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which can be read as The limiting value of z n as n becomes in nitely great is zero, or as z n becomes equal to zero if n becomes in nitely great. A comparison of eqs. (572) and (573) shows that F z represents vs t in the complex plane. This means that a given vs t can be manipulated algebraically in terms of z instead of t, which is found to be a great advantage. In practical applications we work in terms of v nT , the SEQUENCE OF SAMPLES generated by the sampling of an analog signal. Thus, suppose we wish to nd the result of applying a given v nT to the input of a particular digital logic network. To do this, we must rst express v nT in terms of z, which we do by substituting the given v nT into eq. (573). The expression for F z , thus found, is called the z-transform of the sequence v nT . Let us, therefore, begin by nding the z-transforms of some of the most-used forms of v nT encountered in practical work. The simplest v nT signal is called the unit pulse, which consists of just one sample of unit amplitude at t 0, as illustrated in conventional form in Fig. 332.

ArrayList errorList = new ArrayList();

We ll denote the unit pulse by p nT . Note that p nT 1 for n 0, but p nT 0 for all other n. Thus, in eq. (573), for the case of v nT p nT , we have v nT 1 for n 0 but v nT 0 for all other n. Thus, substituting these values into eq. (573), we have that F z 1 that is, the z-transform of the unit pulse is 1. 575

Note that, graphically, the sample values v nT are plotted against nT, where n is the number of the sample counted from the n 0 reference. Next let s consider the very important unit-step sequence, in which all the samples have unit amplitude; that is, v nT 1 for all values of nT, as shown in Fig. 333.

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The errorList is used to hold any error messages generated by the next several lines of code that validate all required search fields have been entered. It goes without saying that the Search Crawler will not function without a URL that specifies the location at which to start crawling. The following code verifies that a starting URL has been entered and that the URL is valid:

We ll denote this unit-step sequence by U nT . Note that U nT 1 for all values of n, including n 0. Thus, substituting v nT U nT 1 into eq. (573) for all values of nT n 0; 1; 2; 3; . . . ; n , we have that F z 1 z 1 z 2 z 3 z n for n ! 1 576

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The above is a valid answer, but can be put in a non-series or closed form as follows. First, multiply both sides of the equation by z 1 , then add the two equations together; doing this will show that (see problem 295 below) F z

// Validate that the start URL has been entered. String startUrl = startTextField.getText().trim(); if (startUrl.length() < 1) { errorList.add("Missing Start URL.");

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