shape simultaneously. Evolutionary algorithms (genetic algorithms, particle swarm) are useful when the problem is non-convex or involves discrete variables like element positions.

3. Data-driven and machine learning approaches are increasingly used for real-time pattern adaptation, though they typically require large training datasets and careful validation.

Trade-offs in Beampattern Design

Synthesis always involves trade-offs:

• Narrowing the main lobe with fixed aperture tends to raise sidelobe levels (unless you add more elements)
• Applying amplitude taper to reduce sidelobes widens the main lobe and reduces directivity
• Adding more elements improves both resolution and directivity but increases hardware cost, calibration complexity, and data throughput requirements
• The choice of synthesis method depends on whether you need a guaranteed global optimum (convex methods), flexibility with non-convex constraints (evolutionary methods), or a quick closed-form solution (analytical methods)

Array Factor

The array factor isolates the contribution of the array geometry and weights from the individual element pattern. It's the spatial filter that the array imposes on incoming signals.

Definition of Array Factor

The array factor is the far-field pattern you'd get if every element were an isotropic radiator:

$AF(\theta, \phi) = \sum_{n=1}^{N} w_n \, e^{j(n-1)kd\sin\theta\cos\phi}$

The total beampattern of a real array equals the product of the array factor and the individual element pattern (pattern multiplication theorem). This separation is powerful because you can design the array factor independently and then account for the element pattern afterward.

Array Factor for Linear Arrays

For a ULA with $N$ elements along one axis, the array factor depends only on $\theta$:

$AF(\theta) = \sum_{n=1}^{N} w_n \, e^{j(n-1)kd\sin\theta}$

With uniform weights ($w_n = 1/N$), this has a closed-form expression:

$AF(\theta) = \frac{1}{N} \cdot \frac{\sin\!\left(\frac{Nkd\sin\theta}{2}\right)}{\sin\!\left(\frac{kd\sin\theta}{2}\right)}$

The pattern is periodic in $kd\sin\theta$. When $d > \lambda/2$, grating lobes appear at angles where $kd\sin\theta = 2\pi m$ for integer $m$. This is why half-wavelength spacing is the standard default for avoiding spatial aliasing.

Array Factor for Planar Arrays

Planar arrays extend beamforming to two dimensions. For a rectangular array with $M \times N$ elements:

$AF(\theta, \phi) = \sum_{m=1}^{M}\sum_{n=1}^{N} w_{mn} \, e^{j(m k d_x \sin\theta\cos\phi \;+\; n k d_y \sin\theta\sin\phi)}$

where $d_x$ and $d_y$ are the element spacings along each axis. Planar arrays can form pencil beams and steer in both azimuth and elevation, giving far more control than a linear array. The trade-off is a much larger number of elements and weights to manage.

Beamsteering

Beamsteering electronically redirects the main lobe without physically rotating the array. This is what makes phased arrays so valuable for applications requiring rapid scanning or target tracking.

Concept of Beamsteering

The idea is straightforward: apply a progressive phase shift across the array elements so that signals from the desired look direction add up coherently. The steering angle is set by the phase gradient, not by any mechanical motion.

Phase Shifting for Beamsteering

For a ULA steered to angle $\theta_0$, the required phase shift on element $n$ is:

$\phi_n = (n-1) k d \sin\theta_0$

Applying these shifts, the steered array factor becomes:

$AF(\theta) = \sum_{n=1}^{N} w_n \, e^{j(n-1)kd(\sin\theta - \sin\theta_0)}$

The main lobe now peaks at $\theta = \theta_0$. The rest of the pattern (sidelobes, nulls) shifts accordingly.

Step-by-step for steering a ULA to angle $\theta_0$:

1. Compute the required phase increment: $\Delta\phi = kd\sin\theta_0$
2. Set element $n$'s phase to $(n-1)\Delta\phi$
3. Multiply each element's signal by $w_n \, e^{-j(n-1)\Delta\phi}$ (the negative sign for receive-mode steering)
4. Sum all weighted signals to form the steered output

Limitations of Beamsteering

• Wideband signals: Phase shifting is equivalent to a frequency-independent time delay only at a single frequency. For wideband arrays, the beam "squints" across frequency, and you need true time delay (TTD) elements instead of simple phase shifters.
• Endfire steering: As $\theta_0$ approaches $\pm 90°$, the effective aperture shrinks (it goes as $\cos\theta_0$), the main lobe broadens, and grating lobes can enter visible space if $d \geq \lambda/2$.
• Scan loss: The element pattern typically has reduced gain at large scan angles, compounding the aperture effect.

Sidelobe Characteristics

Sidelobes are the secondary peaks in the beampattern outside the main lobe. They represent directions where the array has unwanted sensitivity, and managing them is one of the central challenges in array design.

Sidelobe Levels

Sidelobe level (SLL) is measured in dB relative to the main lobe peak. For a uniformly weighted ULA, the first sidelobe is approximately $-13.3$ dB. In many applications, this is too high.

High sidelobes cause problems because:

• Interference and clutter entering through sidelobes degrades SNR
• In radar, sidelobes produce false targets
• In communications, they cause co-channel interference with other users

Sidelobe Reduction Techniques

Amplitude tapering is the most common approach. You reduce the weights on the edge elements relative to the center, which suppresses sidelobes at the cost of a wider main lobe:

• Dolph-Chebyshev weighting produces the narrowest possible main lobe for a specified uniform sidelobe level (e.g., all sidelobes at exactly $-30$ dB)
• Taylor weighting provides a more practical design where sidelobes near the main lobe are controlled and far-out sidelobes decay naturally
• Hamming/Hanning windows from spectral analysis can also be applied, though they aren't optimal in the Chebyshev sense

Spatial tapering techniques include array thinning (removing selected elements from a dense grid) and density tapering (varying element spacing to approximate an amplitude taper with uniform-weight elements).

Optimization-based approaches can minimize peak or integrated sidelobe levels subject to constraints on main lobe width, null placement, or other criteria.

Sidelobe Effects on Performance

• Radar/sonar: Sidelobes increase clutter returns and false alarm rates. Ground clutter entering through sidelobes can mask weak targets.
• Communications: Sidelobes radiate energy toward unintended users, reducing spatial reuse and system capacity.
• Imaging/direction-finding: High sidelobes limit dynamic range and the ability to resolve weak sources near strong ones.

Beamforming Algorithms

Beamforming algorithms determine how the array element signals are combined. The choice of algorithm sets the trade-off between simplicity, interference suppression, and robustness.

Conventional Beamforming

Conventional (delay-and-sum) beamforming applies a fixed set of steering weights and sums the element signals:

$\mathbf{w}_{\text{CBF}} = \frac{1}{N}\left[1, \; e^{-jkd\sin\theta_0}, \; \ldots, \; e^{-j(N-1)kd\sin\theta_0}\right]^T$

This is equivalent to a spatial matched filter for a plane wave arriving from $\theta_0$. It's simple, robust, and computationally cheap. The downside is that its interference rejection is limited to whatever the fixed sidelobe structure provides.

Adaptive Beamforming

Adaptive methods adjust the weight vector based on the statistics of the received data, typically to minimize output power while preserving the signal from the look direction:

• MVDR (Capon) beamformer: Minimizes output power subject to a unity-gain constraint in the look direction. The weight vector is $\mathbf{w}_{\text{MVDR}} = \frac{\mathbf{R}^{-1}\mathbf{a}(\theta_0)}{\mathbf{a}^H(\theta_0)\mathbf{R}^{-1}\mathbf{a}(\theta_0)}$, where $\mathbf{R}$ is the spatial covariance matrix and $\mathbf{a}(\theta_0)$ is the steering vector.
• LCMV beamformer: Extends MVDR by allowing multiple linear constraints (e.g., nulls at known interferer directions).
• GSC (Generalized Sidelobe Canceller): Decomposes the problem into a fixed beamformer plus an adaptive interference canceller, which can simplify implementation.

Adaptive beamformers can place deep nulls on interferers, dramatically improving SINR. However, they require accurate covariance estimation (which needs sufficient snapshots), and they can suffer from signal cancellation if there's a mismatch between the assumed and actual steering vector. Diagonal loading (adding $\sigma^2\mathbf{I}$ to $\mathbf{R}$) is a common robustness technique.

Comparison of Beamforming Methods

Hybrid beamforming, combining analog phase shifting with digital baseband processing, is widely used in massive MIMO and millimeter-wave systems to reduce the number of RF chains while retaining much of the flexibility of fully digital beamforming.

Applications of Beampatterns

Radar and Sonar Systems

Radar and sonar rely on shaped beampatterns for target detection, tracking, and imaging. Narrow beamwidths improve angular resolution, while low sidelobes reduce clutter returns. Beamsteering enables rapid electronic scanning across a surveillance volume. In hostile environments, adaptive beamforming suppresses jamming by placing nulls in the jammer directions.

Wireless Communications

In 5G and beyond, beamforming is central to millimeter-wave communication. Directional beams compensate for the high path loss at these frequencies, and beamsteering tracks mobile users as they move. Massive MIMO base stations use hundreds of elements to serve multiple users simultaneously through spatial multiplexing, with each user receiving a distinct beam. Sidelobe control reduces inter-user interference.

Medical Imaging and Therapy

Diagnostic ultrasound uses beamforming to focus transmitted pulses and process received echoes, directly determining image resolution and contrast. Phased array transducers steer and focus the beam electronically to scan tissue without mechanical motion. In therapy, high-intensity focused ultrasound (HIFU) concentrates acoustic energy at a focal point to ablate tumors non-invasively. Adaptive beamforming algorithms compensate for phase aberrations caused by tissue inhomogeneities, improving both image quality and therapeutic precision.