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It thus follows that if the answer to a DT problem comes out in the basic form of eq. (583), then there is no di culty in expressing the answer in the time domain, because the sample values of y t , that is, y 0 ; y T ; y 2T , and so on, required in eq. (584), appear directly as the coe cients of the powers of z in eq. (583). The practical di culty, however, is that a solution in the z-domain does not generally come out in the basic form of eq. (583); instead, the solution comes out in the form of the ratio of two polynomials in z, in which case the required sample values of y t cannot be found by simple inspection of the solution (as would be the case if the solution were in the form of eq. (583). It is possible, however, to put Y z directly into the form of eq. (583) by the use of ALGEBRAIC LONG DIVISION,* as the following examples will illustrate. Example 1

// Switch search button back to "Search." searchButton.setText("Search"); // Return to default cursor. setCursor(Cursor.getDefaultCursor());

Write the function Y z z= z 0:6) in the time domain, that is, in the form of eq. (584).

Solution The rst step is to put the given Y z into the form of eq. (583), which, as mentioned above, can be done by using algebraic long division. For the given Y z the details are as follows. 1 0:6z 1 0:36z 2 0:216z 3 0:1296z 4 z 0:6 j z z 0:6 0:6 0:6 0:36z 1 0:36z 1 0:36z 1 0:216z 2 0:216z 2 0:216z 2 0:1296z 3 0:1296z 3

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First, the Crawling field is updated to display Done. Second, each of the search controls is reenabled. Third, the Stop button is reverted back to displaying Search. Finally, the cursor is reverted back to the default application cursor. If the search did not yield any matches, the following code displays a dialog box to indicate this fact:

We continue in this manner until the change in the values of the coe cients becomes as small as we wish, at which point we terminate the series. In the above case, we can continue on for several more terms until we have the good approximation that z 1 0:6z 1 0:36z 2 0:216z 3 0:1296z 4 0:0778z 5 0:0467z 6 z 0:6 which is now in the form of eq. (583), and thus, by direct comparison with (583), we see that the actual sampled values of y t , at t 0; T; 2T; . . . ; 6T, are y 0 1:0 y T 0:6 y 2T 0:36 Now, putting these values into eq. (584), we have that the equation for the sampled function in the time domain is ys t t 0:6 t T 0:36 t 2T 0:216 t 3T 0:1296 t 4T 0:0778 t 5T 0:0467 t 6T We can now use the above to pictorially show the form of y nT versus nT in the manner previously described. The result is shown in Fig. 337. y 3T 0:216 y 4T 0:1296 y 5T 0:0778 y 6T 0:0467

z . z 0:6

// Show message if search string not found. if (table.getRowCount() == 0) { JOptionPane.showMessageDialog(SearchCrawler.this, "Your Search String was not found. Please try another.", "Search String Not Found", JOptionPane.WARNING_MESSAGE); }

Solution We rst put Y z into the form of eq. (583) by using algebraic long division, as follows, where, to save space, we ve rounded o to two decimal places.

Natural vitamin E from vegetable oils, nuts, whole grains, and seeds actually consists of eight different forms: d-alpha-, d-beta-, d-gamma-, and d-delta-tocopherol and d-alpha-, d-beta-, d-gamma-, and d-deltatocotrienol. Cooking and processing robs foods of much of their vitamin E because it is easily destroyed by heat, light, and air. Although d-alpha-tocopherol is the most commonly occurring form of vitamin E in nature, the other vitamin E forms are now being found to have comparable, and in some cases even better, specific biological activities. For example, the activity of d-gamma-tocopherol has been found to be superior to d-alpha-tocopherol in destroying one of the very dangerous free radicals that leads to neurological degeneration. In another example, d-gamma-tocotrienol has been found to be about thirty times more active than d-alpha-tocopherol in lowering cholesterol. These and other scientific studies strongly suggest that mixing the various vitamin E tocopherols and tocotrienols would be better than taking d-alphatocopherol alone.

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