MA5242 Wavelets Lecture 4 WT Matrix Factorization

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MA5242 Wavelets Lecture 4 WT Matrix Factorization

• Wayne M. Lawton
• Department of Mathematics
• National University of Singapore
• 2 Science Drive 2
• Singapore 117543

Email matwml_at_nus.edu.sg Tel [phone]
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Even and Odd Subsequences
Define even, odd subseq. of a filter
Theorem If
is any sequence then
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Convolution Representation
If c, d is a pair of wavelet transform filters,
then the wavelet transform of a sequence x is the
pair of sequences in the left side of the
equation below

where
hence
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Even and Odd Subsequences of WT Filters
Theorem If c and d are a pair of wavelet filters
with
then
and
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Even and Odd Subsequences
Define even, odd subseq. of a filter
Theorem
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Matrix Magic
(the polyphase matrix)
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Problem Set 1.
1. Compute the polyphase matrices for the Haar
and Daubechies length wavelet transforms.
2. Show that if
is a unitary space and
then the following conditions are equivalent for
every

3. Show that for every
there exists a unique
that satisfies these conditions.
4. Show that orthogonal projection
is a linear transformation.
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Orthogonal Projectors
Definition. A linear transformation
on a unitary space
is an orthogonal projector if
is an orth. proj.
then
Theorem. If
is an orth. proj. then
If
where
Proof. The first statement is obvious. For all
since P is a projector
hence
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Factorization of Paraunitary Matrices
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Problem Set 2.
1. Show that if
is an orth. proj. then so is
2. Show how to compute the orthogonal projection
onto a subspace using an orthonormal basis for it.
3. Prove that if P is an orth. proj. and Q I-P
then

(ie it is a paraunitary matrix).
4. Compute the factorization of the polyphase
matrices for the Haar and Daubechies length 4 WT
5. Show that if U is a constant unitary matrix
then U(PzQ) (PzQ)U where P and Q are
orthogonal projectors.