Improved computation in terms of accuracy and speed of LTI system response with arbitrary input

Jelicic, Goran; Böswald, Marc; Brandt, Anders

doi:10.1016/j.ymssp.2020.107252

Cited by 4 publications

(3 citation statements)

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“…The coefficients A 11 ; A 12 ; A 21 ; andA 22 are determined by applying the continuity and periodicity conditions as outlined in Equations ( 9) and (10).…”

Section: Results and Demonstration Of Methodsmentioning

confidence: 99%

“…Similarly, if xðtÞ possesses an impulse c 0 δðtÞ at t ¼ 0, (or c 0 δðt − T Þ at t ¼ T ), then the periodicity condition for d n−1 yðtÞ=dt n−1 , the last row in Equation (10), will be modified as:…”

Section: Continuity and Periodicity Conditionsmentioning

confidence: 99%

“…Several authors have used the state-space representation of LTI systems to formulate output solutions to periodic inputs [9][10][11]. The implementation of these methods usually involves matrix calculations or numerical integrations, thus providing little insight into the nature of system response.…”

Section: Introductionmentioning

confidence: 99%

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A new method for calculation of closed‐form response of linear time‐invariant systems to periodic input signals

Safaai‐Jazi

2023

IET Circuits, Devices & Syst

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A new method for finding closed-form time-domain solutions of linear time-invariant (LTI) systems with arbitrary periodic input signals is presented. These solutions, unlike those obtained based on the conventional Fourier-phasor method, have a finite number of terms in one period. To implement the proposed method, the following steps are carried out: (1) For a given system, represented by a transfer function, an impulse response, a block diagram etc., the governing differential equation relating the output of the system, yðtÞ, to its input, xðtÞ, is obtained. (2) An auxiliary differential equation is formed by simply replacing yðtÞ with yðtÞ and equating the input side toxðtÞ alone. The auxiliary differential equation is solved for each time segment of the input signal in one period, leaving the constant coefficients associated with the homogeneous solutions as unknowns. For an nth-order system with an input signal consisting of q segments in one period, there are n � q such unknown coefficients. (3) Continuity of yðtÞ and its derivatives, d k yðtÞ=dt k ; k ¼ 1; ⋅ ⋅ ⋅; n − 1; at the connection points of successive segments and the periodicity conditions for the beginning and end points of the period are implemented. (4) The outcome of step 3 is a system of n � q equations in terms of n � q unknown coefficients described in step 2. Solving this system of equations, the solution for yðtÞ in one period is obtained. (5) Finally, using the linearity and differentiation properties of the system and the coefficients of the input side of the differential equation of the system, the total response, yðtÞ, in one period is constructed in terms of yðtÞ and its derivatives. For stable LTI systems, the proposed method can be used without any limitations. K E Y W O R D S analysis of LTI systems, closed-form response of LTI systems to periodic inputs, periodic inputsThis is an open access article under the terms of the Creative Commons Attribution-NonCommercial-NoDerivs License, which permits use and distribution in any medium, provided the original work is properly cited, the use is non-commercial and no modifications or adaptations are made.

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“…The coefficients A 11 ; A 12 ; A 21 ; andA 22 are determined by applying the continuity and periodicity conditions as outlined in Equations ( 9) and (10).…”

Section: Results and Demonstration Of Methodsmentioning

confidence: 99%

Section: Continuity and Periodicity Conditionsmentioning

confidence: 99%