CHAPTER 13 The Digital Processor

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Our PROBLEM now is to nd the inner details of an encoder circuit capable of converting the equal-amplitude input pulses into corresponding binary numbers in the output of Fig. 329. To do this, let us begin with the following truth table, in which Vq is the amplitude of the quantized pulses being fed into the bank of comparators in Fig. 329. From inspection of Fig. 329 we have that

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Vq 0 1 2 3 4 5 6 7 A 0 1 1 1 1 1 1 1 B 0 0 1 1 1 1 1 1 C 0 0 0 1 1 1 1 1 D 0 0 0 0 1 1 1 1 E 0 0 0 0 0 1 1 1 F 0 0 0 0 0 0 1 1 G 0 0 0 0 0 0 0 1 MSD X3 0 0 0 0 1 1 1 1 NMSD X2 0 0 1 1 0 0 1 1 LSD X1 0 1 0 1 0 1 0 1

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The job of the encoder network is to convert the input signals A; B; C; . . . G into 3-digit binary numbers, X3 ; X2 ; X1 , in accordance with the above truth table. In this regard, let us begin by writing an equation for X1 , this being an equation giving all the conditions for which X1 1. One way to do this is to begin with the basic elemental Boolean equation for X1 . In this particular case, however, it will be much easier to write the required equation from direct inspection of the truth table, as follows. From inspection of the table note that it is not always true that X1 1 when A 1; instead, note that it is always true that X1 1 whenever " AB 1 or when " CD 1 or when " EF 1 or when G 1

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thus the simplest possible equation for X1 is " " " X1 1 A B C D E F G 568

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Next, close inspection of the table shows that X2 will be equal to 1 if the relationship " X2 B D F 569

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" is satis ed. This is true because, as the table shows, X2 will be 1 when B D 1, regardless of the value of C. Also from the table, note that X2 1 if F 1, regardless of the value of G. Equation (569) shows how important it can be to make a close examination of a truth table. Lastly, the table reveals that X3 1 whenever D 1, independent of the values of E; F, and G; hence our nal equation is X3 D 570

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Thus, in Fig. 329, given the seven 1 or 0 input signals (A; B; C; . . . ; E; F; G) to the encoder, we ve found that, to generate the required output PCM signal X3 ; X2 ; X1 , the

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CHAPTER 13 The Digital Processor

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circuitry of the encoder must be such that " " " X1 A B C D E F G " X2 B D F X3 D With these equations as a guide we readily nd that Fig. 330 will correctly serve as an encoder circuit to generate the required output signal X3 ; X2 ; X1 .

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Fig. 330

Problem 294 In the above example, the quantized samples were restricted to 8 di erent voltage levels (Vq 0 to Vq 7 volts) which the encoder circuit then transformed into 8 di erent binary numbers (000 to 111). Now rework the example, this time assuming the quantized pulses were allowed 12 di erent voltage levels (Vq 0 to Vq 11 volts), which the encoder circuit would then have to transform into twelve binary numbers of the form X4 ; X3 ; X2 ; X1 . Your PROBLEM is to (a) write the truth table for the encoder, then (b) write the four simplest Boolean equations that would generate the required truth table.