---
tag: 6C1S8HD38Z
kind: example
name: Shannon wavelet
msc: [42C40, 94A12]
instance_of: [4PZ35C0GV3]
uses: [DPVM34ADZV, 1GHQMBR7DE]
sources:
  - url: https://en.wikipedia.org/wiki/Shannon_wavelet
same_as:
  wikidata: Q7488992
  wikipedia: Shannon wavelet
---

## Example

Write $\operatorname{sinc}(t) = \sin(\pi t)/(\pi t)$ for $t \ne 0$ and $\operatorname{sinc}(0) = 1$.
Use the [[Fourier transform|DPVM34ADZV]] $\hat f(\omega) = \int_{\R} f(t) e^{-i\omega t} \, dt$.
The real *Shannon wavelet* is
$$\psi(t) = 2\operatorname{sinc}(2t) - \operatorname{sinc}(t) = \operatorname{sinc}\left(\frac{t}{2}\right) \cos\left(\frac{3\pi t}{2}\right).$$
Its Fourier transform is the indicator function of the set $\set{\omega : \pi < \abs{\omega} \le 2\pi}$, up to a set of measure zero.

Check.
The transform of $\operatorname{sinc}$ is the indicator of $[-\pi, \pi]$.
The transform of $2\operatorname{sinc}(2t)$ is the indicator of $[-2\pi, 2\pi]$.
Their difference is the indicator of the two bands.
The second formula follows from $\sin(2\pi t) - \sin(\pi t) = 2\cos(3\pi t/2)\sin(\pi t/2)$.

## Remarks

The functions $\psi_{j,k}(t) = 2^{j/2}\psi(2^j t - k)$, $j, k \in \Z$, form an orthonormal basis of $L^2(\R)$.
The reason is that the two bands, shifted by multiples of $2\pi$, tile the line, and their dyadic dilates tile $\R \setminus \set{0}$.
The scaling function of the underlying [[multiresolution analysis|1GHQMBR7DE]] is $\varphi = \operatorname{sinc}$, and its approximation spaces are spaces of band-limited functions.
The Shannon wavelet is perfectly localized in frequency but poorly localized in time: $\abs{\psi(t)}$ decays only like $1/\abs{t}$, so $\psi$ is not integrable.
The [[Meyer wavelet|Q5RN2FNX6Z]] smooths the sharp edges of $\hat\psi$ to gain fast decay in time.

## Explanations

### L1 human

The Shannon wavelet is an ideal band-pass filter turned into a function.
It keeps exactly the frequencies between $\pi$ and $2\pi$ in absolute value and nothing else.
Its stretched copies split the frequency line into octaves with no overlap.
The price for these sharp cuts is a function that ripples on for a long time before it fades.