Finite cosine sums can never yield decaying autocorrelations, forcing a smooth power spectrum.

Evidence

A cosine never dies; it oscillates forever. So too does the sum of a finite number of cosines. It follows that signals having autocorrelations that decay to zero and remain there cannot be represented with a few cosines. Something new is called for. We now show that the power spectrum for such a correlation function is a smooth function, not a set of sharp vertical lines. In the process, we will discover a key to the analysis of sound: the uncertainty principle. — Eric J. Heller, Why You Hear What You Hear- An…, 398

Categorization

• XREF: The uncertainty principle’s appearance in sound analysis connects to its role in quantum mechanics and signal processing.