0.5 The scaling function and scaling coefficients, wavelet and  (Page 3/13)

This theorem shows that, unlike linear constant coefficient differential equations, not just any set of coefficients will support a solution. Thecoefficients must satisfy the linear equation [link] . This is the weakest condition on the $h\left(n\right)$ .

Theorem 2 If $\phi \left(t\right)$ is an $L^1$ solution to the basic recursion equation [link] with $\int \phi \left(t\right)dt=1$ , and

with $\text{Φ}(\pi +2\pi k)\ne 0$ for some $k$ , then

where [link] may have to be a distributional sum. Conversely, if [link] is satisfied, then [link] is true.

Equation [link] is called the fundamental condition , and it is weaker than requiring orthogonality but stronger than [link] . It is simply a result of requiring the equations resulting from evaluating [link] on the integers be consistent. Equation [link] is called a partitioning of unity (or the Strang condition or the Shoenbergcondition).

A similar theorem by Cavaretta, Dahman and Micchelli [link] and by Jia [link] states that if $\phi \in L^p$ and the integer translates of $\phi \left(t\right)$ form a Riesz basis for the space they span, then ${\sum }_{n}h\left(2n\right)={\sum }_{n}h(2n+1)$ .

Theorem 3 If $\phi \left(t\right)$ is an $L^2\cap L^1$ solution to [link] and if integer translates of $\phi \left(t\right)$ are orthogonal as defined by

then

Notice that this does not depend on a particular normalization of $\phi \left(t\right)$ .

If $\phi \left(t\right)$ is normalized by dividing by the square root of its energy $\sqrt{E}$ , then integer translates of $\phi \left(t\right)$ are orthonormal defined by

This theorem shows that in order for the solutions of [link] to be orthogonal under integer translation, it is necessary that thecoefficients of the recursive equation be orthogonal themselves after decimating or downsampling by two. If $\phi \left(t\right)$ and/or $h\left(n\right)$ are complex functions, complex conjugation must be used in [link] , [link] , and [link] .

Coefficients $h\left(n\right)$ that satisfy [link] are called a quadrature mirror filter (QMF) or conjugate mirror filter (CMF), and the condition [link] is called the quadratic condition for obvious reasons.

Corollary 1 Under the assumptions of Theorem  [link] , the norm of $h\left(n\right)$ is automatically unity.

Not only must the sum of $h\left(n\right)$ equal $\sqrt{2}$ , but for orthogonality of the solution, the sum of the squares of $h\left(n\right)$ must be one, both independent of any normalization of $\phi \left(t\right)$ . This unity normalization of $h\left(n\right)$ is the result of the $\sqrt{2}$ term in [link] .

Corollary 2 Under the assumptions of Theorem  [link] ,

This result is derived in the Appendix by showing that not only must the sum of $h\left(n\right)$ equal $\sqrt{2}$ , but for orthogonality of the solution, the individual sums of the even and odd terms in $h\left(n\right)$ must be $1/\sqrt{2}$ , independent of any normalization of $\phi \left(t\right)$ . Although stated here as necessary for orthogonality, the results hold under weaker non-orthogonalconditions as is stated in Theorem  [link] .

Theorem 4 If $\phi \left(t\right)$ has compact support on $0\le t\le N-1$ and if $\phi (t-k)$ are linearly independent, then $h\left(n\right)$ also has compact support over $0\le n\le N-1$ :

Thus $N$ is the length of the $h\left(n\right)$ sequence.

If the translates are not independent (or some equivalent restriction), one can have $h\left(n\right)$ with infinite support while $\phi \left(t\right)$ has finite support [link] .

These theorems state that if $\phi \left(t\right)$ has compact support and is orthogonal over integer translates, $\frac{N}{2}$ bilinear or quadratic equations [link] must be satisfied in addition to the one linear equation [link] . The support or length of $h\left(n\right)$ is $N$ , which must be an even number. The number of degrees of freedom in choosing these $N$ coefficients is then $\frac{N}{2}-1$ . This freedom will be used in the design of a wavelet system developed in  Chapter: Regularity, Moments, and Wavelet System Design and elsewhere.