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# DIGITAL SYSTEMS

###### DIGITAL SYSTEMS 10
DIGITAL SYSTEMSReal world systems and processes Mostly continuous (at the macroscopic level): time, acceleration, chemical reactions Sometimes discrete: quantum states, mass ( of atoms) Mathematics to represent physical systems is continuous (calculus) Mathematics for number theory, counting, approximating physical systems can be discreteRepresentation of information A. Continuous—represented analogously as a value of a continuously variable parameter 1.position of a needle on a meter 2.rotational angle of a gear 3.amount of water in a vessel 4.electric charge on a capacitor B. Discrete—digitized as a set of discrete values corresponding to a finite number of states 1. digital clock 2. painted pickets 3. on/off, as a switchRepresentation of continuous processes Analogous to the process itself • Great Brass Brain—a geared machine to simulate the tides • Slide rule—an instrument which does multiplication by adding lengths which correspond to the logarithms of numbers. • Differential analyzer (Vannevar Bush)—variablesize friction wheels to simulate the behavior of differential equations Tide calculator Vannevar Bush integratorBrass Brain was the equal of 100 mathematicians, weighted a mere 2500 lbs Imagine the fearful gnashings of mathematicians in November, 1928 upon reading this account of the USGS's new "brass brain," which could "do the work of 100 trained mathematicians" in calculating tides: The machine weighs 2,500 pounds. It is 11 feet long, 2 feet wide, and 6 feet high. Its whirring cogs are enclosed in a housing of mahogany and glass. Earthquakes, freshwater floods, and strong winds that cannot be predicted affect the accuracy of the Brass Brain to a degree. Nevertheless 70 of the predicted tides agree within five minutes of the observed tide. The Coast and Geodetic Survey issues an annual bulletin in which it lists the forthcoming tides in 84 ports of the world. The report contains upwards of a million figures, all compiled by the Brass Brain. It has been estimated that the Brass Brain saves the government 125,000 each year in salaries of mathematicians who would be required to take its place. From Instruments of Science: an historical encyclopedia Great Brass Brain “It remained in use until the late 1960s,when an IBM 7090 computer took over the job. Even when digital computers finally took over from analog instruments, the amount of arithmetic needed to properly evaluate the cosine series was so vast that the output had to be limited to simply times of high and low tide for any particular area. This was overcome only when, during the 1970s, digital computers became powerful enough. . .”Discrete representations What is it What is it What is itBabbage difference engine to calculate polynomialsElectronic analog computers—circuitry connected to simulate differential equations • Phonograph record—wiggles in grooves to represent sound oscillations • Electric clocks • Mercury thermometers/barometers Stereo phonograph recordManipulation Analog • adding the lengthequivalents of logarithms to obtain a multiply, e.g., a sliderule • adjusting the volume on a stereo • sliding a weight on a balancebeam scale • adding charge to an electrical capacitor Discrete Marble binary counter • counting—pushbutton counter • digital operations—mechanical calculators • switching—open/closing relays • logic circuits—true/false determination Marchant mechanical calculatorAnalog vs. Discrete Note: "Digital" is a form of representation for discrete Analog • infinitely variableinformation density high • limited resolutionto what resolution can you read a meter • irrecoverable data degradationsandpaper a vinyl record Discrete/Digital • limited statesinformation density low, e.g., one decimal digit can represent only one of ten values • arbitrary resolutionkeep adding states (or digits) • mostly recoverable data degradation, e.g., if information is encoded as painted/not painted pickets, repainting can perfectly restore dataDigital systems • decimal • not so good, because there are few 10state devices that could be used to store information fingers. . . Decimal Binary Hexadecimal 1 4 2 2 • binary 0 0000 0 • excellent for hardware; lots of 2state devices: switches, lights, 1 0001 1 magnetics 2 0010 2 • poor for communication: 2state devices require many digits to represent values with reasonable resolution 3 0011 3 • excellent for logic systems whose states are true and false. But 4 0100 4 binary is king because components are so easy (and cheap) to 5 0101 5 fabricate. 6 0110 6 7 0111 7 • octal base 8: 8 1000 8 • used to conveniently represent binary data; almost as efficient 9 1001 9 as decimal 10 1010 A • hexadecimalbase 16: 11 1011 B • more efficient than decimal; more practical than octal because 12 1100 C of binary digit groupings in computers 13 1101 D 14 1110 E 15 1111 FBinary logic and arithmetic Background • George Boole(1854) linked arithmetic, logic, and binary number systems by showing how a binary system could be used to simplify complex logic problems • Claude Shannon(1938) demonstrated that any logic problem could be represented by a system of series and parallel switches; and that binary addition could be done with electric switches • Two branches of binary logic systems • Combinatorial—in which the output depends only on the present state of the inputs • Sequential—in which the output may depend on a previous state of the inputs, e.g., the “flipflop” circuitAND gate A C B A B C 0 0 0 1 0 0 0 1 0 1 1 1AND gate A C B A B C Simple “AND” Circuit 0 0 0 A B 1 0 0 C 0 1 0 1 1 1 BatteryOR gate A C B A B C 0 0 0 1 0 1 0 1 1 1 1 1OR gate A C B Simple “OR” circuit A B C A 0 0 0 B 1 0 1 C 0 1 1 1 1 1NOT gate A B A B 1 0 0 1NOT gate A B Simple “NOT” circuit A B A 1 0 0 1 BNAND gate A C B A B C 0 0 1 1 0 1 0 1 1 1 1 0NAND gate A C B A B C Simple “NAND” Circuit 0 0 1 A B 1 0 1 C 0 1 1 1 1 0 BatteryPractice Problem Control systems: e.g., car will start only if doors are locked, seat belts are on, key is turned D S K I 0 0 0 0 0 0 1 0 0 1 0 0 0 1 1 0 1 0 0 0 1 0 1 0 1 1 0 0 1 1 1 1Control systems: e.g., car will start only if doors are locked, seat belts are on, key is turned D S K I 0 0 0 0 0 0 1 0 0 1 0 0 I = D AND S AND K 0 1 1 0 1 0 0 0 D 1 0 1 0 I S 1 1 0 0 K 1 1 1 1Binary arithmetic: e.g., adding two binary digits A B R C 0 0 0 0 0 1 1 0 1 0 1 0 1 1 0 1Binary arithmetic: e.g., adding two binary digits A B R C R = (A OR B) AND NOT (A AND B) 0 0 0 0 C = A AND B 0 1 1 0 1 0 1 0 A 1 1 0 1 R B CBoolean algebra properties AND rules OR rules AA = A A +A = A AA' = 0 A +A' = 1 0A = 0 0+A = A 1A = A 1 +A = 1 AB = BA A + B = B+A A(BC) = (AB)C A+(B+C) = (A+B)+C A(B+C) = AB+BC A+BC = (A+B)(A+C) A'B' = (A+B)' A'+B' = (AB)‘ (DeMorgan’s theorem) Notation: = AND + = OR ‘ = NOTComputing power has been growing at an exponential rate Note: graph is a “semilog” plot—the best way to kt. indicate a function y(t)=ae
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