Virtual VNA 2.0: Ambiguity-Free Scattering Matrix Estimation by Terminating Not-Directly-Accessible Ports with Tunable and Coupled Loads

TL;DR

The paper tackles the challenge of reconstructing the full $N\times N$ scattering matrix of a DUT when only $N_A$ ports are directly accessible. It introduces Virtual VNA 2.0, which employs a multi-port load network (MPLN) and, when needed, a simple two-port load network (2PLN) to lift sign ambiguities and a blockwise phase ambiguity, enabling unambiguous end-to-end impedance/scattering matrix estimation from accessible-port measurements. The authors present both complex-valued (closed-form) and intensity-only (gradient-descent) estimation approaches and validate them experimentally on an $8$-port chaotic cavity, achieving accurate reconstruction across the tested frequency range. They show that ambiguity elimination scales linearly with the number of NDA ports and discuss implications for large or embedded antenna arrays and RIS-inspired systems. Overall, the work provides a practical route to non-invasive, globally consistent scattering-matrix characterization in complex environments, with real-world applicability to large-scale antenna arrays.

Abstract

We recently introduced the "Virtual VNA" concept which estimates the $N \times N$ scattering matrix characterizing an arbitrarily complex linear reciprocal system with $N$ monomodal lumped ports by inputting and outputting waves only via $N_\mathrm{A}<N$ ports while terminating the $N_\mathrm{S}=N-N_\mathrm{A}$ remaining ports with known tunable individual loads. However, vexing ambiguities about the signs of the off-diagonal scattering coefficients involving the $N_\mathrm{S}$ not-directly-accessible (NDA) ports remained. If only phase-insensitive measurements were used, an additional blockwise phase ambiguity ensued. Here, inspired by the emergence of "beyond-diagonal reconfigurable intelligent surfaces" in wireless communications, we lift all ambiguities with at most $N_\mathrm{S}$ additional measurements involving a known multi-port load network. We experimentally validate our approach based on an 8-port chaotic cavity, using a simple coaxial cable as two-port load network. Endowed with the multi-port load network, the "Virtual VNA 2.0" is now able to estimate the entire scattering matrix without any ambiguity, even without ever measuring phase information explicitly. Potential applications include the challenging characterization of large and/or embedded antenna arrays.

Figures (4)

• Figure 1: Schematic of the setup of the proposed ambiguity-free "Virtual VNA 2.0" for the characterization of an $N$-element antenna array using an $N_\mathrm{A}$-port VNA (for $N_\mathrm{A}=2$). (A) Concept. (B) Implementation via VNA extension kit. (C) Implementation via integration into antenna array.
• Figure 2: Experimental setup. (A) Reverberation chamber comprising 8 antennas. (B) Setup to electronically switch the termination of four ports between three individual loads. (C) Setup involving the 2PLN (a coaxial cable). (D) Impedance and scattering characteristics of the three individual loads (left column) and the four coefficients characterizing the 2PLN (right column). (E) Measurement procedure for the "Virtual VNA 2.0". $\mathcal{A}$ and $\mathcal{S}$ identify the accessible and NDA ports, respectively. The schematic shows whether a given port is connected to the VNA (black) or terminated by one of the three individual loads (blue, red, yellow) or the 2PLN (purple). The closed-form approach (left column) requires a series of specific configurations, the gradient-descent approach (right column) can be applied to an arbitrarily long sequence of random configurations (except for the last few involving the 2PLN).
• Figure 3: Experimentally achieved frequency-averaged accuracy $\zeta$ (see Eq. (\ref{['eq_zeta']}) for definition) with complex-valued measurements (averaged over 141 linearly spaced frequency points between 740 MHz and 810 MHz).
• Figure 4: Step-by-step procedure for ambiguity-free estimation of the full scattering matrix purely based on intensity-only measurements for 771 MHz. The three columns display $\mathbf{S}_\mathcal{AA}$, $\mathbf{S}_\mathcal{AS}$ and $\mathbf{S}_\mathcal{SS}$ in vectorized form.