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and the same, of course, for x nT in place of y nT . Now apply the last two relations to the preceding equation; doing this gives the relationship Y z b0 X z b1 X z z 1 bp X z z p a1 Y z z 1 a2 Y z z 2 aq Y z z q Now, on the right-hand side of the last equation, factor out X z and Y z and then solve for the ratio of OUTPUT TO INPUT, that is, Y z =X z . Doing this, and using the notation of eq. (585), you should nd that b0 b1 z 1 b2 z 2 bp z p Y z H z X z 1 a1 z 1 a2 z 2 aq z q 589
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Equations (588) and (589) are the basic digital processor equations, (588) being in the time or nT domain and (589) in the z-domain. As we continue, we ll gradually begin to see how they can be applied. As a nal note, remember that the internal operations in a digital processor are performed using binary arithmetic; that is, internally the information is manipulated in the form of strings of 1 s and 0 s. All pulses representing 1 have the same amplitude; a multiplier unit does not multiply the actual amplitudes of the pulses; thus the output of the multiplier unit in Fig. 341 is a binary number a times the binary number at the input to the unit. The resulting e ect is, of course, the same as if the actual amplitudes of the pulses had been multiplied by a. It might seem as if such a procedure would be too time consuming, but we must remember that a digital processor is capable of performing many many millions of operations per second. Problem 304 What is the basic equation for the transfer function, in the z-domain, for the processor in Fig. 345 Problem 305 Write the basic equation, in the z-domain, for the transfer function of a purely nonrecursive digital processor. Problem 306 In the following, x nT and y nT denote, respectively, DT input and output sequences of a DT processor in the time domain. (a) y nT x nT 6x nT T (b) y nT 2x nT 5x nT T 10x nT 2T (c) y nT 6x nT 8x nT T 7y nT T In each case, state whether the processor is a non-recursive or a recursive type. Problem 307 Given that the output y nT of a certain processor with x nT input is y nT 4x nT 7x nT T 5x nT 2T 9x nT 3T (a) Sketch the block diagram of the circuit layout of the processor. (b) For this processor, H z
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CHAPTER 13 The Digital Processor
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Let s conclude this section as follows. The transfer function of a DT processor has already been de ned for the nT-domain (eq. (586) in section 13.5). Now, in the zdomain we have, by eq. (589), that Y z H z X z 590
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so that, correspondingly, H z is now the transfer function expressed in the z-domain (instead of the nT-domain as in eq. (586)). Thus the block diagram form of Fig. 339 in section 13.5 now becomes Fig. 347.
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Fig. 347
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An interesting fact can be discovered as follows. Suppose the INPUT signal to a processor is the UNIT PULSE p nT of Fig. 332. In such a case X z 1, because the z-transform of the unit pulse is 1 ; and hence, for this particular case, eq. (590) becomes Y z H z . The transfer function H z of a linear DT system is equal to the response of the system to UNIT-PULSE INPUT. For this reason the terms transfer function, unit-pulse response, and pulse transfer function are all used interchangeably.
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