Wavelet Filter Coefficients from Scaling Filter Coefficients

I am trying to develop a new type of wavelets and I found out a function that following a particular two scale relation. The function at a scale $t$ say $x(t)$ can be related in a finer scale $x(2t)$ by the following relation:

$$x(t)=0.1x\left(2t\right)+ 0.5x\left(2t-1\right)+0.8x\left(2t-2\right)+0.5x\left(2t-3\right)+0.1x\left(2t-4\right),$$

This is found out by trial and error method by simply changing the values. Now I know that the points $\begin{bmatrix}0.1&0.5&0.8&0.5&0.1\end{bmatrix}$ can be considered as the scaling filter coefficients. Now the problem is finding the wavelet filter coefficients, I am not able to find out a particular relation connecting scaling filter coefficients and wavelet filter coefficients. Is there is a relation connecting the two ??

• You have an extra parenthesis after .5 (?) Apr 5 '16 at 13:56
• typing mistake ..sorry ,edited Apr 5 '16 at 14:01

For perfect reconstruction you need:

$$g \left[ n \right] * \hat{g} \left[ n \right] + h \left[ n \right] * \hat{h} \left[ n \right] = \delta \left[ n \right]$$

Where $g \left[ n \right]$ is the Analysis Wavelet Function, $\hat{g} \left[ n \right]$ is the Synthesis Wavelet Function, $h \left[ n \right]$ Analysis Scaling Function and $\hat{h} \left[ n \right]$ is the Synthesis Scaling Function.

Practically, Scaling Function is the LPF and Wavelet Function is the HPF.

• so,now can you able to tell me how I can get my wavelet coefficients .... Apr 5 '16 at 12:25
• @Abhishek, Could you please mark my answer?
– Royi
Sep 5 '21 at 10:42

If $\{h(n)\}_{n\in\mathbb{N}}$ is the scaling filter, i.e. the sequence such that $$\phi(x) = \sum_{n} h(n) 2^{1/2} \phi(2x-n)$$ where with $\phi(\cdot)$ I indicated the scaling function, then the mother wavelet satisfies $$\psi(x) = \sum_{n} g(n) 2^{1/2} \phi(2x - n)$$ being $$g(n) = (-1)^n \overline{h(1-n)}$$ being $\bar{z}$ the complex conjugate of $z\in\mathbb{C}$. This according to D. F. Walnut, An Introduction to Wavelet Analysis