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Frequency estimation infers a periodic signal's frequency from observed samples. It may use a peak of the discrete Fourier transform, or fit a sinusoidal model with amplitude, phase, and frequency as unknown parameters. Its attainable precision depends on noise, observation duration, sampling rate, and the estimator. Fourier frequency resolution alone does not specify that precision.

A gear vibration spectrum describes how a measured gearbox motion signal is distributed over frequency. An accelerometer on a housing supplies a response spectrum; a rotary encoder supplies a rotation reference. Gear mesh frequency predicts tooth-passage components, but their observed amplitudes also depend on applied loads, sensor direction and the mechanical frequency response. A response peak alone therefore does not identify a structural natural frequency.^[ref]

For an original example, a24-tooth gear at1500 revolutions per minute rotates at $f_r=25\mathrm{Hz}$ and has mesh frequency $f_m=600\mathrm{Hz}$. Consider an illustrative acceleration signal with amplitude modulation:The product-to-sum identity gives a600Hz component of amplitude $A$ and modulation sidebands at575 and625Hz, each of amplitude $\mu A/2$. For $A=1m/{\mathrm{s}}^2$ and $\mu =0.20$, each sideband has peak amplitude $0.10m/{\mathrm{s}}^2$. Gear vibration order expresses these frequencies relative to the chosen shaft: the central order is24 and the neighboring orders23 and25. These positions follow from the constructed signal, without requiring a damaged tooth.

A primary apparatus example makes that ambiguity concrete. NASA's2015 transmission experiment compared rotating accelerometers on a pinion shaft with stationary housing sensors. The rotating radial channel showed sidebands at1854 and2060Hz around a1957Hz mesh frequency, spaced by the103Hz shaft rate. The report derives this shift from the rotating measurement direction and discusses sums and differences of tangential channels. A measurement-frame effect can therefore create sidebands as well as a mechanical defect.^[ref]

A useful test records speed, load, mounting and sensor orientation with each spectrum. If speed changes appreciably within one record, fixed-frequency lines can spread; comparisons should then account for the changing rotation rate. Record duration also controls Fourier frequency resolution. Sideband changes are evidence to interpret alongside a baseline and the test conditions, rather than a unique diagnosis from one spectral peak.

The power linewidth is $\Delta f=f_0/Q\simeq 0.044\mathrm{Hz}$. Fourier frequency resolution is of order $1/T_{\mathrm{obs}}$, so $T_{\mathrm{obs}}\gtrsim Q/f_0\simeq 23\mathrm{s}$, and preferably longer to distinguish the intrinsic linewidth from the observation-window width. zero padding does not improve this physical resolution. A freely decaying signal also needs sufficient microphone dynamic range.

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: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 $f_c^{-2}$.

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.

Graphical corner estimate. Between roughly $1$ and $100\mathrm{Hz}$ the measured power spectral density is approximately the plateau $P_0=10^{-6}{\mathrm{V}}^2\mathrm{H}{\mathrm{z}}^{-1}$. At $10^4$ and $10^5\mathrm{Hz}$,so the descending asymptote is $C/f^2$ with $C\simeq 1{\mathrm{V}}^2\mathrm{Hz}$. The graphical corner frequency of a Lorentzian spectrum is where these asymptotes intersect:Independently, the measured value $P_m(1000\mathrm{Hz})=5\times 10^{-7}{\mathrm{V}}^2\mathrm{H}{\mathrm{z}}^{-1}=P_0/2$ is exactly the half-plateau condition $P_i(f_c)=P_0/2$. Both graphical readings agree. The inferred ideal curve is approximately

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 leastThis is a formal lower bound corresponding to about one cycle; a reliable noise-spectrum estimate normally needs a substantially longer record and averaging.

Sampling. The largest reported frequency is $f_{\max }=10^6\mathrm{Hz}$. The Nyquist–Shannon sampling theorem therefore gives the Nyquist lower boundIn practice choose a rate above $2\mathrm{MS}{\mathrm{s}}^{-1}$, 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.

High-frequency departure and amplifier design. Extrapolating the fitted $f^{-2}$ tail to $10^6\mathrm{Hz}$ predicts approximately $10^{-12}{\mathrm{V}}^2\mathrm{H}{\mathrm{z}}^{-1}$, while the measured value is $10^{-15}{\mathrm{V}}^2\mathrm{H}{\mathrm{z}}^{-1}$. 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 $P_{\mathrm{out}}=|H{|}^2{P}_{\mathrm{in}}$, 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.

For the $110\mathrm{Hz}$ fundamental alone, the Nyquist–Shannon sampling theorem places the limiting sampling rate at twice that frequency:Use a rate strictly above this boundary for a sinusoid of unknown phase: exactly two samples per cycle can, for a particular phase, all fall at zero. If higher Fourier harmonics are retained, the sampling rate must instead exceed twice the highest retained frequency, or those components must be removed to prevent aliasing.

For a straightforward discrete Fourier transform estimate, Fourier frequency resolution obeys $\Delta f\sim 1/T_{\mathrm{obs}}$. Resolving $0.07\mathrm{Hz}$ therefore suggestsAt the limiting sampling rate this is about $f_s{T}_{\mathrm{obs}}\simeq 3143$ samples. These are the usual sampling and spectral-resolution estimates. The $14.3\mathrm{s}$ value is not a universal lower bound on single-tone frequency estimation: fitting a sinusoidal model with sufficiently low noise can estimate its frequency more precisely than the spacing of discrete Fourier transform bins. The question gives no noise level or estimator, so a sharper universal minimum time cannot be deduced.

The consistent matching is:

Series C stops after about six months, identifying machine 2. Its short record gives spectrum W a coarser Fourier frequency resolution. Series A resolves the annual variation but misses the changing daily phase: the regular three-day interval samples the daily oscillation at the same phase, so its daily contribution aliases to a constant. Its spectrum is Y. Machine 3 produces the half-daily-frequency alias in X, leaving D as its time series. The fully sampled two-year record B and spectrum Z agree with the supplied first row.

These identifications use the record lengths, frequency locations and sampling intervals. They do not require assuming that the plotted spectral heights have an identical normalization.