Inversion Z transforms

Find the inverse z-transform of every partial fraction. Then we can claim that the sequence of sampled values of the unknown function y(t) with the given z-transform is... 

The inverse transforms of the partial fractions can be found easily from Table 28.1 or similar but more inclusive tables. Such tables yield the inverse z-transform in terms of continuous functions. For example, if... 

We should remember, though, that the inverse z-transform cannot... 

Example 28.4 Inverse z Transform of a More Complex Expression... 

VII.15 Find the inverse z-transform of the following expressions, using long division. 

With the inverse z-transform we attempt to take back the values of a function at the sampling instants, given its z-transform. The inverse z-transform operation is symbolized as follows ... 

The inverse z-transform does not even help us to determine the sampling period T for the computed sampled values y(0), y(T), y(2T),. ... 

If we attempt to find a continuous function y(t) which coincides with the sampled values j/(0), y(T), y(2T), produced by the inverse z-transform, we should remember that these sampled values could be derived from two distinct functions (see also Remark 3 in Section 28.1). It follows, then, that the inverse transform of a function y(z) does not necessarily yield a unique continuous function y(t). 

Let us now proceed with the mechanics of two methods used to determine inverse z-transforms. The first is the partial-fractions expansion and the second is based on the long division of two polynomials. 

Example 28.3 Inverse z-Transform by Partial-Fractions Expansion... 

The procedure of the long division for computing the inverse z-transform is very simple and can be used for any expression. The expansion to partial fractions, on the other hand, could be a very difficult procedure, if it is very hard to determine the partial fractions or the partial fractions cannot be found among the terms of Table 28.1 or other equivalent. 

Does the inverse z-transform yield unique values at the sampling instants Explain why or why not. 

Does the inverse z-transform help you find the sampling period Explain. 

Equation (29.3) yields the discrete transfer function of the velocity PID control algorithm. For given changes in the sampled values of the error signal, the resulting discrete-time control action can be found from the inverse z-transform ... 

Take the kth term within the braces from eq. (29.27) and find its inverse z-transform. From Table 28.1 we can easily find that... 

VII.16 Compute the inverse z-transforms of the expressions given in Problem VII. 14. 

The z transformation of an impulse-sampled function is unique i.e., there is only one F(f.) that is the z transform of a given f y The inverse z transform of any is also unique i.e., there is only one f(nT,) that corresponds to a given F(,). 

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