數(shù)字設(shè)計(jì)基礎(chǔ)雙語課件(第1章)

1、 1. Digital systems & representation 1.1 Digital systems 1.2 Numbers representation1.3 Performing arithmetic with binary numbers1.4 Representation of alphanumeric symbols1 1.1 Digital Systems1. The realm of digital systemsA digital system is often designed to satisfy two kinds of tasks: （1）To contro

2、l apparatus（2）To perform calculationsA digital system can perform some calculations and on the basis of the result, take certain control actions.2 1.1 Digital Systems2. Two-valued logic signals The input and output signals have only two values. Hence, the digital systems might use 5 volts and 0 volt

3、s to represent the two values. 3. Positive and negative logic representation（1）Positive logic representationThe higher voltage represents a logic 1, and the lower voltage represents a logic 0. 3 1.1 Digital Systems（2）Negative logic representationThe higher voltage represents a logic 0, and the lower

4、 voltage represents a logic 1. （3）Mixed logic representationUsing both positive and negative representations in the same system.4 1.1 Digital Systems4. Logic functionsIn order to develop the algorithm or logic function for a digital system, logic signals must be represented by names as in algebra.Lo

5、gic function can be written in a two-valued algebra called Boolean algebra.5. Three fundamental logic operationsAND, OR, and NOT are three fundamental logic operations, from which any complicated logic function can be created.5 1.2 Numbers Representation1. Decimal numbersDecimal number system has te

6、n digits 0,1,2,3,4,5,6,7,8 and 9. It is a positional number system, which uses digits multiplied by powers of 10 that depend upon the position of the digit. 10 is the base of the number system.(235)10 = 2102+ 3101+ 5100Ten different voltages are needed for each possible value of the digit in the dec

7、imal number system. Then electronic circuits will have to be designed which can accept ten different voltages and produce ten different voltages, one for each digit of the number.6 1.2 Numbers Representation2. Binary numbersBinary number system has only two digits 0 and 1. They are called bits (bina

8、ry digits). 2 is the base of the binary system. Binary number can be represented as:(110001)2 = 125+ 124+ 023+ 022+ 021+ 120Numbers can be a fraction or have a fractional part.(0.101)2 = 12-1+ 02-2+ 12-3The binary number system fits in well with digital systems. There are only two digits used, 0 and

9、 1, in all numbers, so only two different voltages are needed for each digit.7 1.2 Numbers Representation3. Hexadecimal and octal numbersThe positional number system using the base of 16 is called the hexadecimal number system. The 16 different numbers are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E

10、 and F.The positional number system using the base of 8 is called the octal number system. The 8 different numbers are 0, 1, 2, 3, 4, 5, 6 and 7.8 1.2 Numbers Representation4. Number conversionNumbers are likely to be entered using the decimal representation, but dealt in binary within the digital s

11、ystems. Therefore, there needs to be some way of converting from binary to decimal and vice versa.Small decimal numbers can be easily converted into binary by considering the values of powers of 2.Exp1: Convert the decimal number 25 into binary. (25)10 = 16+8+1 =10000+01000+00001 = (11001)2（1）Decima

12、l to binary9 1.2 Numbers Representation（2） Binary to decimalExp2: Convert the binary number 110001 to decimal. (110001)2 = 125+ 124+ 023+ 022+ 021+ 120 = (49)10（3） Binary to hexadecimalTo convert a binary number into hexadecimal is a matter of dividing the binary number into groups of four digits an

13、d converting each group into one hexadecimal digit.Exp3: Convert the binary number 110001 to hexadecimal. (110001)2 = (31)1610 1.2 Numbers Representation（4） Hexadecimal to binaryExp4: Convert the hexadecimal number 1AC to binary. (1AC)16 = (0001 1010 1100)2 （5） Binary to octalExp5: Convert the binar

14、y number 001010011 to octal. (001010011)2 = (123)8（6） Octal to binaryExp6: Convert the octal number 456 to binary. (456)8 = (100 101 110)2 11 1.3 Performing arithmetic with binary numbers1. AdditionThe rules are essentially the same for decimal and binary. Pairs of digits are added together, startin

15、g with the least significant digits. When a result digit is equal to or greater than the base, a carry is generated which is added to the next pair of digits.12 1.3 Performing arithmetic with binary numbers2. Negative numbers and subtractionBinary subtraction use the similar approach to the decimal

16、approach of borrowing from the next column of digits.In decimal, we have to borrow a digit from the next column if the subtrahend is greater than the minuend.For implementation reasons, subtraction is usually done by the “addition of complements”. Such method can be applied to any number system.13 1

17、.3 Performing arithmetic with binary numbersThe 2s complement of a binary number N is defined by:2s complement2n -NPositive numbers always start with a leading 0; Negative numbers with a leading 1.Negative numbers are represented within most digital systems and computers in 2s complement because it

18、simplifies addition and subtraction.14 1.3 Performing arithmetic with binary numbersSubtractionTo perform subtraction of Y from X, the 2s complement of Y is formed to create Y, then we add Y to X.The method to create the 2s complement of any number is to invert the digits and add 1.1s complementInverting the digits of a binary number is called forming the 1s complement which is defined as: -X = (2n -1) - Y15 1.3 Performing arithmetic with binary numbers3. Binary-coded decimal numbersIn BCD, each decimal digit of a number is represen