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x 2 = P x 2 + ( I - P ) x 2 .

For any give X ( z ) , X ( z ) and z - 1 X ( z ) have the same energy. Using the above facts, we find that for any projection matrix, P ,

D p ( z ) = I - P + z - 1 P X p ( z ) = def T ( z ) X p ( z )

has the same energy as X p ( z ) . This is equivalent to the fact that T ( z ) is unitary on the unit circle (one can directly verify this). Therefore (from [link] ) it follows that the subband signals have the same energy as the original signal.

In order to make the free parameters explicit for filter design, we now describe V 0 and v i using angle parameters. First consider v i , with v i = 1 . Clearly, v i has ( M - 1 ) degrees of freedom. One way to parameterize v i using ( M - 1 ) angle parameters θ i , k , k 0 , 1 , ... , M - 2 would be to define the components of v i as follows:

( v i ) j = l = 0 j - 1 sin ( θ i , l ) cos ( θ i , j ) for j 0 , 1 , ... , M - 2 l = 0 M - 1 sin ( θ i , l ) for j = M - 1 .

As for V 0 , it being an M × M orthogonal matrix, it has M 2 degrees of freedom. There are two well known parameterizations of constant orthogonal matrices,one based on Givens' rotation (well known in QR factorization etc. [link] ), and another based on Householder reflections. In the Householder parameterization

V 0 = i = 0 M - 1 I - 2 v i v i T ,

where v i are unit norm vectors with the first i components of v i being zero. Each matrix factor I - 2 v i v i T when multiplied by a vector q , reflects q about the plane perpendicular to v i , hence the name Householder reflections. Since the first i components of v i is zero, and v i = 1 , v i has M - i - 1 degrees of freedom. Each being a unit vector, they can be parameterized as before using M - i - 1 angles. Therefore, the total degrees of freedom are

i = 0 M - 1 ( M - 1 - i ) = i = 0 M - 1 i = M 2 .

In summary, any orthogonal matrix can be factored into a cascade of M reflections about the planes perpendicular to the vectors v i .

Notice the similarity between Householder reflection factors for V 0 and the factors of H p ( z ) in [link] . Based on this similarity, the factorization of unitary matrices and vectors in this section is called the Householder factorization.Analogous to the Givens' factorization for constant unitary matrices, also one can obtain a factorization of unitary matrices H p ( z ) and unitary vectors V ( z ) [link] . However, from the points of view of filter bank theory and wavelet theory,the Householder factorization is simpler to understand and implement except when M = 2 .

Perhaps the simplest and most popular way to represent a 2 × 2 unitary matrix is by a rotation parameter (not by a Householder reflection parameter). Therefore,the simplest way to represent a unitary 2 × 2 matrix H p ( z ) is using a lattice parameterization using Given's rotations. Since two-channel unitary filter banks play an important role in the theory and designof unitary modulated filter banks (that we will shortly address), we present the lattice parameterization [link] . The lattice parameterization is also obtained by an order-reduction procedurewe saw while deriving the Householder-type factorization in [link] .

Theorem 45 Every unitary 2 × 2 matrix H p ( z ) (in particular the polyphase matrix of a two channel FIR unitary filter bank) is of the form

H p ( z ) = 1 0 0 ± 1 R ( θ K - 1 ) Z R ( θ K - 2 ) Z ... Z R ( θ 1 ) Z R ( θ 0 ) ,

where

R ( θ ) = cos θ sin θ - sin θ cos θ and Z = 1 0 0 z - 1

Equation [link] is the unitary lattice parameterization of H p ( z ) . The filters H 0 ( z ) and H 1 ( z ) are given by

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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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