Past exam of the year 2 of the earth sciences (geology) course of the University of Oxford / 2024 / Paper 3 / 7 / g

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- Solution g

Solution

Most of the mean-square signal power lies at low frequencies. Each nonzero coefficient has magnitude

2/ (π ∣ r ∣) V

, so its power contribution decreases as

r^{- 2}

. The pair at the fundamental frequencies

\pm 0.25 Hz

alone contains

\frac{8}{π ^{2}} ≃ 81.1%

of the total. The first three positive odd harmonics, together with their negative-frequency partners, contain about

93.3%

. 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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