Fig. A1.

Frequency characteristics of truncated sinusoids. A: temporal representation of an infinite-duration sinusoid with frequency f = 1 Hz and period T = 1 s. Acceleration is shown, but the analysis is general for any sinusoid. B–E: finite-duration sinusoids ranging between 1 and 9 cycles; these are mathematically realized by multiplying an infinite-duration sinusoid (thin black line) with a square window (gray line). F: the frequency -domain representation for an infinite-duration sinusoid has all its energy represented by impulses at +1 and −1 Hz. G–J: the square window in time corresponds to the sinc function (gray line) in frequency. Multiplication in time corresponds to convolution in frequency; convolving the impulse (thin black line) with the sinc function (gray line) yields the frequency spectrum of the finite-duration sinusoid (thick black line). These finite-duration sinusoids exhibit small but significant frequency distortions.