Window Functions

Window functions taper finite signal blocks before FFT processing to prevent spectral leakage — the spreading of energy into neighbouring frequency bins caused by discontinuities at block edges.

The DFT assumes the input is one period of a periodic signal. When the signal doesn’t have an integer number of cycles in the block, the endpoints are mismatched. A window smoothly tapers the signal to zero at the edges, greatly reducing this leakage.

Four window types

Hanning (Hann) — the default choice for FFT analysis.

\[w[n] = 0.5\bigl(1 - \cos(2\pi n / (N-1))\bigr)\]

import pyminidsp as md
win = md.hann_window(256)

Hamming — similar to Hanning but with a lower first sidelobe.

\[w[n] = 0.54 - 0.46\cos(2\pi n / (N-1))\]

win = md.hamming_window(256)

Blackman — strongest sidelobe suppression, widest main lobe.

\[w[n] = 0.42 - 0.5\cos(2\pi n/(N-1)) + 0.08\cos(4\pi n/(N-1))\]

win = md.blackman_window(256)

Rectangular — all ones (no tapering). Narrowest main lobe but maximum sidelobe leakage.

win = md.rect_window(256)

Comparison

Window	Edge values	Sidelobe level	Main lobe width
Rectangular	1.0	Highest	Narrowest
Hanning	0.0	Low	Medium
Hamming	0.08	Lower first sidelobe	Medium
Blackman	0.0	Lowest	Widest

Rule of thumb: start with Hanning. Use Blackman when minimising leakage matters more than frequency resolution.