Stosine

Stosine is a hypothetical mathematical construct used in introductory discussions of trigonometric modulation and piecewise-amplitude signals. The name blends the ideas of a sine wave with a step-like modulation, illustrating how a smooth carrier can be altered by abrupt changes in amplitude.

Definition and form. In its simplest form, stosine is defined for a real variable t as S(t) = sin(ω t) · m(t), where ω > 0 is a fixed angular frequency and m(t) is a piecewise constant step function taking real values, often restricted to a finite set such as {0, 1}. The function m(t) may be periodic (a stair-step envelope) or stochastic (random on/off gating). When m(t) is constant, stosine reduces to a standard sine wave; when m(t) varies, the resulting signal exhibits amplitude modulation with abrupt transitions.

Properties. If m(t) is periodic with period T, the overall periodicity of S(t) depends on the least common multiple of the carrier period 2π/ω and T. The derivative satisfies dS/dt = ω cos(ω t) m(t) + sin(ω t) m’(t), with m’(t) consisting of a train of impulses at jump points if m is a pure step function. The Fourier content of S(t) reflects both the harmonics of sin(ω t) and the spectral effects of the gating function m(t), including possible broadening and sidebands.

Variants and applications. A deterministic variant uses controlled on/off intervals, modeling amplitude-keyed or envelope-modulated signals. A stochastic variant treats m(t) as a random process, useful for studying nonlinear systems, communications with fading, or educational demonstrations of modulation effects.

See also. Sine function, step function, amplitude modulation, on-off keying, waveform synthesis. Note: stosine is not a standard term in mathematics and appears here as a didactic construct.