Question from the book, Wavelet and filter banks by Nguy and Strang

I have been spending some time analysis Daubechies 22, and I read the book.
(Wavelet and Filter Bank by Nguy and Strang)
There is one part that I did not understand. On page 169, it did one
example to find Daubechies coefficients at p = 2. I understood it till it
said "Then the 2p-1 roots of C(z) are -1, -1, 2-squroots(3)" Then it went
on to find the 4 coefficients of D4. I did not know how he got to that
result and what equations he used to find it.

Can someone please explain the process of getting the coefficients?
Thank you

Huang Andy wrote:
> I have been spending some time analysis Daubechies 22, and I read the book.
> (Wavelet and Filter Bank by Nguy and Strang)
> There is one part that I did not understand. On page 169, it did one
> example to find Daubechies coefficients at p = 2. I understood it till it
> said "Then the 2p-1 roots of C(z) are -1, -1, 2-squroots(3)" Then it went
> on to find the 4 coefficients of D4. I did not know how he got to that
> result and what equations he used to find it.
>
> Can someone please explain the process of getting the coefficients?
> Thank you

Jumping into this book after long absence...

The numerical coefficients cited in the example are found by found by
expanding the factored polynomial, C(z) on the next line. The factors
are [1 - (2-sqrt(3))z^-1], for the root inside the unit circle, and
(1+z^-1)^2 because we need p=2 roots at -1. As for the factor alpha = 1
/ (4*sqrt(2)), I guess that's a normalization constant, but I haven't
paged back to see exactly how it's defined. Anyway, multiply out these
factors as on the next line (remember the factor of alpha), and convert
the resulting coefficients of z^-k to decimal. I tried a couple of them
and got the same numbers Strang gives.

David L. Rick
Hach Company

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