Inversion of z-transforms

Inversion of z transforms by long division is very easily accomplished numerically by a digital computer. The FORTRAN subroutine LONGD given in Table 18.2 performs this long division. The output variable X is calculated for NT sampling times, given the coefficients AQ, X 1), A(2), A M) of the numerator and the coefficients 6(1), 6(2),..., B N) of the denominator. 

Inversion of z transforms by long division is very easily accomplished numerically by a digital computer. The FORTRAN subroutine LONGD given in Table 14.2 performs this long division. The output variable Y is calculated for NT sampling... 

We assume that a molecule, such as NH3, belongs to the point group Csv, and that it has three equivalent bonds represented b3rfunctions 1, 4 2, and Si s, as well as a lone-pair orbital 4 which is not equivalent to the bond orbitals. The z axis is the principal axis. If we act on the i j with the projection operator for the Xth rep, the result will be a linear combination of the functions in 4, that transform like the Xth rep. In this manner we can project new functions d> each of which is a linear combination of the "iTj. The inverse of the transformation that takes the 4 y into the is the transformation that gives us the in terms of the To keep the computation down to a minimum we will treat the projected functions 4> not as linear combinations of s, p, p , and p but as the combinations of the base functions of the axial rotation group so, pi, po, and p i. Thus, there are only three functions involved, namely, 00, 01, and 0 i. These functions transform according to the A(0o) and E [Pg.317]

C. LONG DIVISION. The most interesting and most useful z-transform inversion technique is simple long division of the numerator by the denominator of The ease with which z transforms can be inverted by this technique is one of the reasons why z transforms are often used. 

A combination of Appendix 7.1 and partial fractions may also be used to derive the inverse of a z-transform (i.e. to determine the corresponding time domain function) in a similar manner to that for Laplace transforms. This is illustrated by the following example. 

However, the z-transform f(z) represents a sampled data signal of which f(0 is the continuous form. Thus the inverse of f(z) is the sampled data signal f (t) corresponding to f(f), where, from equation 7.206 ... 

As will be seen, the line generates a corresponding line of F values by inversion of a given transformation and one chooses the maximum value to be on the safe side. That value, for all four transformations, lies at the point (R, Z) = (1 1 L, 0), as will now be shown. 

The inversion of transform (5.3.2) and the determination of L (t ), when the analytical dependence Y j is known, have been considered previouslyThe results were based on the well-known inversion equations of Kirkwood and Fuoss (see also Ref. ) establishing a relationship between L(t ) and the corresponding dynamic compliance function Z(ico). Indeed, the reduced complex dynamic compliance corresponding to a distribution L j) is given by... 

A minimal surface can be represented (locally) by a set of three integrals. They represent the inverse of a mapping from the minimal surface to a Riemann surface. The mapping is a composite one first the minimal surface is mapped onto the unit sphere (the Gauss map), then the sphere itself is mapped onto the complex plane by stereographic projection. Under these operations, the minimal surface is transformed into a multi-sheeted covering of the complex plane. Any point on the minimal surface (except flat points), characterised by cartesian coordinates (x,y,z) is described by the complex number (o, which... 

Red-Al) or the Z-olefin 341 after benzylation. Stereoselective dihydroxylation and regiose-lective protection of the diol provides protected polyol 342 that is transformed in three steps to terminal epoxide 343. Removal of the p-methoxybenzyl group followed by acid-catalyzed epoxide opening yields exclusively tetrahydropyran 344 after protection of the primary alcohol and deacetylation. Inversion of the C2 stereocenter by an oxidation-reduction sequence and deprotection furnishes the C-mimic of a,a-trehalose 345. 

Inversion of Olefin Stereochemistry The preparation of alkenes via inversion of the double bond geometry is an important synthetic transformation. For example, interconversion of the (Z)-alkene to the (ff-isomer depicted below involves treatment of the (Z)-epoxide with the nucleophilic LiPPh2 followed by phosphorus alkylation to furnish the betaine,which undergoes 5yn-elimination to produce the (Ef-alkene. The alkene inversion works for di-, tri-, and tetra-substituted olefins. 

We note that the wave packet (x, t) is the inverse Fourier transform of A K). The mathematical development and properties of Fourier transforms are presented in Appendix B. Equation (1.11) has the form of equation (B.19). According to equation (B.20), the Fourier transform A k) is related to (x, z) by... 

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