The recurrent-continuant method of transfer function synthesis

It is shown that the ratio of input voltage to output voltage for any passive ladder type low-pass filter network is always a polynomial in the complex frequency s. This polynomial can always be written by inspection as a single determinant in the form of a recurrent. The recurrent determinant is then converted by elementary manipulation into an alternate form known as a continuant. The transfer function of any passive ladder type low-pass filter network can also be written by inspection as a single determinant in continuant form. The synthesis of such networks is thus reduced to the simple procedure of writing the given transfer function in recurrent form by inspection, use of simple arithmetic operations to change the recurrent into continuant form, and reading the numerical values of the network elements directly from the continuant.