Technology / Technical guide
FFT Frequency Resolution: Record Length & Zero Padding
Frequency-bin spacing is a grid property. The ability to distinguish nearby components also depends on the observed duration, window, amplitudes, noise and model. Zero padding provides a finer set of spectral points from the same observations, so it should not be described as a longer measured record.
More plotted points are not more observed information.
Keep sample count, transform length and resolving assumptions as separate quantities.
01 / Measured duration
The retained observations define the time boundary.
For N uniformly spaced samples with interval Ts, the DFT record period is N Ts and sample rate is fs = 1/Ts.
For N uniformly spaced samples with interval Ts, the DFT record period is N Ts and sample rate is fs = 1/Ts. The timestamp difference from first to last sample is (N−1)Ts, a related but different interval.
Stating the period convention prevents an off-by-one duration from entering a frequency calculation. Gaps or nonuniform sampling also require their own interpretation; an arbitrary list of N timestamps is not automatically the uniform sequence assumed by this relation.
02 / Bin grid
Unpadded spacing follows fs/N.
An N-point DFT of those N samples has bin spacing Δf = fs/N.
An N-point DFT of those N samples has bin spacing Δf = fs/N. A hypothetical 128-sample sequence at 32 samples per second has a four-second DFT period and 0.25 Hz spacing.
The bin coordinate identifies where the transform is evaluated. It does not assert that physical frequencies must be exact multiples of 0.25 Hz, or that every pair separated by that amount will produce two unambiguous peaks.
03 / Padding example
Added zeros refine the same spectral representation.
Padding that invented record to a 512-point transform gives grid spacing 32/512 = 0.0625 Hz.
Padding that invented record to a 512-point transform gives grid spacing 32/512 = 0.0625 Hz. The measured sequence still contains 128 observations over the same four-second DFT period. No additional physical time was observed.
MathWorks explains padding as spectral interpolation rather than improved component resolution. More points can depict the existing peak shape more closely, but they cannot reconstruct an unobserved change occurring after the original record ended.
04 / Separation
Peak width is not determined by grid spacing alone.
A finite window spreads a component across a frequency range.
A finite window spreads a component across a frequency range. Two components can have overlapping lobes, and a stronger one can obscure a weaker one. A finer grid samples that overlap more densely without necessarily separating its causes.
The window guide covers this shaping mechanism. There is no universal claim that a particular Δf guarantees discrimination for every amplitude ratio, noise level, observation history or analysis method.
05 / Frequency estimate
An estimate need not be restricted to a bin centre.
A mathematical estimator with additional signal assumptions may infer an off-bin frequency.
A mathematical estimator with additional signal assumptions may infer an off-bin frequency. Its uncertainty depends on the model and evidence. That is different from declaring a narrow displayed grid to be the uncertainty of a physical measurement.
Likewise, transform length does not automatically set the amplitude denominator. An unchanged finite tone contributes the same underlying measured samples after padding. Using the padded count as though it were additional signal observations changes a normalisation assumption.
06 / Scope
Duration and stationarity answer different questions.
A longer actual record can contain more cycles, but it can also combine conditions that changed during observation.
A longer actual record can contain more cycles, but it can also combine conditions that changed during observation. Better nominal spacing does not prove that one stationary spectrum accurately represents the whole interval. Time variation remains part of the signal meaning.
No MARSTON spectral resolution or frequency-estimation accuracy is asserted. The coherence guide examines relationships between defined records; this page supplies no acquisition settings or field analysis procedure.
Related technical guides: FFT Windows & Spectral Leakage: Finite Records · Power Spectral Density: Units, Integration & RMS · Sampling Jitter: Timing Error, Signal Slope & SNR.
General background: MathWorks: DFT spacing and interpolation by zero padding. 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 |
|---|---|---|
| N | Count of measured uniform samples | Different from padded transform length. |
| N Ts | DFT record period, seconds | Not the first-to-last timestamp difference. |
| fs/N | Unpadded bin spacing, Hz | A grid quantity, not guaranteed discrimination. |
| fs/M | Spacing after padding to M points | The measured duration remains unchanged. |
| Frequency estimate | Model-dependent inference | Uncertainty is not simply a displayed bin interval. |
Engineering questions.
Does padding create a longer observed record?
No. The extra entries are zeros, not new measurements.
Must a physical tone lie on a DFT bin?
No. The grid describes transform evaluation points.
Does 0.0625 Hz spacing guarantee separated tones?
No. Window, amplitudes, noise and model also matter.
Can a frequency estimate lie between bins?
Yes, under its additional signal assumptions and uncertainty.
Are N Ts and (N−1)Ts identical?
No. They describe different interval conventions.
Does this page specify actual sample rates?
No. All numerical record values are hypothetical.
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