⚡ electrical engineering

Analysis and Approximation of a Spatially Wideband Antenna Array Factor

This paper presents a universal approximation for the spatially wideband antenna array factor by modeling it as a spatially variant convolution of the narrowband factor and a bandwidth-dependent kernel, demonstrating how increased bandwidth suppresses sidelobes in fully populated arrays and grating lobes in sparse arrays.

Original authors: Marcin Wachowiak, André Bourdoux, Sofie Pollin

Published 2026-08-14

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Marcin Wachowiak, André Bourdoux, Sofie Pollin

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to shout a message to a friend across a crowded, noisy field. If you just stand still and yell a single note, your voice travels in a tight, focused beam. But if you start shouting a whole song with many different pitches at once, the sound behaves differently; it spreads out, and the "echoes" or "ghost voices" that usually bounce off the crowd (interference) might get blurred or smoothed out. This is the world of antenna arrays, the invisible tools that let our phones, radars, and satellites "see" and "hear" in specific directions.

In the science of antennas, engineers usually design systems to work with signals that are "narrowband," meaning they use a very tight slice of the radio spectrum, like a single laser beam. In this world, the antenna array acts like a flashlight, creating a bright main beam and some annoying, weaker "sidelobes" or "grating lobes" that point in the wrong directions and cause interference. However, as we push for faster internet and sharper radar images, we are forced to use "wideband" signals—huge chunks of the radio spectrum all at once. When you combine a very wide signal with a large antenna array, the old rules break down. The "flashlight" starts to smear, and the annoying ghost voices behave in complex, unpredictable ways. Understanding exactly how these wideband signals reshape the antenna's view is crucial for building the next generation of communication and sensing systems.

This paper, titled "Analysis and Approximation of a Spatially Wideband Antenna Array Factor," dives deep into this smearing effect. The authors, Marcin Wachowiak, André Bourdoux, and Sofie Pollin, set out to create a new mathematical "map" to predict how antenna arrays behave when they are hit with these massive, wideband signals. They aren't just guessing; they are deriving precise formulas that act like a universal translator, converting the complex behavior of wideband signals into something engineers can actually calculate and use.

The core of their discovery is a clever way of looking at the problem. Instead of trying to solve the entire messy equation from scratch every time, they show that the wideband signal's effect is essentially a "convolution." To use a playful analogy, imagine the antenna's standard, narrowband pattern as a sharp, crisp photograph. The wideband signal acts like a special, fuzzy lens placed over that photo. The authors found that the final image (the wideband pattern) is simply the original sharp photo being "smeared" or "averaged" by this fuzzy lens. The shape and size of the lens depend on the bandwidth of the signal and how the signal's energy is distributed across frequencies.

One of the most exciting findings in their simulations is that this "smeared" effect isn't always bad. In fact, for certain types of antennas—specifically "sparse" arrays where the antenna elements are spaced far apart—this smearing is a superpower. Usually, sparse arrays suffer from "grating lobes," which are like strong, unwanted ghost beams that appear at regular intervals and ruin the signal. The authors show that by using a wideband signal, these ghost beams get stretched out and their energy is spread over a wider angle. It's like taking a bright, blinding spotlight and turning it into a soft, wide floodlight; the peak intensity of the annoying ghost beam drops significantly, effectively suppressing the interference.

The paper provides several specific formulas to describe this. For example, they show that if you have a uniform signal (one that uses all frequencies equally), the "fuzzy lens" becomes a specific shape called a sinc function. They also demonstrate that if you try to "tame" the signal by using filters (windows) to smooth out the edges of the frequency band, you actually make the lens sharper again, which reduces the helpful smearing effect on the ghost beams. In other words, a raw, wideband signal is better at cleaning up the mess than a filtered one.

The authors tested these ideas through rigorous simulations across a wide range of scenarios, including arrays with very large apertures (size) and different element spacings. They found their new "universal approximation" works accurately even when the antennas are spaced far apart, a situation where older models fail. They even derived a specific formula to predict the height of those annoying grating lobes based on the product of the bandwidth and the aperture size. If this product is large enough, the grating lobes are suppressed, but if the spacing is too wide relative to the bandwidth, the suppression isn't perfect, and the "ghosts" remain visible, just slightly dimmer.

In short, this paper hands engineers a new toolkit. It explains that when you turn up the bandwidth, you aren't just getting more data; you are fundamentally changing the shape of the antenna's vision. By treating the wideband effect as a spatial smearing process, the authors provide a clear, mathematical way to design antennas that can handle the massive bandwidths of the future, turning what was once a chaotic problem into a predictable, manageable feature. Their work suggests that for sparse arrays, embracing the wideband nature of signals is a powerful strategy to silence the unwanted echoes that have plagued radar and communication systems for so long.

Technical Summary: Analysis and Approximation of a Spatially Wideband Antenna Array Factor

Problem Statement

The array factor (AF) is a critical component in modeling multi-antenna systems, dictating spatial selectivity, beam patterns, and interference characteristics. Traditionally, antenna arrays are designed under the spatially narrowband (SNB) assumption, where the product of the array aperture and signal bandwidth is small ( $DB/c \ll 1$ ). However, emerging applications such as joint communication and sensing, high-throughput communications, and high-resolution radar imaging require arrays with both large apertures and wide bandwidths. Under these conditions, the SNB assumption fails, and spatially wideband (SWB) effects—such as beam squint and the modification of sidelobe and grating lobe structures—must be accurately modeled.

While existing literature has investigated SWB effects in specific contexts (e.g., millimeter-wave MIMO, radar ambiguity functions, and ultra-wideband sparse arrays), there is a lack of a general, closed-form expression for the SWB AF. Such an expression is necessary to understand how signal bandwidth and amplitude spectra modify the SNB AF to produce the resulting SWB AF.

Methodology

The paper derives and analyzes approximations for the SWB AF of uniform linear arrays (ULAs) through a progression of mathematical formulations:

Fundamental Definition: The SWB AF is defined via a matched filter formulation, integrating the SNB AF over the signal bandwidth weighted by the signal energy spectral density $|S(f)|^2$ .
Narrowband Signal Approximation: For large apertures with element spacing satisfying the Nyquist criterion ( $d_s \le \lambda_c/2$ ) and narrowband signals, the SWB AF is approximated as the product of the SNB AF and the signal's autocorrelation function. This separates the effects of spatial windowing (shaping the SNB AF) and spectral windowing (modifying sidelobe decay).
Wideband Signal Approximation (Dense Arrays): For fully populated, dense arrays with uniform spatial and spectral weighting, the discrete sum of antenna elements is approximated by a continuous integral. This yields a closed-form expression involving the sine integral function ($Si$), valid when the fractional bandwidth is small relative to the inverse of the normalized element spacing.
Universal Approximation (Convolution Formulation): The core methodological contribution is reformulating the SWB AF as a spatially variant convolution. By recognizing the SWB AF as the convolution of the SNB AF and a spatially wideband kernel (derived from the signal autocorrelation), the authors decouple the array geometry from the signal bandwidth effects.
- The SNB AF acts as a spatial filter.
- The SWB kernel acts as a weighting function that spreads energy.
- This formulation allows for the analysis of arbitrary element spacings (including sparse arrays) and bandwidths.
Grating Lobe Analysis: For sparse arrays ( $d_s > \lambda_c/2$ ), the paper analyzes how the widening of the SWB kernel with increasing bandwidth suppresses grating lobe amplitudes by spreading their energy over a wider angular range. A closed-form approximation for the grating lobe envelope is derived as a function of the bandwidth-aperture product.

Key Contributions

Derivation of SWB AF Approximations: The paper provides closed-form approximations for the SWB AF under various conditions, including narrowband signals with large apertures and wideband signals with uniform weighting.
Universal Convolution Framework: A novel formulation is introduced where the SWB AF is expressed as the convolution of the SNB AF and a SWB kernel. This framework offers an intuitive interpretation of how bandwidth and spectral weighting shape the final beampattern.
Validation Across Parameters: The proposed approximations are shown to be accurate across a wide range of fractional bandwidths, element spacings (from Nyquist-compliant to sparse), and aperture sizes.
Grating Lobe Envelope Approximation: For sparse arrays, the paper provides a specific approximation for the envelope of grating lobe amplitudes. This quantifies the trade-off where increasing bandwidth suppresses peak grating lobe levels at the cost of broadening their angular spread and raising the sidelobe floor.
Impact of Windowing: The analysis demonstrates that while spatial windowing (e.g., Hamming, Hann) suppresses sidelobes in the SNB AF, spectral windowing can degrade the bandwidth's ability to suppress grating lobes by narrowing the SWB kernel. Conversely, a uniform spectrum offers the best-case scenario for bandwidth-induced grating lobe suppression.

Results

Sidelobe Suppression: For fully populated arrays, increasing the bandwidth effectively suppresses the AF sidelobes. The approximation accurately predicts this behavior for various bandwidth-aperture products.
Sparse Array Behavior: In sparse arrays, increasing the bandwidth spreads the energy of grating lobes over a wider angular range, thereby reducing their peak amplitudes. The paper provides a formula to estimate this suppression based on the bandwidth-aperture product ( $B_f D_\lambda$ ).
Accuracy of Approximations: The universal convolution approximation remains accurate even for large fractional bandwidths and element spacings, provided the number of antennas is sufficiently large ( $M \gg 1$ ). The SNB AF acts as a spatial anti-aliasing filter, mitigating errors that might arise from the continuous approximation of the SWB kernel.
Trade-offs: The results highlight a trade-off: while wider bandwidths reduce the peak amplitude of grating lobes, they broaden the lobes and can increase the overall sidelobe floor, potentially removing deep nulls in the pattern.

Significance

The paper claims that its primary significance lies in providing a general closed-form expression for the SWB AF, which has been absent in previous literature. This expression enables a deeper understanding of how signal bandwidth and amplitude spectra modify the traditional SNB AF. By offering a universal approximation based on spatially variant convolution, the work provides a practical tool for system designers to evaluate performance and optimize array configurations (including sparse arrays) for wideband applications without relying solely on computationally intensive numerical simulations. The derived grating lobe envelope approximation further aids in predicting performance limits for sparse wideband arrays.