window-function

Collection of window functions for signal processing and spectral analysis.

npm install window-function

import { hann, kaiser, generate, apply } from 'window-function'

// Every window: fn(i, N, ...params) → number
hann(50, 101)                    // single sample → 1.0

// Generate full window as Float64Array
let w = generate(hann, 1024)

// Apply window to a signal in-place
let signal = new Float64Array(1024).fill(1)
apply(signal, hann)              // signal *= hann

// Parameterized windows pass extra args
generate(kaiser, 1024, 8.6)     // Kaiser with β = 8.6

Or import individual windows directly:

import hann from 'window-function/hann'
import kaiser from 'window-function/kaiser'

Reference

rectangular · triangular · bartlett · welch · connes · hann · hamming · cosine · blackman · exactBlackman · nuttall · blackmanNuttall · blackmanHarris · flatTop · bartlettHann · lanczos · parzen · bohman

kaiser · gaussian · generalizedNormal · tukey · planckTaper · powerOfSine · exponential · hannPoisson · cauchy · rifeVincent · confinedGaussian

dolphChebyshev · taylor · kaiserBesselDerived · dpss · ultraspherical

Simple — no parameters

rectangular(i, N)

$w(n) = 1$

No windowing. Best frequency resolution, worst spectral leakage. Use for transient signals already zero at edges, or harmonic analysis with integer cycles. -13 dB sidelobe · -6 dB/oct rolloff

triangular(i, N)

$w(n) = 1 - \left|\frac{2n - N + 1}{N}\right|$

Linear taper, nonzero endpoints. Simple smoothing, 2nd-order B-spline. -27 dB sidelobe · -12 dB/oct rolloff

bartlett(i, N)

$w(n) = 1 - \left|\frac{2n - N + 1}{N - 1}\right|$

Linear taper, zero endpoints. Bartlett's method PSD estimation.[^bartlett1950] -27 dB sidelobe · -12 dB/oct rolloff

welch(i, N)

$w(n) = 1 - \left(\frac{2n - N + 1}{N - 1}\right)^2$

Parabolic taper. Welch's method PSD estimation.[^welch1967] -21 dB sidelobe · -12 dB/oct rolloff

connes(i, N)

$w(n) = \left[1 - \left(\frac{2n - N + 1}{N - 1}\right)^2\right]^2$

Welch squared (4th power parabolic). FTIR spectroscopy, interferogram apodization.[^connes1961] -24 dB/oct rolloff

hann(i, N)

$w(n) = 0.5 - 0.5\cos!\left(\frac{2\pi n}{N-1}\right)$

Raised cosine, zero endpoints. The default general-purpose choice. STFT with 50% overlap (COLA). Also called "Hanning" (misnomer).[^blackman1958] -32 dB sidelobe · -18 dB/oct rolloff

hamming(i, N)

$w(n) = 0.54 - 0.46\cos!\left(\frac{2\pi n}{N-1}\right)$

Raised cosine, nonzero endpoints. Optimized for first sidelobe cancellation. FIR filter design, speech processing.[^hamming1977] -43 dB sidelobe · -6 dB/oct rolloff

cosine(i, N)

$w(n) = \sin!\left(\frac{\pi n}{N-1}\right)$

Half-period sine. MDCT audio codecs: MP3, AAC, Vorbis.[^princen1987] -23 dB sidelobe · -12 dB/oct rolloff

blackman(i, N)

$w(n) = 0.42 - 0.5\cos!\left(\frac{2\pi n}{N-1}\right) + 0.08\cos!\left(\frac{4\pi n}{N-1}\right)$

3-term cosine sum. Better leakage than Hann at the cost of wider main lobe.[^blackman1958] -58 dB sidelobe · -18 dB/oct rolloff

exactBlackman(i, N)

$w(n) = 0.42659 - 0.49656\cos!\left(\frac{2\pi n}{N-1}\right) + 0.076849\cos!\left(\frac{4\pi n}{N-1}\right)$

Blackman with exact zero placement at 3rd and 4th sidelobes.[^harris1978] -69 dB sidelobe · -6 dB/oct rolloff

nuttall(i, N)

$w(n) = 0.355768 - 0.487396\cos!\left(\frac{2\pi n}{N!-!1}\right) + 0.144232\cos!\left(\frac{4\pi n}{N!-!1}\right) - 0.012604\cos!\left(\frac{6\pi n}{N!-!1}\right)$

4-term cosine sum, continuous 1st derivative. High-dynamic-range analysis without edge discontinuity.[^nuttall1981] -93 dB sidelobe · -18 dB/oct rolloff

blackmanNuttall(i, N)

$w(n) = 0.3635819 - 0.4891775\cos!\left(\frac{2\pi n}{N!-!1}\right) + 0.1365995\cos!\left(\frac{4\pi n}{N!-!1}\right) - 0.0106411\cos!\left(\frac{6\pi n}{N!-!1}\right)$

4-term cosine sum, lowest sidelobes among 4-term windows.[^nuttall1981] -98 dB sidelobe · -6 dB/oct rolloff

blackmanHarris(i, N)

$w(n) = 0.35875 - 0.48829\cos!\left(\frac{2\pi n}{N!-!1}\right) + 0.14128\cos!\left(\frac{4\pi n}{N!-!1}\right) - 0.01168\cos!\left(\frac{6\pi n}{N!-!1}\right)$

4-term minimum sidelobe. ADC testing, measurement instrumentation, >80 dB dynamic range.[^harris1978] -92 dB sidelobe · -6 dB/oct rolloff

flatTop(i, N)

$w(n) = 1 - 1.93\cos!\left(\frac{2\pi n}{N!-!1}\right) + 1.29\cos!\left(\frac{4\pi n}{N!-!1}\right) - 0.388\cos!\left(\frac{6\pi n}{N!-!1}\right) + 0.028\cos!\left(\frac{8\pi n}{N!-!1}\right)$

5-term cosine sum, near-zero scalloping. Peak ~4.64 (by design). Amplitude calibration, transducer calibration (~0.01 dB accuracy). ISO 18431.[^heinzel2002] -93 dB sidelobe · -6 dB/oct rolloff

bartlettHann(i, N)

$w(n) = 0.62 - 0.48\left|\frac{n}{N!-!1} - 0.5\right| - 0.38\cos!\left(\frac{2\pi n}{N-1}\right)$

Bartlett-Hann hybrid. Balanced near/far sidelobe levels.[^ha1989] -36 dB sidelobe

lanczos(i, N)

$w(n) = \text{sinc}!\left(\frac{2n}{N-1} - 1\right)$

Sinc main lobe. Image resampling, interpolation (FFmpeg, ImageMagick).[^duchon1979] -26 dB sidelobe

parzen(i, N)

$w(n) = 1 - 6a^2(1-a)$ for $|a| \le 0.5$,   $w(n) = 2(1-a)^3$ for $|a| > 0.5$,   $a = |(2n-N+1)/(N-1)|$

4th-order B-spline. Always-positive spectrum. Kernel density estimation.[^parzen1961] -53 dB sidelobe · -24 dB/oct rolloff

bohman(i, N)

$w(n) = (1-|a|)\cos(\pi|a|) + \frac{\sin(\pi|a|)}{\pi}$,   $a = \frac{2n-N+1}{N-1}$

Autocorrelation of cosine window. Fast sidelobe decay, spectral estimation. -46 dB sidelobe · -24 dB/oct rolloff

Parameterized — adjustable tradeoff

kaiser(i, N, beta)

beta: shape — 0 = rectangular, 5.4 = Hamming, 8.6 (default) = Blackman.

$w(n) = \frac{I_0!\left(\beta\sqrt{1 - \left(\frac{2n-N+1}{N-1}\right)^2}\right)}{I_0(\beta)}$

Near-optimal DPSS approximation via Bessel I₀. The standard parameterized window for FIR filter design.[^kaiser1974]

gaussian(i, N, sigma)

sigma: width, default 0.4.

$w(n) = \exp!\left[-\frac{1}{2}\left(\frac{2n-N+1}{\sigma(N-1)}\right)^2\right]$

Gaussian bell, minimum time-bandwidth product. STFT/Gabor transform, frequency estimation via parabolic interpolation.[^gabor1946]

generalizedNormal(i, N, sigma, p)

sigma: width (default 0.4), p: shape — 2 = Gaussian, large = rectangular.

$w(n) = \exp!\left[-\frac{1}{2}\left|\frac{2n-N+1}{\sigma(N-1)}\right|^p\right]$

Continuous family between Gaussian and rectangular. Adjustable time-frequency tradeoff.

tukey(i, N, alpha)

alpha: taper fraction — 0 = rectangular, 0.5 (default), 1 = Hann.

$w(n) = \tfrac{1}{2}[1+\cos(\pi(n/(\alpha(N!-!1)/2)-1))]$ in tapered edges,   $w(n) = 1$ in flat center.

Flat center with cosine-tapered edges. Preserves signal amplitude while tapering. Vibration analysis, LIGO.

planckTaper(i, N, epsilon)

epsilon: taper fraction, default 0.1.

C∞-smooth bump function (infinitely differentiable). Gravitational wave analysis (LIGO/Virgo).[^mckechan2010]

powerOfSine(i, N, alpha)

alpha: exponent — 0 = rectangular, 1 = cosine, 2 (default) = Hann.

$w(n) = \sin^\alpha!\left(\frac{\pi n}{N-1}\right)$

$\sin^\alpha$ family. Codec design, parameterized spectral analysis.

exponential(i, N, tau)

tau: time constant, default 1.

$w(n) = \exp!\left(\frac{-|2n-N+1|}{\tau(N-1)}\right)$

Exponential decay from center. Modal analysis, impact testing.[^harris1978]

hannPoisson(i, N, alpha)

alpha: decay, default 2. At α ≥ 2 the transform has no sidelobes.

$w(n) = \frac{1}{2}\left(1-\cos\frac{2\pi n}{N!-!1}\right)\exp!\left(\frac{-\alpha|2n!-!N!+!1|}{N-1}\right)$

Hann × exponential. Unique no-sidelobe property enables frequency estimators using convex optimization.

cauchy(i, N, alpha)

alpha: width, default 3.

$w(n) = \frac{1}{1+\left(\frac{\alpha(2n-N+1)}{N-1}\right)^2}$

Lorentzian shape. Matches spectral line shapes in spectroscopy.[^harris1978]

rifeVincent(i, N, order)

order: 1 (default) = Hann, 2, 3. Throws for other values.

$w(n) = \frac{1}{Z}\sum_{k=0}^{K}(-1)^k a_k\cos\frac{2\pi kn}{N-1}$

Class I cosine-sum optimized for sidelobe fall-off. Power grid harmonic analysis, interpolated DFT.[^rife1970]

confinedGaussian(i, N, sigmaT)

sigmaT: temporal width, default 0.1.

Optimal RMS time-frequency bandwidth. Time-frequency analysis, audio coding.[^starosielec2014]

Array-computed — cached

Compute the full window on first call, cache the result. Recomputed when parameters change.

dolphChebyshev(i, N, dB)

dB: sidelobe attenuation, default 100.

$W(k) = (-1)^k T_{N-1}!\left(\beta\cos\frac{\pi k}{N}\right)$,   $w = \text{IDFT}(W)$

Optimal: narrowest main lobe for given equiripple sidelobe level. Antenna design, radar.[^dolph1946]

taylor(i, N, nbar, sll)

nbar: constant-level sidelobes (default 4), sll: level in dB (default 30).

$w(n) = 1 + 2\sum_{m=1}^{\bar{n}-1} F_m \cos\frac{2\pi m(n-(N!-!1)/2)}{N}$

Monotonically decreasing sidelobes. The radar community standard for SAR image formation.[^taylor1955]

kaiserBesselDerived(i, N, beta)

beta: shape, default 8.6. N must be even.

$w(n) = \sqrt{\frac{\sum_{j=0}^{n} K(j)}{\sum_{j=0}^{N/2} K(j)}}$,   $K(j) = I_0!\left(\beta\sqrt{1-\left(\frac{2j-N/2}{N/2}\right)^2}\right)$

Princen-Bradley condition for perfect MDCT reconstruction. AAC, Vorbis, Opus audio codecs.[^princen1987]

dpss(i, N, W)

W: half-bandwidth [0, 0.5], default 0.1.

$\mathbf{T}\mathbf{v} = \lambda\mathbf{v}$,   $T_{jk} = \frac{\sin 2\pi W(j-k)}{\pi(j-k)}$

Dominant eigenvector of sinc Toeplitz matrix — provably optimal energy concentration. Also called Slepian window. Multitaper spectral estimation, neuroscience, climate science.[^slepian1978]

ultraspherical(i, N, mu, xmu)

mu: 0 = Dolph-Chebyshev, 1 (default) = Saramaki. xmu: sidelobe control (default 1).

$W(k) = C_n^\mu!\left(x_\mu\cos\frac{\pi k}{N}\right)$,   $w = \text{IDFT}(W)$

Gegenbauer polynomial window. Independent control of sidelobe level and taper rate. Antenna design, beamforming.[^streit1984]

Choosing a window

Every window trades frequency resolution (narrow main lobe), spectral leakage (low sidelobes), and amplitude accuracy (flat top). No single window wins all three.

Metrics

Three functions for quantitative window comparison:

import { hann, enbw, scallopLoss, cola } from 'window-function'

enbw(hann, 1024)                 // 1.5 — noise bandwidth (bins)
scallopLoss(hann, 1024)          // 1.42 — worst-case amplitude error (dB)
cola(hann, 1024, 512)            // 0 — perfect STFT reconstruction

• enbw(fn, N, ...params) — equivalent noise bandwidth in frequency bins. Rectangular = 1.0, Hann = 1.5, Blackman-Harris = 2.0. Lower = less noise.
• scallopLoss(fn, N, ...params) — worst-case amplitude error in dB between DFT bins. Rectangular = 3.92, Hann = 1.42, flat-top ≈ 0.
• cola(fn, N, hop, ...params) — COLA deviation. 0 = perfect STFT reconstruction at given hop size.

[^dolph1946]: C.L. Dolph, "A Current Distribution for Broadside Arrays," Proc. IRE 34, 1946. [^gabor1946]: D. Gabor, "Theory of Communication," J. IEE 93, 1946. [^bartlett1950]: M.S. Bartlett, "Periodogram Analysis and Continuous Spectra," Biometrika 37, 1950. [^taylor1955]: T.T. Taylor, "Design of Line-Source Antennas," IRE Trans. Antennas Propag. AP-4, 1955. [^blackman1958]: R.B. Blackman & J.W. Tukey, The Measurement of Power Spectra, Dover, 1958. [^connes1961]: J. Connes, "Recherches sur la spectroscopie par transformation de Fourier," Revue d'Optique 40, 1961. [^parzen1961]: E. Parzen, "Mathematical Considerations in the Estimation of Spectra," Technometrics 3, 1961. [^welch1967]: P.D. Welch, "The Use of FFT for Estimation of Power Spectra," IEEE Trans. Audio Electroacoustics AU-15, 1967. [^rife1970]: D.C. Rife & G.A. Vincent, "Use of the DFT in Measurement of Frequencies and Levels of Tones," Bell Syst. Tech. J. 49, 1970. [^kaiser1974]: J.F. Kaiser, "Nonrecursive Digital Filter Design Using the Sinh Window Function," IEEE Int. Symp. Circuits and Systems, 1974. [^hamming1977]: R.W. Hamming, Digital Filters, Prentice-Hall, 1977. [^harris1978]: F.J. Harris, "On the Use of Windows for Harmonic Analysis with the DFT," Proc. IEEE 66, 1978. [^slepian1978]: D. Slepian, "Prolate Spheroidal Wave Functions — V," Bell Syst. Tech. J. 57, 1978. [^duchon1979]: C.E. Duchon, "Lanczos Filtering in One and Two Dimensions," J. Applied Meteorology 18, 1979. [^nuttall1981]: A.H. Nuttall, "Some Windows with Very Good Sidelobe Behavior," IEEE Trans. ASSP 29, 1981. [^streit1984]: R.L. Streit, "A Two-Parameter Family of Weights for Nonrecursive Digital Filters and Antennas," IEEE Trans. ASSP 32, 1984. [^princen1987]: J.P. Princen, A.W. Johnson & A.B. Bradley, "Subband/Transform Coding Using Filter Bank Designs Based on TDAC," ICASSP, 1987. [^ha1989]: Y.H. Ha & J.A. Pearce, "A New Window and Comparison to Standard Windows," IEEE Trans. ASSP, 1989. [^heinzel2002]: G. Heinzel, A. Rudiger & R. Schilling, "Spectrum and Spectral Density Estimation by the DFT," Max Planck Institute, 2002. [^mckechan2010]: D.J.A. McKechan et al., "A Tapering Window for Time-Domain Templates," Class. Quantum Grav. 27, 2010. [^starosielec2014]: S. Starosielec & D. Hagemeier, "Discrete-Time Windows with Minimal RMS Bandwidth," Signal Processing 102, 2014.