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(a) Write the elemental equation that will satisfy the table. (b) Simplify the answer to part (a) to a form that requires and and or terms, but just one not term. (c) Using standard symbols, sketch the answer to part (b) in block diagram form. Problem 293 Let A; B; C; D denote four input binary signals that must be switched to satisfy the following truth table where, as usual, Z is the state of the output binary signal: A B C D Z 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 1 0 1 1 A B C 0 1 0 0 0 0 1 1 1 0 1 1 D 0 1 0 1 Z 0 0 0 1 A B C D Z 1 0 0 0 0 1 0 1 0 1 0 0 1 1 1 0 1 0 1 0 A B 1 1 1 1 1 1 1 1 C 0 0 1 1 D Z 0 0 1 0 1 0 0 1
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(a) Write the elemental equation that will satisfy the table. (b) Simplify the answer to (a) into a form that requires only one not operation. (c) Using standard symbols, sketch the answer to (b) in block diagram form.
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The Digital Processor. Digital Filters
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13.1 Bandwidth Requirements for Digital Transmission. Sampling Theorem. PAM and PCM
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We have learned that in a digital system information is given, and transmitted, in the form of short rectangular-type pulses of voltage or current, the presence or absence of a pulse denoting 1 or 0 (as illustrated in Fig. 311 in Chap. 12). It should be noted that the transmission of information in pulse form requires that the equipment be able to uniformly amplify and pass a wide range of frequencies; that is, it must possess a relatively WIDE BANDPASS characteristic. This is because a rectangular pulse type of signal is composed of a large number of harmonic frequencies (note 18 in Appendix). Consider now Figs. 323 and 324, in which T is the uniform amount of time allotted to the appearance of each pulse, T having the same value in both gures.
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Fig. 323
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Fig. 324
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Now imagine two streams of pulses, one composed of those of Fig. 323, the other of Fig. 324. Then note that the frequency F of the fundamental component will be the
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same in both cases; thus
CHAPTER 13 The Digital Processor
F 1=T eq: 91 ; Chap: 5
because T is given to be the same in both cases (but the fundamental waves would not, in general, have equal amplitudes; that is, they would not have equal peak values ). The point, however, that we wish to emphasize here is that, for practical purposes, it will require more higher-order harmonics of F to get a su ciently good representation of Fig. 323 than Fig. 324. This is because the ideal rectangular pulse of Fig. 323 has what are called points of discontinuity, that is, times at which the voltage or current would have to INSTANTLY jump from one value to a di erent value, which is, of course, impossible in the real world. Another, but less severe, type of discontinuity would occur if a voltage or current were required to instantly alter its rate of change, such as instantly changing from, say, an increasing value to a decreasing value. To get an almost exact representation of Fig. 323 would require the inclusion of a large number of higher-order harmonics of the fundamental frequency; thus, to transmit a nearly exact form of Fig. 323 through a system would require that the system have a relatively WIDE BANDPASS characteristic. On the other hand, note that the pulses depicted in Fig. 324 are relatively smooth, having virtually no points of discontinuity. Thus, pulses in the form of Fig. 324 could be transmitted through a system having a considerably NARROWER BANDPASS than that required for Fig. 323. The point we wish to make is that, if the pulse train of Fig. 324 is good enough to do the job (of representing 1 s and 0 s), then we need not try to make the train more closely resemble the ideal case of Fig. 323. The amount of circuit bandwidth required must especially be considered if the information is to be transmitted by wireless; that is, if the information, in digital form, is used to modulate a high-frequency carrier wave. This is because, in the case of a modulated wave, the information is not actually contained in the carrier wave itself but, instead, is contained in a cluster of side-band waves, with the carrier in the center of the cluster (see Fig. 31-A, note 24 in Appendix). For this reason the total bandwidth required to transmit a modulated wave depends only upon the HIGHEST FREQUENCY COMPONENT present in the information being transmitted. Thus, if fh is the highest frequency component of importance in the information signal, then, if the carrier is amplitude-modulated (AM), it would require a total bandwidth of 2fh to transmit the information without distortion. (For practical purposes the same bandwidth requirement, 2fh , applies to a frequency-modulated carrier.) In general, to prevent interference between stations transmitting on adjacent frequencies, it s necessary to limit the amount of bandwidth allocated to each station. This means that we must decide what constitutes the highest frequency of importance in a given case. This is, of course, an engineering judgement that must be tempered by the restriction on maximum allowable bandwidth. In the previous chapter we dealt with digital switching systems, and how such systems can be used to make purely arithmetic calculations. Let s now consider another very important application of digital signals, in which information in ANALOG FORM is converted into and transmitted in DIGITAL FORM. (Then generally, at the end of the transmission system, the nal step is to convert the digital signal back into its original analog form.) Let us, at this point, simply state that the reason for using such a system is that it makes it possible to GREATLY REDUCE THE EFFECTS OF ALL TYPES OF NOISE. The actual conversion of an analog signal into a corresponding digital signal is accomplished by the process of SAMPLING the analog signal. The success of the system
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