Relationship between wavelet shape and filter points

Question

MATLAB has a library of wavelet functions, showing their "continuous forms" as well as the the decomposition and reconstruction filters.

In decimated wavelet transform the filter size remains the same and the data points are downsampled by $2^j$ at every level $j$. A question that I cannot find an answer is how are the wavelet filters and the actual continuous forms are related? I am chemist, exploring wavelets for some applications.

For example, MATLAB has db9. Its continuous estimation, wavelet function, psi or phi do not look similar to the decomposition filter points. One text says that some wavelets start out as filter and later their continuous forms are estimated, but why in the case of db9 and many other, the filter points do not match the shape of the wavelet function?

1. Although this has been discussed in DSP, the point which is not clear is why don't the filter points exactly match the wavelet shape? The envelope sort of does but it is not exact.

2. Those who design wavelets, what comes first, the wavelets or the filters? If the filter is generated first, how do we get a continuous form, as shown in green "wavelet function, psi".

Thanks.

Answer 1

I found A note on a practical relationship between filter coefficients and scaling and wavelet functions of Discrete Wavelet Transforms quite instructive. It explains why for the implementation of the wavelet transform all you need are the filter coefficients, but if you wish to have a qualitative understanding of the reconstruction of a perturbed signal you will want to know the corresponding scaling and wavelet functions as well.

The calculation of the scaling and wavelet functions from the filter coefficients is carried out by means of the Cascade algorithm. The successive $k+1$-th approximation to the scaling function $\phi(t)$ is determined from the filter coefficients $h[n]$ by the equation $$\phi^{(k+1)}(t)=\sqrt 2\sum_{n}h[n]\phi^{(k)}(2t-n).$$ The $k\rightarrow\infty$ limit of the iteration can be readily obtained numerically, but in general there is no closed form, which is another reason to prefer the filter coefficients.