Closed-form expressions of discrete-time transfer-function responses and optimal choice of numerator coefficients (zeros)

General closed-form expressions of discrete-time transfer-function responses, valid for real as well as complex eigenvalues, are derived in this paper. The closed-form expressions include the numerator coefficients of the transfer function, a Vandermonde-like matrix and a vector containing the transfer-function eigenresponses, the latter two only depending on the eigenvalues. The closed-form expressions are used to calculate the numerator coefficients optimally, by minimizing the discrete pulse transient, the discrete step transient, the discrete ramp transient, etc. Simulations are presented, clearly demonstrating excellent transfer function response properties based on the optimal choices of the numerator coefficients