[time-nuts] Window functions Was: A simple sampling DMTD

[Gregory Maxwell]

On Wed, Nov 27, 2019 at 6:22 PM Joseph Gwinn <joegwinn at comcast.net> wrote:
> in two batches.  The window functions largely eliminate the splice
> error due to the FFT, which is fact splices each batch into a circle,
> causing a discontinuity at the splice.  If the window function is very
> small at the splice, there is little discontinuity power sprayed into
> innocent FFT bins.
>
> In radar, Taylor windows are used, as well as Dolph-Chebyshev if very
> low sidelobes are needed.

Related to but distinct from low-sidelobe window functions are
_sidelobeless_ windows, which I think may be particularly useful in
control-loop applications like PLLs.

The gaussian window is obviously sidelobeless, but it has infinite
support so it's not useful.

But it turns out that there exist finite windows that are completely
sidelobeless (a fourier transform of their kernel is monotone).

A triangular window convolved with a truncated gaussian window is
sidelobeless (with an appropriate choice in parameters), as is a
hann-poisson window.

While working on the problem of sinusodial modelling (
https://arxiv.org/abs/1602.05900 ), which is essentially the same
problem as PLL tracking tones in a noisy signal, I found sidelobeless
windows to work much better than even low sidelobe windows like
Dolph-Chebyshev ... but I never found a lot of coverage in the
literature.