Toeplitz matrix

Solution to the above Mth order equation always exists if the M + 1 by M + 1 Toeplitz matrix has a non-negative definite. In a typical experimental situation, lm f i-6-. the nonresonant contribution, dominates the CARS... 

Under these conditions, the spread function is termed shift invariant and s is called a Toeplitz matrix. When Eq. (17) does not hold, the spread function is called shift variant, and the spreading can no longer be described by simple convolution. 

I. INTRODUCTION AND EXAMPLES A Toeplitz matrix of order N has the form... 

The (i, )th element of a Toeplitz matrix depends onfy on This properly of the DAF propagator yields an efficient... 

See Ref. [148] for a physically appealing derivation.) Equation (12) yields an efficient, accurate and formally exact scheme to perform quantum dynamics. It involves the action of a handed, sparse and Toeplitz matrix on a vector. The propagation scheme has been shown to accurately represent [147] all quantum dynamical features including zero-point effects, tunneling as well as over-barrier reflections and in this sense differs from standard semi-classical treatments. The approach also substantially differs from other formalisms such as centroid dynamics [97,98,168-171], where the Feynman path centroid is propagated in a classical-like... 

A Toeplitz matrix T = [t,y] has equal elements along diagonals parallel to the principal diagonal. A n x n Toeplitz matrix is defined as... 

We now generate a filter matrix, W, as an infinite block Toeplitz matrix ... 

Unlike the square matrix in Equation 12.22, this matrix is symmetric and all the elements in its diagonals are symmetric. This is known as a Toeplitz matrix, and because of the properties just mentioned, it is significantly easier to invert. 

Many scientific computations (for signal processing, for partial differential equations, in statistics, for approximation of functions by polynomials, and so on) are reduced to solving Toeplitz or Hankel systems Ax = b, having Toeplitz or Hankel matrix A. A Hankel matrix becomes a Toeplitz matrix by appropriate row interchange (reflecting the rows about the median row), so we shall consider only Toeplitz systems. Anmxn Toeplitz matrix is defined by its first row and its first column, which requires only m + n 1 units of storage space. 

The description used here also includes the casting function f 1() on the input samples. Then, each subset of N elements (produced with the k iterator in node 4) is compared to the subset of the first N elements (produced in node 3). This comparison uses an accumulated multiplication of the two sets (node 5) for each value of the iterator k, resulting in the auto-correlation coefficients r [] []. In the CD interpolation application, these coefficients are considered as Toeplitz matrix elements after some further postprocessing embedded in the functions f2()-f5(). 

A square matrix Q of size N x N is called Toeplitz matrix of order N if Qij = Bi-j, i.e. matrix elements on any line parallel to the main diagonal are the same [52],... 

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