Comparing two FFT spectra with different frame length or window compares apples and oranges. The power spectral density (PSD) solves this: it normalises spectral energy to 1 Hz bandwidth, making measurements under different conditions comparable.

From energy spectrum to power density

Sxx(k) = (1 / (fs · N · U)) · |X(k)|2 (0 < k < N/2)
U = (1 / N) · Σn=0N−1 w(n)2

Unit: (signal unit)² / Hz, e.g. (m/s²)²/Hz for acceleration. PSD is an energy density – integrated over a band you get energy.

Parseval's theorem

Σn=0N−1 |x(n)|2 = (1 / N) · Σk=0N−1 |X(k)|2

Energy is conserved – it redistributes but does not disappear. PSD is honest: integration in frequency equals total energy in time.

Welch's method

A single FFT is a noisy estimate. Averaging FFTs over overlapping frames reduces estimation noise – Welch (1967):

  1. Split signal into M overlapping frames of length N (typically 50 % overlap).
  2. Window and FFT each frame.
  3. Square magnitudes.
  4. Average over all M frames.

PSD vs. amplitude – when?

ViewWhen to use
Amplitude spectrumquantify discrete tonal components (orders)
Energy spectrumenergy per bin (rare, intermediate step)
PSDbroadband stochastic signals (bearings, flow, friction)

Application: bearing monitoring

A healthy bearing produces broadband background with a characteristic PSD shape. Incipient damage typically lifts PSD energy around cage frequency or roller pass frequency – as a band-wide rise, not a sharp line. PSD is the right tool.