Technology / Technical guide
Power Spectral Density: Units, Integration & RMS
A spectrum’s vertical axis determines what its peaks mean. Amplitude, squared magnitude and power spectral density describe different quantities, even when plotted against the same frequency. For an engineering signal, the word power in PSD does not automatically mean electrical output in watts.
A density must be interpreted through a frequency interval.
Follow the input units into the spectral units and integration before comparing levels or linking them to RMS.
01 / Different axes
Amplitude and density are not the same ordinate.
An amplitude spectrum has a convention based on the original signal unit.
An amplitude spectrum has a convention based on the original signal unit. A squared spectrum uses squared units. A PSD expressed per hertz uses squared signal units divided by hertz; MathWorks states this distinction for its spectral estimates.
For displacement, that density could have units m²/Hz. For voltage it can be V²/Hz, without being W/Hz. Converting a voltage statistic to resistive power also needs the applicable resistance and physical boundary, rather than a label change.
02 / Frequency integral
The density contributes mean-square value over a band.
Under the appropriate stationary-statistics and normalisation convention, integrating a PSD over its frequency domain gives the represented mean-square quantity.
Under the appropriate stationary-statistics and normalisation convention, integrating a PSD over its frequency domain gives the represented mean-square quantity. For a zero-mean signal this corresponds to variance. A retained mean or DC contribution needs its own stated treatment.
A peak density is therefore not itself an RMS amplitude. Its frequency width matters to the integral. The RMS guide explains the whole-interval statistic; PSD describes how a squared quantity is distributed over frequency.
03 / Band illustration
A flat invented density has a simple area.
Hypothetical one-sided PSD: suppose a zero-mean signal has density 2 units²/Hz throughout 5–8 Hz and zero elsewhere.
Hypothetical one-sided PSD: suppose a zero-mean signal has density 2 units²/Hz throughout 5–8 Hz and zero elsewhere. Its integrated mean-square value is 2×3 = 6 units², giving RMS √6, approximately 2.45 units.
The interval width is three hertz, not the number of displayed points inside it. These invented values describe a mathematical density, not a measured vibration level, an equipment limit or electrical generation.
04 / Sidedness
Combining positive and negative frequencies changes levels.
For a real signal, a two-sided representation has paired positive and negative frequency contributions.
For a real signal, a two-sided representation has paired positive and negative frequency contributions. A one-sided representation combines those contributions at positive interior frequencies while preserving the corresponding integral.
DC and, in a discrete spectrum where present, the Nyquist endpoint require their own handling. Uniformly doubling every ordinate changes the result. Frequency expressed in radians rather than cycles also requires the matching density convention to retain the same integral.
05 / Estimated density
Finite records add bandwidth and variability.
A finite window gives a tone a represented width, so its peak PSD can change with the window and observation length even when its integrated contribution is comparable.
A finite window gives a tone a represented width, so its peak PSD can change with the window and observation length even when its integrated contribution is comparable. A narrower plotted feature is not automatically a smaller signal mean-square value.
Welch estimation averages segment spectra to reduce estimate variability, with segment duration affecting the frequency representation. Averaging does not manufacture new independent physical observations. The resolution guide separates record information from a denser displayed grid.
06 / Interpretation
A spectral feature is not a unique mechanical cause.
Sensor response, coordinate direction and represented time can influence the spectrum.
Sensor response, coordinate direction and represented time can influence the spectrum. A peak can have several possible physical or processing causes. The coherence guide addresses a different relationship between two signals, without proving a cause from shared frequency.
No MARSTON PSD, vibration threshold or power-quality result is assigned. This explanation identifies units and mathematical boundaries without selecting acquisition parameters, performing measurements or prescribing a diagnosis.
Related technical guides: FFT Windows & Spectral Leakage: Finite Records · FFT Frequency Resolution: Record Length & Zero Padding · Frequency Coherence: Shared Content & Evidence Limits.
Extend the vibration-signal quantity review with vibration displacement, velocity and acceleration and vibration envelopes and sidebands.
General background: MathWorks: PSD units, sidedness and segment-averaged estimation. These references explain industry concepts; they are separate from the applicable MARSTON product and project documents.
Key concepts and conditions.
Read each quantity with its reference, units and stated conditions. A general relation does not establish a model-specific value.
| Concept / quantity | Meaning / relation | Conditions / limits |
|---|---|---|
| Signal unit | Original measured quantity | Establish it before interpreting power terminology. |
| PSD | Squared signal unit/Hz | A density, not amplitude per bin. |
| Integrated PSD | Represented mean-square quantity | Frequency range and normalisation must match. |
| One-sided spectrum | Combined real-signal frequency contributions | Special endpoints are not doubled uniformly. |
| PSD estimate | Finite-record statistical representation | Window and segmentation affect its meaning. |
Engineering questions.
Is voltage PSD automatically W/Hz?
No. A physical resistance and boundary are needed for that conversion.
Is the highest PSD ordinate an RMS value?
No. Integration over frequency is a separate operation.
Does more frequency points increase the band area?
Not when a consistent density and integration convention are retained.
Is variance always the same as total mean-square?
Only for a zero-mean signal; otherwise mean treatment matters.
Can a taller narrow line represent similar total magnitude?
Yes. Peak density and integrated contribution are different quantities.
Is √6 a product vibration result?
No. It follows from an invented flat density.
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