7.6 Conditions on h[n] and g[n]

Here we derive sufficient conditions on the coefficients used in the scaling equation and wavelet scaling equation thatensure, for every $k\in \mathbb{Z}$ , that the sets $\{{\phi }_{k,n}(t)\colon n\in \mathbb{Z}\}$ and $\{{\psi }_{k,n}(t)\colon n\in \mathbb{Z}\}$ have the orthonormality properties described in The Scaling Equation and The Wavelet Scaling Equation .

For $\{{\phi }_{k,n}(t)\colon n\in \mathbb{Z}\}$ to be orthonormal at all $k$ , we certainly need orthonormality when $k=1$ . This is equivalent to

There is an interesting frequency-domain interpretation of the previous condition. If we define

Given orthonormality at level $k=0$ , we have now derived a condition on $h(n)$ which is necessary and sufficient for orthonormality at level $k=1$ . Yet the same condition is necessary and sufficient for orthonormality at level $k=2$ :

To find conditions on $\{g(n)\}$ ensuring that the set $\{{\psi }_{k,n}(t)\colon n\in \mathbb{Z}\}$ is orthonormal at every $k$ , we can repeat the steps above but with $g(n)$ replacing $h(n)$ , ${\psi }_{k,n}(t)$ replacing ${\phi }_{k,n}(t)$ , and the wavelet-scaling equation replacing the scaling equation. This yields

Note that the two conditions $$\forall \text{odd P}\colon G(z)=\pm (z^{-P}H(-z^{-1}))$$ $$2=H(z)H(z^{-1})+H(-z)H(-z^{-1})$$ are sufficient to ensure that both $\{{\phi }_{k,n}(t)\colon n\in \mathbb{Z}\}$ and $\{{\psi }_{k,n}(t)\colon n\in \mathbb{Z}\}$ are orthonormal for all $k$ and that $\perp (W_k, V_k)$ for all $k$ , since they satisfy the condition $2=G(z)G(z^{-1})+G(-z)G(-z^{-1})$ automatically.