= Solution
\b[Ideal spectrum.] Write $P_i(f)=C/(f^2+f_c^2)$, with $C>0$. The <overdamped harmonic-trap power spectrum> is a <Lorentzian line shape>:
$$
P_i\simeq\frac C{f_c^2}\quad(f\ll f_c),\qquad
P_i\simeq\frac C{f^2}\quad(f\gg f_c).
$$
Thus its <log-log plot> is horizontal at low <frequency> and approaches slope $-2$ at high <frequency>, changing around $f_c$. The left panel below compares two characteristic frequencies with the same illustrative $C$; a common numerator is explicitly assumed for this sketch. Their high-frequency tails coincide and their low-frequency plateaux differ as $f_c^{-2}$.
\Image[/past-exam-of-the-physics-a-course-of-the-university-of-cambridge/2017/paper-1-d12-noise-spectra.png]
{title=Ideal Lorentzian spectra at two corner frequencies and the measured spectrum with its graphical corner estimate}
The right panel plots all nine original data points, including the very-low-frequency excess and the highest-frequency suppression. It also shows the ideal curve inferred only from the requested intermediate range.
\b[Graphical corner estimate.] Between roughly $1$ and $100\,\mathrm{Hz}$ the measured <power spectral density> is approximately the plateau $P_0=10^{-6}\,\mathrm{V^2\,Hz^{-1}}$. At $10^4$ and $10^5\,\mathrm{Hz}$,
$$
f^2P_m(f)\simeq1\,\mathrm{V^2\,Hz},
$$
so the descending asymptote is $C/f^2$ with $C\simeq1\,\mathrm{V^2\,Hz}$. The <graphical corner frequency of a Lorentzian spectrum> is where these asymptotes intersect:
$$
\boxed{f_c=\sqrt{\frac C{P_0}}\simeq10^3\,\mathrm{Hz}.}
$$
Independently, the measured value $P_m(1000\,\mathrm{Hz})=5\times10^{-7}\,\mathrm{V^2\,Hz^{-1}}=P_0/2$ is exactly the half-plateau condition $P_i(f_c)=P_0/2$. Both graphical readings agree. The inferred ideal curve is approximately
$$
P_i(f)=\frac{10^{-6}\,\mathrm{V^2\,Hz^{-1}}}{1+(f/1000\,\mathrm{Hz})^2}.
$$
\b[Low-frequency excess and recording duration.] The two lowest-frequency measurements increase approximately as $1/f$, whereas the ideal trapped oscillator tends to a constant plateau. This resembles <flicker noise> or slow apparatus drift superposed on the motion. The <transimpedance amplifier>, detector or environment could contribute; the spectrum alone does not identify the source uniquely. It would be incorrect to attribute that whole excess to the ideal Brownian oscillator.
A finite record of duration $T$ has ordinary <Fourier frequency resolution> of order $1/T$. Resolving the lowest tabulated <frequency> $0.01\,\mathrm{Hz}$ therefore requires at least
$$
\boxed{T_{\min}\simeq\frac1{0.01\,\mathrm{Hz}}=100\,\mathrm s.}
$$
This is a formal lower bound corresponding to about one cycle; a reliable noise-spectrum estimate normally needs a substantially longer record and averaging.
\b[Sampling.] The largest reported <frequency> is $f_{\max}=10^6\,\mathrm{Hz}$. The <Nyquist–Shannon sampling theorem> therefore gives the Nyquist lower bound
$$
\boxed{f_s\ge2f_{\max}=2.0\times10^6\,\mathrm{samples\,s^{-1}}.}
$$
In practice choose a rate above $2\,\mathrm{MS\,s^{-1}}$, with an analogue <anti-aliasing filter> suppressing unmeasured content above half the <sampling rate>. Exact equality cannot recover an arbitrary pure sinusoid precisely at the boundary for every phase; it is the limiting lower-bound answer rather than a comfortable practical sampling design.
\b[High-frequency departure and <amplifier> design.] Extrapolating the fitted $f^{-2}$ tail to $10^6\,\mathrm{Hz}$ predicts approximately $10^{-12}\,\mathrm{V^2\,Hz^{-1}}$, while the measured value is $10^{-15}\,\mathrm{V^2\,Hz^{-1}}$. The observed drop from $10^5$ to $10^6\,\mathrm{Hz}$ has effective logarithmic slope $-5$, much steeper than the ideal $-2$.
A detector or <amplifier> bandwidth limit, or intentional filtering before digitization, can explain such suppression. For a <linear filter> the <spectral density transformation under a linear filter> is $P_{\rm out}=|H|^2P_{\rm in}$, so the last point corresponds approximately to an additional relative voltage gain $\sqrt{10^{-3}}=0.0316$, if attributed entirely to the readout transfer function. One additional point does not determine a unique filter order or identify the circuit responsible.
Rolling off an <amplifier>'s gain beyond the useful <signal> band reduces integrated high-frequency <noise>, suppresses interference, and helps prevent <aliasing> in the <analogue-to-digital converter>. Limiting high-frequency loop gain can also improve feedback stability and suppress unwanted oscillation. The cutoff must retain the frequencies needed for the measurement; excessive filtering would distort the desired fluctuation spectrum itself.
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