Enveloping and Multiplication

Question

In the context of the Fourier Transform, when we multiply the rotating vector tracing out a circle, $e^{-2\pi i f t}$, by an input function $g(t)$ in the complex plane, the output graph is wound around the circle, but does not form a circumscription.

However, when we multiply the exponential factor of the solution to a linear IVP with constant coefficients, i.e., $y = .2524e^{-1.6t}$, by the trig factor of the solution, i.e., $\cos(6.3t - 1.1634)$, the full solution $y = .2524e^{-1.6t}\cos(6.3t - 1.1634)$ is bound by $y = \pm .2524e^{-1.6t}$ as an envelope.

Why is this tangency enveloping behavior created in one context of multiplication but not the other?

Answer 1

You can think of this in terms of the modulus:

• In the first case, the unit circle is modulated by some function $g(t)$, where it is not necessarily true that $g$ satisfies $|g(t)|\le1.$ Hence, it is possible for the product to 'overshoot' the unit circle.
• In the second case, the product is bounded by the envelope function, since the modulus of $\sin$ and $\cos$ are unity.