9.3 Conventional beamforming

Conventional beamforming basics

Conventional beamforming enhances signals arriving from a desired direction while suppressing interference and noise, all by exploiting the spatial diversity of a sensor array. Each sensor's received signal gets multiplied by a complex weight, and the weighted signals are summed to form a directional "beam." The result is improved signal-to-noise ratio (SNR) and spatial selectivity.

The core idea: if you know where your signal is coming from, you can align the array's response to favor that direction and attenuate everything else.

Uniform linear arrays

A uniform linear array (ULA) consists of equally spaced sensor elements arranged along a straight line. The inter-element spacing $d$ is typically set to half the signal wavelength ($d = \lambda/2$). This choice avoids spatial aliasing (more on that under grating lobes) and maximizes spatial resolution.

ULAs dominate beamforming literature because their geometry leads to analytically tractable array manifold vectors and straightforward beam steering. More complex geometries (circular, planar) exist, but the ULA is the starting point for understanding all of them.

Far-field assumptions

Conventional beamforming assumes far-field conditions: the signal source is far enough from the array that incoming wavefronts are effectively planar rather than curved. Quantitatively, the source distance should satisfy $R \gg 2L^2/\lambda$, where $L$ is the total array aperture.

With planar wavefronts, the path-length difference between adjacent elements depends only on the angle of arrival, not on range. This means you can steer the beam using simple phase shifts instead of computing element-specific time delays from a point-source model.

Narrowband signals

Conventional beamforming works best with narrowband signals, where the signal bandwidth $B$ is much smaller than the center frequency $f_c$ (i.e., $B \ll f_c$). Under this condition, the time delay $\tau$ between elements can be approximated as a phase shift $e^{-j\omega_c \tau}$, since the signal envelope changes negligibly over $\tau$.

This approximation is what makes frequency-domain beamforming efficient. For wideband signals, the phase-shift approximation breaks down, and you need techniques like true time-delay beamforming or subband decomposition, where the signal is split into narrowband subbands that are each beamformed independently.

Beampattern analysis

The beampattern (or array factor) describes the array's spatial response as a function of angle of arrival (AoA). Analyzing it tells you about the main beam's width, sidelobe levels, null locations, and overall directivity. It's the spatial analog of a frequency response in temporal filtering.

Array manifold vectors

The array manifold vector (also called the steering vector) captures the phase relationship across array elements for a signal arriving from angle $\theta$. For a ULA:

$a(\theta) = [1,\; e^{-j2\pi d\sin(\theta)/\lambda},\; \ldots,\; e^{-j2\pi(N-1)d\sin(\theta)/\lambda}]^T$

where $d$ is the element spacing, $\lambda$ is the wavelength, and $N$ is the number of elements. Each entry represents the relative phase at that element compared to the reference element (element 0).

The set of all $a(\theta)$ as $\theta$ varies forms the array manifold, a curve in $\mathbb{C}^N$. Beamformer weights and beampattern analysis both build directly on these vectors.

Beam steering vectors

To steer the main beam toward a desired direction $\theta_d$, you set the weight vector equal to the (normalized) steering vector for that direction:

$w = \frac{1}{N} a(\theta_d)$

The beamformer output is then $y = w^H x$, where $x$ is the received signal vector. Signals arriving from $\theta_d$ add coherently (all phase differences cancel), while signals from other directions add with residual phase offsets and are attenuated.

The normalization by $1/N$ ensures unity gain in the look direction. Without it, the output scales with the number of elements.

Directivity vs. beamwidth

Directivity measures how well the beamformer concentrates its sensitivity in the look direction relative to an omnidirectional response. For a ULA with $N$ isotropic elements at half-wavelength spacing, the directivity is approximately:

$D \approx N$

(Note: the value $2N$ sometimes cited assumes a specific convention for counting both sides of a linear array or a particular normalization. For a standard ULA with isotropic elements and $d = \lambda/2$, the array gain against spatially white noise is $N$, i.e., $10\log_{10}(N)$ dB.)

The half-power beamwidth (HPBW) is the angular width of the main beam between the $-3$ dB points. For a ULA broadside beam, HPBW is approximately:

$\text{HPBW} \approx \frac{0.886\lambda}{Nd}$

(in radians, valid for small beamwidths). More elements or a larger aperture $Nd$ yields a narrower beam and higher directivity. There's a direct tradeoff: narrower beams give better spatial resolution and interference rejection, but require more elements or a larger array.

Delay-and-sum beamforming

Delay-and-sum (DAS) beamforming is the simplest and most intuitive beamforming method. You compensate for the propagation delay differences across the array so that the desired signal aligns in time at every element, then sum. The desired signal adds coherently (amplitude grows as $N$), while uncorrelated noise adds incoherently (power grows as $N$), yielding an SNR improvement of $N$ (or $10\log_{10}(N)$ dB).

Time-domain implementation

1. Choose the desired steering direction $\theta_d$.
2. Compute the required time delay for each element: $\tau_n = \frac{n \cdot d \sin(\theta_d)}{c}$, where $c$ is the propagation speed and $n = 0, 1, \ldots, N-1$.
3. Apply the delay to each element's signal (via interpolation or fractional-delay filters).
4. Sum the time-aligned signals:

$y(t) = \frac{1}{N}\sum_{n=0}^{N-1} x_n(t - \tau_n)$

Time-domain DAS handles wideband signals naturally since it applies true time delays rather than frequency-dependent phase shifts. The cost is that fractional-delay implementation requires interpolation, which adds computational overhead.

Frequency-domain implementation

For narrowband signals, replace time delays with phase shifts:

1. Compute the DFT (or use the analytic signal) of each element's received signal to get $X_n(\omega)$.
2. Apply the phase shift: $X_n(\omega) e^{j\omega \tau_n}$.
3. Sum across elements:

$Y(\omega) = \frac{1}{N}\sum_{n=0}^{N-1} X_n(\omega)\, e^{j\omega \tau_n}$

1. Inverse-transform if a time-domain output is needed.

For a truly narrowband signal at center frequency $\omega_c$, you only need to evaluate this at $\omega = \omega_c$, making the computation very efficient.

Computational complexity

For narrowband signals where $K \ll L$, the frequency-domain approach is significantly cheaper. For wideband signals, the time-domain approach may be preferable since you'd need many frequency bins anyway.

Spatial filtering

Beamforming is fundamentally a spatial filter: it shapes the array's directional response just as a temporal filter shapes a frequency response. The weight vector $w$ determines which spatial frequencies (angles) pass through and which are attenuated.

Interference rejection

A conventional beamformer rejects interference primarily through its beampattern shape. Signals arriving outside the main beam are attenuated by the sidelobe structure. However, conventional DAS beamforming cannot place deliberate nulls at specific interferer locations.

To actively null interferers, you need adaptive methods. The MVDR (Capon) beamformer minimizes output power subject to a unity-gain constraint in the look direction:

$w_{\text{MVDR}} = \frac{R^{-1} a(\theta_d)}{a^H(\theta_d) R^{-1} a(\theta_d)}$

where $R$ is the spatial covariance matrix of the received data. This automatically places nulls toward dominant interferers. The LCMV beamformer generalizes this by allowing multiple linear constraints simultaneously.

Sidelobe levels

Sidelobes are the secondary peaks in the beampattern outside the main beam. A uniform-weight DAS beamformer (rectangular window) produces the highest sidelobes, with the first sidelobe about $-13.3$ dB below the main beam peak for a ULA.

You can reduce sidelobes by applying window functions (tapering) to the weights:

• Hamming window: first sidelobe at roughly $-43$ dB, moderate main beam broadening
• Hann window: first sidelobe at roughly $-32$ dB
• Chebyshev window: equiripple sidelobes at a user-specified level

The tradeoff is always the same: lower sidelobes come at the cost of a wider main beam (reduced resolution). This is the spatial equivalent of the spectral leakage vs. resolution tradeoff in windowed DFT analysis.

Grating lobes

Grating lobes are full-amplitude replicas of the main beam that appear when the element spacing $d$ exceeds $\lambda/2$. They arise because the array cannot distinguish between angles that produce phase shifts differing by integer multiples of $2\pi$.

The grating lobe condition for a ULA steered to $\theta_d$ is:

$d(\sin\theta_g - \sin\theta_d) = m\lambda, \quad m = \pm 1, \pm 2, \ldots$

To guarantee no grating lobes for any steering angle within visible space ($\theta \in [-90°, 90°]$), you need:

$d \leq \frac{\lambda}{2}$

This is the spatial Nyquist criterion. If a larger aperture is needed without adding more elements, sparse or non-uniform array designs (e.g., minimum redundancy arrays, coprime arrays) can suppress grating lobes at the cost of increased sidelobe levels.

Robust beamforming techniques

Real-world arrays suffer from calibration errors, element position uncertainties, mutual coupling, and limited training data. These mismatches degrade conventional and adaptive beamformers, sometimes severely. Robust techniques address this by trading a small amount of optimal performance for much greater resilience to model errors.

Diagonal loading

Diagonal loading adds a scaled identity matrix to the sample covariance matrix before computing adaptive weights:

$\hat{R}_{\text{loaded}} = \hat{R} + \epsilon I$

where $\epsilon > 0$ is the loading factor. This regularizes the covariance estimate, preventing the beamformer from placing deep nulls based on noisy eigenvectors.

How to choose $\epsilon$:

• Rule of thumb: set $\epsilon$ proportional to the noise power (e.g., $\epsilon = \sigma_n^2$)
• Data-driven: use shrinkage estimators like the Ledoit-Wolf method, which optimally balances the sample covariance with the identity matrix
• Constraint-based: choose $\epsilon$ so that the white noise gain stays above a specified threshold

Diagonal loading is equivalent to norm-constrained beamforming with an $\ell_2$-norm constraint (they're Lagrangian duals of each other).

Norm-constrained beamforming

Norm-constrained beamforming directly limits the weight vector's norm to prevent excessive noise amplification:

$\min_w \; w^H R w \quad \text{s.t.} \quad w^H a(\theta_d) = 1, \quad \|w\|_2 \leq \alpha$

The constraint $\|w\|_2 \leq \alpha$ bounds the white noise gain (WNG), ensuring the beamformer doesn't become overly sensitive to uncorrelated noise. Smaller $\alpha$ means more robustness but less ability to null interferers.

Other norm choices serve different purposes:

• $\ell_1$-norm: promotes sparse weight vectors (fewer active elements)
• $\ell_\infty$-norm: limits the maximum weight magnitude on any single element

These constrained problems are convex and solvable via standard methods (Lagrange multipliers, second-order cone programming, or ADMM).

Adaptive beamforming vs. conventional

Conventional beamforming is a solid baseline: predictable, stable, and cheap. Adaptive methods outperform it when interference is present, but they need sufficient training data and accurate models. In practice, adaptive beamformers almost always include robustness mechanisms (diagonal loading, norm constraints) to avoid catastrophic performance loss from model mismatch.

Applications of beamforming

Beamforming is used wherever you have an array of sensors and want to exploit spatial information. The core principles are the same across domains; what changes is the propagation medium, signal characteristics, and system constraints.

Radar and sonar systems

In radar and sonar, beamforming serves both transmit and receive functions. On transmit, focusing energy toward a target increases the SNR at the receiver. On receive, spatial filtering suppresses clutter (ground reflections, reverberation) and jamming.

Modern radar systems use space-time adaptive processing (STAP), which jointly filters in the spatial and Doppler domains to separate slow-moving targets from ground clutter. Sonar systems face additional challenges from multipath propagation and a speed of sound that varies with depth and temperature, requiring environment-adaptive beamforming.

Wireless communications

Beamforming is central to modern wireless standards (5G NR, Wi-Fi 6/7). Transmit beamforming focuses energy toward intended users, increasing received SNR and reducing interference to others. Receive beamforming suppresses co-channel interference from other users or cells.

In massive MIMO systems (arrays with 64+ antennas), beamforming enables simultaneous service to many users on the same time-frequency resource through spatial multiplexing. Multi-user beamforming techniques like zero-forcing or regularized zero-forcing (which adds diagonal loading to the channel Gram matrix) balance inter-user interference suppression with noise enhancement.

Microphone arrays for speech enhancement

Microphone arrays in conferencing systems, hearing aids, and smart speakers use beamforming to isolate a target speaker from background noise and competing talkers. The acoustic environment is challenging: room reverberation creates multipath, and sources are often in the near field.

Common architectures include the Generalized Sidelobe Canceller (GSC), which decomposes the problem into a fixed beamformer (maintaining the desired signal) and an adaptive noise canceller (estimating and subtracting interference). Post-filtering stages using Wiener filtering or spectral subtraction further clean up the beamformer output, improving both speech quality and intelligibility.