Find support of a wavelet

Question

I am a complete novice in wavelet analyses, but I was given a task to find a support set of a given wavelet (in 1D, e.g. db4). Please, point me a way of doing this. (Since, how I understood from some research, the support is shift-able, it is needed to find only its length, am I right?)

Answer 1

You have to study the refinement equation. Assume finite support and compare the support intervals on both sides.

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For D4, the refinement equation reads as

$$\phi(x)=a_0\phi(2x)+a_1\phi(2x-1)+a_2\phi(2x-2)+a_3\phi(2x-3)$$

Assume that for some finite interval $\operatorname{supp} \phi\subset [a,b]$. Then interesting things in $\phi(2x-k)$ only happen when $x\in[\frac12(a+k),\frac12(b+k)]$. For equality of the enclosing intervals on both sides, $a=\frac12 a$ and $b=\frac12(b+3)$ are necessary, from which one finds $[a,b]=[0,3]$.

Answer 2

For orthogonal wavelet systems, length of the support always is one less than the number of coefficients in the refinement equation see, page 67.

For the case of D4, your wavelet family has the support $[0,3]$ and its length is 3. If you want to find the support of $\phi(2^jx-k)$ then it is: $$supp(\phi)=\left [\frac{0+k}{2^j},\frac{3+k}{2^j}\right]$$