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CHAPTER 13 The Digital Processor
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As always in our discussions it s understood that  2:71828 . . . (eq. (146) in Chap. 6). If we wish to make use of Euler s formula (also in Chap. 6) we can write that z A j!T A cos !T j sin !T which emphasizes the fact that z is a complex number. Thus the real time function of eq. (572) is now being expressed in terms of a complex number z. The advantage is that the algebraic operations, if conducted in the complex plane, are simpler than if we were restricted to the use of real values only. Next, let s raise the given equation, z A j!T , to the n power ; thus z n A n  j!nT that is, z n 1 cos !nT j sin !nT An
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in which n is a positive whole number (the number of terms in the series of eq. (573)). Note, however, that in accordance with eq. (573) we must allow n to become in nitely great (which we indicate by writing n ! 1). But A is a number greater than 1; thus 1=An becomes equal to zero when n becomes in nitely great. Hence inspection of the last equation for z n , above, shows that as n becomes in nitely great z n becomes equal to zero; thus
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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.
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Fig. 332. The unit pulse.
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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
CHAPTER 13 The Digital Processor
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.
Fig. 333
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
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