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Sensitivity analysis is crucial in microwave imaging and design procedures. We proposed a time-domain self- adjoint method for the computation of response Jacobians. The responses and their derivatives are computed with a single time-domain analysis. The overhead of the Jacobian computation is negligible compared to the time required by the simulation even when the number of optimizable parameters exceeds thousands. However, two drawbacks have become obvious: 1) memory requirements may become excessive when the number of perturbation grid points is large and the simulation time is long and 2) Jacobian accuracy may degrade due to the intrinsic inaccuracy of the local numerical field solution at dielectric interfaces of high contrast. Here, we propose an improved method for the self-adjoint computation of the Jacobian. It drastically reduces the memory requirements by implementing a novel spectral sensitivity formula, which operates on the spectral components of the E-field rather than on its time waveforms. It significantly improves the accuracy of the Jacobian by departing from the conventional finite-difference Yee cell and employing its own independent central-node finite-difference grid. The proposed approach is validated by 2-D and 3-D examples with lossy dielectric inhomogeneous structures. This study aims at the acceleration of wideband microwave image reconstruction via efficient Jacobian calculation.