If you ever look at an acoustic inspection result, sooner or later you will see a spectrum: a curve showing how strongly each frequency is present in the signal. Behind this curve lies one of the most consequential mathematical discoveries of the 20th century: the Fast Fourier Transform.

From time to frequency

A microphone delivers a voltage sequence over time – a time signal x(t). This view is intuitive but useless for evaluation: which part of the noise comes from gear meshing? Which from the bearings? Which from the cooling fan?

The Fourier transform answers this question. It decomposes any periodic or transient signal into a sum of sinusoids of different frequency. The frequency domain reveals what is hidden in the time domain.

The Discrete Fourier Transform (DFT)

In practice we work with samples, not continuous signals:

X(k) = Σn=0N−1 x(n) · e−j 2π k n / N

In words: for every frequency bin k we sum all samples x(n), weighted by a complex oscillation of frequency k. The result X(k) is a complex number – its magnitude is the amplitude, its argument the phase.

Why the DFT is too slow

Direct computation requires N multiplications per frequency bin, in total O(N²) operations. For a typical NVH frame with N = 16,384 that is ~268 million complex multiplications per spectrum. Inline inspection would be impossible.

The FFT trick: divide and conquer

In 1965 Cooley and Tukey showed that the DFT can be split recursively into two smaller DFTs (for even and odd indices). Complexity drops to O(N · log N) – for N = 16,384 only ~230 thousand operations, a 1000× speed-up.

X(k) = Σm=0N/2−1 x(2m) · W2mk + Wk · Σm=0N/2−1 x(2m+1) · W2mk

with W = e−j 2π / N (twiddle factor)

The data flow is visualised in the famous butterfly diagram – every butterfly pair combines two inputs into two outputs with exactly one complex multiplication.

Spectral resolution and frame length

Δf = fs / N

Example: at fs = 51,200 Hz and N = 16,384 samples, resolution is Δf = 3.125 Hz. Finer resolution requires longer observation – the product of resolution and observation time is physically bounded (Heisenberg-like uncertainty).

Example: e-motor spectrum

An e-drive runs at 3,000 rpm (= 50 Hz rotation). The FFT spectrum typically shows:

  • 50 Hz – fundamental rotation (1st order)
  • 100, 150, 200 Hz – higher orders from imbalance and asymmetry
  • 600 Hz – 12th order, e.g. switching frequency of power electronics
  • 3,000 Hz – stator/rotor mesh frequency
  • Broadband 8–12 kHz – bearing noise, flow components

SonicTC.NVH evaluates exactly this spectrum, part by part. Deviations from the reference – e.g. an elevated 12th order – mark a NOK part.

What to remember

The FFT is no black box: it is a clever computation path that speeds up the DFT by orders of magnitude. It is the prerequisite for every inline inspection with acoustic methods – and has the same expressive power as the DFT, just much faster. In upcoming articles we cover why signals must not be "cut off" (window functions) and when order analysis beats classical FFT.