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From a signal to a spectrum
A signal is a sequence of values over time, such as a measured vibration. A spectrum shows which frequency components it contains. Fourier decomposition describes the signal using oscillating building blocks.
Amplitude describes the strength of a component. Phase describes its position within an oscillation cycle. A power spectrum alone therefore does not contain everything needed to reconstruct the original time series.
Four methods, the same input
| Method | What it does | What to watch for |
|---|---|---|
| Identity | Leaves the input unchanged. | A reference for the unfiltered condition. |
| Moving average | Averages neighbouring samples. | Can weaken short impulses and amplitude. |
| FFT low-pass | Removes components above a cutoff frequency. | May remove useful high frequencies; boundary artefacts are possible. |
| X6 phase filter | Selects Fourier components by their angle. | Is not an ordinary low-pass filter and can remove a clean target signal. |
These comparisons operate on complete stored signals. The centred moving average also uses later samples, so this implementation is not a directly causal real-time filter. A good offline plot does not establish suitability for fast machine control.
Which numbers help?
| Metric | Plain-language meaning |
|---|---|
| RMSE | A typical error on the signal’s scale; usually smaller is better for the same target. |
| NRMSE | Error relative to the specified reference norm; the precise definition matters. |
| PSD | How signal power is distributed across frequencies. |
| Coherence | Frequency-dependent linear agreement; not a reconstruction guarantee by itself. |
| Phase and delay | Whether oscillations line up and whether an output arrives earlier or later. |
A useful counterexample is y = −x: the curve is inverted. Its power spectrum is unchanged, and magnitude-squared coherence can be one. The original signal is still not reconstructed correctly. This is why X6 checks several criteria together.
Make the experiment informative
- Initially change one parameter at a time.
- Compare every method against the same known reference.
- Check short events and phase alignment as well as noise.
- Keep the seed, settings, raw data and evidence report together.
- Use new tests before transferring a synthetic result to real sensors.