Digital Logic

1. Digital Logic http://www.pds.ewi.tudelft.nl/~iosup/Courses/2011_ti1400_1.ppt

2. Outline Basics of Boolean algebra and digital implementation Sum of products form and digital implementation Functional Units Repeated Operations Other Building Blocks

3. Unit of Information • Computers consist of digital (binary) circuits • Unit of information: bit(Binary digIT), e.g. 0and 1 • There are two interpretations of 0and 1: • as data values • as truth values (true andfalse)

4. Bit Strings • By grouping bits together we obtain bit strings • e.g<10001> which can be given a specific meaning • For instance, we can represent non-negative numbers by bitstrings:

5. Boolean Logic • We want a computer that can calculate, i.e transform strings into other strings: 1 +2 = 3 <01>  <10> = <11> • To calculate we need an algebrabeing able to use only two values • George Boole (1854) showed that logic (or symbolic reasoning) can be reduced to a simple algebraic system

6. Boolean algebra • Rules are the same as school algebra: • There is, however, one exception: ! Commutative Law Distributive Law Associative Law

7. Boolean algebra • To see this we have to find out what the operations “+” and “.” mean in logic • First the “.” operation: x.y(orx y) • Suppose x means “black” and y means “cows”. Then, x.y means “black cows” • Hence “.” implies the class of objects that has both properties. Also called ANDfunction.

8. Boolean algebra • The “+” operation merges independent objects: x + y (or x  y) • Hence, ifx means “man” and y means “woman” • Then x+y means “man or woman” • Also called OR function

9. Boolean algebra • Now suppose both objects are identical, for example x means “cows” • Then x.x comprises no additional information • Hence

10. Boolean algebra • Next, we select “0” and “1” as the symbols in the algebra • This choice is not arbitrary, since these are the only number symbols for which holds x2 = x • What do these symbols mean in logic? • “0” : Nothing • “1”: Universe • So 0.y = 0 and 1.y = y

11. Boolean algebra • Also, if x is a class of objects, then 1-x is the complement of that class • It holds that x(1-x) = x -x2 = x-x =0 • Hence, a class and its complement have nothing in common • We denote 1-x as x

12. Boolean algebra • A nice property of this system that we write any function f(x) as • We can show this by observing that virtually every mathematical function can be written in polynomial form, i.e

13. Boolean algebra • Now • Hence, • Let b = a0 and a = a0 + a1 • Then we have • From this it follows that

14. Boolean algebra • So • More dimensional functions can be derived in an identical way:

15. Binary addition • We apply this on the modulo-2 addition

16. Binary multiplication • Same for modulo-2 multiplication

17. Functions • Let X denote bitstring, e.g., <x4x3x2 x1> • Any polynomial functionY=f(X)can be constructed using Boolean logic • Also holds for functions with more arguments • Functions can be put in table form or in formula form

18. Gates • We use basic components to represent primary logic operations (called gates) • Components are made from transistors x x x+y x.y y y OR AND x x INVERT

19. Networks of gates • We can make networks of gates x y EXOR

20. Outline • Basics of Boolean algebra and digital implementation • Sum of products form and digital implementation • Functional Units • Repeated Operations • Other Building Blocks

21. simplify f x y Sum of product form

22. Minimization of expressions • Logic expressions can often be minimized • Saves components • Example:

23. Karnaugh maps (1) x y • Alternative geometrical method v w

24. Karnaugh maps (2) Different drawing y w v x

25. Don’t Cares • Some outputs are indifferent • Can be used for minimization

26. de Morgan’s Laws NAND and NOR gates • NAND and NOR gates are universal • They are easy to realize

27. Outline • Basics of Boolean algebra and digital implementation • Sum of products form and digital implementation • Functional Units • Repeated Operations • Other Building Blocks

28. Delay • Every network of gates has delays transition time 1 input 0 1 propagation delay output 0 time

29. Packaging Vcc Gnd

30. nandgates A Y ADD B Y A,B  time delay Making functions

31. Functional Units • It would be very uneconomical to construct separate combinatorial circuits for every function needed • Hence, functional units are parameterized • A specific function is activated by a special control stringF

32. A Y F B F A B F F Y Arithmetic and Logic Unit

33. Outline • Basics of Boolean algebra and digital implementation • Sum of products form and digital implementation • Functional Units • Repeated Operations • Other Building Blocks

34. Repeated operations • Y : = Y + Bi, i=1..n • Repeated addition requires feedback • Cannot be done without intermediate storage of results Y F B F

35. Registers Y F B F = storage element

36. SR flip flop • Storage elements are not transient and are able to hold a logic value for a certain period of time R Qa Qb S

37. Clocks • In many circuits it is very convenient to have the state changed only at regular points in time • This makes design of systems with memory elements easier • Also, reasoning about the behavior of the system is easier • This is done by a clock signal clockperiod

38. Qn C Qn D D flip flop • D flip flop samples at clock is high and stores if clock is low D Qn+1 0 0 1 1

39. state change Edge triggered flip flops • In reality most systems are built such that the state only changes at rising edge of the clock pulse • We also need a control signal to enable a change

40. enables a state change I R/W C C I C D R/W Q O O time Basic storage element

41. Outline • Basics of Boolean algebra and digital implementation • Sum of products form and digital implementation • Functional Units • Repeated Operations • Other Building Blocks

42. 4-bit register I R/W I I I C C D C D C D C D Q Q Q Q O O O O

43. A B Y = A if m=1 Y = B if m=0 m MPLEX Y Y Only output yA= 1, rest is 0 Decoder A Some basic circuits

44. a1 3 a1 a2 #y 0 0 0 0 1 1 1 0 2 1 1 3 2 1 0 a2 Decoder Y Only output yA= 1, rest is 0 Decoder A

45. A B Y = A if m=1 Y = B if m=0 m MPLEX Y b y a m Multiplexer

46. End of Lecture • Comments? • Questions?

47. Memory Din mplex REG1 Dout REG2 decoder Address REG3 REG4 R/W

48. Counter preset MPLEX R/W REG INC 0001 output

49. Sequential circuits • The counter example shows that systems have state • The state of such systems depend on the current inputs and the sequence of previous inputs • The state of a system is the union of the values of the memory elements of that system

50. code S0 S1 S2 State diagrams • We call the change from one state to another a statetransition • Can be represented as a state diagram