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For any give , and have the same energy. Using the above facts, we find that for any projection matrix, ,
has the same energy as . This is equivalent to the fact that 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 and using angle parameters. First consider , with . Clearly, has degrees of freedom. One way to parameterize using angle parameters , would be to define the components of as follows:
As for , it being an orthogonal matrix, it has 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
where are unit norm vectors with the first components of being zero. Each matrix factor when multiplied by a vector , reflects about the plane perpendicular to , hence the name Householder reflections. Since the first components of is zero, and , has degrees of freedom. Each being a unit vector, they can be parameterized as before using angles. Therefore, the total degrees of freedom are
In summary, any orthogonal matrix can be factored into a cascade of reflections about the planes perpendicular to the vectors .
Notice the similarity between Householder reflection factors for and the factors of 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 and unitary vectors [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 .
Perhaps the simplest and most popular way to represent a unitary matrix is by a rotation parameter (not by a Householder reflection parameter). Therefore,the simplest way to represent a unitary matrix 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 matrix (in particular the polyphase matrix of a two channel FIR unitary filter bank) is of the form
where
Equation [link] is the unitary lattice parameterization of . The filters and are given by
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