Cut a piece and join its ends
Picture a continuous sine wave, rising and falling like a vibrating string. Record just one second of it. An FFT expresses that finite piece as a sum of repeating oscillations. Each oscillation joins smoothly to itself at the record boundary, so the calculation treats the recorded piece as if it repeated.
A cut after a whole number of cycles can join neatly. A cut at an awkward point may join a rising voltage to a very different value. Repeating that piece creates a jump. Describing the jump requires additional frequencies, and small peaks appear beside the strong one. This spread into other frequencies is called spectral leakage.
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Gentler ends reduce distant sidelobes
Multiply each recorded sample by a weight. Keep more of the middle and less of each end. The shape of these weights is a window function. A Hann window resembles a smooth hill, gradually reducing the voltage near the join. A rectangular window gives every sample the same weight and retains the abrupt cut.
Hann reduces distant sidelobes substantially. A weak tone hidden beneath leakage from a strong one may then emerge. Tektronix’s FFT application note illustrates the trade with window-response diagrams: the Hann response concentrates more energy near the main peak, whose central peak becomes wider than with a rectangular window.
If the weak tone is very close, that wider lobe can cover both signals. Window choice therefore depends on the problem: widely unequal tones some distance apart, or two similarly sized tones very close together. Comparing windows on the same samples helps reveal which features move with the processing.
A longer observation records a finer distinction
Extend the observation from one second to two and the instrument receives more real motion of the voltage. Two slightly different tones have longer to slip out of step, making their difference easier to record. The unpadded frequency interval also shrinks from one hertz to half a hertz.
Appending zeros adds no observed motion. It evaluates the same recording on a denser frequency grid, a process called zero padding. The broad peak keeps its width. Adding more plotted points cannot give an unresolved pair new physical resolution.
The original frequency interval is not a universal threshold for separating any pair of peaks either. The window and the relative signal sizes matter. Keeping the actual observation time and window with a result is more useful than reporting the number of plotted points alone.
Noise height depends on the bandwidth behind a point
White noise spreads voltage fluctuations over a broad frequency range. Observe for longer and each original spectral point collects noise from a narrower band. Its noise floor falls when displayed as power per frequency point. To compare different observation times, express mean-square voltage per hertz: power spectral density. For the same white noise, its average density stays approximately constant.
The window sets how much surrounding noise contributes to a point. Equivalent noise bandwidth compares its response with an ideal rectangular frequency band that passes the same white-noise power. A Hann window applied to one second of data has a noise bandwidth of about 1.5 hertz. FFT and swept superheterodyne analyzers can use different definitions of their reported resolution bandwidth, so the definition belongs with the number.
A real measurement first needs a voltage scale and a clock. Separate known voltage levels check scale and offset; an independent time reference checks sample rate. Integrating the calibrated density across a selected band then gives its mean-square voltage. Leakage from a strong tone, a weak signal and noise all contribute when they lie inside that band.