Basic Digital Logic

1. Basic Digital Logic Chapter 2 Basic Digital Logic

2. Outlines • Basic Digital Logic Gates • Two types of digital logic circuits • Combinational logic circuits • Sequential logic circuits • Combinational logic circuit design • Using sum of product • Using product of sum • Karnaugh map: Minimization of logic circuits • Without don’t care term • With don’t care term • Construction from smaller components • Sequential logic circuit Basic Digital Logic

3. B A A B A Switches • Abstraction of building block of digital computers • Basic logic functions • AND • OR • NOT Basic Digital Logic

4. Basic Digital Logic Gates NOT gate AND gate OR gate XOR gate NAND gate NOR gate XNOR gate Basic Digital Logic

5. Combinational Logic Circuit Sequential Logic Circuit Two types of digital logic circuits • Combinational logic circuits • Output depends only on its input at that time. • Sequential logic circuits • Output depends both on • its previous output and • its input at that time Basic Digital Logic

6. Combinational Logic Circuit Basic Digital Logic

7. How to design a combinational logic circuit • Decide how to encode input and output in 0 and 1 • Describe each bit of the output in term of input • Truth table • Logical function • Construct a logic circuit from the logical function Basic Digital Logic

8. Truth table and logical function out = f (in1, in2, …) Basic Digital Logic

9. Boolean Logic • Basic Boolean operations • AND: X Y, X &Y, X Y, X  Y • OR: X Y, X + Y • NOT: ~X, X’, X • Any Boolean expressions can be described in: • Disjunctive normal form (DNF) /sum of product • Conjunctive normal form (CNF) / product of sum Basic Digital Logic

10. Example: Exclusive OR (XOR) • output = X Y + XY Basic Digital Logic

11. Construct Sum of Products • Find out all conditions when the function is true • X  Y = T(1) when • X=0, Y=1 => X Y =1 • X=1, Y=0 => XY = 1 • OR the conditions • X Y + XY minterm Sum of products Basic Digital Logic

12. Exclusive OR (XOR): Sum of products • output =X Y +XY X output Y Basic Digital Logic

13. Construct Product of Sums • Find out all conditions when the negation of the function is true • X  Y = F(0) when • X=0, Y=0 => X + Y = 0 • X=1, Y=1 =>X +Y = 0 • AND the conditions • ( X + Y )(X +Y) maxterm Product of sum Basic Digital Logic

14. Exclusive OR (XOR) • output = ( X + Y )(X +Y ) X output Y Basic Digital Logic

15. Karnaugh Map (K-map) Basic Digital Logic

16. 2-variable K-map Basic Digital Logic

17. 3-variable K-map Basic Digital Logic

18. 4-variable K-map Basic Digital Logic

19. Full Adder Full adder A B Ci S Co Basic Digital Logic

20. S=A’B’C+A’BC’+ABC+AB’C’ Full Adder Co=AB+BC+AC Basic Digital Logic

21. Encoder A3 Encoder X1 A2 A1 X0 A0 Basic Digital Logic

22. Encoder X1 = A1A0 Basic Digital Logic

23. Encoder X0 = A1+A3 Basic Digital Logic

24. 2 - 4 decoder b3 b2 b1 b0 a1 a0 Decoder • If a1 a0 is a binary i, • bi is 1and • bj is 0 whenji. Basic Digital Logic

25. a1 b3 a0 b2 b1 b0 Decoder b3 = a1 a0 b2 = a1a0 b1 =a1 a0 b0 =a1a0 Basic Digital Logic

26. Multiplexor 2-1 multiplexor D1 D0 x S X = Ds Basic Digital Logic

27. Multiplexor X = S D0 + S D1 Basic Digital Logic

28. D1 D0 S X Multiplexor X = S D0 + S D1 Basic Digital Logic

29. Construction from Smaller Components Basic Digital Logic

30. C S1 S0 Co S Co S FA1 FA0 A B Ci A B Ci A1 B1 A0 B0 Ci=0 2-bit Full Adder 1 1 A1 A0 0 1 + B1 B0 + 1 0 0C S1 S0 2-bit full adder Basic Digital Logic

31. 2 2 2 2 2 2 4-bit Full Adder 1 0 1 1 0 1 1 1 + 1 0 0 1 0 A1 A0 B1 B0 + C S1 S0 C S1 S0 Co S Co S 2FA1 2FA0 A B Ci A B Ci A1 B1 A0 B0 Ci=0 4-bit full adder Basic Digital Logic

32. Sequential Circuit Basic Digital Logic

33. Falling edge Rising edge clock cycle time / clock period Clock Signal oscillating between 0 and 1 Edge-triggered clocking The state of the sequential circuits changes on the clock edge. Basic Digital Logic

34. Types of sequential circuits • Synchronous circuits • With clock • Use in digital computers • Asynchronous circuits • Without clock Basic Digital Logic

35. Q Q 0 1 1 R R 0 0 1 1 0 Q’ Q’ S S 0 1 1 0 reset set Q 0 Q 1/0 1 R ? R 0/1 ? 1/0 ? Q’ Q’ S 0/1 S 0 ? hold 1 unstable SR Latch Basic Digital Logic

36. 0 0 0 0 C C C C Q0 Q0 Q Q Q Q Q0 Q0 Q0 Q0 Q Q Q Q 0 1 D D D D 0 0 1 1 1 0 0 1 0 0 Q0 0 C D Q C=0 hold data C=1 load data 0 1 0 1 D Latch Basic Digital Logic

37. QD xD xx QQ D C Q Q D C Q Q D latch D latch D latch D latch Q Q Q Q D C D C D C D C D latch D latch D latch D latch Q Q Q Q D C D C D C D C 10 01 xx xx xD DD D C Q Q D C Q Q 00 11 C D Q C=1 0 load data D Flip-flop Basic Digital Logic

38. Registers D7 D6 D5 D4 D3 D2 D1 D0 clk D flip-flop D flip-flop D flip-flop D flip-flop D flip-flop D flip-flop D flip-flop D flip-flop O7 O6 O5 O4 O3 O2 O1 O0 Hold time C D Q 8 8 8-bit REGISTER Set-up time Din Dout clk Basic Digital Logic

39. Register Files n-1 n-2 0 n-2n decoder Register number n Write n-1 MUX Register n-1 8 Datain Dataout 8 Register n-2 . . . Register 0 Basic Digital Logic

40. Finite State Machines Next-state function (Combinational) Current State (registers) Output function (Combinational) Outputs Inputs Basic Digital Logic

41. a b clk D Q Q D Q Q clk a b Counter 0 -> 3 A = a  b B =b Basic Digital Logic

42. D Q Q D Q Q R clk a b Counter with reset a b R clk A = R (a  b) B = Rb Basic Digital Logic