Windows and spectral leakage

Use this page when a finite observation must be tapered and the resulting amplitude or noise normalization must remain scientifically interpretable.

The scientific question¶

Observing only a finite record multiplies an underlying signal by an implicit rectangular window. In frequency space, multiplication becomes convolution, so power from a component need not remain in one Fourier bin. A taper can suppress distant leakage at the cost of a wider main lobe, reduced coherent amplitude, and changed noise bandwidth.

There is no universally best window. The choice depends on whether the target is line amplitude, nearby-feature separation, broadband power, or another observable.

Mathematical objects¶

Let $w_n$ be a real window and $y_n=w_nx_n$ the windowed record. Its sum controls the response to a bin-centered coherent sinusoid. Its sum of squares controls white-noise power. Scaling a window changes both sums, so every estimator must state which normalization is applied.

The equivalent noise bandwidth (ENBW) expresses how much white-noise bandwidth an estimator admits relative to an ideal one-bin filter. It has frequency units when multiplied by the sample frequency.

Core derivation¶

For sample frequency $f_s=1/\Delta t$ , define coherent gain and ENBW by:

G_{\mathrm{coh}}=\frac{1}{N}\sum_{n=0}^{N-1}w_n, \qquad B_{\mathrm{ENBW}} =f_s\frac{\sum_{n=0}^{N-1}w_n^2} {\left(\sum_{n=0}^{N-1}w_n\right)^2}.

(1)

The coherent amplitude of a bin-centered tone is reduced by $G_{\mathrm{coh}}$ , while white-noise density conversion depends on $B_{\mathrm{ENBW}}$ . Therefore (1) shows why an amplitude correction and a noise-power correction are not interchangeable.

Let $W_m$ and $Y_k$ use the same DFT pair as $X_k$ on the signal-axes page. Multiplication in the sample domain then gives:

Y_k = \frac{1}{N}\sum_{m=0}^{N-1} W_m X_{(k-m)\bmod N}.

(2)

Thus (2) says that the DFT of $y_n$ is $1/N$ times the circular convolution of the two spectra under the documented unnormalized-forward DFT convention. This redistribution explains leakage but does not guarantee that tapering resolves overlapping components.

What the ecosystem already owns¶

JAX FFT owns FFT mechanics, and JAX owns convolution mechanics. Applying a supplied array as a window is ordinary array computation. JAX does not assign the window a scientific name, normalization contract, ENBW, or provenance.

What Jaxstro may add¶

The proposed jaxstro.signal may own a small, explicit catalog of window definitions with normalization mode, coherent gain, sum-of-squares gain, ENBW, axis units, and provenance. A spectral estimator could require this metadata rather than accepting an unlabeled multiplier.

The planned layer would not claim that one window is optimal or hide correction factors inside an ambiguous output.

Evidence required before implementation¶

Required evidence would include:

exact coefficient fixtures for every supported window and length convention;
DC and bin-centered sinusoid tests for coherent-gain correction;
white-noise experiments for ENBW and power-density normalization;
off-bin sinusoid tests for main-lobe width and leakage behavior;
scale-invariance tests showing which reported quantities change under window rescaling;
odd, even, and short-length edge cases; and
provenance for window family, parameters, length, and normalization.

Claim boundary¶

No runtime catalog, performance result, or universally preferred window is claimed.

Connected foundations and methods¶

First establish cadence and DFT normalization in Signal axes, cadence, and units. Then use Power and cross-spectral estimation for one-sided and two-sided power conventions and Phase and delay for cross-spectral phase. The finite-data viewpoint in One-dimensional interpolation and scale concepts in Functions, units, and scales provide additional context.