The Z Transform

The z-transform is the prineipal analytieal tool for single-input-single-output dis-erete-time systems, and is analogous to the Laplaee transform for eontinuous systems. 

Coneeptually, the symbol z ean be assoeiated with diserete time shifting in a differenee equation in the same way that. v ean be assoeiated with differentiation in a differential equation. 

Taking Laplaee transforms of equation (7.1), whieh is the ideal sampled signal, gives 

We now introduce another important transform, the z-transform The z-transform is perhaps the most useful of all transforms in practical digital signal processing. It is defined as  

An important property of the z-transform is its affect on a delay. If we define a signal 

The summation in m is just the z-transform of x defined in m rather, than n and so we can write  

Hence delaying a signal by nj multiplies its z-transform by z . This important property of the z-transform and is discussed further in section 10.3.5. 

Why use yet another transform The reason is that by setting z to we greatly simplify the DTFT, and this simplification allows us to perform sophisticated operations in analysis and synthesis with relatively simple mathematieal operations. The key to the ease of use of the z-transform, is that equation 10.34, can be re-written as  

A common analytical tool for digital filters is the Z transform representation. As we said before, weTl define Z (Z to the minus 1) as a single sample of delay, and in fact, Z is sometimes called the Delay Operator. To transform a filter using the Z transform, simply capitalize all variables x andy, and replace all time indices ( - a) with the appropriate time delay operator Z . Thus, the Z transformed version of Equation 3.3 would be written  

We ll see in subsequent sections and chapters how the Z transform can be used for analyzing and manipulating signals and filters. 

Armed with this fact, it becomes trivial to see 

The triviality of the equivalence applies only to finite sequences. We take on trust that the results apply also to products where one or both factors extend indefinitely in both directions. We then need to use the concept of 

8 Not quite. The individual polynomial factors can be multiplied by arbitrary scalar factors provided that the product of all those factors is equal to 1. We shall resolve this later by using only polynomials whose coefficients sum to 1. 

Laurent Polynomial where the exponent in a power of z can be negative as well as positive or zero. 

The Laurent polynomial corresponding to a given sequence is called its Generating Function, its z-transform, or its symbol. 

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See Refs. 198, 218, and 256. The z -transform is useful when data is available at only discrete points. Let... 

The z -transform is used in process control when the signals are at intervals of At. A brief table (Table 3-2) is provided here. 

The z -transform can also be used to solve difference equations, just like the Laplace transform can be used to solve differential equations. 

Figure 8-5 illustrates the concept of samphng a continuous function. At integer values of the saiTmling rate. At, the value of the variable to be sampled is measured and held until the next sampling instant. To deal with sampled data systems, the z transform has been developed. The z transform of the function given in Fig. 8-5 is defined as... 

To obtain the z-transform of a first-order sampled data system in cascade with a zero-order hold (zoh), as shown in Figure 7.10. 

See Ogunnaike, Babatunde A., and W. Harmon Ray, Process Dynamics, Modeling, and Control, Oxford University Press (1994) Seborg, D., T. F. Edgar, and D. A. Mellichamp, Process Dynamics and Control, 2d ed., Wiley, New York (2003). The z-transform is useful when data is available at only discrete points. Let... 

In Chap. 18 we will define mathematically the sampling process, derive the z transforms of common functions (learn our German vocabulary) and develop transfer functions in the z domain. These fundamentals are then applied to basic controller design in Chap. 19 and to advanced controllers in Chap. 20. We will find that practically all the stability-analysis and controller-design techniques that we used in the Laplace and frequency domains can be directly applied in the z domain for sampled-data systems. 

Sequences of impulses, such as the output of an impulse sampler, can be z-transformed. For a specified sampling period 7, the z transformation of an impulse-sampled signal/,, is defined by the equation... 

The notation ] means the z-transformation operation. The values are the magnitudes of the continuous function / ) (before impulse-sampling) at the sampling periods. We will use the notation that the z transform of/J, is F,, . 

You should now be able to guess what the z transformation of would be. We know there would be an term in the denominator of the Laplace transformation of this function. So we can extrapolate our results to predict that there would be a (z — 1) in the denominator of the z transformation. 

The z transformation of this function after impulse sampling is... 

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

But if/( ) must be equal to just, the term /,q in the equation above must be equal to 1 and all the other termsniust be equal to zero. Therefore the z transformation of the unit impulse is unity. 

Example 18.3. Suppose we want to take the z transformation of the function... 

We know how to find the z transformations of functions. Let us now turn to the problem of expressing input-output transfer-function relationships in the z domain. Figure 18.9a shows a system with samplers on the input and on the output of the process. Time, Laplace, and z-domain representations are shown. G(2, is called a pulse transfer function. It will be defined below. 

Defining in this way permits us to use transfer functions in the z domain [Eq. (18.57)] just as we use transfer functions in the Laplace domain. G,, is the z transform of the impulse-sampled response of the process to a unit impulse function <5( . In z-transforming functions, we used the notation =... 

To generate the Nyquist plots discussed above, the z transform of the appropriate transfer functions must first be obtained. Then is substituted for z, and 0) is varied from 0 to a>J2. 

The big advantage of this method is that the analytical step of taking the z transformation is eliminated. You just deal with the original continuous transfer functions. For complex, high-order systems, this can eliminate a lot of messy algebra. 

The transfer function approach will be used where appropriate throughout the remainder of this chapter. Transfer functions of continuous systems will be expressed as functions of s, e.g. as G(j) or H(s). In the case of discrete time systems, the transfer function will be written in terms of the z-transform, e.g. as G(z) or H(z) (Section 7.17). An elementary knowledge of the Laplace transformation on the part of the reader is assumed and a table of the more useful Laplace transforms and their z-transform equivalents appears in Appendix 7.1. 

It can be seen from the above that the output of a sampler and its transform can both be represented by infinite series. The use of the z-transform simplifies the treatment of such systems, and relationships for sampled data processes can be derived in terms of z-transforms which are similar to those obtained for equivalent continuous systems employing the Laplace transform. 

For simple functions the z-transforms can be obtained easily from their equivalent Laplace transforms by the use of partial fractions and Appendix 7.1. 

Determine the z-transform equivalent of the transfer function of the heat exchanger system described in Example 7.3. 

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 ... 

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