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It s apparent that, in this case, the value of L will depend upon the value of the constant multiplier a. As a matter of fact, after some thought we realize that we must consider three separate possibilities for the value of a, these being the cases for a GREATER than 1, a EQUAL to 1, and a LESS than 1. Let us consider each of the three cases as follows. (a > 1): Here the values of the output samples, an , theoretically become in nitely great for n 1. Thus in this case L is certainly not equal to zero, so that Fig. 348 is unstable for a > 1. This is illustrated in Fig. 349. Case II. (a 1): Since 1n 1 we have that L 1, and thus Fig. 348 is unstable for a 1, as illustrated in Fig. 350. Case III. (a < 1): The integral power of a number less than 1 is less than the given number; for instance, if a 1=3, then a2 1=9; a3 1=27, and so on. Thus for this case L becomes equal to zero L 0 as n becomes in nitely great; hence Fig. 348 is stable for a < 1, as illustrated in Fig. 351. In regard to Fig. 349, the output of an actual processor could not, of course, become in nitely great ; instead, in such a case the output would cease increasing and stall when the maximum holding capacity of the digital circuits was exceeded. Now let s return to Fig. 348 and this time apply the z-transform. Let us begin by writing down the nT-domain equation for the gure, which, since it s given that x nT p nT , is y nT p nT ay nT T Case I.

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Now take the z-transform of the above equation. Doing this, and remembering that the z-transform of the unit pulse is 1 , we have Y z 1 aY z z 1 thus, Y z 1 1 az 1 z z a

Folic acid, together with vitamins B12 and B6, is important in the process of methylation Biotin is a water-soluble vitamin essential for the metabolism of carbohydrates It is a coenzyme in the synthesis of fatty acids and amino acids and it stimulates the production of proteins Biotin has been found to enhance insulin sensitivity and increase glucose utilization in the liver Choline is not a vitamin, but it is related in function to the vitamin B-complex Proper fat metabolism in the liver depends on it A deficiency can lead to liver cancer Choline is essential for making acetylcholine and important components of cell membranes such as lecithin (phosphatidylcholine) and an important molecule called sphingomyelin Choline is also needed as a methyl donor in some methylation reactions, especially in the liver Although choline can be made in the body, it is nevertheless considered to be an essential micronutrient.

Now consider the following. We ve agreed to de ne the stability or instability of a DT processor in terms of its response Y z to unit pulse input. For unit pulse input, however, it is true that Y z H z , as was pointed out at the end of section 13.6. Hence (always assuming unit pulse input) we can just as well write H z instead of Y z . With this understood, eq. (593) becomes H z z z a 594

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