Multiplexed De-embedding in Wireless Sensing

• Multiplexed de-embedding is a wireless sensing technique that extracts multi-port scattering parameters from DUTs without direct electrical connections.
• It employs a reconfigurable tunable load network to generate programmable fixture realizations, enhancing measurement diversity and computational identifiability.
• Experimental validations demonstrate that increasing PF realizations and accessible antennas improves accuracy in recovering complex scattering matrices across various MIMO setups.

Searching arXiv for the cited wireless multi-port sensing and Virtual VNA papers to ground the article in current preprints. arxiv_search(query="Wireless Multi-Port Sensing Virtual VNA de-embedding over-the-air fixture del Hougne", max_results=10) Multiplexed de-embedding is a wireless multi-port sensing methodology for remotely retrieving the scattering matrix of a multi-port device under test (DUT) when its ports are not directly accessible. In the formulation developed for over-the-air (OTA) sensing, each DUT port is connected to a distinct not-directly-accessible (NDA) antenna, a separate set of accessible antennas is connected to the measurement instrument, and the intervening wireless propagation environment together with the antennas constitutes an OTA fixture. The central problem is to characterize that fixture and then de-embed it so that the DUT scattering parameters can be estimated without any direct electrical connection to the DUT ports. In the more recent low-complexity formulation, the tunable load network (TLN) between the OTA fixture and the DUT is also used to generate multiple programmable fixture (PF) realizations, so that measurement diversity accumulated across sequential configurations compensates for limited hardware parallelism at the accessible side (Hougne, 17 Jul 2025, Hougne, 29 Sep 2025).

1. Problem setting and core definitions

The basic system contains a multi-port DUT with $N_\mathrm{S}$ lumped, monomodal ports, none of which are directly accessible. Each DUT port is connected to a distinct NDA antenna. A set of $N_\mathrm{A}$ accessible antennas is connected to the vector network analyzer (VNA) or equivalent measurement chain, and these accessible antennas are the only ports where waves can be injected and captured. The OTA fixture is the network formed by the wireless propagation environment, all antennas, and the physical separations between accessible and NDA sides. The objective is to estimate the complete scattering matrix $\mathbf{S}^\mathrm{D}$ of the DUT without any direct electrical connection to its ports (Hougne, 17 Jul 2025).

A known, reconfigurable TLN provides the crucial controllability. In one mode, the NDA antennas are connected to the DUT. In another mode, they are connected to known tunable loads that provide multiple distinct terminations, including coupled 2-port loads. In the low-complexity extension, the series connection of the OTA fixture and the TLN is termed a programmable fixture. By altering the TLN configuration, the PF changes in a known way as seen from the accessible ports. Measurements taken across an ensemble of PF realizations can then be processed jointly, which is the operation referred to as multiplexed de-embedding (Hougne, 29 Sep 2025).

This framework differs from classical de-embedding in which a fixture is probed through direct electrical access. Here, the “fixture” is OTA, the inaccessible side is mediated by NDA antennas, and identifiability depends jointly on fixture characterization and on the number of independent measurements available at the accessible antennas.

2. Network-theoretic formulation

At a given load configuration, the complete system is described by a multi-port scattering relation. With $\mathcal{A}$ denoting accessible-antenna ports, $\mathcal{S}$ NDA-antenna ports, $\mathbf{S}^\mathrm{F}$ the OTA-fixture scattering matrix, and $\mathbf{S}^\mathrm{L}$ the scattering matrix of the loads terminating the NDA antennas, the accessible-side measurement obeys

$$\mathbf{S} = \mathbf{S}^\mathrm{F}_\mathcal{AA} + \mathbf{S}^\mathrm{F}_\mathcal{AS} \left( \left(\mathbf{S}^\mathrm{L}\right)^{-1} - \mathbf{S}^\mathrm{F}_\mathcal{SS}\right)^{-1} \mathbf{S}^\mathrm{F}_\mathcal{SA}.$$

If the accessible antennas are partitioned into transmit and receive subsets, $\mathcal{T}$ and $\mathcal{R}$, and only the transmission matrix is measured, then

$N_\mathrm{A}$0

These relations place the OTA fixture and the termination network in a standard multi-port S-parameter formalism and define the nonlinear inverse problem that de-embedding must solve (Hougne, 17 Jul 2025).

In the PF formulation, the system is modeled as a cascade of three multi-port scattering matrices: $N_\mathrm{A}$1, $N_\mathrm{A}$2, and $N_\mathrm{A}$3. The PF scattering matrix $N_\mathrm{A}$4 is obtained from $N_\mathrm{A}$5 and $N_\mathrm{A}$6 via the Redheffer star product. The measurable accessible-side coefficients then become

$N_\mathrm{A}$7

or, for transmission-only measurements,

$N_\mathrm{A}$8

For $N_\mathrm{A}$9 PF realizations, the measurements constitute a nonlinear system $\mathbf{S}^\mathrm{D}$0, $\mathbf{S}^\mathrm{D}$1, where $\mathbf{S}^\mathrm{D}$2 parameterizes the DUT. For an $\mathbf{S}^\mathrm{D}$3-port reciprocal DUT, the number of independent unknowns is $\mathbf{S}^\mathrm{D}$4 (Hougne, 29 Sep 2025).

3. OTA-fixture characterization and the de-embedding workflow

The original wireless multi-port sensing procedure has three stages. First, the OTA fixture is characterized with the DUT disconnected and the NDA antennas attached to the TLN. Multiple measurements are acquired at the accessible ports for various known termination configurations. Each NDA port must be connected in turn to at least three mutually-distinct individual loads; for $\mathbf{S}^\mathrm{D}$5, coupled 2-port loads between pairs of NDA antennas are needed. The unknown OTA fixture parameters are then estimated with the “Virtual VNA” technique by using all measurements together in a gradient-descent fit of the fixture model to the measured scattering responses (Hougne, 17 Jul 2025).

The paper gives a normalized mismatch objective of the form

$\mathbf{S}^\mathrm{D}$6

This stage is reported to be robust against noise, and the only unresolvable ambiguity is an overall sign ambiguity on cross terms between accessible and NDA ports; that ambiguity does not affect final DUT estimation (Hougne, 17 Jul 2025).

Second, the NDA antennas are connected to the unknown DUT, and the accessible-side scattering parameters $\mathbf{S}^\mathrm{D}$7 or transmission matrix $\mathbf{S}^\mathrm{D}$8 are measured. Third, the previously estimated OTA fixture is de-embedded by inverting the scattering relation for the only remaining unknown, $\mathbf{S}^\mathrm{D}$9. The corresponding optimization again minimizes a normalized discrepancy between measured and predicted accessible-side responses and can enforce reciprocity of the DUT. In the low-complexity formulation, the same logic is extended from one fixture to an ensemble of PF realizations, and the joint loss is minimized across all realizations using gradient descent with Adam and automatic differentiation (Hougne, 29 Sep 2025).

4. Multiplexing through programmable fixture realizations

The defining innovation of multiplexed de-embedding is to use the TLN not only for OTA-fixture characterization but also to generate measurement diversity during DUT sensing. When the DUT characteristics cannot be identified from a single PF realization, additional PF realizations are introduced by reconfiguring the TLN. Each realization can connect different subsets of NDA antennas to the DUT and terminate the rest with known loads, thereby changing the PF response seen from the accessible side in a known manner. The DUT is then estimated jointly from the ensemble of measurements rather than from any single realization in isolation (Hougne, 29 Sep 2025).

This strategy addresses the case in which the number of accessible antennas is too small to supply enough independent measurements in parallel. The paper formulates the trade-off in terms of $\mathcal{A}$0, the number of independent measurements per PF realization, and $\mathcal{A}$1, the number of PF realizations. In principle, $\mathcal{A}$2 should at least match, and preferably exceed, $\mathcal{A}$3, the number of independent DUT unknowns. For a 4-port reciprocal DUT, $\mathcal{A}$4. For SISO measurements with $\mathcal{A}$5, the effective-rank analysis of the Jacobian $\mathcal{A}$6 shows that $\mathcal{A}$7 is required to approach full identifiability; for larger $\mathcal{A}$8, fewer PFs are needed (Hougne, 29 Sep 2025).

The resulting interpretation is that multiplexed de-embedding exchanges spatial measurement parallelism for sequential configurational diversity. This suggests a direct hardware–identifiability trade-off: additional accessible channels and additional PF realizations serve partly substitutable roles, but only insofar as the PF ensemble provides sufficient diversity and the inverse problem remains numerically stable.

5. Experimental demonstrations

The first experimental validation was performed at $\mathcal{A}$9 for 1-port DUTs and 5-port DUTs, using a rich-scattering OTA fixture inside a reverberation chamber. For 1-port DUTs, the experiments considered various loads, including a phase shifter with 64 different states, with 5 NDA antennas of which 4 were fixed in reference loads, and with 2–8 accessible antennas. The full method yielded highly accurate estimation with MSE $\mathcal{S}$0 and visually indistinguishable reflection-coefficient circles. Benchmarks using only two loads, or a simplified model with no coupling and perfect matching, produced MSEs 100× larger and obvious errors. Increasing the number of accessible antennas and using all S-matrix blocks improved performance, although $\mathcal{S}$1 remained workable (Hougne, 17 Jul 2025).

For 5-port DUTs, two classes of devices were tested: an array of 5 phase shifters with no inter-port transmission, and a full 5-port unknown coupled circuit. Accurate recovery of all diagonal and off-diagonal scattering entries required at least 6–8 accessible ports for a 5-port DUT, reflecting the requirement that the number of measurable parameters must equal or exceed the number to be recovered. When coupled loads were absent from the Virtual VNA stage, sign ambiguities appeared in off-diagonal elements; this was acceptable if only diagonal terms or magnitudes were required. The study also reported that other hardware simplifications, specifically only two known loads, are acceptable for $\mathcal{S}$2 provided coupled loads are present, and that a simplistic model assuming perfect matching yields large errors unless the antennas are extremely well matched (Hougne, 17 Jul 2025).

The low-complexity extension demonstrated remote estimation of a reciprocal, non-unitary 4-port DUT with 10 complex-valued unknowns via a rich-scattering OTA fixture using only a single transmission coefficient between two accessible antennas across 30 different PF realizations. The experiments examined SISO with 2 accessible ports, $\mathcal{S}$3 MIMO with $\mathcal{S}$4, $\mathcal{S}$5 MIMO with $\mathcal{S}$6, and $\mathcal{S}$7 MIMO with $\mathcal{S}$8. The mean squared error in recovering $\mathcal{S}$9 dropped sharply as the number of PF realizations increased, and with $\mathbf{S}^\mathrm{F}$0 the DUT became identifiable when measured across approximately 30 PF realizations. The TLN in that experiment was realized with SP8T switches and characterized for all used configurations using a VNA (Hougne, 29 Sep 2025).

6. Applications, assumptions, and terminological scope

The reported application areas are wireless multi-port sensor grids, RFID sensor arrays, wireless bioelectronics, and, more generally, remote characterization of multi-port circuits or sensor networks with no direct electrical access. The stated advantages are that no physical reconnections or cabling are required for multi-port characterization, the method is robust to arbitrary propagation environments without any free-space or joined modeling requirement, and full multi-port S-parameter estimation is possible rather than only diagonal or reflection terms (Hougne, 17 Jul 2025).

The limitations are equally explicit. The OTA fixture must be linear, passive, time-invariant, and reciprocal, because otherwise the underlying S-parameter formalism and Virtual VNA approach are not directly applicable. All NDA ports must not radiate directly to free space, and the DUT must not couple to the wireless propagation environment except via their lumped ports. Sufficiently many accessible antennas, or alternatively sufficient PF diversity, are needed to ensure a well-posed estimation problem. The TLN must provide reasonably rich configurational diversity; coupled loads are essential for resolving mutual coupling and off-diagonal S-matrix elements unless only diagonal terms or magnitudes are needed. Fully autonomous load cycling for untethered operation was deferred to future work, and the maximal number of useful, non-redundant PF configurations is finite and set by the physical TLN design (Hougne, 17 Jul 2025, Hougne, 29 Sep 2025).

The low-complexity paper frames the hardware consequence sharply: multi-port VNAs or MIMO receivers become unnecessary for full characterization, and two SDRs suffice, because measurement diversity is generated electronically through the TLN rather than by increasing the number of simultaneously measured channels. This suggests that multiplexed de-embedding is best understood as a sequentially diversified inverse problem rather than merely a reduced-port version of conventional de-embedding (Hougne, 29 Sep 2025).

A separate usage of related terminology appears in de novo peptide sequencing, where DIANovo addresses multiplexed DIA spectra by disentangling coeluted peptide signals with a dilated CNN, a Transformer encoder using FlashAttention 2, RoPE, and coelution-aware pretraining. That problem is also described as de-embedding in the provided technical summary, but it is not an S-parameter or fixture-de-embedding formalism. This suggests that, across domains, “multiplexed de-embedding” denotes recovery of latent component responses from multiplexed observations, while the underlying mathematics and physical assumptions remain domain-specific (Ma et al., 2024).