---
tag: 90V8S5CXE6
kind: example
name: Daubechies wavelet
msc: [42C40, 65T60]
instance_of: [4PZ35C0GV3]
uses: [1GHQMBR7DE, YH4BERB952, 7QERXM0C52]
sources:
  - url: https://en.wikipedia.org/wiki/Daubechies_wavelet
same_as:
  wikidata: Q1172774
  wikipedia: Daubechies wavelet
---

## Example

For each integer $N \ge 1$, Ingrid Daubechies constructed an orthonormal [[wavelet|4PZ35C0GV3]] $\psi$ from a [[multiresolution analysis|1GHQMBR7DE]].
Its [[scaling function|YH4BERB952]] $\varphi$ satisfies a refinement equation $\varphi(x) = \sqrt2 \sum_{k=0}^{2N-1} h_k \varphi(2x - k)$ with $2N$ real coefficients.
Both $\varphi$ and $\psi$ have compact support of length $2N - 1$, and $\psi$ has $N$ [[vanishing moments|7QERXM0C52]].
For $N = 1$ this is the Haar wavelet.

The case $N = 2$ has the filter
$$h_0 = \frac{1 + \sqrt3}{4\sqrt2}, \quad h_1 = \frac{3 + \sqrt3}{4\sqrt2}, \quad h_2 = \frac{3 - \sqrt3}{4\sqrt2}, \quad h_3 = \frac{1 - \sqrt3}{4\sqrt2}.$$
Check of the filter conditions.
The sum is $\frac{8}{4\sqrt2} = \sqrt2$, as needed for $\int \varphi = 1$.
The sum of squares is $\frac{(4 + 2\sqrt3) + (12 + 6\sqrt3) + (12 - 6\sqrt3) + (4 - 2\sqrt3)}{32} = 1$.
And $h_0 h_2 + h_1 h_3 = \frac{(1 + \sqrt3)(3 - \sqrt3) + (3 + \sqrt3)(1 - \sqrt3)}{32} = \frac{2\sqrt3 - 2\sqrt3}{32} = 0$.
So the filter is orthogonal to its shift by $2$, which reflects the orthogonality of the translates $\varphi(\cdot - k)$.

The wavelet is $\psi(x) = \sqrt2 \sum_k g_k \varphi(2x - k)$ with $g_k = (-1)^k h_{1-k}$, so its coefficients are $g_{-2} = h_3$, $g_{-1} = -h_2$, $g_0 = h_1$, $g_1 = -h_0$.
Check of two vanishing moments: $\sum_k g_k = h_3 - h_2 + h_1 - h_0 = 0$ and $\sum_k k g_k = -2h_3 + h_2 - h_0 = \frac{-2 + 2\sqrt3 + 3 - \sqrt3 - 1 - \sqrt3}{4\sqrt2} = 0$.
These two discrete conditions give $\int \psi = 0$ and $\int x \psi(x) \, dx = 0$.
The scaling function of this case is continuous but not smooth, and it has no closed formula; it is computed from the refinement equation.

## Explanations

### L1 human

Daubechies wavelets are a family of wavelets that live on short intervals and are smoother as the index grows.
Each one is fixed by a short list of numbers, its filter, and the discrete wavelet transform uses only these numbers.
The four numbers above give the simplest one after the Haar wavelet.
Their shape looks jagged, almost like a fractal, but they are widely used in signal and image compression.