Impulse response and power spectrum are interchangeable representations of a linear system

Evidence

For linear systems, we may either strike the system impulsively, or sinusoidally drive it at first one sinusoidal frequency, then another, and so on, always waiting for it to settle into a steady state. The energy imparted by the sudden impulse is proportional to sum of all the power deposited by sinusoidal driving. … The power at each frequency—that is, the energy deposited if actually driven at that frequency divided by the time the sinusoid was on—is just the cosine transform of the impulse velocity response. … Therefore, we need only the impulse response to calculate a power spectrum, or conversely a power spectrum to calculate the impulse response. — Eric J. Heller, Why You Hear What You Hear- An…, 1086

Categorization

• XREF: The impulse-response/power-spectrum equivalence is the same duality appearing in EQ design: an impulse is a broadband excitation whose response reveals the system’s full frequency content.