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This paper presents a comparative theoretical study of amplitude, phase, and frequency modulation. In the first part, the fundamental mathematical expressions for the three types of modulation are derived. They are expressed in three different forms: as amplitude equations, side band equations and modulation vector equations. The amplitude equations indicate the envelope of the radio-frequency directly. The side band equations refer to the number, amplitude, and phase of the side bands produced by modulation. In the modulation vector equations, corresponding side bands are combined in pairs to form a "modulation vector." This is a r-f magnitude, rotating with the angular velocity of the carrier and its amplitude is simultaneously being changed at an audio rate. The main results derived from a discussion of these equations are: In phase and frequency modulation an infinite number of side bands is produced. Amplitude modulation produces but one pair of side bands. In amplitude modulation the modulation vector, representing the first pair of side bands is in phase with the carrier. In phase and frequency modulation, it is 90 degrees out of phase with respect to the carrier. Frequency modulation is equivalent to a phase modulation in which the phase shift is inversely proportional to the audio frequency. By means of the modulation vector, a new vector diagram of the phase modulation is given. In the second part, amplitude modulation is considered in which undesired phase or frequency modulation or a combination of the two takes place simultaneously.