  Recent advances in counter-adversarial systems have garnered significant
research attention to inverse filtering from a Bayesian perspective. For
example, interest in estimating the adversary's Kalman filter tracked estimate
with the purpose of predicting the adversary's future steps has led to recent
formulations of inverse Kalman filter (I-KF). In this context of inverse
filtering, we address the key challenges of non-linear process dynamics and
unknown input to the forward filter by proposing an inverse extended Kalman
filter (I-EKF). The purpose of this paper and the companion paper (Part II) is
to develop the theory of I-EKF in detail. In this paper, we assume perfect
system model information and derive I-EKF with and without an unknown input
when both forward and inverse state-space models are non-linear. In the
process, I-KF-with-unknown-input is also obtained. We then provide theoretical
stability guarantees using both bounded non-linearity and unknown matrix
approaches and prove the I-EKF's consistency. Numerical experiments validate
our methods for various proposed inverse filters using the recursive
Cram\'{e}r-Rao lower bound as a benchmark. In the companion paper (Part II), we
further generalize these formulations to highly non-linear models and propose
reproducing kernel Hilbert space-based EKF to handle incomplete system model
information.
