Direct and Regularized Inverse De-Embedding for Single-Carrier Signal Recovery in Measurement Front-Ends

To address the degradation of recovery accuracy caused by amplitude fluctuation, phase distortion, delay distortion, and noise amplification in single-carrier signal measurement chains, this paper investigates direct inverse and regularized inverse de-embedding compensation methods. Based on a linear time-invariant system model, single-carrier signal de-embedding is formulated as an ill-conditioned inverse problem that is sensitive to weak-response frequency points and observation noise. A unified frequency-domain compensation framework is then established, including the Direct method, Tikhonov regularized inverse compensation, Wiener-type inverse compensation, and truncated inverse compensation. To evaluate the applicability of these methods, a narrowband single-carrier signal and four measurement-chain models are constructed, including a smooth reference chain, a passband-edge attenuation chain, a multiple local-fading ill-conditioned chain, and a measured S-parameter-based chain. The simulation results show that the compensation gain is closely related to the magnitude response of the measurement chain. The Direct, Tikhonov, and Truncated methods produce similar results when the chain response is relatively flat or when the regularization constraint is weak, whereas the Wiener-type method achieves better NMSE performance under the tested conditions. Parameter-sweep and SNR experiments further show that the effectiveness of regularized inverse compensation depends on the ill-conditioning degree of the measurement chain, the noise level, and the parameter settings. Measured single-carrier signal experiments verify the feasibility of the proposed framework. Frequency-domain de-embedding compensation based on the measured S21 improves the NMSE from -18.7808 dB before compensation to -37.9458 dB after compensation. The measured results also show that, when the measurement-chain response is relatively flat, the additional improvement of Tikhonov and Truncated methods over the Direct method is limited, while the Wiener-type method provides a slight NMSE improvement. Overall, the proposed framework provides a practical approach for single-carrier signal recovery and clarifies the applicability of different inverse compensation methods under different measurement-chain and noise conditions.