In
signal processing
Signal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing ''signals'', such as audio signal processing, sound, image processing, images, and scientific measurements. Signal processing techniq ...
, a polyphase matrix is a matrix whose elements are
filter mask
A respirator is a device designed to protect the wearer from inhaling hazardous atmospheres including fumes, vapours, gases and particulate matter such as dusts and airborne pathogens such as viruses. There are two main categories of respir ...
s. It represents a
filter bank
In signal processing, a filter bank (or filterbank) is an array of bandpass filters that separates the input signal into multiple components, each one carrying a single frequency Sub-band coding, sub-band of the original signal. One application of ...
as it is used in
sub-band coders alias
discrete wavelet transform
In numerical analysis and functional analysis, a discrete wavelet transform (DWT) is any wavelet transform for which the wavelets are discretely sampled. As with other wavelet transforms, a key advantage it has over Fourier transforms is temporal ...
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
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An example ...
; i.e.,
convolution
In mathematics (in particular, functional analysis), convolution is a operation (mathematics), mathematical operation on two function (mathematics), functions ( and ) that produces a third function (f*g) that expresses how the shape of one is ...
and
means
downsampling In digital signal processing, downsampling, compression, and decimation are terms associated with the process of ''resampling'' in a multi-rate digital signal processing system. Both ''downsampling'' and ''decimation'' can be synonymous with ''comp ...
.
If the above formula is implemented directly, you will compute values that are subsequently flushed by the down-sampling. You can avoid their computation by splitting the filters and the signal into even and odd indexed values before the wavelet 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
:
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
filterbank
In signal processing, a filter bank (or filterbank) is an array of bandpass filters that separates the input signal into multiple components, each one carrying a single frequency sub-band of the original signal. One application of a filter bank is ...
s,
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
In mathematics, a matrix (plural matrices) is a rectangular array or table of numbers, symbols, or expressions, arranged in rows and columns, which is used to represent a mathematical object or a property of such an object.
For example,
\begi ...
and
module theory
In mathematics, a module is a generalization of the notion of vector space in which the field of scalars is replaced by a ring. The concept of ''module'' generalizes also the notion of abelian group, since the abelian groups are exactly the mod ...
. 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
In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if and ...
of the polyphase matrix is a
Kronecker delta
In mathematics, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually just non-negative integers. The function is 1 if the variables are equal, and 0 otherwise:
\delta_ = \begin
0 &\text i \neq j, \\
1 &\ ...
, which is zero everywhere except for one value.
:
By
Cramer's rule
In linear algebra, Cramer's rule is an explicit formula for the solution of a system of linear equations with as many equations as unknowns, valid whenever the system has a unique solution. It expresses the solution in terms of the determinants o ...
the inverse of
can be given immediately.
:
Orthogonality
Orthogonality means that the
adjoint matrix
In mathematics, the conjugate transpose, also known as the Hermitian transpose, of an m \times n complex matrix \boldsymbol is an n \times m matrix obtained by transposing \boldsymbol and applying complex conjugate on each entry (the complex c ...
is also the inverse matrix of
. The adjoint matrix is the
transposed matrix
In linear algebra, the transpose of a matrix is an operator which flips a matrix over its diagonal;
that is, it switches the row and column indices of the matrix by producing another matrix, often denoted by (among other notations).
The tr ...
with
adjoint filter In signal processing, the adjoint filter mask h^* of a filter mask h is reversed in time and the elements are complex conjugated.
:(h^*)_k = \overline
Its name is derived from the fact that the convolution with the adjoint filter is the adjoint ope ...
s.
:
It implies, that the
Euclidean norm
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean s ...
of the input signals is preserved. That is, the according wavelet transform is an
isometry
In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: ἴσος ''isos'' mea ...
.
:
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
In mathematics, the operator norm measures the "size" of certain linear operators by assigning each a real number called its . Formally, it is a norm defined on the space of bounded linear operators between two given normed vector spaces.
Introdu ...
.
: