Introduction Number Systems and Conversion

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8 Unit 1timing diagrams, state tables, and graphs is presented in Unit 13. The first step indesigning a sequential switching circuit is to construct a state table or graph whichdescribes the relationship between the input and output sequences (Unit 14).Methods for going from a state table or graph to a circuit of gates and flip-flops aredeveloped in Unit 15. Methods of implementing sequential circuits using programmablelogic are discussed in Unit 16. In Unit 18, combinational and sequentialdesign techniques are applied to the realization of systems for performing binaryaddition, multiplication, and division. The sequential circuits designed in this textare called synchronous sequential circuits because they use a common timing signal,called a clock, to synchronize the operation of the memory elements.Use of a hardware description language, VHDL, in the design of combinationallogic, sequential logic, and digital systems is introduced in Units 10, 17, and 20.VHDL is used to describe, simulate, and synthesize digital hardware. After writingVHDL code, the designer can use computer-aided design software to compile thehardware description and complete the design of the digital logic. This allows thecompletion of complex designs without having to manually work out detailed circuitdescriptions in terms of gates and flip-flops.The switching devices used in digital systems are generally two-state devices,that is, the output can assume only two different discrete values. Examples ofswitching devices are relays, diodes, and transistors. A relay can assume twostates—closed or open—depending on whether power is applied to the coil or not.A diode can be in a conducting state or a nonconducting state. A transistor can bein a cut-off or saturated state with a corresponding high or low output voltage. Ofcourse, transistors can also be operated as linear amplifiers with a continuousrange of output voltages, but in digital applications greater reliability is obtainedby operating them as two-state devices. Because the outputs of most switchingdevices assume only two different values, it is natural to use binary numbersinternally in digital systems. For this reason binary numbers and number systemswill be discussed first before proceeding to the design of switching circuits.1.2 Number Systems and ConversionWhen we write decimal (base 10) numbers, we use a positional notation; each digitis multiplied by an appropriate power of 10 depending on its position in the number.For example,953.78 10 9 10 2 5 10 1 3 10 0 7 10 1 8 10 2Similarly, for binary (base 2) numbers, each binary digit is multiplied by the appropriatepower of 2:1011.11 2 1 2 3 0 2 2 1 2 1 1 2 0 1 2 1 1 2 212 8 0 2 1 11 11.75 101434
Number Systems and Conversion 9Note that the binary point separates the positive and negative powers of 2 just asthe decimal point separates the positive and negative powers of 10 for decimalnumbers.Any positive integer R (R 1) can be chosen as the radix or base of a number system.If the base is R, then R digits (0, 1, . . . , R1) are used. For example, if R 8, thenthe required digits are 0, 1, 2, 3, 4, 5, 6, and 7. A number written in positional notationcan be expanded in a power series in R. For example,N (a 4 a 3 a 2 a 1 a 0 .a 1 a 2 a 3 ) R a 4 R 4 a 3 R 3 a 2 R 2 a 1 R 1 a 0 R 0 a 1 R 1 a 2 R 2 a 3 R 3where a i is the coefficient of R i and 0 a i R1. If the arithmetic indicated in thepower series expansion is done in base 10, then the result is the decimal equivalentof N. For example,147.3 8 1 8 2 4 8 1 7 8 0 3 8 1 64 32 7 103.375 10The power series expansion can be used to convert to any base. For example,converting 147 10 to base 3 would be written as147 10 1 (101) 2 (11) (101) 1 (21) (101) 0where all the numbers on the right-hand side are base 3 numbers. (Note: Inbase 3, 10 is 101, 7 is 21, etc.) To complete the conversion, base 3 arithmeticwould be used. Of course, this is not very convenient if the arithmetic is beingdone by hand. Similarly, if 147 10 is being converted to binary, the calculationwould be147 10 1 (1010) 2 (100) (1010) 1 (111) (1010) 0Again this is not convenient for hand calculation but it could be done easily in acomputer where the arithmetic is done in binary. For hand calculation, use thepower series expansion when converting from some base into base 10.For bases greater than 10, more than 10 symbols are needed to represent thedigits. In this case, letters are usually used to represent digits greater than 9. Forexample, in hexadecimal (base 16), A represents 10 10 , B represents 11 10 , C represents12 10 , D represents 13 10 , E represents 14 10 , and F represents 15 10 . Thus,A2F 16 10 16 2 2 16 1 15 16 0 2560 32 15 2607 10Next, we will discuss conversion of a decimal integer to base R using the divisionmethod. The base R equivalent of a decimal integer N can be represented asN (a n a n1 ··· a 2 a 1 a 0 ) R a n R n a n1 R n1 · · · a 2 R 2 a 1 R 1 a 0
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