In
applied mathematics
Applied mathematics is the application of mathematical methods by different fields such as physics, engineering, medicine, biology, finance, business, computer science, and industry. Thus, applied mathematics is a combination of mathemati ...
, the transfer matrix is a formulation in terms of a
block-Toeplitz matrix of the two-scale equation, which characterizes
refinable function In mathematics, in the area of wavelet analysis, a refinable function is a function which fulfils some kind of self-similarity. A function \varphi is called refinable with respect to the mask h if
:\varphi(x)=2\cdot\sum_^ h_k\cdot\varphi(2\cdot x-k ...
s. Refinable functions play an important role in
wavelet
A wavelet is a wave-like oscillation with an amplitude that begins at zero, increases or decreases, and then returns to zero one or more times. Wavelets are termed a "brief oscillation". A taxonomy of wavelets has been established, based on the num ...
theory and
finite element
The finite element method (FEM) is a popular method for numerically solving differential equations arising in engineering and mathematical modeling. Typical problem areas of interest include the traditional fields of structural analysis, heat t ...
theory.
For the mask
, which is a vector with component indexes from
to
,
the transfer matrix of
, we call it
here, is defined as
:
More verbosely
:
The effect of
can be expressed in terms of the
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 ''com ...
operator "
":
:
Properties
*
.
* If you drop the first and the last column and move the odd-indexed columns to the left and the even-indexed columns to the right, then you obtain a transposed
Sylvester matrix In mathematics, a Sylvester matrix is a matrix associated to two univariate polynomials with coefficients in a field or a commutative ring. The entries of the Sylvester matrix of two polynomials are coefficients of the polynomials. The determinant ...
.
* The determinant of a transfer matrix is essentially a resultant.
:More precisely:
:Let
be the even-indexed coefficients of
(
) and let
be the odd-indexed coefficients of
(
).
:Then
, where
is the
resultant
In mathematics, the resultant of two polynomials is a polynomial expression of their coefficients, which is equal to zero if and only if the polynomials have a common root (possibly in a field extension), or, equivalently, a common factor (ove ...
.
:This connection allows for fast computation using the
Euclidean algorithm.
* For the
trace
Trace may refer to:
Arts and entertainment Music
* ''Trace'' (Son Volt album), 1995
* ''Trace'' (Died Pretty album), 1993
* Trace (band), a Dutch progressive rock band
* ''The Trace'' (album)
Other uses in arts and entertainment
* ''Trace'' ...
of the transfer matrix of
convolved
In mathematics (in particular, functional analysis), convolution is a mathematical operation on two functions ( and ) that produces a third function (f*g) that expresses how the shape of one is modified by the other. The term ''convolution'' ...
masks holds
:
* For 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 a ...
of the transfer matrix of convolved mask holds
:
:where
denotes the mask with alternating signs, i.e.
.
* If
, then
.
: This is a concretion of the determinant property above. From the determinant property one knows that
is
singular
Singular may refer to:
* Singular, the grammatical number that denotes a unit quantity, as opposed to the plural and other forms
* Singular homology
* SINGULAR, an open source Computer Algebra System (CAS)
* Singular or sounder, a group of boar, ...
whenever
is singular. This property also tells, how vectors from the
null space
In mathematics, the kernel of a linear map, also known as the null space or nullspace, is the linear subspace of the Domain of a function, domain of the map which is mapped to the zero vector. That is, given a linear map between two vector space ...
of
can be converted to null space vectors of
.
* If
is an eigenvector of
with respect to the eigenvalue
, i.e.
:
,
:then
is an eigenvector of
with respect to the same eigenvalue, i.e.
:
.
* Let
be the eigenvalues of
, which implies
and more generally
. This sum is useful for estimating the
spectral radius
In mathematics, the spectral radius of a square matrix is the maximum of the absolute values of its eigenvalues. More generally, the spectral radius of a bounded linear operator is the supremum of the absolute values of the elements of its spectru ...
of
. There is an alternative possibility for computing the sum of eigenvalue powers, which is faster for small
.
:Let
be the periodization of
with respect to period
. That is
is a circular filter, which means that the component indexes are
residue class
In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" when reaching a certain value, called the modulus. The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his bo ...
es with respect to the modulus
. Then with the
upsampling
In digital signal processing, upsampling, expansion, and interpolation are terms associated with the process of resampling in a multi-rate digital signal processing system. ''Upsampling'' can be synonymous with ''expansion'', or it can describe a ...
operator
it holds
:
:Actually not
convolutions are necessary, but only
ones, when applying the strategy of efficient computation of powers. Even more the approach can be further sped up using the
Fast Fourier transform.
* From the previous statement we can derive an estimate of the
spectral radius
In mathematics, the spectral radius of a square matrix is the maximum of the absolute values of its eigenvalues. More generally, the spectral radius of a bounded linear operator is the supremum of the absolute values of the elements of its spectru ...
of
. It holds
:
:where
is the size of the filter and if all eigenvalues are real, it is also true that
:
,
:where
.
See also
*
Hurwitz determinant
References
*
* {{cite thesis
, first=Henning
, last=Thielemann
, url=http://nbn-resolving.de/urn:nbn:de:gbv:46-diss000103131
, title=Optimally matched wavelets
, type=PhD thesis
, year=2006
(contains proofs of the above properties)
Wavelets
Numerical analysis