Windows And Atoms

WDM is built from a cosine-tapered frequency window phi and a family of shifted atoms g_{n,m}.

These are the ingredients that make the coefficient grid interpretable.

The Window phi

The package uses a compactly supported frequency-domain window:

• flat in the middle
• smoothly tapered at the edges
• exactly zero outside its support

The roll-off is controlled by the parameter a.

• smaller a: narrower flat region, broader taper
• larger a: broader flat region, narrower taper

In this repository the default choice is a = 1/3.

The easiest way to read a is:

• smaller a: the flat part of the window is narrower and the taper occupies more of the band
• larger a: the flat part is wider and the taper is tighter

That is visible directly in frequency space, but it also changes the corresponding time-domain localization. The figure below shows both effects side by side.

How to interpret it:

• Left panel: the actual cosine-tapered window shape in normalized frequency coordinates
• Right panel: a normalized time-domain shape implied by the same phi, included only to show the localization trend as a changes

This is the core tradeoff:

• a broader and flatter frequency window usually gives a more concentrated band selection in frequency
• but that also changes how extended the corresponding shape is in time

So a is one of the main knobs for how sharply the transform separates nearby frequencies versus how localized the atoms remain in time.

Shifted Windows Define The Channels

The same base window is shifted to different channel locations m. That is how WDM separates the signal into localized frequency bands.

The figure below overlays a few shifted windows on top of a sample spectrum.

The interior channels are centered away from DC and Nyquist, while the two edge channels have a special form because they sit at the boundaries of the sampled frequency range.

The Atoms g_{n,m}

Once the window is placed at channel m, the transform also modulates it in time.

• m changes where the atom lives in frequency
• n changes where the atom lives in time

That is the origin of the two-dimensional WDM grid.

Why Orthogonality Matters

If two different atoms overlap too strongly, then coefficients stop having a clean interpretation. A large coefficient could be partly due to leakage from several neighboring atoms rather than one localized feature.

Near-orthogonality fixes that:

• one atom corresponds to one localized time-frequency pattern
• coefficients can be read more independently
• reconstruction remains stable and well-behaved

The study notebook includes overlap maps that visualize this directly.

Atom Shift Animation

The animation below keeps one frequency channel fixed and shifts the atom across time bins. The frequency support stays in the same band, while the time-domain shape moves across the signal duration.

This is the key intuition behind the n index: it is not a sample number, but a location on the coarser WDM time grid.

Channel Shift Animation

The complementary animation below keeps one time bin fixed and shifts the atom through the WDM channels.

This shows the role of m:

• the active frequency band moves up and down the spectrum
• the time-localization stays in roughly the same place
• the time-domain atom oscillates faster or slower depending on the channel

So, at a high level:

• changing n at fixed m moves an atom in time
• changing m at fixed n moves an atom in frequency

Implementation Surface

The shared helpers that define these pieces live in:

• wdm_transform.windows.phi_unit
• wdm_transform.windows.phi_window
• wdm_transform.windows.gnmf
• wdm_transform.windows.cnm