Toward a unified theory of modulation part I: Phase-envelope relationships

Classical modulation theory deals mainly with single-parameter proportional modulation techniques, i.e., with processes such as AM and FM, wherein the amplitude or angle of a sinusoidal carrier wave is varied as a linear function of an information signal. Classical modulation theory has been immensely useful, but it is a restrictive theory because it can deal tractably with only a limited class of possible modulation processes. In this paper we seek to take the first few steps toward a broader and more unified theory of modulation--a theory which will encompass several recently proposed hybrid modulation schemes as well as conventional techniques, which will suggest new modulation processes, and which might lead toward unification of modulation theory and coding theory. The paper is divided into two parts. Part I is devoted to theoretical development as summarized below. The results of Part I are applied in Part II to study modulation processes per se. The paper as a whole is in the nature of a wide ranging survey rather than a rigorously detailed analysis. The most general form of modulated wave exhibits simultaneous phase and envelope fluctuations. In the first half of Part I, equations are developed to describe phase-envelope relationships when the wave is band-limited. These equations prove to be intimately dependent on the real and complex zeros of the wave. Real zeros ("zero crossings") are familiar physical attributes. Complex zeros, however, are physically rather covert and are a concept foreign to many engineers. Both kinds of zeros are easy to interpret mathematically in terms of the factorization of a Fourier series. In the second half of Part I, three simple examples are used to illustrate relationships between zeros, spectra, envelopes, and phase functions. These examples lead to operational procedures for representing signals in terms of zeros and, most importantly, to a viewpoint wherein zeros are regarded as fundamental informational attributes of signals. This viewpoint leads, in Part II, to the concept of modulation as zero manipulation.