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

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 terms in one period. To implement proposed following steps are carried out: (1) For given system, represented by transfer function, an impulse response, block diagram etc., governing differential equation relating output y ( t ) $y(t)$ , to its input, x $x(t)$ obtained. (2) An auxiliary formed simply replacing ‾ $\overline{y}(t)$ and equating side alone. The solved each time segment signal period, leaving constant coefficients associated homogeneous as unknowns. nth-order system consisting q segments there n × $n\times q$ such unknown coefficients. (3) Continuity derivatives, d k / ${d}^{k}\overline{y}(t)/d{t}^{k},$ = 1 ⋅ − $k=1,\cdot \cdot ,n-1,$ at connection points successive periodicity conditions beginning end period implemented. (4) outcome step 3 equations described 2. Solving this equations, solution (5) Finally, using linearity differentiation properties total constructed derivatives. stable LTI systems, can be used without any limitations.