In signal processing

Signal processing

Signal processing is an area of systems engineering, electrical engineering and applied mathematics that deals with operations on or analysis of signals, in either discrete or continuous time...

,
a polyphase matrix is a matrix whose elements are filter mask

Linear filter

Linear filters in the time domain process time-varying input signals to produce output signals, subject to the constraint of linearity.This results from systems composed solely of components classified as having a linear response....

s.
It represents a filter bank

Filter bank

In signal processing, a filter bank is an array of band-pass filters that separates the input signal into multiple components, each one carrying a single frequency subband of the original signal. One application of a filter bank is a graphic equalizer, which can attenuate the components...

 as it is used
in sub-band coders alias discrete wavelet transform

Discrete wavelet transform

In numerical analysis and functional analysis, a discrete wavelet transform is any wavelet transform for which the wavelets are discretely sampled...

s.
If are two filters,
then one level the traditional wavelet transform
maps an input signal to two output signals ,
each of the half length:
Note, that the dot means polynomial multiplication, i.e. convolution

Convolution

In mathematics and, in particular, functional analysis, convolution is a mathematical operation on two functions f and g, producing a third function that is typically viewed as a modified version of one of the original functions. Convolution is similar to cross-correlation...

 and
means downsampling

Downsampling

In signal processing, downsampling is the process of reducing the sampling rate of a signal. This is usually done to reduce the data rate or the size of the data....

.

If the above formula is implemented directly, you will compute values
that are subsequently flushed by the down-sampling.
You can avoid that by splitting the filters and the signal
into even and odd indexed values before the transformation.
The arrows and
denote left and right shifting, respectively.
They shall have the same precedence like convolution,
because they are in fact convolutions with a shifted discrete delta impulse.
The wavelet transformation reformulated to the split filters is:
This can be written as matrix-vector-multiplication

Matrix multiplication

In mathematics, matrix multiplication is a binary operation that takes a pair of matrices, and produces another matrix. If A is an n-by-m matrix and B is an m-by-p matrix, the result AB of their multiplication is an n-by-p matrix defined only if the number of columns m of the left matrix A is the...

This matrix is the polyphase matrix.

Of course, a polyphase matrix can have any size,
it need not to have square shape.
That is, the principle scales well to any filterbanks,
multiwavelets,
wavelet transforms based on fractional refinements.

Properties

The representation of sub-band coding by the polyphase matrix
is more than about write simplification.
It allows the adaptation of many results from matrix theory and module theory.
The following properties are explained for a matrix,
but they scale equally to higher dimensions.

Invertibility / Perfect reconstruction

The case that a polyphase matrix allows reconstruction of a processed signal from the filtered data,
is called perfect reconstruction property.
Mathematically this is equivalent to invertibility.
According to the theorem of invertibility
of a matrix over a ring,
the polyphase matrix is invertible if and only if
the determinant

Determinant

In linear algebra, the determinant is a value associated with a square matrix. It can be computed from the entries of the matrix by a specific arithmetic expression, while other ways to determine its value exist as well...

 of the polyphase matrix is a Kronecker delta,
which is zero everywhere except of one value.

By Cramer's rule

Cramer's rule

In linear algebra, Cramer's rule is a theorem, which gives an expression for the solution of a system of linear equations with as many equations as unknowns, valid in those cases where there is a unique solution...

 the inverse of
can be given immediately.

Orthogonality

Orthogonality means that the adjoint matrix 
is also the inverse matrix of .
The adjoint matrix is the transposed matrix with adjoint filters.

It implies, that the Euclidean norm of the input signals is preserved.
That is, the according wavelet transform is an isometry

Isometry

In mathematics, an isometry is a distance-preserving map between metric spaces. Geometric figures which can be related by an isometry are called congruent.Isometries are often used in constructions where one space is embedded in another space...

.

The orthogonality condition
can be written out

Operator norm

For non-orthogonal polyphase matrices the question arises
what Euclidean norms the output can assume.
This can be bounded by the help of the operator norm

Operator norm

In mathematics, the operator norm is a means to measure the "size" of certain linear operators. Formally, it is a norm defined on the space of bounded linear operators between two given normed vector spaces.- Introduction and definition :...

.
For the polyphase matrix
the Euclidean operator norm can be given explicitly
using the Frobenius norm  and the z transform :


This is a special case of the matrix
where the operator norm can be obtained via z transform
and the spectral radius

Spectral radius

In mathematics, the spectral radius of a square matrix or a bounded linear operator is the supremum among the absolute values of the elements in its spectrum, which is sometimes denoted by ρ.-Matrices:...

 of a matrix or the according spectral norm.
A signal, where these bounds are assumed
can be derived from the eigenvector
corresponding to the maximizing and minimizing eigenvalue.

Lifting scheme

The concept of the polyphase matrix allows matrix decomposition

Matrix decomposition

In the mathematical discipline of linear algebra, a matrix decomposition is a factorization of a matrix into some canonical form. There are many different matrix decompositions; each finds use among a particular class of problems.- Example :...

.
For instance the decomposition into addition matrices

Triangular matrix

In the mathematical discipline of linear algebra, a triangular matrix is a special kind of square matrix where either all the entries below or all the entries above the main diagonal are zero...

leads to the lifting scheme

Lifting scheme

The lifting scheme is a technique for both designing wavelets and performing the discrete wavelet transform.Actually it is worthwhile to merge these steps and design the wavelet filters while performing the wavelet transform....

.

However, classical matrix decompositions like LU

LU decomposition

In linear algebra, LU decomposition is a matrix decomposition which writes a matrix as the product of a lower triangular matrix and an upper triangular matrix. The product sometimes includes a permutation matrix as well. This decomposition is used in numerical analysis to solve systems of linear...

and QR decomposition

QR decomposition

In linear algebra, a QR decomposition of a matrix is a decomposition of a matrix A into a product A=QR of an orthogonal matrix Q and an upper triangular matrix R...

 cannot be applied immediately,
because the filters form a ring with respect to convolution,
not a field.