On the Differential Equation for the Generation of Frequency Modulation

Representations are presented for the general solution of the linear differential equation associated with the electronic generation of frequency modulated signals. The frequency modulation component is determined explicitly by an orthogonal fundamental solution matrix, while the amplitude modulation component, which arises as an impurity in the generation process, is discussed with the aid of elementary aspects of the theory of Lie algebras. The Lie algebra thus defined is three dimensional and algebraically simple. Asymptotic representations of the amplitude modulation component to the first-order in the absolute frequency deviation are shown to be completely prescribed by the basis vectors chosen for the Lie algebra.