课件案例数电lec

1、1Digital Logic Design and ApplicationJin YanhuaLecture #2General Positional-Number-System ConversionAddition and Subtraction of Nondecimal NumbersRepresentation of Negative NumbersUESTC, Spring 2014Jin. UESTC2Chapter 2Number Systems and CodesHow can numeric quantities be represented and manipulated?

2、How can nonnumeric data, events, and conditions be represented?Jin. UESTC32.1 Positional Number SystemsThe traditional number system is called a positional number system. For example: 3567 = 3*1000 + 5*100 + 6*10 + 7*1A number is represented as a string of digits.Each digit position has a associated

3、 weight.Numbers value = a weighted sum of the digitsDecimal fractions: weights are negative powers of 10Jin. UESTC42.1 Positional Number SystemsIn general, a number D can be represented asD = dp-1 dp-2 . d1 d0 . d-1 d-2 . d-nD2 = d i 2i D16= d i 16i r: Base or Radix of r number system（计数的基数）ri: Weig

4、ht corresponding to the digits position（第i位的权）Jin. UESTC52.2 Octal and Hexadecimal NumbersBinary-to-Octal or Hexadecimal conversion Octal- or Hexadecimal- to-Binary conversion1000110010012 = ( )8 = ()16RadixDigitsBinary20,1Octal807Hexadecimal1609,AF(11011.10101)2= ( 23.52 )8= ( 1B.A8 )16How to conve

5、rt to decimalJin. UESTC62.3 General Positional-Number-System ConversionRadix-r-to-DecimalExample 1：( 101.01 )2 = ( )10 ( 7F.8 )16 = ( )105.25127.5A shortcut? ( F1AC )16 = ( ( ( F16 ) +1 ) 16 + A ) 16 + CMethod: use the formula nested expansion formula 嵌套形式Jin. UESTC7Decimal-to-Radix-r Conversions Co

6、nvert the integer and fractional parts separately and add the results afterwards.Integer part: Successively divide number by r, taking remainder as result.Example: Convert 5710 to binary57 / 2 = 28 remainder 1 (LSB) /2 = 14 remainder 0 /2 = 7 remainder 0 /2 = 3 remainder 1 /2 = 1 remainder 1 /2 = 0

7、remainder 1 (MSB)Ans: 1110012Jin. UESTC8Decimal-to-Radix-r ConversionsFractional PartExample: convert .310 to binary.3 * 2 = .6 integer part = 0.6 * 2 = 1.2 integer part = 1.2 * 2 = .4 integer part = 0.4 * 2 = .8 integer part = 0.8 * 2 = 1.6 integer part = 1.6 * 2 = 1.2 integer part = 1, etc.Ans: .0

8、100112Successively multiply number by r, taking integer part as result and chopping off integer part before next iteration.May be unending!Jin. UESTC9Decimal-to-Radix-rMethod: Radix Multiplication or Division 1001 11000.01011 2-52.3 General Positional-Number-System ConversionInteger Parts 整数部分：除r取余，

9、逆序排列 Example 2：( 156 )10 = ( )2Decimal Fraction 小数部分：乘r取整，顺序排列 Example 3：( 0.37 )10 = ( )2Jin. UESTC102.3 General Positional-Number-System ConversionExample 4：Require 10-2 ，complete the following conversion ( 617.28 )10 = ( )210 0110 1001 . 0100 0112-n = 10-2 n = 7思考：任意两种进位计数制之间的转换 以十进制（二进制）作为桥梁Jin.

10、 UESTC112.4 Addition and Subtraction of Nondecimal NumbersBinary Addition: Cin（carry in）Cout（carry out）S（sum）：Binary Subtraction: Bin（Borrow in）Bout（Borrow out）D（difference） 1011 1110+ 1000 1101 1010 1010 0101 0101Jin. UESTC12Binary addition tableInputOutputXYCinCoutS00000001010100101110100011011011

11、01011111inputoutputXYBinBoutD0000000111010110111010001101001100011111Binary subtraction tableJin. UESTC132.5 Representation of Negative NumbersSigned-Magnitude RepresentationMSB as the sign bit ( 0 = plus, 1 = minus ) 000010012 = = 910 011111112 = = 12710 000000002 = = 010There are two possible repr

12、esentations of Zero.An equal number of positive and negative integers.An n-bit signed-magnitude integer range is ( 2n-11) + ( 2n-11) Adders for singed-magnitude systems are more complex.Jin. UESTC142.5 Representation of Negative NumbersComplement Number Systems plement Representation The complement

13、of an n-digit number is obtained by subtracting it from rn。 If a number D is complemented twice, the result is D.Taking the complement is more difficult than changing the sign, but two numbers can be added directly. How to make it easy?Jin. UESTC152.5 Representation of Negative NumbersComplement Num

14、ber Systems plement Representation The complement of an n-digit number is obtained by subtracting it from rn。Diminished plement The Diminished plement of an n-digit number is obtained by subtracting it from rn -1.Jin. UESTC16Two plement Representation2n number(2n 1) number Onescomplement Onescomplem

15、ent + 1 Only one representations of Zero An n-bit two plement range is 2n-1 +(2n-11)Expanding the sign bit 2.5 Representation of Negative NumbersJin. UESTC17Example 5. Write the 8-bit two plement representations for 119.+119 的补码和反码是？解：写出相应正数的二进制表示：11901110111 则其补码可以通过下式算法得到： 全1码：1 1 1 1 1 1 1 1 减去+1

16、19：0 1 1 1 0 1 1 1119的反码：1 0 0 0 1 0 0 0 加1： 1 119的补码：1 0 0 0 1 0 0 1Jin. UESTC18NOTEPositive number has the same： Sign-Magnitude, Ones- Complement, and Twos- Complement 正数的原码、反码、补码相同Jin. UESTC19QuizConvert from decimal to binary: 73.42Write the 8-bit Signed-magnitude, Two plement, and Ones- complement representations for each decimal number: Jin. UESTC20Only one representations of Zero ( 零只有一种表示 ) 00 0 0 0 0