Virtual VNA: Minimal-Ambiguity Scattering Matrix Estimation With a Fixed Set of “Virtual” Load-Tunable Ports

We estimate the scattering matrix of an arbitrarily complex linear, passive, reciprocal, time-invariant system with N monomodal lumped ports by inputting and outputting waves only via a fixed set of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$N_{\text {A}}\lt N$ </tex-math></inline-formula> ports while terminating the remaining <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$N_{\text {S}}=N-N_{\text {A}}$ </tex-math></inline-formula> “not-directly-accessible” (NDA) ports with tunable individual loads. First, we present a closed-form approach requiring at least three arbitrary, distinct, and known loads at each NDA port; sign ambiguities on off-diagonal scattering coefficients associated with NDA ports are inevitable. Being matrix-valued, our approach is ideally suited to mitigate noise sensitivity using more accessible ports. It also yields <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$1 \; + \; 2N_{\text {S}} \; + \; N_{\text {S}}(N_{\text {S}}-1)/2$ </tex-math></inline-formula> as upper bound on the number of required measurements <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$N_{\mathrm {cal}}$ </tex-math></inline-formula> for <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$N_{\text {A}}\gt 1$ </tex-math></inline-formula> in the low-noise regime. Second, we present a gradient-descent approach using (potentially opportunistic) random load configurations, enabling flexible adjustments of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$N_{\mathrm {cal}}$ </tex-math></inline-formula> to further mitigate noise. Third, we present an intensity-only gradient-descent approach that dispenses with phase-sensitive detectors at the expense of an additional blockwise phase ambiguity. Then, we discuss in what applications the inevitable remaining ambiguities are problematic and how they can be lifted. Finally, we experimentally validate all three approaches with an eight-port reverberation chamber and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$N_{\text {A}}=N_{\text {S}}=4$ </tex-math></inline-formula>, systematically assessing the sensitivity to noise and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$N_{\mathrm {cal}}$ </tex-math></inline-formula>. We coin our technique “virtual vector network analyzer (VNA)” because it implies that suitably tunable and characterized individual loads can essentially be interpreted as additional “virtual” (as opposed to actual) VNA ports. Our method can not only characterize static large antenna systems and circuits with many and/or embedded ports but also characterize reconfigurable wave systems (smart radio environments, physical neural networks, and programmable photonic integrated circuits); it may furthermore enable wireless sensing paradigms in indoor surveillance, nondestructive testing, and bioelectronics.