FFT / Auto Spectrum

The FFT (Fast Fourier Transform) analysis converts time-domain signals into the frequency domain, revealing the spectral content of acoustic and vibration measurements. The Auto Spectrum is the magnitude-squared representation of the FFT, providing a power-proportional view of each frequency component.

Theory

Discrete Fourier Transform (DFT)

The DFT decomposes a finite-length time signal into a sum of complex sinusoids at discrete frequency bins. For a block of N samples, the transform yields N/2 + 1 unique spectral lines (single-sided spectrum) with a frequency resolution of:

Δf = f_s / N

where f_s is the sampling rate and N is the block size. Increasing the block size improves frequency resolution but reduces time resolution.

Windowing

Because the DFT assumes a periodic signal, applying it to a finite block introduces spectral leakage. Window functions taper the signal edges to reduce this effect. Each window represents a trade-off between main-lobe width (frequency resolution) and side-lobe level (leakage suppression).

Window	Main Lobe Width	Side Lobe Level	Typical Use
Rectangular (None)	Narrowest	−13 dB	Transient signals fully contained in block
Hanning	Moderate	−31 dB	General-purpose continuous signals
Hamming	Moderate	−43 dB	Slightly better leakage than Hanning
Flat Top	Wide	−93 dB	Amplitude-accurate calibration measurements
Kaiser-Bessel	Adjustable	Adjustable	Parametric control via β

The Hanning window is the default and recommended choice for most acoustic and vibration analyses. Use Flat Top only when precise amplitude accuracy at known frequencies is required.

Averaging

Averaging multiple FFT blocks reduces the variance of the spectral estimate. V-Listen supports the following averaging modes:

Linear (RMS): Averages the power spectra of all blocks equally. The variance decreases as 1/n, where n is the number of averages.
Exponential: Applies an exponential weighting so that newer blocks contribute more heavily, useful for slowly varying signals.
Peak Hold: Retains the maximum value at each frequency bin across all blocks.

Parameters

Parameter	Range / Options	Description
Spectrum Size (N)	256 – 65536	Number of samples per FFT block. Determines frequency resolution.
Overlap	0% – 95%	Percentage of overlap between consecutive blocks. 50% or 66.7% is typical for Hanning.
Window Type	Hanning, Hamming, Flat Top, Rectangular, Kaiser-Bessel	Window function applied to each block.
Averaging Mode	Linear / Exponential / Peak Hold	Method for combining multiple blocks.
Number of Averages	1 – unlimited	Number of blocks to average. Higher values yield smoother spectra.

Output Formats

Magnitude Representations

Linear Magnitude — amplitude in engineering units (Pa, m/s², etc.)
RMS — root-mean-square amplitude per frequency bin
Power (Auto Spectrum) — magnitude squared, proportional to signal power
dB — logarithmic scale referenced to a standard value:
L = 20 · log₁₀(X / X_ref)
where X_ref = 20 μPa for sound pressure and 1 μm/s² for acceleration.

Phase

Phase output displays the angle of each complex frequency bin in degrees (−180° to +180°) or radians. Phase is meaningful only when a coherent reference is available.

Variants

FFT vs Time

Computes successive FFT blocks over the duration of the recording and displays them as a spectrogram (time-frequency map). Color mapping represents amplitude in dB. Useful for identifying transient events and time-varying spectral content.

FFT vs RPM

When a tacho signal is available, the FFT can be computed at defined RPM intervals during a run-up or coast-down. The result is a waterfall diagram with RPM on the horizontal axis and frequency on the vertical axis, allowing identification of speed-dependent resonances.

For RPM-based analysis, ensure that the tacho channel is correctly configured and the trigger level is appropriate for the pulse signal. See Order Spectrum for order-domain analysis.

Practical Tips

Choose a spectrum size that gives adequate frequency resolution for the frequency range of interest. A common starting point is N = 4096 at 48 kHz (giving Δf ≈ 11.7 Hz).
Use at least 50% overlap with a Hanning window to avoid loss of data at the tapered edges.
Increase the number of averages when measuring broadband noise to obtain a smoother estimate.
Compare your FFT result with the PSD when energy density per Hz is important — see Power Spectral Density.

Ensure the input signal does not clip. Clipping introduces harmonic distortion that is indistinguishable from genuine signal content in the spectrum.