{{Short description|In wavelet theory}} {{about|the transfer matrix in wavelet theory|the transfer matrix in control systems|Transfer function matrix|the transfer matrix method in statistical mechanics|Transfer-matrix method (statistical mechanics)|the transfer matrix method in optics|Transfer-matrix method (optics)|the transfer matrix in dynamical systems theory|Transfer operator|the transfer matrix in combinatorics|Adjacency matrix|a single scalar|Transfer coefficient (disambiguation)}}
In applied mathematics, 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 <math>h</math>, which is a vector with component indexes from <math>a</math> to <math>b</math>, the transfer matrix of <math>h</math>, we call it <math>T_h</math> here, is defined as :<math> (T_h)_{j,k} = h_{2\cdot j-k}. </math> More verbosely :<math> T_h = \begin{pmatrix} h_{a } & & & & & \\ h_{a+2} & h_{a+1} & h_{a } & & & \\ h_{a+4} & h_{a+3} & h_{a+2} & h_{a+1} & h_{a } & \\ \ddots & \ddots & \ddots & \ddots & \ddots & \ddots \\ & h_{b } & h_{b-1} & h_{b-2} & h_{b-3} & h_{b-4} \\ & & & h_{b } & h_{b-1} & h_{b-2} \\ & & & & & h_{b } \end{pmatrix}. </math> The effect of <math>T_h</math> can be expressed in terms of the downsampling operator "<math>\downarrow</math>": :<math>T_h\cdot x = (h*x)\downarrow 2.</math>
==Properties== {{unordered list | <math>T_h\cdot x = T_x\cdot h</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. {{pb}} More precisely: {{pb}} Let <math>h_{\mathrm{e}}</math> be the even-indexed coefficients of <math>h</math> (<math>(h_{\mathrm{e}})_k = h_{2k}</math>) and let <math>h_{\mathrm{o}}</math> be the odd-indexed coefficients of <math>h</math> (<math>(h_{\mathrm{o}})_k = h_{2k+1}</math>). {{pb}} Then <math>\det T_h = (-1)^{\lfloor\frac{b-a+1}{4}\rfloor}\cdot h_a\cdot h_b\cdot\mathrm{res}(h_{\mathrm{e}},h_{\mathrm{o}})</math>, where <math>\mathrm{res}</math> is the resultant. {{pb}} This connection allows for fast computation using the Euclidean algorithm. | For the trace of the transfer matrix of convolved masks holds {{pb}} <math>\mathrm{tr}~T_{g*h} = \mathrm{tr}~T_g \cdot \mathrm{tr}~T_h</math> | For the determinant of the transfer matrix of convolved mask holds {{pb}} <math>\det T_{g*h} = \det T_g \cdot \det T_h \cdot \mathrm{res}(g_-,h)</math> {{pb}} where <math>g_-</math> denotes the mask with alternating signs, i.e. <math>(g_-)_k = (-1)^k \cdot g_k</math>. | If <math>T_{h}\cdot x = 0</math>, then <math>T_{g*h}\cdot (g_-*x) = 0</math>. {{pb}} This is a concretion of the determinant property above. From the determinant property one knows that <math>T_{g*h}</math> is singular whenever <math>T_{h}</math> is singular. This property also tells, how vectors from the null space of <math>T_{h}</math> can be converted to null space vectors of <math>T_{g*h}</math>. | If <math>x</math> is an eigenvector of <math>T_{h}</math> with respect to the eigenvalue <math>\lambda</math>, i.e. {{pb}} <math>T_{h}\cdot x = \lambda\cdot x</math>, {{pb}} then <math>x*(1,-1)</math> is an eigenvector of <math>T_{h*(1,1)}</math> with respect to the same eigenvalue, i.e. {{pb}} <math>T_{h*(1,1)}\cdot (x*(1,-1)) = \lambda\cdot (x*(1,-1))</math>. | Let <math>\lambda_a,\dots,\lambda_b</math> be the eigenvalues of <math>T_h</math>, which implies <math>\lambda_a+\dots+\lambda_b = \mathrm{tr}~T_h</math> and more generally <math>\lambda_a^n+\dots+\lambda_b^n = \mathrm{tr}(T_h^n)</math>. This sum is useful for estimating the spectral radius of <math>T_h</math>. There is an alternative possibility for computing the sum of eigenvalue powers, which is faster for small <math>n</math>. {{pb}} Let <math>C_k h</math> be the periodization of <math>h</math> with respect to period <math>2^k-1</math>. That is <math>C_k h</math> is a circular filter, which means that the component indexes are residue classes with respect to the modulus <math>2^k-1</math>. Then with the upsampling operator <math>\uparrow</math> it holds {{pb}} <math>\mathrm{tr}(T_h^n) = \left(C_k h * (C_k h\uparrow 2) * (C_k h\uparrow 2^2) * \cdots * (C_k h\uparrow 2^{n-1})\right)_{[0]_{2^n-1}}</math> {{pb}} Actually not <math>n-2</math> convolutions are necessary, but only <math>2\cdot \log_2 n</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 <math>\varrho(T_h)</math>. It holds {{pb}} <math>\varrho(T_h) \ge \frac{a}{\sqrt{\# h}} \ge \frac{1}{\sqrt{3\cdot \# h}}</math> {{pb}} where <math>\# h</math> is the size of the filter and if all eigenvalues are real, it is also true that {{pb}} <math>\varrho(T_h) \le a</math>, {{pb}} where <math>a = \Vert C_2 h \Vert_2</math>. }}
==See also== * Hurwitz determinant
==References== * {{cite journal |first=Gilbert|last=Strang |author-link=Gilbert Strang |title=Eigenvalues of <math>(\downarrow 2){H}</math> and convergence of the cascade algorithm |journal=IEEE Transactions on Signal Processing |volume=44 |pages=233–238 |year=1996 |doi=10.1109/78.485920 }} * {{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)
Category:Wavelets Category:Numerical analysis