Theoretical investigation and experimental verification of the nonanalytic form of the conversion equations in a frequency divider by two

The conversion matrix, usually represented in a conventional complex-number form, assumes a nonanalytic, real-imaginary form when the frequency of the small signal fs is equal to one-half the local oscillator (LO) frequency f0. This is due to interference phenomena between the input small signal at fs = f0/2 and the converted signal again at f0 − fs = fs = f0/2; the interference is dependent on the phase relation between the input small signal and the LO. The conversion matrix with fs = f0/2 is used in the design of frequency dividers by two. The conversion equations have been rewritten for the case of fs = f0/2, and the conversion matrix has been found to assume a real-imaginary form. Consequently, Barkhausen's criterion has been reassessed for this case. Experiments are performed that confirm this formalism, and computer simulations based on a nonlinear model for the nonlinear device are presented, showing similar results.