pvx Window Reference

Generated from commit dff2a4f (commit date: 2026-04-21T21:20:38-04:00).

pvx currently supports 50 analysis windows. This file defines each one mathematically and explains it in plain English.

Notation

• Window length: $N$
• Sample index: $n \in \{0,\dots,N-1\}$
• Center index: $m=(N-1)/2$
• Normalized center coordinate: $x_n=(n-m)/m$

Formula Key

(W0) Rectangular

$w[n]=1$

(W1) Cosine series

$w[n]=\sum_{k=0}^{K} a_k\cos\left(\frac{2\pi k n}{N-1}\right)$

(W2) Sine/Cosine

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

(W3) Bartlett

$w[n]=1-\left|\frac{n-m}{m}\right|$

(W4) Triangular

$w[n]=\max\left(1-\left|\frac{n-m}{(N+1)/2}\right|,0\right)$

(W5) Bartlett-Hann

$x=\frac{n}{N-1}-\frac{1}{2},\quad w[n]=0.62-0.48|x|+0.38\cos(2\pi x)$

(W6) Tukey

$$x=\frac{n}{N-1},\quad w[n]=\begin{cases} \frac{1}{2}\left(1+\cos\left(\pi\left(\frac{2x}{\alpha}-1\right)\right)\right), & 0\le x<\frac{\alpha}{2} \\ 1, & \frac{\alpha}{2}\le x<1-\frac{\alpha}{2} \\ \frac{1}{2}\left(1+\cos\left(\pi\left(\frac{2x}{\alpha}-\frac{2}{\alpha}+1\right)\right)\right), & 1-\frac{\alpha}{2}\le x\le 1 \end{cases}$$

Special cases in pvx: $\alpha\le 0$ gives rectangular behavior, and $\alpha\ge 1$ collapses to Hann.

(W7) Parzen

$$u=\left|\frac{2n}{N-1}-1\right|,\quad w[n]=\begin{cases} 1-6u^2+6u^3, & 0\le u\le \frac{1}{2} \\ 2(1-u)^3, & \frac{1}{2}<u\le 1 \\ 0, & u>1 \end{cases}$$

(W8) Lanczos

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

(W9) Welch

$w[n]=\max\left(1-x_n^2,0\right)$

(W10) Gaussian

$w[n]=\exp\left(-\frac{1}{2}\left(\frac{n-m}{\sigma}\right)^2\right),\quad \sigma=r_\sigma m$

(W11) General Gaussian

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

(W12) Exponential

$w[n]=\exp\left(-\frac{|n-m|}{\tau}\right),\quad \tau=r_\tau m$

(W13) Cauchy

$w[n]=\frac{1}{1+\left(\frac{n-m}{\gamma}\right)^2},\quad \gamma=r_\gamma m$

(W14) Cosine power

$w[n]=\sin\left(\frac{\pi n}{N-1}\right)^p$

(W15) Hann-Poisson

$w[n]=w_{\text{Hann}}[n]\exp\left(-\alpha\frac{|n-m|}{m}\right)$

(W16) General Hamming

$w[n]=\alpha-(1-\alpha)\cos\left(\frac{2\pi n}{N-1}\right)$

(W17) Bohman

$x=\left|\frac{2n}{N-1}-1\right|,\quad w[n]=(1-x)\cos(\pi x)+\frac{\sin(\pi x)}{\pi}$

(W18) Kaiser-Bessel

$w[n]=\frac{I_0\left(\beta\sqrt{1-r_n^2}\right)}{I_0(\beta)},\quad r_n=\frac{n-m}{m}$

Each supported pvx window maps to one of the formula families above with the per-window constants shown below.

Quantitative Metrics

• Coherent gain: $CG=\frac{1}{N}\sum_{n=0}^{N-1}w[n]$
• Equivalent noise bandwidth (bins): $ENBW=\frac{N\sum_n w[n]^2}{(\sum_n w[n])^2}$
• Scalloping loss (dB): response ratio at a half-bin sinusoid offset
• Main-lobe width (bins): measured from the first post-DC local minimum in zero-padded FFT magnitude
• Peak sidelobe (dB): maximum sidelobe level outside the main lobe

Complete Window Catalog

Window	Family	Parameters	Formula	Coherent gain	ENBW (bins)	Scalloping loss (dB)	Main-lobe width (bins)	Peak sidelobe (dB)	Plots	Pros	Cons	Usage advice
hann	Cosine series	coeffs=(0.5, -0.5)	W1	0.499756	1.500733	-1.422	4.000	-31.468	time / freq	Balanced leakage suppression and frequency resolution.	Not optimal for amplitude metering or extreme sidelobe rejection.	Default choice for most pvx time-stretch and pitch-shift workflows.
hamming	Cosine series	coeffs=(0.54, -0.46)	W1	0.539775	1.363305	-1.750	4.000	-42.675	time / freq	Balanced leakage suppression and frequency resolution.	Not optimal for amplitude metering or extreme sidelobe rejection.	Default choice for most pvx time-stretch and pitch-shift workflows.
blackman	Cosine series	coeffs=(0.42, -0.5, 0.08)	W1	0.419795	1.727601	-1.098	6.000	-58.109	time / freq	Strong sidelobe suppression for cleaner spectral separation.	Wider main lobe than Hann/Hamming.	Use for dense harmonic material when leakage artifacts dominate.
blackmanharris	Cosine series	coeffs=(0.35875, -0.48829, 0.14128, -0.01168)	W1	0.358575	2.005332	-0.825	8.000	-92.011	time / freq	Strong sidelobe suppression for cleaner spectral separation.	Wider main lobe than Hann/Hamming.	Use for dense harmonic material when leakage artifacts dominate.
nuttall	Cosine series	coeffs=(0.355768, -0.487396, 0.144232, -0.012604)	W1	0.355594	2.022220	-0.811	8.000	-93.325	time / freq	Strong sidelobe suppression for cleaner spectral separation.	Wider main lobe than Hann/Hamming.	Use for dense harmonic material when leakage artifacts dominate.
flattop	Cosine series	coeffs=(1, -1.93, 1.29, -0.388, 0.0322)	W1	0.999514	3.772117	-0.016	10.000	-68.311	time / freq	Very accurate amplitude estimation in FFT bins.	Very wide main lobe and reduced frequency discrimination.	Use for measurement-grade magnitude tracking, not fine pitch separation.
blackman_nuttall	Cosine series	coeffs=(0.363582, -0.489177, 0.1366, -0.0106411)	W1	0.363405	1.977073	-0.850	8.000	-98.174	time / freq	Strong sidelobe suppression for cleaner spectral separation.	Wider main lobe than Hann/Hamming.	Use for dense harmonic material when leakage artifacts dominate.
exact_blackman	Cosine series	coeffs=(0.426591, -0.496561, 0.0768487)	W1	0.426386	1.694500	-1.149	6.000	-68.236	time / freq	Strong sidelobe suppression for cleaner spectral separation.	Wider main lobe than Hann/Hamming.	Use for dense harmonic material when leakage artifacts dominate.
sine	Sinusoidal	none	W2	0.636309	1.234304	-2.096	3.000	-22.999	time / freq	Smooth endpoint behavior and straightforward implementation.	Less configurable than Kaiser/Tukey families.	Use for stable, low-complexity alternatives to Hann.
bartlett	Triangular	none	W3	0.499756	1.333985	-1.822	4.000	-26.523	time / freq	Cheap linear taper with intuitive behavior.	Higher sidelobes than stronger cosine-sum windows.	Use for low-cost processing or quick exploratory runs.
boxcar	Rectangular	none	W0	1.000000	1.000000	-3.922	2.000	-13.264	time / freq	Narrowest main lobe and maximal bin sharpness.	Highest sidelobes and strongest leakage/phasiness on non-bin-centered content.	Use only for controlled test tones or when leakage is acceptable.
triangular	Triangular	none	W4	0.500244	1.332683	-1.826	4.000	-26.523	time / freq	Cheap linear taper with intuitive behavior.	Higher sidelobes than stronger cosine-sum windows.	Use for low-cost processing or quick exploratory runs.
bartlett_hann	Hybrid taper	fixed coefficients	W5	0.499756	1.456559	-1.517	4.000	-35.874	time / freq	Moderate leakage suppression with lightweight computation.	Less common and less interpretable than standard Hann/Hamming.	Use as a middle-ground taper when Bartlett feels too sharp.
tukey	Tukey	alpha=0.5	W6	0.749634	1.222819	-2.236	2.656	-15.123	time / freq	Interpolates between rectangular and Hann behavior.	Behavior changes strongly with alpha and can be inconsistent across presets.	Use when you need a controllable flat center with soft edges.
tukey_0p1	Tukey	alpha=0.1	W6	0.949536	1.039289	-3.505	2.094	-13.309	time / freq	Interpolates between rectangular and Hann behavior.	Behavior changes strongly with alpha and can be inconsistent across presets.	Use when you need a controllable flat center with soft edges.
tukey_0p25	Tukey	alpha=0.25	W6	0.874573	1.102579	-2.960	2.281	-13.601	time / freq	Interpolates between rectangular and Hann behavior.	Behavior changes strongly with alpha and can be inconsistent across presets.	Use when you need a controllable flat center with soft edges.
tukey_0p75	Tukey	alpha=0.75	W6	0.624695	1.360664	-1.728	3.188	-19.395	time / freq	Interpolates between rectangular and Hann behavior.	Behavior changes strongly with alpha and can be inconsistent across presets.	Use when you need a controllable flat center with soft edges.
tukey_0p9	Tukey	alpha=0.9	W6	0.549731	1.446988	-1.521	3.625	-24.972	time / freq	Interpolates between rectangular and Hann behavior.	Behavior changes strongly with alpha and can be inconsistent across presets.	Use when you need a controllable flat center with soft edges.
parzen	Polynomial	piecewise cubic	W7	0.374817	1.918397	-0.897	8.000	-53.046	time / freq	Smooth high-order taper with good sidelobe control.	Broader main lobe than lightweight windows.	Use for spectral denoising/analysis where sidelobe cleanup is critical.
lanczos	Sinc	none	W8	0.589202	1.299668	-1.889	3.281	-26.405	time / freq	Sinc-derived shape with useful compromise behavior.	Can exhibit oscillatory spectral behavior versus cosine-sum defaults.	Use for interpolation-adjacent analysis experiments.
welch	Quadratic	none	W9	0.666341	1.200587	-2.223	2.875	-21.295	time / freq	Center-emphasizing parabola with simple form.	Not as strong at sidelobe suppression as Blackman-family windows.	Use when center weighting is desired with minimal complexity.
gaussian_0p25	Gaussian	sigma_ratio=0.25	W10	0.313156	2.258144	-0.668	11.406	-87.677	time / freq	Smooth bell taper with low ringing and predictable decay.	Frequency resolution changes noticeably with sigma.	Use sigma presets to tune transient locality versus spectral leakage.
gaussian_0p35	Gaussian	sigma_ratio=0.35	W10	0.436580	1.626488	-1.268	6.719	-52.077	time / freq	Smooth bell taper with low ringing and predictable decay.	Frequency resolution changes noticeably with sigma.	Use sigma presets to tune transient locality versus spectral leakage.
gaussian_0p45	Gaussian	sigma_ratio=0.45	W10	0.548949	1.320556	-1.869	4.125	-35.366	time / freq	Smooth bell taper with low ringing and predictable decay.	Frequency resolution changes noticeably with sigma.	Use sigma presets to tune transient locality versus spectral leakage.
gaussian_0p55	Gaussian	sigma_ratio=0.55	W10	0.641515	1.171861	-2.350	3.000	-28.906	time / freq	Smooth bell taper with low ringing and predictable decay.	Frequency resolution changes noticeably with sigma.	Use sigma presets to tune transient locality versus spectral leakage.
gaussian_0p65	Gaussian	sigma_ratio=0.65	W10	0.713490	1.097657	-2.705	2.625	-22.575	time / freq	Smooth bell taper with low ringing and predictable decay.	Frequency resolution changes noticeably with sigma.	Use sigma presets to tune transient locality versus spectral leakage.
general_gaussian_1p5_0p35	Generalized Gaussian	power=1.5, sigma_ratio=0.35	W11	0.393587	2.016584	-0.784	5.156	-24.895	time / freq	Additional shape control beyond standard Gaussian.	Extra parameterization increases tuning complexity.	Use for research/tuning tasks where shoulder steepness matters.
general_gaussian_2p0_0p35	Generalized Gaussian	power=2, sigma_ratio=0.35	W11	0.377081	2.230016	-0.633	5.281	-19.717	time / freq	Additional shape control beyond standard Gaussian.	Extra parameterization increases tuning complexity.	Use for research/tuning tasks where shoulder steepness matters.
general_gaussian_3p0_0p35	Generalized Gaussian	power=3, sigma_ratio=0.35	W11	0.364287	2.445593	-0.532	5.469	-16.299	time / freq	Additional shape control beyond standard Gaussian.	Extra parameterization increases tuning complexity.	Use for research/tuning tasks where shoulder steepness matters.
general_gaussian_4p0_0p35	Generalized Gaussian	power=4, sigma_ratio=0.35	W11	0.359267	2.552433	-0.496	5.562	-15.060	time / freq	Additional shape control beyond standard Gaussian.	Extra parameterization increases tuning complexity.	Use for research/tuning tasks where shoulder steepness matters.
exponential_0p25	Exponential	tau_ratio=0.25	W12	0.245310	2.075492	-1.022	15.562	-31.885	time / freq	Simple edge decay with fast computation.	Can produce less-uniform center weighting than cosine families.	Use for experiments emphasizing center-heavy weighting.
exponential_0p5	Exponential	tau_ratio=0.5	W12	0.432187	1.313323	-2.032	3.625	-19.190	time / freq	Simple edge decay with fast computation.	Can produce less-uniform center weighting than cosine families.	Use for experiments emphasizing center-heavy weighting.
exponential_1p0	Exponential	tau_ratio=1	W12	0.631991	1.082055	-2.854	2.594	-20.141	time / freq	Simple edge decay with fast computation.	Can produce less-uniform center weighting than cosine families.	Use for experiments emphasizing center-heavy weighting.
cauchy_0p5	Cauchy / Lorentzian	gamma_ratio=0.5	W13	0.553402	1.229775	-2.209	3.562	-22.733	time / freq	Heavy tails preserve more peripheral samples than Gaussian.	Tail energy can increase leakage relative to steeper tapers.	Use when you want softer attenuation of far-window samples.
cauchy_1p0	Cauchy / Lorentzian	gamma_ratio=1	W13	0.785259	1.041963	-3.108	2.375	-19.034	time / freq	Heavy tails preserve more peripheral samples than Gaussian.	Tail energy can increase leakage relative to steeper tapers.	Use when you want softer attenuation of far-window samples.
cauchy_2p0	Cauchy / Lorentzian	gamma_ratio=2	W13	0.927233	1.004393	-3.648	2.094	-14.689	time / freq	Heavy tails preserve more peripheral samples than Gaussian.	Tail energy can increase leakage relative to steeper tapers.	Use when you want softer attenuation of far-window samples.
cosine_power_2	Cosine power	power=2	W14	0.499756	1.500733	-1.422	4.000	-31.468	time / freq	Simple power parameter controls edge steepness.	Higher powers can overly narrow effective support.	Use to smoothly increase edge attenuation versus basic sine.
cosine_power_3	Cosine power	power=3	W14	0.424206	1.735739	-1.074	5.000	-39.295	time / freq	Simple power parameter controls edge steepness.	Higher powers can overly narrow effective support.	Use to smoothly increase edge attenuation versus basic sine.
cosine_power_4	Cosine power	power=4	W14	0.374817	1.945394	-0.862	6.000	-46.741	time / freq	Simple power parameter controls edge steepness.	Higher powers can overly narrow effective support.	Use to smoothly increase edge attenuation versus basic sine.
hann_poisson_0p5	Hann-Poisson	alpha=0.5	W15	0.432946	1.609944	-1.258	5.188	-35.245	time / freq	Combines smooth Hann center with exponential edge suppression.	Can over-attenuate edges for high alpha values.	Use when you need stronger edge decay than Hann without full flattop cost.
hann_poisson_1p0	Hann-Poisson	alpha=1	W15	0.378797	1.734176	-1.112	103.750	-80.017	time / freq	Combines smooth Hann center with exponential edge suppression.	Can over-attenuate edges for high alpha values.	Use when you need stronger edge decay than Hann without full flattop cost.
hann_poisson_2p0	Hann-Poisson	alpha=2	W15	0.297878	2.023174	-0.870	164.719	-80.005	time / freq	Combines smooth Hann center with exponential edge suppression.	Can over-attenuate edges for high alpha values.	Use when you need stronger edge decay than Hann without full flattop cost.
general_hamming_0p50	General Hamming	alpha=0.50	W16	0.499756	1.500733	-1.422	4.000	-31.468	time / freq	Tunable Hamming-style cosine weighting.	Less standardized than classic Hann/Hamming choices.	Use to sweep sidelobe-vs-width behavior near Hamming family defaults.
general_hamming_0p60	General Hamming	alpha=0.60	W16	0.599805	1.222475	-2.179	3.469	-31.600	time / freq	Tunable Hamming-style cosine weighting.	Less standardized than classic Hann/Hamming choices.	Use to sweep sidelobe-vs-width behavior near Hamming family defaults.
general_hamming_0p70	General Hamming	alpha=0.70	W16	0.699854	1.091920	-2.762	2.656	-24.078	time / freq	Tunable Hamming-style cosine weighting.	Less standardized than classic Hann/Hamming choices.	Use to sweep sidelobe-vs-width behavior near Hamming family defaults.
general_hamming_0p80	General Hamming	alpha=0.80	W16	0.799902	1.031273	-3.227	2.312	-18.649	time / freq	Tunable Hamming-style cosine weighting.	Less standardized than classic Hann/Hamming choices.	Use to sweep sidelobe-vs-width behavior near Hamming family defaults.
bohman	Bohman	none	W17	0.405087	1.786613	-1.022	6.000	-45.997	time / freq	Continuous-slope taper with good qualitative leakage control.	Less common in audio tooling and harder to tune by intuition.	Use for exploratory spectral work needing smooth derivatives.
cosine	Sinusoidal	none	W2	0.636309	1.234304	-2.096	3.000	-22.999	time / freq	Smooth endpoint behavior and straightforward implementation.	Less configurable than Kaiser/Tukey families.	Use for stable, low-complexity alternatives to Hann.
kaiser	Kaiser-Bessel	beta from --kaiser-beta	W18	0.331708	2.162181	-0.712	9.125	-105.921	time / freq	Continuously tunable width/sidelobe tradeoff via beta.	Needs beta tuning; poor beta choices can over-blur or under-suppress sidelobes.	Use when you need explicit control over leakage versus resolution.
rect	Rectangular	none	W0	1.000000	1.000000	-3.922	2.000	-13.264	time / freq	Narrowest main lobe and maximal bin sharpness.	Highest sidelobes and strongest leakage/phasiness on non-bin-centered content.	Use only for controlled test tones or when leakage is acceptable.

Complete Plot Gallery (All Windows)

Window	Time-domain shape	Magnitude spectrum
hann
hamming
blackman
blackmanharris
nuttall
flattop
blackman_nuttall
exact_blackman
sine
bartlett
boxcar
triangular
bartlett_hann
tukey
tukey_0p1
tukey_0p25
tukey_0p75
tukey_0p9
parzen
lanczos
welch
gaussian_0p25
gaussian_0p35
gaussian_0p45
gaussian_0p55
gaussian_0p65
general_gaussian_1p5_0p35
general_gaussian_2p0_0p35
general_gaussian_3p0_0p35
general_gaussian_4p0_0p35
exponential_0p25
exponential_0p5
exponential_1p0
cauchy_0p5
cauchy_1p0
cauchy_2p0
cosine_power_2
cosine_power_3
cosine_power_4
hann_poisson_0p5
hann_poisson_1p0
hann_poisson_2p0
general_hamming_0p50
general_hamming_0p60
general_hamming_0p70
general_hamming_0p80
bohman
cosine
kaiser
rect

Practical Selection Guidance

• Use hann or hamming for balanced everyday phase-vocoder work.
• Use kaiser when you need explicit sidelobe control with --kaiser-beta.
• Use flattop for amplitude-accuracy-focused spectral measurement.
• Use tukey_* when you want a controllable flat center region.
• Use Gaussian/Cauchy/Exponential families to experiment with edge-decay shape and time-locality.

Attribution

See ATTRIBUTION.md.