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Theorem 51 H p ( z ) forms an order K , unitary filter bank with PCS symmetry iff

H p ( z ) = I 0 0 J i = 1 K A i z - 1 B i - B i z - 1 A i A 0 B 0 - B 0 A 0 I 0 0 J P

where A i B i - B i A i are constant orthogonal matrices. H p ( z ) is characterized by 2 K M / 2 2 parameters.

Characterization of unitary H p ( z ) — linear-phase symmetry

For the linear-phase case,

H p ( z ) = Q W 0 ( z ) W 0 R ( z ) W 1 ( z ) - W 1 R ( z ) I 0 0 J = 1 2 Q I I I - I W 0 ( z ) + W 1 ( z ) W 0 R ( z ) - W 1 R ( z ) W 0 ( z ) - W 1 ( z ) W 0 R ( z ) + W 1 R ( z ) I 0 0 J = def 1 2 Q I I I - I W 0 ' ( z ) W 1 ' ( z ) ( W 1 ' ) R ( z ) ( W 0 ' ) R ( z ) I 0 0 J .

Therefore, we have the following Theorem:

Theorem 52 H p ( z ) of order K , forms a unitary filter bank with linear-phase filters iff

H p ( z ) = 1 2 Q I I I - I i = 1 K A i z - 1 B i B i z - 1 A i A 0 B 0 B 0 A 0 I 0 0 J ,

where A i B i B i A i are constant orthogonal matrices. H p ( z ) is characterized by 2 K M / 2 2 parameters.

Characterization of unitary H p ( z ) — linear phase and pcs symmetry

In this case, H p ( z ) is given by

H p ( z ) = I 0 0 J W 0 ( z ) D W 0 R ( z ) J D W 0 ( z ) V ( - 1 ) M / 2 W 0 R ( z ) J V = def 1 2 I 0 0 J W 0 ' ( z ) D ( W 0 ' ) R ( z ) J - D W 0 ' ( z ) ( W 0 ' ) R ( z ) J P = 1 2 I 0 0 J I D - D I W 0 ' ( z ) 0 0 ( W 0 ' ) R ( z ) I 0 0 J P .

Therefore we have proved the following Theorem:

Theorem 53 H p ( z ) of order K forms a unitary filter bank with linear-phase and PCS filters iff there exists a unitary, order K , M / 2 × M / 2 matrix W 0 ' ( z ) such that

H p ( z ) = 1 2 I D - J D J W 0 ' ( z ) 0 0 ( W 0 ' ) R ( z ) I 0 0 J P .

In this case H p ( z ) is determined by precisely ( M / 2 - 1 ) L + M / 2 2 parameters where L K is the McMillan degree of W 0 ' ( z ) .

Characterization of unitary H p ( z ) — linear phase and ps symmetry

From the previous result we have the following result:

Theorem 54 H p ( z ) of order K forms a unitary filter bank with linear-phase and PS filters iff there exists a unitary, order K , M / 2 × M / 2 matrix W 0 ' ( z ) such that

H p ( z ) = 1 2 I D - J J D W 0 ' ( z ) 0 0 ( W 0 ' ) R ( z ) I 0 0 J P .

H p is determined by precisely ( M / 2 - 1 ) L + M / 2 2 parameters where L K is the McMillan degree of W 0 ' ( z ) .

Notice that Theorems  "Characterization of Unitary H p (z) — PS Symmetry" through Theorem  "Characterization of Unitary H p (z) — Linear Phase and PS Symmetry" give a completeness characterization for unitary filter banks with thesymmetries in question (and the appropriate length restrictions on the filters). However, ifone requires only the matrices W 0 ' ( z ) and W 1 ' ( z ) in the above theorems to be invertible on the unit circle (and notunitary), then the above results gives a method to generate nonunitary PR filter banks with the symmetries considered. Notice however, thatin the nonunitary case this is not a complete parameterization of all such filter banks.

Linear-phase wavelet tight frames

A necessary and sufficient condition for a unitary (FIR) filter bank to give rise to a compactly supported wavelet tight frame (WTF) isthat the lowpass filter h 0 in the filter bank satisfies the linear constraint [link]

n h 0 ( n ) = M .

We now examine and characterize how H p ( z ) for unitary filter banks with symmetries can be constrained to give rise to wavelet tight frames (WTFs).First consider the case of PS symmetry in which case H p ( z ) is parameterized in [link] . We have a WTF iff

first row of H p ( z ) z = 1 = 1 / M ... 1 / M .

In [link] , since P permutes the columns, the first row is unaffected. Hence [link] is equivalent to the first rows of both W 0 ' ( z ) and W 1 ' ( z ) when z = 1 is given by

2 / M ... 2 / M .

This is precisely the condition to be satisfied by a WTF of multiplicity M / 2 . Therefore both W 0 ' ( z ) and W 1 ' ( z ) give rise to multiplicity M / 2 compactly supported WTFs. If the McMillan degree of W 0 ' ( z ) and W 1 ' ( z ) are L 0 and L 1 respectively, then they are parameterized respectively by M / 2 - 1 2 + ( M / 2 - 1 ) L 0 and M / 2 - 1 2 + ( M / 2 - 1 ) L 1 parameters. In summary, a WTF with PS symmetry can be explicitly parameterized by 2 M / 2 - 1 2 + ( M / 2 - 1 ) ( L 0 + L 1 ) parameters. Both L 0 and L 1 are greater than or equal to K .

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Source:  OpenStax, Wavelets and wavelet transforms. OpenStax CNX. Aug 06, 2015 Download for free at https://legacy.cnx.org/content/col11454/1.6
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