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Frequency Domain Analysis
Published in Anastasia Veloni, Nikolaos I. Miridakis, Erysso Boukouvala, Digital and Statistical Signal Processing, 2018
Anastasia Veloni, Nikolaos I. Miridakis, Erysso Boukouvala
Let the periodic sequence x˜[n] with period N. Obviously it is true that x˜[n]=x˜[n+rN] (where n, r are integers). As in the case of continuous-time signals, a periodic sequence can be represented by a Fourier series, that is, by a sum of complex exponential terms whose frequency is an integer multiple of the fundamental frequency, 2π/N, as shown in the following relations: SynthesisEquation:x˜[n]=1N∑k=0N−1X˜[k]ej2πkNn
Feature Generation and Feature Engineering for Sequences
Published in Guozhu Dong, Huan Liu, Feature Engineering for Machine Learning and Data Analytics, 2018
Guozhu Dong, Lei Duan, Jyrki Nummenmaa, Peng Zhang
Different from frequent sequence patterns, closed sequential patterns, and partial order patterns, periodic sequence patterns are discovered from one single sequence only instead of a set of more sequences [11, 16, 38]. A periodic sequence pattern occurs in a sequence repeatedly. Finding periodic sequence patterns is useful in long activity logs.
Ordinary differential equations defined by a trigonometric polynomial field: behaviour of the solutions
Published in Dynamical Systems, 2023
Let be . A function is weakly almost periodic of slope r if it is and if there exists a uniformly bounded sequence for the sup-norm of functions that are periodic modulo and there exists a sequence such that We call the sequence the -periodic sequence of the function h.
Growth of number of periodic orbits of one family of skew product maps
Published in Dynamical Systems, 2018
We denote by the periodic sequence of period m + 1, , defined by . Note that if ξ ∈ Σ2 is a periodic sequence of period m + 1, then ξ is a fixed point of σm + 1, that is, ξ is a periodic point of period m + 1 of σ. Let be a periodic point of G of period m + 1. We define the stable and unstable sets as
Reduced-complexity interpolating control with periodic invariant sets
Published in International Journal of Control, 2023
Sheila Scialanga, Sorin Olaru, Konstantinos Ampountolas
We want to prove that for each initial state in the feasible set, x converges to Ω in finite time with control action (16). Consider , as candidate Lyapunov function for obtained by solving the LP problem (5) in order to determine the state decomposition, and , as local candidate Lyapunov function defined at each periodic cycle and obtained by solving the LP problem (15) when a new local state is defined. A new function is considered at each periodic sequence. The motivation to consider two different Lyapunov functions is that the interpolating coefficient (candidate Lyapunov function) is updated in two different ways: (a) for the initial state of the periodic sequence; (b) during the periodic cycle of period p. For this reason two Lyapunov functions are required. We first prove that the function defines a local Lyapunov function within a periodic sequence, i.e. for steps. Figure 4 depicts the state decomposition at time and . If the local state is not updated, the proportion between the segments that connect the inner and outer states to current state do not change. That is, the interpolating coefficient s is constant and equal to and, Consider now the updated decomposition, which follows from the solution of the LP problem (15). Note that the updated inner state is closer to the current state compared to its previous value while the outer state is kept constant. Then the updated interpolating coefficient provides a smaller value compared to the interpolating coefficient in the previous time step . This argument can be applied at each time step of the periodic sequence, and thus it verifies a Lyapunov function that guarantees stability to the system within a periodic sequence.