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What Is The WDM Domain?

The Wilson-Daubechies-Meyer (WDM) transform is a time-frequency representation for a one-dimensional sampled signal.

It sits between two familiar extremes:

  • A time series tells you exactly when something happens, but not which frequency band it belongs to.
  • An FFT tells you exactly which frequencies are present, but not where in time they are active.

WDM tries to keep some of both.

The Core Idea

Instead of representing a signal with only global sine waves, WDM represents it with a collection of localized atoms g_{n,m}:

  • n indexes a time bin
  • m indexes a frequency channel

Each coefficient W[n, m] says how much the signal looks like that particular localized atom.

That is why a WDM coefficient grid can be read as a packed time-frequency map.

Why The Grid Has Shape (nt, nf + 1)

For a signal of length

N = nt * nf

the transform stores coefficients in a matrix of shape

(nt, nf + 1).

The channels are packed as:

  • m = 0: DC edge channel
  • m = 1 .. nf - 1: interior frequency channels
  • m = nf: Nyquist edge channel

So the extra + 1 comes from carrying both edge channels explicitly.

Reading The Packed Grid

  • Moving horizontally changes the time-bin index n
  • Moving vertically changes the channel index m
  • Bright coefficients indicate that the signal resembles that localized oscillatory atom

In practice, narrow-band features often occupy only a few nearby WDM channels, while transients stay localized in time instead of being smeared across the entire FFT.

Frequency Packetization Animation

The animation below shows one way to interpret the forward transform:

  • start from the FFT of the data
  • select one active WDM channel window at a time
  • compute the corresponding coefficient column
  • fill the packed WDM grid channel by channel

WDM frequency packetization

Why This Matters

This view is useful when you want:

  • better time localization than an FFT
  • a structured coefficient grid that still supports exact reconstruction
  • a domain where localized signals can be separated from broadband or stationary noise more naturally

For a full worked example, see the study notebook: