Transfer Matrix

Information about Transfer Matrix

The transfer matrix is a formulation in terms of a block-Toeplitz matrix of the two-scale equation, which characterizes refinable functions. Refinable functions play an important role in wavelet theory and finite element theory.

For the mask $\text{[math]}$ , which is a vector with component indexes from $\text{[math]}$ to $\text{[math]}$ , the transfer matrix of $\text{[math]}$ , we call it $\text{[math]}$ here, is defined as

$\text{[math]}$ .

More verbosely

$\text{[math]}$

The effect of $\text{[math]}$ can be expressed in terms of the downsampling operator " $\text{[math]}$ ":

$\text{[math]}$ .

Properties

$\text{[math]}$ .
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.
The determinant of a transfer matrix is essentially a resultant.

More precisely:

Let $\text{[math]}$ be the even indexed coefficients of $\text{[math]}$ ( $\text{[math]}$ ) and let $\text{[math]}$ be the odd indexed coefficients of $\text{[math]}$ ( $\text{[math]}$ ).

Then $\text{[math]}$ , where $\text{[math]}$ is the resultant.

This connection allows for fast computation using the Euclidean algorithm.

For the trace of the transfer matrix of convolved masks holds

$\text{[math]}$

For the determinant of the transfer matrix of convolved mask holds

$\text{[math]}$

where $\text{[math]}$ denotes the mask with alternating signs, i.e. $\text{[math]}$ .

If $\text{[math]}$ , then $\text{[math]}$ .

This is a concretion of the determinant property above. From the determinant property one knows that $\text{[math]}$ is singular whenever $\text{[math]}$ is singular. This property also tells, how vectors from the null space of $\text{[math]}$ can be converted to null space vectors of $\text{[math]}$ .

If $\text{[math]}$ is an eigenvector of $\text{[math]}$ with respect to the eigenvalue $\text{[math]}$ , i.e.

$\text{[math]}$ ,

then $\text{[math]}$ is an eigenvector of $\text{[math]}$ with respect to the same eigenvalue, i.e.

$\text{[math]}$ .

Let $\text{[math]}$ be the eigenvalues of $\text{[math]}$ , which implies $\text{[math]}$ and more generally $\text{[math]}$ . This sum is useful for estimating the spectral radius of $\text{[math]}$ . There is an alternative possibility for computing the sum of eigenvalue powers, which is faster for small $\text{[math]}$ .

Let $\text{[math]}$ be the periodization of $\text{[math]}$ with respect to period $\text{[math]}$ . That is $\text{[math]}$ is a circular filter, which means that the component indexes are residue classes with respect to the modulus $\text{[math]}$ . Then with the upsampling operator $\text{[math]}$ it holds

$\text{[math]}$

Actually not $\text{[math]}$ convolutions are necessary, but only $\text{[math]}$ 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 of $\text{[math]}$ . It holds

$\text{[math]}$

where $\text{[math]}$ is the size of the filter and if all eigenvalues are real, it is also true that

$\text{[math]}$ ,

where $\text{[math]}$ .

References

Gilbert Strang: Eigenvalues of $\text{[math]}$ and convergence of the cascade algorithm. IEEE Transactions on Signal Processing, 44:233-238, 1996.
Henning Thielemann: Optimally matched wavelets, 2006 (contains proofs of the above properties)

In mathematics, in the area of wavelet analysis, a refinable function is a function which fulfills some kind of self-similarity. A function is called refinable with respect to the mask if

This condition is called refinement equation,
..... Click the link for more information.

A wavelet is a kind of mathematical function used to divide a given function into different frequency components and study each component with a resolution that matches its scale. A wavelet transform is the representation of a function by wavelets.
..... Click the link for more information.

Finite element analysis (FEA) is a computer simulation technique used in engineering analysis. It uses a numerical technique called the finite element method (FEM). There are many finite element software packages, both free and proprietary.
..... Click the link for more information.

For the computer graphics technique, see .

In signal processing, downsampling (or "subsampling") 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.
..... Click the link for more information.

In mathematics, a Sylvester matrix is a matrix associated to two polynomials that gives us some information about those polynomials. It is named for James Joseph Sylvester.
..... Click the link for more information.

resultant of two monic polynomials and over a field is defined as the product

of the differences of their roots, where and take on values in the algebraic closure of .
..... Click the link for more information.

Euclidean algorithm (also called Euclid's algorithm) is an algorithm to determine the greatest common divisor (GCD) of two elements of any Euclidean domain (for example, the integers).
..... Click the link for more information.

In linear algebra, the trace of an n-by-n square matrix A is defined to be the sum of the elements on the main diagonal (the diagonal from the upper left to the lower right) of A, i.e.
..... Click the link for more information.

convolution is a mathematical operator which takes two functions f and g and produces a third function that in a sense represents the amount of overlap between f and a reversed and translated version of g.
..... Click the link for more information.

In algebra, a determinant is a function depending on n that associates a scalar, det(A), to every nÃ—n square matrix A. The fundamental geometric meaning of a determinant is as the scale factor for volume when A
..... Click the link for more information.

invertible or non-singular if there exists an n-by-n matrix such that

where denotes the n-by-n identity matrix and the multiplication used is ordinary matrix multiplication.
..... Click the link for more information.

Null space may refer to:

kernel (matrix) In linear algebra, the null space of a matrix.
kernel (linear operator) In advanced linear algebra and functional analysis, the null space of a linear operator.

..... Click the link for more information.

In mathematics, the spectral radius of a matrix or a bounded linear operator is the supremum among the moduli of the elements in its spectrum, which is sometimes denoted by ρ(·).
..... Click the link for more information.

Upsampling is the process of increasing the sampling rate of a signal.

The upsampling factor (commonly denoted by L) is usually an integer or a rational fraction greater than unity.
..... Click the link for more information.

fast Fourier transform (FFT) is an efficient algorithm to compute the discrete Fourier transform (DFT) and its inverse. FFTs are of great importance to a wide variety of applications, from digital signal processing and solving partial differential equations to algorithms for
..... Click the link for more information.

This article is copied from an article on Wikipedia.org - the free encyclopedia created and edited by online user community. The text was not checked or edited by anyone on our staff. Although the vast majority of the wikipedia encyclopedia articles provide accurate and timely information please do not assume the accuracy of any particular article. This article is distributed under the terms of GNU Free Documentation License.

Herod_Archelaus

iPedia Free Information Center

Transfer Matrix

Information about Transfer Matrix

Properties

References