The FFT and analysis windows

The FFT turns a block of samples into a spectrum, but the result depends as much on the signal as on the window used to analyse it. Understanding spectral leakage and windows is the difference between reading a spectrum and misreading it. This tool shows the effect of windows on a test signal; this page explains what happens and why.

The FFT and resolution

The FFT computes the Discrete Fourier Transform of N samples and returns N/2+1 frequency bins equally spaced between 0 and Fs/2 (Nyquist). The bin spacing — the frequency resolution — depends only on Fs and N:

$$\Delta f=\frac{f_s}{N}X[k]=\sum _{n=0}^{N-1}x[n]e^{-j2\pi kn/N}$$

At a given Fs, more samples (large N) give finer bins and therefore the ability to tell apart closer lines. But N samples at rate Fs are N/Fs seconds of observation: resolution is bought with measurement time. There is no shortcut — it is a direct consequence of the transform, not a limitation of the algorithm.

Spectral leakage

The FFT treats the N samples as one period repeating forever. If a sinusoid completes an integer number of cycles in the window, the repetition is seamless and the energy lands in a single bin. But if it does not — the normal case — a discontinuity appears at the edges of the block, and that discontinuity has spectral content at all frequencies: the line’s energy spreads onto neighbouring bins. This is spectral leakage.

The practical consequence is serious: a strong line produces skirts that can bury a weak nearby line, even when their frequency spacing is well beyond the resolution Δf. Leakage is not noise — it is the signal itself, poorly localised.

Windows and their trade-offs

An analysis window multiplies the N samples by a function that tapers smoothly to zero at the edges, cancelling the edge discontinuity. The price is an unavoidable compromise, fully described in the field’s reference work — F. J. Harris, “On the Use of Windows for Harmonic Analysis with the Discrete Fourier Transform” (Proc. IEEE, 1978) — which tabulates and compares dozens of windows.

A window’s spectrum has two parts: a main lobe (the line, widened) and the side-lobes (the residual skirts). The two trade off:

• a window with a narrow lobe resolves close lines but has high side-lobes → more leakage;
• a window with low side-lobes suppresses leakage but has a wide lobe → worse resolution.

The classic windows are sums of cosines; each extra term lowers the side-lobes at the cost of the lobe:

$$w[n]=\sum _{k=0}^K(-1{)}^k{a}_k\cos \left(\frac{2\pi kn}{N}\right)$$

• Rectangular (no window) — the narrowest lobe of all, but side-lobes at only −13 dB.
• Hann / Hamming — one cosine term: side-lobes at −31 / −43 dB, moderate lobe. The general compromise.
• Blackman / Blackman-Harris — two/three terms: side-lobes down to −92 dB, a markedly wider lobe.
• Flat-top — five coefficients (four cosine terms), optimised for amplitude: a very wide lobe, but almost no scalloping. Note a little-known property: this window is not non-negative. With the SRS/Matlab coefficients used here the minimum is −0.0706 and the first sample is −4.2·10⁻⁴, not zero — so the general description “goes smoothly to zero at the edges”, true for all the others, has to be corrected for the flat-top to “crosses zero and goes negative”.

ENBW and scalloping

Two metrics summarise a window’s behaviour beyond the lobe shape.

The Equivalent Noise Bandwidth (ENBW) is the width, in bins, of the ideal rectangular filter that passes the same noise power as the window. It measures how much noise each bin collects: high ENBW = worse signal-to-noise ratio for a line. It depends on the coefficients alone:

$$\text{ENBW}=N\cdot \frac{{\sum }_n{w}_n^2}{{\left({\sum }_n{w}_n\right)}^2}$$

The rectangular window has ENBW = 1 bin (the minimum possible); softer windows rise (Hann ≈ 1.5, flat-top ≈ 3.8). It is the equivalent-noise-bandwidth price of suppressing leakage.

The scalloping loss is instead the worst-case amplitude error: a line falling exactly halfway between two bins is read attenuated (up to −3.9 dB for the rectangular). The flat-top is designed precisely to cancel it (scalloping ≈ 0 dB): that is why it is the window for calibrated amplitude measurement, even though its wide lobe makes it unsuited to resolving close lines.

The windows compared

The lobe columns are the total width at the −3 and −6 dB levels, in bins: it is the direct measure of resolution, and it runs from the rectangular’s 0.88 bin to the flat-top’s 3.72. The rest of the table reads as the announced trade-off — lower side-lobes paid for with growing ENBW and lobe — with one glaring exception: Hamming has lower side-lobes and lower ENBW and a narrower lobe than Hann. It dominates on every metric in the table. So why is Hann half the world’s default?

The Hann/Hamming paradox: roll-off is what counts

Because the first side-lobe is not the whole story. What matters, when a weak line has to be seen, is the leakage level at the distance where that line sits, and the two windows behave in opposite ways as you move away:

Hann starts worse (−31.5 dB) but falls by 17.8 dB per octave; Hamming starts better (−43.6) and falls by only 5.9. The crossover happens very early: by 10 bins away Hann is already 24 dB better (−71.1 against −47.4), and at 200 bins the gap is 76 dB. The reason is in the coefficients: Hann goes exactly to zero at the edges with zero derivative, Hamming stops at 0.08 and that residual step generates a tail decaying as 1/f.

So the correct criterion is not “which has the lowest first side-lobe” but “how far away is the line I want to see”: Hamming only wins in the immediate neighbourhood, Hann everywhere else in the spectrum. Blackman-Harris is a third case: an excellent first side-lobe (−92 dB) but almost no slope (0.9 dB/octave), because its four coefficients are optimized to minimize the peak side-lobe, not to make them decay. It is the right choice when a uniform floor is needed, not when depth far from the line is.

The flat-top remains the extreme case for amplitude: scalloping measured at −0.01 dB against the rectangular’s −3.92, at the price of a 3.72-bin lobe that resolves nothing nearby.

Periodic or symmetric: which convention

There is a detail from which differences between tools arise. An N-point window can be built with denominator N or N − 1:

• periodic (denominator N) — the window is one exact period of a periodic function: w[0] = 0 but w[N−1] ≠ 0. This is the right convention for FFT analysis, because the DFT assumes precisely that the block repeats;
• symmetric (denominator N − 1) — the window is symmetric about its centre: w[0] = w[N−1] = 0. This is the right convention for FIR filter design, where the symmetry guarantees linear phase.

On a Hann at N = 16 the difference is visible by eye: the last sample is 0.0381 in the periodic version and 0 in the symmetric one. This tool uses periodic, consistent with its purpose. How much the choice weighs on the metrics:

At N = 1024 the ENBW difference is a tenth of a percentage point: irrelevant. At N = 8 it is 14 %, that is decisive. The practical rule: for FFT blocks of normal size the two conventions are interchangeable, for short windows (filter banks, small-block analysis, precomputed tables on firmware) they are not — and that is the case where two different tools disagree and an afternoon goes into looking for the mistake somewhere else.

When to use which window

• Synchronous / transient signal (integer number of cycles, or an impulse decaying within the window): rectangular — no leakage to correct, maximum resolution.
• Generic analysis: Hann or Hamming — the sensible default, contained leakage without giving up too much resolution.
• Weak line next to a strong one (high dynamic range): Blackman-Harris — very low side-lobes so the weak line is not lost under the strong one’s skirts.
• Precise amplitude measurement of a line: flat-top — zero scalloping, the read amplitude is correct regardless of where the line falls relative to the bins.

Limitations

• Radix-2 FFT: N must be a power of 2.
• Synthetic test signal (sinusoids + seeded Gaussian noise), not a real acquisition.
• Single-sided spectrum calibrated in peak amplitude (coherent-gain corrected); it is not a power spectral density estimate.
• Demonstrative tool, not a measurement-grade spectrum analyser.

References

• F. J. Harris, “On the Use of Windows for Harmonic Analysis with the Discrete Fourier Transform”, Proceedings of the IEEE 66(1), 1978 — the classic reference on windows, ENBW, scalloping and side-lobes, with the comparative tables still used today.
• Related tool — the FFT / Window Explorer puts this page into practice: compose a signal, pick the window and read the spectrum and metrics.