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Characteristics of different window functions

Short answer

Windows trade off main-lobe width (transition width) and side-lobe level (ripple/attenuation). Rectangular: 4π/M width, −13 dB side lobe, 21 dB attenuation. Bartlett, Hanning, Hamming, Blackman provide increasing attenuation (25, 44, 53, 74 dB) and wider main lobes (8π/M or 12π/M). All are symmetric for linear phase. Kaiser allows adjustable β for any attenuation.

Discuss the characteristics of different types of window functions.

A window w(n) of length M truncates the ideal impulse response. In frequency the result is the ideal response convolved with W(e^jω), so two features of W matter: the main-lobe width (sets the transition width) and the side-lobe level (sets the ripple/attenuation).

Windoww(n), 0 ≤ n ≤ M−1Main-lobe widthPeak side lobeMin. stop-band attenuation
Rectangular14π/M−13 dB21 dB
Bartlett (triangular)1 − 2|n − α|/(M−1)8π/M−25 dB25 dB
Hanning0.5 − 0.5 cos(2πn/(M−1))8π/M−31 dB44 dB
Hamming0.54 − 0.46 cos(2πn/(M−1))8π/M−41 dB53 dB
Blackman0.42 − 0.5 cos(2πn/(M−1)) + 0.08 cos(4πn/(M−1))12π/M−57 dB74 dB
KaiserI₀(β√(1−((n−α)/α)²)) / I₀(β)depends on βadjustableadjustable

(α = (M−1)/2 in all windows.)

General characteristics

  • All are symmetric about α, so the filter has linear phase.
  • Tapered windows (Bartlett, Hanning, Hamming, Blackman) fall smoothly towards the ends, which lowers the side lobes (less Gibbs ripple) but widens the main lobe (wider transition band). The rectangular window is the opposite.
  • The ripple in the pass band and stop band is almost equal and set by the window, not by M. Increasing M only narrows the transition band.
  • Hamming has the lowest peak side lobe among the 8π/M windows; Blackman has the best attenuation but needs the longest filter.
  • The Kaiser window gives continuous control of the trade-off through β.

✓ Answer checked by Claude Opus 5.5 (high effort)

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