Definition of z-Transforms

A. LINEARITY. The linearity property is easily proved from the definition of z transformation. 

Passing the step function through an impulse sampler gives = Kun(t)I t)-, where /(,) is the sequenee of unit impulses defined in Eq. (14.3). Using the definition of z transformation [Eq. (14.22)J gives... 

The notation 2 ) [] means z transforming using the Z variable where Z = ze This theorem is proved by going back to the definition of z transformation. 

F. UNIT IMPULSE FUNCTION. By definition, the z transformation of an impulse-sampled function is... 

Using the definition of the modified z transformation given in Eq. (18.127) gives... 

In order to extract some more information from the csa contribution to relaxation times, the next step is to switch to a molecular frame (x,y,z) where the shielding tensor is diagonal (x, y, z is called the Principal Axis System i.e., PAS). Owing to the properties reported in (44), the relevant calculations include the transformation of gzz into g x, yy, and g z involving, for the calculation of spectral densities, the correlation function of squares of trigonometric functions such as cos20(t)cos20(O) (see the previous section and more importantly Eq. (29) for the definition of the normalized spectral density J((d)). They yield for an isotropic reorientation (the molecule is supposed to behave as a sphere)... 

From the definition of the Laplace transform, Eq. (11.28), it is straightforward to show that it replaces a differential operator d/dt by the Laplace variable s (see Appendix G for details). The feedback circuit is typically an amplifier with an RC network, as shown in Fig. 11.6. The RC network is used for compensation, which will be explained here. By denoting the Laplace transform of the voltage on the z piezo, Vz(t), by U(s), the Laplace transform of the feedback circuit is... 

Consider first the series junction of N waveguides containing transverse force and velocity waves. At a series junction, there is a common velocity while the forces sum. For definiteness, we may think of A ideal strings intersecting at a single point, and the intersection point can be attached to a lumped load impedance Rj (s), as depicted in Fig. 10.11 for TV= 4. The presence of the lumped load means we need to look at the wave variables in the frequency domain, i.e., V(s) = C v for velocity waves and F(s) = C / for force waves, where jC denotes the Laplace transform. In the discrete-time case, we use the z transform instead, but otherwise the story is identical. 

At this point it should be observed that even if it is natural to assume that all the functions F = F(Z) in the definition space A = F are analytic functions of the composite complex variable Z = zt, z2,..., zN, one has still to be very careful in the definition of the complex transformation U. Some of the complications which may occur are well illustrated by the simple example of complex scaling as defined in Eq. (2.26). 

Pq, is expressed in Cartesian coordinates. These polar tensors T), can be derived from experimental intensities by elementary coordinate transformation. If the axes x, y, and z are chosen such that the bonds are oriented along one of the axes, then the derivatives can be used to interpret the changes of the electron clouds during a vibration. Besides, considering the definitions of the axes, it is possible to transfer atomic polar tensors between similar molecules and to estimate their intensities (Person and Newton, 1974 Person and Overend, 1977). 

The application of the z-transform and of the coherence theory to the study of displacement chromatography were initially presented by Helfferich [35] and later described in detail by Helfferich and Klein [9]. These methods were used by Frenz and Horvath [14]. The coherence theory assumes local equilibrium between the mobile and the stationary phase gleets the influence of the mass transfer resistances and of axial dispersion (i.e., it uses the ideal model) and assumes also that the separation factors for all successive pairs of components of the system are constant. With these assumptions and using a nonlinear transform of the variables, the so-called li-transform, it is possible to derive a simple set of algebraic equations through which the displacement process can be described. In these critical publications, Helfferich [9,35] and Frenz and Horvath [14] used a convention that is opposite to ours regarding the definition of the elution order of the feed components. In this section as in the corresponding subsection of Chapter 4, we will assume with them that the most retained solute (i.e., the displacer) is component 1 and that component n is the least retained feed component, so that... 

Although the definition above refers to the sequence of sampled values of a continuous function y(t), it is customary to talk about the z-transform of the function y(t), and it is depicted by... 

Let, then be a list allowable starting materials. Each set L of materials drawn from determines a definite EH(JL), represented by a r-matrix B L). If there is a reaction matrix R such that B h) + R represents an EM that contains the molecole Z, we shall say that the ordered pair (L,R) is a synthesis of Z from L. Since each reaction matrix (as we shall see) is a unique combination of redox and homoaptic-homol rtic reactions performed in a certain order, each pair (L,J7) gives a synthetic pathway in which each intermediate transformation is one of four elementary types.h We shall let... 

Table 9.6 summarizes many definitions of mean diameter. Note that the definition for number can be substituted for mass. The number length mean becomes the mass length mean—the symbol would be Z)m[l,0]. Most often, data is reported as mass fraction and, therefore, to apply the number definitions requires a transformation from the mass fractions ... 

Let us consider the extent of reaction and 2 (for first and second reactions, respectively). Then, by definition of yield (pA or selectivity Sa, the mole number of A transformed in R at the time (t) will be a (batch) or position (z) (continuous), and therefore. 

---