Multi-Phase Power-Sinusoids 1: Introduction

Multi-phase power-sinusoids are not yet in the textbooks. They generalize , , and to by introducing extra parameters: the number of phases , the phase index , the power , together with the quantity . The factor is the selector for the phase (just as selects and selects ). The numbers and are integration constants. They are kept at 0 in this Demonstration.

Sinusoids are developed as solutions to differential equations in Multi-Phase Power Sinusoids 2: Differential Equations, and as modified exponential series in Multi-Phase Power-Sinusoids 3: Exponential Series.

When and , the traditional functions and are the differences between pairs of phases (Thumbnail, Fig. 1), that is, ; as the second term is the negation of the first, the amplitude is twice that of the phases and the integration constants cancel (giving a static zero mean). The functions and are not symmetrical; and are "missing phases". This is because is a cyclotomic number with and ; in contrast, the powers are independent unsigned directions so there are four different phases.

Now look at the first snapshot (Fig. 2.3), which shows the 3-phase sinusoid. Change . Note that odd powers always sum to zero. Do you think that their absolute values can sum to a constant? When do you think that even powers sum to a constant? This topic is explored in Multi-Phase Power-Sinusoid 4: Summations.

Now look at snapshot 2 (Fig. 2.4). (Half the phases are displaced slightly when and are both even to show superimposition.) Higher numbers of phases and powers are shown in later snapsots. They follow the convention of projecting the phases onto the same - plane. Multi-Phase Power-Sinusoids 5: Helices and Pulses shows other projections; three-dimensional -- helices, -dimensional trajectories, and wave-packets (modulations of travelling pulses).