Closed form Expressions of Linear Continuous-and Discrete-Time Filter Responses

Closed-form expressions of linear system responses are presented in a general framework that emphasizes efficient computational implementation. The closed-form expressions depend linearly on the numerator coefficients of the transfer function. They also depend on a matrix including partial fraction expansion coefficients and the system's eigenvalues, and a vector containing linearly independent time basis functions. The closed forms facilitates various possibilities in the optimization of the numerator coefficients or zeros. One application involving the tracking of a reference impulse response, or equivalently the tracking of a filter in the frequency domain, is presented