Narrow time signals yield wide frequency spectra via beat cancellation

Evidence

The Fourier construction of the power spectrum reveals why a wide time pulse gives a narrow power spectrum, and vice versa. Choosing a cosine with a frequency shifted from the maximum of the power spectrum peak by Δf = fmax – f = d, we find (figure 4.11) that the product of the cosine and the autocorrelation oscillates between positive and negative values with period 1/d. This is quite apparent in panel B in figure 4.11. If the autocorrelation is decaying with time, the question arises, how many such beats are seen before the decay is significant? If the autocorrelation lasts several such beats (wide time signal), the negative and positive contributions will nearly cancel when forming the average, meaning the spectrum must be small at that frequency difference (narrow frequency spectrum). If at the same frequency difference d, the autocorrelation dies before the positive-negative cancellation can take place (narrow time signal), say, after just the initial positive part, the averaging out will not occur and the power spectrum will be larger (wider frequency spectrum). More power survives at off-center frequencies—that is, the spectrum is wider for a narrow or thin correlation function (see the annotations in figure 4.11). It is clear from this that the shorter the duration of the autocorrelation, the broader the resulting power spectrum. — Eric J. Heller, Why You Hear What You Hear- An…, 414

Categorization

• XREF: This beat-cancellation intuition is the same mechanism behind diffraction — aperture width vs. diffraction spread in optics.