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The problem of linear time-variant filtering is addressed in the fraction-of-time (FOT) probability framework. The adopted approach, which is an alternative to the classical stochastic one, provides a statistical characterization of the system in terms of time averages of functions of time rather than ensemble averages of stochastic processes. Thus, it is particularly useful when stochastic systems transform ergodic input signals into nonergodic output signals, as it happens with several channel models encountered in practice. The analysis is carried out with reference to the wide class of the generalized almost-cyclostationary signals, which includes, as,a special case, the class of almost-cyclostationary signals. In this paper, systems are classified as deterministic or random in the FOT probability framework. Moreover, the new concept of expectation in the FOT probability framework of the impulse-response function of a system is introduced. For the linear time-variant systems, the higher order system characterization in the time domain is provided in terms of the system temporal moment function, which is the kernel of the operator that transforms the additive sinewave components contained in the input lag product into the additive sinewave components contained in the output lag product. Moreover, the higher order characterization in the frequency domain is also provided, and input/output relationships are derived in terms of temporal and spectral moment and cumulant functions. Developments and examples of application of the theory introduced here are presented in part II of this two-part paper.