Foundations of the functional approach for signal analysis

Cyclostationarity: New trends and applications

“…By properly building numerical sequences that are not relatively measurable [207,247,Section 7], a class of signals is introduced in [208] such that their measured statistical functions (Section 3) are not convergent as the measurement interval is increased. Thus, these signals are suitable to be adopted when a secure communication channel needs to be established.…”

Time average estimation in the fraction-of-time probability framework

“…2.11] or fraction-of-time (FOT) probability approach [17,Chap. 8] and was first adopted in [44] for correlation and spectral analysis (see [29] for an extensive historical perspective).…”

“…Such a measure plays an equivalent role to the theoretical probability P in the stochastic approach where the object (Ω, F, P ) is considered. Once the concept of relative measurability is introduced for a signal x(t), a valid distribution function can be constructed whose expectation is the infinite time average [29]. The distribution at point ξ represents the fraction-of-time that the signal is below the level ξ [17,Chap.…”

“…Using such distribution and expectation, all the familiar probabilistic functions as mean, moments and cumulants can be constructed. Moreover, considering two or more time-shifted versions of the same signal, concepts can be extended to the multivariate case leading to the definition of joint distributions, autocorrelation, cross moments, and cumulants [18], [19], [29]. Independence of functions in the functional approach can be defined [27], [26], and a central limit theorem (CLT) proved [12].…”

“…Consequently, the relative measure is not σ-additive and not continuous. Moreover, the infinite time average, unlike the ensemble expectation, is not σ-linear [29]. Such lack of σ-additivity and σ-linearity is not an obstacle using the FOT approach in practice when dealing with time averages.…”