The optimum design of non-uniform transmission-lines for a given bandwidth is discussed, and it is stated that the shorter the non-uniform transmission line the more violently its characteristic impedance varies. The concept of complexity of this characteristic impedance function is introduced and a design method is suggested for minimizing the complexity.Numerical examples are given to illustrate the method, and it is shown that it is possible to design tapers even shorter than the Chebyshev type.The method is capable of further extension.