Hann window

A Hann window tapers a record with a raised cosine. For the endpoint-symmetric convention with $N>2$,It vanishes at the two stored endpoints. A periodic convention instead uses denominator $N$, corresponding to sampling one cosine period without storing its repeated endpoint. SciPy's primary documentation distinguishes these with sym=True and sym=False; the default is symmetric.^[ref] The adjacent Hanning window alias also supports the terminology used in older instrument documentation.

An original short-array example makes the difference concrete. At $N=4$, the symmetric coefficients are $(0,3/4,3/4,0)$, while the periodic coefficients are $(0,1/2,1,1/2)$. Their means are $3/8$ and $1/2$. A symmetric even-length array need not contain a coefficient equal to one, despite the nominal raised-cosine peak.

The coherent gain of a window is $G_c=N^{-1}{\sum }_n{w}_n$. It equals $1/2$ for the periodic Hann convention with $N>1$, and $(N-1)/(2N)$ for the symmetric convention with $N>2$. A coherent complex tone gives windowed DFT magnitude $N|A|G_c$ at its exact bin, so amplitude recovery must use the chosen coefficient sum.

For the periodic convention, its discrete-time Fourier transform is obtained from the rectangular window transform by modulation:The first central zeros lie at $\nu =\pm 2/N$ for the usual $N>3$ case, twice as far from the center as the rectangular window's first zeros. Smooth tapering reduces distant sidelobes while widening the central feature. Nearby frequencies and a weak component far from a strong one can therefore favor different window choices.

Noise normalization is another distinction. For $N>2$, the periodic Hann coefficients obey ${\sum }_n{w}_n=N/2$ and ${\sum }_n{w}_n^2=3N/8$, giving equivalent noise bandwidthin DFT bins. The manufacturer's table lists both coherent gain and noise bandwidth.^[ref] Correcting a coherent amplitude does not make noise behave as though the rectangular weights had been used.

The original comparison below evaluates the finite 64-sample periodic Hann and rectangular coefficients at a dense frequency grid, normalizing each response by its own coefficient sum. The extra frequency samples display the same finite-record transform more finely; they do not create a longer observation.