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

Abstract

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, $y\left(t\right)$, to its input, $x\left(t\right)$, is obtained. (2) An auxiliary differential equation is formed by simply replacing $y\left(t\right)$ with $\bar{y}\left(t\right)$ and equating the input side to$x\left(t\right)$ 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 $n\times q$ such unknown coefficients. (3) Continuity of $\bar{y}\left(t\right)$ and its derivatives, $d^k\bar{y}\left(t\right)/d{t}^k,$ $k=1,\cdot \cdot \cdot ,n-1,$ 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 $n\times q$ equations in terms of $n\times q$ unknown coefficients described in step 2. Solving this system of equations, the solution for $\bar{y}\left(t\right)$ 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, $y\left(t\right)$, in one period is constructed in terms of $\bar{y}\left(t\right)$ and its derivatives. For stable LTI systems, the proposed method can be used without any limitations.