Solution

ID: past-exam-of-the-year-2-of-the-earth-sciences-geology-course-of-the-university-of-oxford/2024/paper-3/7/g/solution

Most of the mean-square signal power lies at low frequencies. Each nonzero coefficient has magnitude , so its power contribution decreases as . The pair at the fundamental frequencies alone contains
of the total. The first three positive odd harmonics, together with their negative-frequency partners, contain about . The harmonic power of a square wave still has a nonzero tail: higher harmonics are needed to reproduce sharp edges even though their combined power is small.
Figure 1.
Odd Fourier harmonics reconstruct a unit square wave and concentrate most power in the fundamental
. Original analytic calculations. The reconstruction and paired-harmonic powers use the derived exact coefficients, rather than the supplied noisy spectral illustration. The horizontal axis is time in seconds; voltage and mean-square voltage units are retained. Details.

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