... may I introduce myself?...

Transcription

1 Digital Systems Prof. Mafred Schimmler Lehrstuhl für Techische Iformatik Istitut für Iformatik Christia Albrechts Uiversität zu Kiel Tel.:

2 Prof. Dr. Mafred Schimmler... may I itroduce myself? Siemes AG Berli Assistet, CAU Kiel Research Fellow Uiversity Aarhus Research Fellow ANU Caberra Geschäftsführer, ISATEC GmbH, Kiel Fachhochschulprofessor, FH Stralsud Professor, TU Brauschweig 2004-? Professor, CAU Kiel 2

3 Materials for Lecture 1. Script: The script is ew. Therefore, there are still some mistakes i it. These mistakes will be foud ad corrected durig the semester. For every mistake that you have show me (first time), I will doate a chocolate bar. 2. Assigmets are ormally available i the website o Wedesday from 20:00 hrs. These assigmets are to be submitted up to the secod of the followig Modays till 08:00 hrs i the cabiet, situated at the groud floor of the buildig i Herma-Rodewald-Str Books ad www-liks: Will be give i the lecture 3

4 Assigmets 1. Every week, you must work o the assigmets by yourselves (o copyig). The solutios will be marked with up to 100 Poits. 2. The assigmet series are give i fixed groups of two. Both parters should thereby, have equal share of the homework. 3. Please esure that the ames, matriculatio umbers ad the group umber are clearly stated i your assigmets. If the submitted homework cosists of several pages, they have to be clipped or stapled together. 4

5 Exam Requiremets 1. At the ed of the semester, there is a oral examiatio. 2. Registratio for the examiatio: iformally 5

6 Examiatio Materials Allowed materials for the examiatio are: Pe, plai paper Not allowed are: Pocket calculators, scripts, books, mobiles, PDAs, computers, tables, prited formula collectios 6

7 Topics Overview 1. Number Represetatios i Computers 2. Fudametals of Digital Circuits 3. The MOS Trasistor 4. CMOS Techology ad CMOS Gate 5. Combiatorial Circuit 6. Computer Arithmetic 7. Flip-Flops 8. Sequetial Circuit 9. Arithmetic Logic Uit 10. Memories 11. The DLX Processor 12. Assembler 7

8 1. Number represetatios i computers Literatur: Waldschmidt, K.: Schaltuge der Dateverarbeitug, Teuber, 1980, ISBN Klar, R.: Digitale Recheautomate, de Gruyter, 1976, ISBN Leohard, E.: Grudlage der Digitaltechik, Haser Verlag, 1976, ISBN

9 Polyadic represetatio of umbers N-1 = Σ b i * B i i=0 = b N-1 B N-1 + b N-2 B N b 1 B 1 + b 0 B 0 kow as B-adic represetatio of b i {0,1,...,B-1} kow as digits 9

10 Polyadic represetatio of umbers Abbreviatios for B-adic represetatios: (b N-1 b N-2...b 1 b 0 ) B or, whe it is clear o which basis it is about: b N-1 b N-2...b 1 b 0 10

11 Clause: The N-digit B-adic represetatio eables every whole umber of {0,1,...,B N -1} to be represeted i exactly oe way. Proof: Every umber ca be represeted i at least oe way (iductio o N) i at most oe way (coutig the combiatio of umbers) 11

12 Horerscheme N-1 = Σ b i * B i i=0 = bn-1 B N-1 + b N-2 B N b 1 B 1 + b 0 B 0 = ((..(bn-1 B+ b N-2 )*B+ b N-3 )*B...+b 1 )*B+ b 0 12

13 Coversio of umbers betwee polyadic systems 1. The method of iterative divisio with remaider: : B = q 1 Rem b 0 q 1 : B = q 2 Rem b 1 q 2 : B = q 3 Rem b 2 q 3 : B = q 4 Rem b 3 q 4 : B = q 5 Rem b q N-2 : B = q N-1 Rem b N-2 q N-1 : B = 0 Rem b N-1 13

14 Coversio of umbers betwee polyadic systems Calculatio i the rootsystem: (x) B (y) B 1. Represet the basis B' of the desired system i the root system. 2. q 0 = 3. Repeat for ascedig i: q i+1 = q i div B ; r i = q i mod B till q i+1 =0. 4. The r i are the B -adic represetatio of y 14

15 Coversio of umbers from B-adic i the B -adic umber system through calculatios i the root system (B-adic umber system) q 0 := i := 0 q i+1 := q i div B b i := q i mod B Output the Digit b i i := i+1 q i =0? ei ja 15

16 Coversio of umbers betwee polyadic systems 2. Processig Horerschemes from left to right: ((..(b N-1 B+ b N-2 )*B+ b N-3 )*B...+b 1 )*B+ b 0 Calculatio i the target system: 1. Coversio of all b i i the B -adic system 2. Coversio of B i the B -adic system 3. Compute i the B -adic system 16

17 Coversio of umbers from the B-adic i the B -adic umber system through calculatios i the target system (B -adic umber system) Wadle alle Ziffer b i ud die Basis B id B -adische System um i := N := 0 := * B + b i-1 i := i - 1 i = 0? ei ja Gib aus 17

18 Coversio of umbers betwee polyadic Systems whose bases are powers of 2 1. Coversio of all digits ito the biary system 2. Covert the source umber (digit by digit) ito a biary umber 3. Group up appropriate bits together (LSB first) for always oe digit at the desired system 4. Produce all digits of the target system like that (LSB least sigificat bit, hece LSB first meas: start with the bit with the least priority) 18

19 Biary 2-adisch Terary 3-adisch Octal 8-adisch Decimal 10-adisch Hexadecimal 16-adisch A B C D E F

20 Defiitio: Let be a atural umber, represeted as N- digits B-adic umber. The B-complemet of is the N-digits B-adic umber formed from the last N digit of B N -. The B-complemet will be iterpreted as - 20

21 Coversio of a umber ito egative umber (Bcomplemet): (b N-1 b N-2...b 1 b 0 ) B Each digit b i will be replaced by the digit (B-1-b i ). After that, oe is added to the umber that was produced. 21

22 4-digits 2's complemetumbers Decimal 10-adisch Biary 2-adisch

23 Represetable rage of N-digits B-adic umbers i B-complemet for eve B {-(B/2)B N-1,...,+(B/2)B N-1-1} Exactly those umbers begiig with a digit B/2 are egative. 23

24 Clause: The result of a additio of two N-digits 2-adic umbers will agai be (with N positios) withi the represetable rage, if for the sum after the additio, the pre-sig positio (positio N-1) is i accordace with the securig positio (positio N). 24

25 Represetatio of Ratioal Numbers i the Fixed Poit Format N-1 = Σ b i * B i i=-m = b N-1 B N-1 + b N-2 B N b 1 B 1 + b 0 B 0 + b -1 B -1 + b -2 B b -M+1 B -M+1 +b -M B -M Represetable rage (if B is eve): [-(B/2)*B N-1..+(B/2)*B N-1 -B -M ] 25

26 Horerscheme for ratioal umbers -1 = Σ b i * B i i=-m = b-1 B -1 + b -2 B b -M+1 B -M+1 + b -M B -M = ((..(b-m B -1 + b -M+1 )*B -1 + b -M+2 )*B b -1 )*B -1 26

27 27 Coversio of umbers betwee polyadic systems 1. Method of the iterative multiplicatio with trucatio: : ' : : b B : ' : ' : b B : ' : ' : b B M M M M M M M b B : ' : ' : 1

28 Floatig poit umbers = V * 0,Matissa * 2 Expoet Whe V = +1 if the sig is + ad V = -1, if the sig is. The rage of represetable umbers at m matissa bits ad e expoet bits is e1 2 1 e1 m m

29 Floatig poit umbers have the advatage that they cover a much bigger umber rage as compared to fixed poit umbers of the same legth. Furthermore, they are much more precise aroud or ear to zero. 29

30 Multiplicatio of Floatig Poit Numbers N 1 = V 1 * 0,M 1 * 2 E1, N 2 = V 2 * 0,M 2 * 2 E2 N 1 *N 2 = (V 1 *V 2 ) * 0,(M 1 *M 2 )* 2 E1+E2 30

31 Additio of Floatig Poit Numbers N 1 = V 1 * 0,M 1 * 2 E1, N 2 = V 2 * 0,M 2 * 2 E2 1. Calculate expoet differeces. (e.g. E1>E2). d=e1-e2 2. Shiftig of the matissa M 2 by d positios to the right. M 2 =M 2 >> d 3. Additio of the matissas M 1 ad M 2 4. Calculatio of the sig of the result 5. Normalizatio N 1 +N 2 =(V)*0,(M 1 +M 2 )*2 E1 31

32 IEEE 754 Format 32-Bit (float, sigle) 1 sig bit 8 Expoet bits (MSB first) 23 Matissa bits (MSB first) The value w of such a umber ca be calculated by: w = (-1) V *(1,M)*2 E-127, whe E>0 ad E<255 w = (-1) V *(0,M)*2-126, whe E=0 ad M 0 w = (-1) V *0, whe E=0 ad M=0 w = (-1) V *Ifiity ( ), whe E=255 ad M=0 w = NaN (Not a umber), whe E=255 ad M 0 Represetable rage approximately [ ] 32

33 IEEE 754 Format 64-Bit (double) 1 sig bit 11 Expoet bits (MSB first) 52 Matissa bits (MSB first) The value w of such a umber ca be calculated by: w = (-1) V *(1,M)*2 E-1023, whe E>0 ad E<2047 w = (-1) V *(0,M)*2-1022, whe E=0 ad M 0 w = (-1) V *0, whe E=0 ad M=0 w = (-1) V *Ifiity ( ), whe E=2047 ad M=0 w = NaN (Not a umber), whe E=2047 ad M 0 Represetable rage approximately [ ] 33

34 IEEE 754 Format 80-Bit (exteded) 1 sig bit 15 Expoet bits (MSB first) 64 Matissa bits (MSB first) The value w of such a umber ca be calculated by: w = (-1) V *(0,M)*2 E-16383, whe E>0 ad E<32767 w = (-1) V *Ifiity ( ), whe E=32767 ad M=0 w = NaN (Not a umber), whe E=32767 ad M 0 Represetable Rage approximately [ ] 34

35 Codig of the Decimal Numbers Decimal- Biary Aike 3-Excess 2aus5 Digit Weights Noe

36 EBCDIC (Exteded Biary Coded Decimal Iterchage Code) A B C D E F blak. < ( + l &! $ ) ; / ^, % >? : * a b c d e f g h i j k l m o p q r A 1010 s t u v w x y z B 1011 C 1100 A B C D E F G H I D 1101 J K L M N O P Q R E 1110 S T U V W X Y Z F

37 37 ASCII (America Stadard Code for Iformatio Iterchage) A B D E F NUL SOH STX ETX EOT ENQ ACK BEL BS SKIP LF VT FF CR SO SI DLE DC1 DC2 DC3 DC4 NAK SYN ETB CAN EM SUB ESC FS GS HOME NL 001 SP! # $ % & ( ) * +, -. / : ; < = >? A B C D E F G H I J K L M N O P Q R S T U V W X Y Z [ \ ] _ 101 a b c d e f g h i j k l m o 110 C p q r s t u v w x y z { } ~ DEL 111