Perfect reconstruction of 2 channel filter bank

I am studying Perfect reconstruction of 2 channel filter bank for using Biorthogonal Spline wavelet for decomposing ECG signal. I am using Wavelets and Filter Banks By Gilbert Strang, Truong Nguyen to study perfect reconstruction. I am trying to understand the remove distortion condition, and it is given as:

\begin{equation}\label{eqn:updated_no_dist} {P_0(z) - P_0(-z) = 2z^{-k}} \end{equation}

Now,it states, they need to design a low pass filter which satisfy the above equation, and to design a low-pass filter, they use this equation.

\begin{equation}\label{eqn:updated_no_distortion} {P_0(z) = (1 + z^{-1})^{2p}Q(z)} \end{equation} where p can be any integer and $Q(z)$ a polynomial of degree $2p-2$.

They use $p = 2$ and $Q(z) = -1 + 4z^{-1} - z^{-2}$. I am unable to find the reason why they use $p=2$ and where from the equation $Q(z)$ comes and what is the reason to use that exact equation. I read other papers as well, but I didn't find any reason for choosing $p=2$.

The other question is, what is the relationship between perfect reconstruction and Biorthogonal Spline wavelet.

It would be great if someone can explain this.

• As far as I understand, the author just uses $p=2$ and $Q(z) = -1 + 4z^{-1} - z^{-2}$ to get Daubechies maxflat product filter with $N=2$. It's just an example, not a generic choice. Jan 6 '18 at 15:34
• @Tendero Thanks for the comment. I want to understand, what will happen if I choose $p=1$ or $p=3$? and how the $Q(z)$ will be changed. What is $N$ here by the way? Order? Jan 6 '18 at 15:57
• Sorry, I mixed up the nomenclature. $2N$ would be the number of zeros at $z=-1$. Regarding how $Q(z)$ changes with $p$, if you want to get the same filter that specific examples shows then you just have to solve the equation $$\frac{1}{16}(-1+9z^{-2}+16z^{-3}+9z^{-4}-z^{-6}) = (1 + z^{-1})^{2p}Q(z)$$ Fix a value of $p$, and then find $Q(z)$ such that the equality holds. Jan 6 '18 at 16:09
• Will the left-hand side remain same for all the cases? I thought it is only when $p=2$ and $Q(z)=-1+4z^{-1}-z^{-2}$ have chosen, then it will yields that result. Jan 6 '18 at 16:42
• As I said before, the left side of the equation corresponds to a specific filter known as Daubechies maxflat product filter (judging by the book you provided). It's just an example the author uses. Jan 6 '18 at 17:00