Technology / Technical guide
FFT Windows & Spectral Leakage: Finite Records
An FFT describes a finite sequence through a discrete frequency representation. Taking a finite record already imposes a window, even when its weights are all one. The window shapes the spectrum of each component, so a broad feature or a neighbouring peak cannot be interpreted without that representation boundary.
The record boundary changes the spectral picture.
Distinguish a real frequency component from the spread produced by finite duration and weighting.
01 / Finite record
Truncation multiplies the represented waveform.
A finite record can be viewed as the underlying waveform multiplied by a time window.
A finite record can be viewed as the underlying waveform multiplied by a time window. Multiplication in time corresponds to convolution in frequency, spreading the representation of a tone according to the window’s spectrum.
The DFT also describes a periodic continuation of the retained sequence. A mismatch across that continuation boundary gives an intuitive view of leakage. It does not imply that the physical source suddenly developed all the frequencies visible in the finite-record transform.
02 / Coherent example
A tone’s relation to the record grid matters.
Hypothetical rectangular record: 200 samples at 100 samples per second define a two-second DFT interval.
Hypothetical rectangular record: 200 samples at 100 samples per second define a two-second DFT interval. A 3 Hz tone completes six cycles in that interval; a 3.2 Hz tone completes 6.4 cycles. Both are invented signal frequencies.
For the ideal bin-centred tone, its discrete representation fits the periodic grid. An off-grid tone spreads across bins even without random noise or a physical change of frequency. This is distinct from the sampling-aliasing question.
03 / Lobe tradeoff
Less distant leakage can mean a broader peak.
A window spectrum has a main lobe and side lobes.
A window spectrum has a main lobe and side lobes. NI explains the tradeoff between suppressing leakage through smaller side lobes and broadening the main lobe. A strong component’s side lobes can obscure a weaker nearby component.
The width and height describe different properties. A cleaner-looking background does not prove that two closely spaced components are now separable. No one named window simultaneously eliminates every leakage, width and amplitude interpretation problem.
04 / Amplitude scale
Weighting changes coherent tone amplitude.
For an ideal complex tone exactly on the DFT grid, multiplying by weights w[n] gives a matched-bin contribution proportional to A times the sum of those weights.
For an ideal complex tone exactly on the DFT grid, multiplying by weights w[n] gives a matched-bin contribution proportional to A times the sum of those weights. Dividing by the unweighted sample count would therefore retain the window’s coherent-amplitude factor.
That algebra assumes the stated complex tone and transform convention. A real off-grid waveform can involve overlapping positive/negative contributions and leakage. A single correction factor is not a universal recovery of every original peak or transient amplitude.
05 / Noise scale
Squared weighting enters a different quantity.
For independent equal-variance sample errors, the variance of their weighted sum depends on the sum of squared weights.
For independent equal-variance sample errors, the variance of their weighted sum depends on the sum of squared weights. Coherent tone amplitude instead involves the square of the weight sum when converted to power. These normalisations are not interchangeable.
The PSD guide follows squared quantities per frequency. Applying a tone-amplitude convention to broadband noise can change the reported meaning even when both plots use the same FFT points and window name.
06 / Scope
Windowing does not add missing observations.
A tapered record reduces the influence of its edges.
A tapered record reduces the influence of its edges. It cannot restore information omitted before sampling or reveal an event outside the retained interval. Zero padding changes the displayed frequency grid, while windowing changes the weighted record itself.
No MARSTON waveform, window implementation or spectral fault signature is provided. These ideal concepts explain representation without selecting field acquisition settings, performing vibration tests or changing sensor processing.
Related technical guides: FFT Frequency Resolution: Record Length & Zero Padding · Power Spectral Density: Units, Integration & RMS · Frequency Coherence: Shared Content & Evidence Limits.
General background: NI: Finite-record leakage and window-lobe tradeoffs. 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 |
|---|---|---|
| Record window | Weights multiplying finite observations | Even uniform weights define a window. |
| Main lobe | Central spread around a component | Width affects close-component interpretation. |
| Side lobes | Additional spectral spread | A strong component can obscure another. |
| Sum of weights | Coherent-tone amplitude scale | Depends on the stated transform convention. |
| Sum of squared weights | Independent-error variance scale | Different from coherent amplitude normalisation. |
Engineering questions.
Does leakage prove random measurement noise?
No. A finite off-grid tone can produce it.
Does no selected taper mean no window?
No. A finite uniformly weighted record is rectangular windowing.
Do lower side lobes always mean narrower peaks?
No. Main-lobe width can increase.
Can an amplitude correction restore every transient?
No. Its assumptions and retained interval constrain it.
Does windowing resolve aliasing?
No. Prior sampling ambiguity is a separate mechanism.
Are the two-second tones real equipment records?
No. They are hypothetical finite-sequence examples.
Technical enquiry
Discuss the technical question.
An FFT describes a finite sequence through a discrete frequency representation. Share the model and applicable operating or design information for an engineering discussion.