Non-uniform Antenna Loading Effect on Embedded Element Patterns and Application to Fault Detection

TL;DR

This work presents an iterative, rank-one-update-based framework to transform Embedded Element Patterns (EEPs) from uniform to non-uniform antenna loading and to invert these transformations to estimate faulty termination impedances. Grounded in loaded admittance theory, it extends single-fault results to $N$ faults via a recursive algorithm that updates the admittance matrix and EEPs, expressed in a compact forward relation and solvable through least-squares with far-field samples. Numerical validation on a 16-element MWA tile demonstrates accurate fault identification with ~2% error in noiseless conditions and ~4% under realistic additive and multiplicative noise, with performance strongly influenced by the choice of reference antenna. The method supports measurement-based fault diagnostics for large arrays (e.g., SKA) using minimal EEP measurements and remains computationally efficient due to the recursive formulation.

Abstract

A new, iterative algorithm is presented to calculate the Embedded Element Pattern (EEP) tranformation from a set of patterns computed for a uniform antenna port loading (scaled identinty matrix) to a set of those computed for a non-uniform one (arbitrary diagonal matrix). This method proves particularly useful when inverting the computations to derive the non-uniform entries of the arbitrary load, given the minimum number of EEPs necessary, which disposes of the redundancy of other matrix-based computations and leads to numerically stable impedance fault calculation. As the EEPs are envisioned to be obtained primarily through measurement, our method is also tested with the inclusion of various noise components and its convergence is evaluated, suggesting the minimum SNR and fading level of the measurement apparatus, as well as the optimal choice of reference antenna to minimise the estimation error.

Figures (5)

• Figure 1: Network schematic of a $Z_{\rm L}$-loaded antenna array of N elements described by its impedance matrix $\mathbf{Z}_{\rm A}$, where a faulty element at position $k$ is modelled with an extra series impedance $\Delta Z$ along the $Z_{\rm L}$ termination. The gray lines show the real excitation and termination conditions, while the red lines show the referenced ones for the voltage transfer function of the second term in Eq. (\ref{['eq:circuit_interp']}) (figure adapted from kyriakou2024).
• Figure 2: Schematic representation of the recursion with respect to its progression using impedance matrices. The encircled blocks are defined as impedances, while also expressed as the inverse of the respective admittance matrix presented in the text. Element $n$ radiates, while element $k$ is a sequential element of the recursion.
• Figure 3: Discrete color grid of the absolute error value using the far-field sampling $\Omega_\theta\times\Omega_\phi$, of the estimations of the faulty termination impedance of antenna $n$ using the reference EEP of antenna $k$.
• Figure 4: RMS error versus SNR of true with respect to algorithmically estimated termination impedance, normalised over the mean of all impedances and smoothed with a 10-point moving average across the SNR range, for a 16-element MWA tile over 1000 realisations of additive Gaussian noise. Results are either shown averaged over computations for each reference element $n$, for the case of $n=4$ or for the optimum EEP $n$ that most frequently minimises the RMS across all 1000 realisations. A brief inset (with no moving average) also shows the high-SNR trend of the last two cases.
• Figure 5: RMS error versus SNR of true with respect to algorithmically estimated termination impedance, normalised over the mean of all impedances and smoothed with a 10-point moving average across the SNR range, for a 16-element MWA tile over 1000 realisations of additive Gaussian noise and multiplicative Rician fading. The fading level ${\rm K}$ of each curve is reported in the legend. Only the 'optimum $n$' curves are shown here.