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1 數位系統導論蔡宗漢 (Tsung-Han Tsai) Dept. of E.E., N.C.U. 1 教學目標 : 1 了解數位電子電路的基本原理, 例如資訊的二進位系統布林代數 2 了解數位電子電路的基本原件, 如 : 組合電路循序電路加法器比較器等等授課大綱 : 1 數位邏輯的原理 2 元件的認識( 如 AND/OR 閘, 加法器 ) 3 數位邏輯功能單元 4 數位邏輯的設計 2

2 Grading & Textbook Homework & Class Participation:30% Midterm exam:30% Final exam:40% Textbook: M. Morris Mano, Digital Design, 3rd, 2002 Prentice-Hall, Inc. 參考書籍 : Randy H. Katz, Contemporary Logic Design, 1995 The Benjamin/Cummings Publishing Company, Inc. 台北圖書代理 M. Morris Mano & Charles R. Kime, Logic and Computer Design Fundamentals, 1997 Prentice-Hall, Inc. 新月代理 3 Outlines Digital computers and information Binary numbers Numbers base conversions Binary arithmetic Complements Binary logic Binary storage and registers 4

3 Digital Computers & Digital Systems Information familiar to us I/P MEM CU CPU O/P Computer 5 數位電路之實現 Logic Standard Logic ASIC PLD Gate Array Cell-Based IC Full Custom IC PLD CPLD FPGA ASIC:Application-Specific Integrated Circuit PLD:Programmable Logic Device SPLD:Small/Simple Programmable Logic Device CPLD:Complex Programmable Logic Device FPGA:Field Programmable Gate Array 6

4 常見名詞 TRANSISTOR: 電晶體 Chip: 矽晶片, 微晶片 Integrated Circuit [IC]: 積體電路 Small Scale IC[SSI] Medium Scale IC[MSI] Large Scale IC[LSI] Very Large Scale IC[VLSI] Ultra Large Scale IC[ULSI] Giga Scale IC[GSI] SOC 小型 IC 中型 IC 大型 IC 超大型 IC? 大型 IC?? 型 IC 單晶片系統 Kilo:K 仟 Mega:M 百萬 Giga:G 十億 Tera:T 兆 7 A= B + C IF (A) THEN X =Y Algorithm Level Gate Level RAM ALU ROM CONTROL Functional Level Geometric Level Circuit Level SUBMITTING YOUR LAYOUT DESIGN IN CIF OR GDSII VIA MAGNETIC TAPE OR 8

5 9 10

6 11 12

7 TTL Gate and ECL Gate 13 TTL Gate and ECL Gate 14

8 CMOS Gate 15 Binary Numbers Binary number 2 symbols (0, 1) Binary signal 2 states (Low, High) Decimal numbers: A10 = a n-1 a n-2 a 1 a 0.a -1 a -2 a -m = Binary numbers: A 2 = n 1 i= m n 1 i= m i ai *10 0 ai i ai *2 ai = 0,1 9 16

9 Binary Numbers In general, base r (radix r) numbers: A r = n 1 i= m ai * r i 0 ai r 1 EX:( ) 2 =1 x x x x x x x 2-2 =(26.75) Powers of Two n 2 n n 2 n n 2 n

10 Numbers Base Conversions Let A r = B 10 ai r i = j bj * 10 0 ai r 1 0 bj 9 Conversion from A r to B 10 is straightforward, since it conforms to out experience Conversion from B 10 to A r, however, requires a little bit of extra knowledge:successive divisions (for the integer part ) and multiplications (for the fraction) of B 10 by r, and observe the remainders and carries 19 Numbers Base Conversions Conversion procedure (1) Let B 10 = B1 10 +B2 10 (integer & fraction, respectively), ie.. q 1 j= p bj *10 j q 1 j= 0 (2) B1=B1`*r +a0 B1`=B1``*r+a1 B1 (n-2) =B1 (n-1) *r+a n-2, and B1 (n-1) =a n-1 = bj *10 j + 1 j= p bj *10 j 20

11 Numbers Base Conversions (3) B2 * r = a -1 * B2` B2` * r = a -2 * B2`` B2 (m-1) * r = a -m * B2 (m) Note that B2 (m) is not necessarily zero, in which case there is a truncation error associated with the conversion 21 Number Base Conversions Convert decimal 41 to binary 41/2 = 20 remainder = 1 20/2 = 10 = 0 10/2 = 5 = 0 5/2 = 2 = 1 2/2 = 1 = 0 1/2 = 0 = 1 Convert decimal to binary x 2 = x 2 = x 2 = x 2 = (41) 10 =(101001) 2 (0.6875) 10 =(0.1011) 2 22

12 Octal & hexadecimal numbers Base 10 Base2 Base 8 Base16 Base10 Base2 Base 8 Base A B C D E F 23 Binary Arithmetic Binary Addition 0+0=0 sum of 0 with a carry of 0 0+1=1 sum of 1 with a carry of 0 1+0=1 sum of 1 with a carry of 0 1+1=10 sum of 0 with a carry of 1 Binary Addition with carry 1+0+0=01 sum of 1 with a carry of =10 sum of 0 with a carry of =10 sum of 0 with a carry of =11 sum of 1 with a carry of 1?carry bit 24

13 Binary Subtraction Binary Arithmetic 0 0 = = = = 1 borrow 1 from the higher bit EX : Binary Arithmetic Binary Multiplication 0 x 0 = 0 0 x 1 = 0 1 x 0 = 0 1 x 1 = 1 EX : x

14 Binary Division Binary Arithmetic EX : = COMPLEMENTS Complements are used in digital computers for simplifying the subtraction operation and logical manipulation There are two types of complements for each base r system:the radix complement and the diminished radix complement. 28

15 Diminished radix complement Given a number N in base r having n digits, the (r-1) s complement of N is defined as (r n -1)-N The 9 s complement of is = The 9 s complement of is = The 1 s complement of is The 1 s complement of is Radix complement The r s complement of an n-digit number N in base r is defined as r n -N, for N 0 and 0 for N=0 The 10 s complement of is The 10 s complement of is The 2 s complement of is The 2 s complement of is

16 Subtraction with complements The subtraction of two n-digit unsigned numbers M N in base r can be done as follows: (1):Add the minuend, M, to the subtrahend, N. This performs M+(r n N)=M N + r n (2):if M N, the sum will produce an end carry, r n, which can be discarded:what is left is the result M N (3):if M < N, the sum does not produce an end carry and is equal to r n (N M), which is the r s complement of (N M). To obtain the answer in a familiar form, take the r s complement of the sum and placed a negative sign in front 31 Signed Binary Numbers Decimal Signed 2 s complement Signed 1 s complement Signed magnitude

17 Arithmetic addition Note that negative numbers must be initially in 2 s complement and that if the sum obtained after the addition is negative, it is in 2 s complement form BCD Code Decimal symbol BCD Digit

18 BCD Addition When the binary sum is equal to or less than 1001 (without a carry), the corresponding BCD digit is correct Other Decimal Codes DECIMAL DIGIT BCD EXCESS Unused bit combinations

19 Gray Code Gray code Decimal equivalent Gray code Decimal equivalent ASCII Character Code 38

20 ASCII Character Code 39 Binary Storage And Registers MEMORY UNIT J O H N Memory Register PROCESSOR UNIT 8 cells 8 cells 8 cells 8 cells Processor Register INPUT UNIT KEYBOARD J O H N 8 cells CONTROL Input Register 40

21 Binary Logic 41 Binary Logic 42

22 Binary Logic 43

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<4D6963726F736F667420506F776572506F696E74202D20B5DAD2BBD5C228B4F2D3A1B0E6292E707074205BBCE6C8DDC4A3CABD5D> Homeworks ( 第三版 ):.4 (,, 3).5 (, 3).6. (, 3, 5). (, 4).4.6.7 (,3).9 (, 3, 5) Chapter. Number systems and codes 第一章. 数制与编码 . Overview 概述 Information is of digital forms in a digital system, and