A seismologist wants the power spectral density of background noise at a monitoring…

PHYSICS · Multi-part · Fermi University Physics · Oscillations, Waves & Acoustics · Electronics, Instrumentation & Communication · Coupled oscillators, normal modes and Fourier analysis

Modes, Packets and Media, Ch The continuum limit, Fourier series and the spectrum of a measured signal

Problem

A seismologist wants the power spectral density of background noise at a monitoring station, with the fractional scatter (one standard deviation, relative to the mean) of the estimate at each frequency no larger than

5\%

. A single periodogram, however long the record, scatters by

100\%

of its mean at every frequency; averaging

K

independent, equal-length segments (Bartlett's method) cuts the fractional scatter to

1/\sqrt K

while coarsening the frequency resolution to

K

times the single-segment bin spacing. ((a)) Find the minimum number of segments

K

needed to bring the scatter down to

5\%

. ((b)) The available continuous record is

T_{\mathrm{tot}} = 3600\,\mathrm s

(one hour), divided into

K

equal non-overlapping segments of length

T_{\mathrm s}=T_{\mathrm{tot}}/K

. Find

T_{\mathrm s}

and the resulting frequency resolution

\Delta\nu = 1/T_{\mathrm s}

. ((c)) Two suspected noise peaks lie at

1.000\,\mathrm{Hz}

and

1.080\,\mathrm{Hz}

. Check whether the averaging scheme of part (b) can resolve them, and if not, find the minimum total recording time that meets both the

5\%

scatter target and resolves this pair, keeping the segments non-overlapping.

Browse the problem library