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Related Concept Videos

Sampling Continuous Time Signal01:11

Sampling Continuous Time Signal

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In signal processing, a continuous-time signal can be sampled using an impulse-train sampling technique, followed by the zero-order hold method. Impulse-train sampling involves the use of a periodic impulse train, which consists of a series of delta functions spaced at regular intervals determined by the sampling period. When a continuous-time signal is multiplied by this impulse train, it generates impulses with amplitudes corresponding to the signal's values at the sampling points.
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In an underdamped second-order system, where the damping ratio ζ is between 0 and 1, a unit-step input results in a transfer function that, when transformed using the inverse Laplace method, reveals the output response. The output exhibits a damped sinusoidal oscillation, and the difference between the input and output is termed the error signal. This error signal also demonstrates damped oscillatory behavior. Eventually, as the system reaches a steady state, the error diminishes to zero.
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A system is linear if it displays the characteristics of homogeneity and additivity, together termed the superposition property. This principle is fundamental in all linear systems. Linear time-invariant (LTI) systems include systems with linear elements and constant parameters.
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In any LTI (Linear Time-Invariant) system, the convolution of two signals is denoted using a convolution operator, assuming all initial conditions are zero. The convolution integral can be divided into two parts: the zero-input or natural response and the zero-state or forced response, with t0 indicating the initial time.
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Signal processing techniques are essential for accurately converting continuous signals to digital formats and vice versa. When a continuous signal is sampled with a period T, the resulting sampled signal exhibits replicas of the original spectrum in the frequency domain, spaced at intervals equal to the sampling frequency. To handle this sampled signal, a zero-order hold method can be applied, which creates a piecewise constant signal by retaining each sample's value until the next...
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The Discrete-Time Fourier Series (DTFS) is a fundamental concept in signal processing, serving as the discrete-time counterpart to the continuous-time Fourier series. It allows for the representation and analysis of discrete-time periodic signals in terms of their frequency components. Unlike its continuous counterpart, which utilizes integrals, the calculation of DTFS expansion coefficients involves summations due to the discrete nature of the signal.
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    Area of Science:

    • Control Systems Engineering
    • Stochastic Systems Theory
    • Signal Processing

    Background:

    • Challenges in filtering for 2-D discrete-time periodic Markov jump systems with multiplicative noise.
    • The problem of missing measurements, modeled as a Bernoulli random variable.
    • The need to address nonsynchronous phenomena between system and filter due to mode information loss.

    Purpose of the Study:

    • To design and implement an asynchronous H-infinity filter for 2-D discrete-time periodic Markov jump systems.
    • To incorporate a periodic Hidden Markov Model (HMM) to handle mode information loss and nonsynchronous issues.
    • To guarantee mean-square asymptotic stability and H-infinity disturbance attenuation for the filtering error system.

    Main Methods:

    • Utilizing a periodic Hidden Markov Model (HMM) to manage system and filter asynchrony.
    • Formulating filter design conditions based on linear matrix inequalities (LMIs).
    • Considering general transition rate matrices and conditional probability matrices for various known/unknown transition probability cases.

    Main Results:

    • Sufficient conditions for the existence of the asynchronous H-infinity filter are derived.
    • A constructive solution for the filter parameters is provided.
    • The effectiveness of the proposed filter design is validated using the Darboux equation.

    Conclusions:

    • The developed asynchronous H-infinity filter effectively addresses missing measurements and nonsynchronous operation in 2-D periodic Markov jump systems.
    • The LMI-based approach provides a robust framework for designing such filters, even with partially or fully unknown system parameters.
    • The study demonstrates a practical method for enhancing filtering performance in complex stochastic systems.