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Window Function Coefficient Calculator

Choose a window type and length to get its coefficient weights.

How it is calculated

For a window of length N (2 to 256; a decimal is cut to a whole number, and anything outside the range gives no result) the tool uses the symmetric form with N − 1 in the denominator. With i = 0 … N−1: Hann w = 0.5 − 0.5 cos(2πi/(N−1)), Hamming w = 0.54 − 0.46 cos(2πi/(N−1)), Blackman w = 0.42 − 0.5 cos(2πi/(N−1)) + 0.08 cos(4πi/(N−1)), rectangular w = 1. Coherent gain is Σw ÷ N, and equivalent noise bandwidth is N × Σw² ÷ (Σw)² in FFT bins. Only the first 8 coefficients are listed, followed by a count of the rest. A window name the tool does not know falls back to Hann without any warning. Values show four decimals.

Worked example

Choose Hamming with N = 8. The tool lists w[0] to w[7]: 0.0800, 0.2532, 0.6424, 0.9544, then the same four values mirrored (0.9544, 0.6424, 0.2532, 0.0800). Their sum is 3.86, so the coherent gain is 3.86 ÷ 8 = 0.4825. The equivalent noise bandwidth works out to 1.4971 bins. For comparison, a rectangular window of the same length gives gain 1.0000 and 1.0000 bins, so Hamming gives up some amplitude and widens the noise bandwidth in exchange for lower spectral leakage.

Try this example →

Common mistakes

  • Using these symmetric coefficients as a periodic window for spectral analysis. Many FFT tools put N rather than N − 1 in the denominator, so the values differ slightly, and for analysis windows that difference can matter.
  • Expecting the full list for large N. Only the first 8 weights are printed and the rest are just counted; because the window is symmetric, you can mirror the first half.
  • Forgetting the coherent gain when reading amplitudes. A windowed spectrum is scaled down by that factor (0.4825 here), so peaks must be divided by it to recover amplitude.
  • Misspelling the window name. An unknown name quietly becomes a Hann window instead of showing an error.
  • Entering N = 1 or a length above 256, which gives no result.

About this calculator

When you take a finite chunk of a signal for an FFT, the abrupt cut at each edge creates spectral leakage; window functions taper the signal smoothly toward zero at both ends to reduce that leakage. Common windows have closed-form coefficient formulas — the Hann window is w(n) = 0.5 − 0.5·cos(2πn/(N−1)), the Hamming window is w(n) = 0.54 − 0.46·cos(2πn/(N−1)) — and each window trades off main-lobe width against side-lobe suppression, summarized by its coherent gain and equivalent noise bandwidth (ENBW).

This calculator takes a window type and length and returns the per-sample coefficient weights along with coherent gain and ENBW. DSP and audio engineers use these coefficients when implementing FFT-based spectral analysis, filter design, or spectrogram generation, choosing a window (Hann for general-purpose analysis, Hamming for narrower main-lobe needs, Blackman for lower side lobes, etc.) that matches whether they need frequency resolution or amplitude accuracy more.

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