CprE 281: Digital Logic

Transcription

1 CprE 281: Digital Logic Instructor: Alexander Stoytchev

2 Design Examples CprE 281: Digital Logic Iowa State University, Ames, IA Copyright Alexander Stoytchev

3 Administrative Stuff HW3 is out It is due on Monday Sep 4pm. Please write clearly on the first page (in BLOCK CAPITAL letters) the following three things: Your First and Last Name Your Student ID Number Your Lab Section Letter Also, please Staple your pages

4 Administrative Stuff TA Office Hours: 11:00am-1:00pm on Wednesdays (Jinyuan Jia) Location: TLA (Coover Hall - first floor) 9:50am-11:50am on Thursday (Siyuan Lu) Location: TLA (Coover Hall - first floor)

5 Administrative Stuff Homework Solutions will be posted on BlackBoard

6 Quick Review

7 The Three Basic Logic Gates x x x 1 x x 1 x 2 2 x 1 x x 1 + x 2 2 NOT gate AND gate OR gate You can build any circuit using only these three gates [ Figure 2.8 from the textbook ]

8 (a) Dual-inline package V DD Gnd (b) Structure of 7404 chip Figure B.21. A 7400-series chip.

9 V DD x 1 x 2 x 3 Figure B.22. An implementation of f = x 1 x 2 + x 2 x 3. f

10 NAND Gate x 1 x 2 f

11 NOR Gate x 1 x 2 f

12 AND followed by NOT = NAND x 1 x x 1 2 x x 1 2 x 1 x x 1 2 x 2 x 2 x 1 x 2 f f x 1 x 2 f

13 NAND followed by NOT = AND x 1 x x 1 2 x x 1 2 x 1 x x 1 2 x 2 x 2 x 1 x 2 f f x 1 x 2 f

14 OR followed by NOT = NOR x x x 2 x 1 + x 2 x 2 x 1 x 2 f f x 1 x 2 f

15 NOR followed by NOT = OR x 1 x 1 + x 2 x 2 x 1 + x 2 x 1 x 1 + x 2 x 2 x 1 x 2 f f x 1 x 2 f

16 Why do we need two more gates? They can be implemented with fewer transistors. (more about this later)

17 Building a NOT Gate with NAND x x x x x 0 1 x 1 0 x x f impossible combinations Thus, the two truth tables are equal!

18 Building an AND gate with NAND gates [

19 Building an OR gate with NAND gates [

20 Implications Any Boolean function can be implemented with only NAND gates!

21 Implications Any Boolean function can be implemented with only NAND gates! The same is also true for NOR gates!

22 Another Synthesis Example

23 Truth table for a three-way light control [ Figure 2.31 from the textbook ]

24 Minterms and Maxterms (with three variables) [ Figure 2.22 from the textbook ]

25 Let s Derive the SOP form

26 Let s Derive the SOP form

27 Sum-of-products realization f x 1 x 2 x 3 [ Figure 2.32a from the textbook ]

28 Let s Derive the POS form [ Figure 2.31 from the textbook ]

29 Let s Derive the POS form

30 Product-of-sums realization x 3 x 2 x 1 f [ Figure 2.32b from the textbook ]

31 Multiplexers

32 2-1 Multiplexer (Definition) Has two inputs: x 1 and x 2 Also has another input line s If s=0, then the output is equal to x 1 If s=1, then the output is equal to x 2

33 Graphical Symbol for a 2-1 Multiplexer s x 1 x f [ Figure 2.33c from the textbook ]

34 Analogy: Railroad Switch

35 Analogy: Railroad Switch x 1 x 2 select f

36 Analogy: Railroad Switch x 1 x 2 select f This is not a perfect analogy because the trains can go in either direction, while the multiplexer would only allow them to go from top to bottom.

37 Truth Table for a 2-1 Multiplexer [ Figure 2.33a from the textbook ]

38 Let s Derive the SOP form

39 Let s Derive the SOP form

40 Let s Derive the SOP form Where should we put the negation signs? s x 1 x 2 s x 1 x 2 s x 1 x 2 s x 1 x 2

41 Let s Derive the SOP form s x 1 x 2 s x 1 x 2 s x 1 x 2 s x 1 x 2

42 Let s Derive the SOP form s x 1 x 2 s x 1 x 2 s x 1 x 2 s x 1 x 2 f (s, x 1, x 2 ) = s x 1 x 2 + s x 1 x 2 + s x 1 x 2 + s x 1 x 2

43 Let s simplify this expression f (s, x 1, x 2 ) = s x 1 x 2 + s x 1 x 2 + s x 1 x 2 + s x 1 x 2

44 Let s simplify this expression f (s, x 1, x 2 ) = s x 1 x 2 + s x 1 x 2 + s x 1 x 2 + s x 1 x 2 f (s, x 1, x 2 ) = s x 1 (x 2 + x 2 ) + s (x 1 +x 1 )x 2

45 Let s simplify this expression f (s, x 1, x 2 ) = s x 1 x 2 + s x 1 x 2 + s x 1 x 2 + s x 1 x 2 f (s, x 1, x 2 ) = s x 1 (x 2 + x 2 ) + s (x 1 +x 1 )x 2 f (s, x 1, x 2 ) = s x 1 + s x 2

46 Circuit for 2-1 Multiplexer x 1 s s x 2 f x 1 x f (b) Circuit (c) Graphical symbol [ Figure 2.33b-c from the textbook ]

47 More Compact Truth-Table Representation s x 1 x 2 f (s, x 1, x 2 ) (a) Truth table s 0 1 f (s, x 1, x 2 ) x 1 x 2 [ Figure 2.33 from the textbook ]

48 4-1 Multiplexer (Definition) Has four inputs: w 0, w 1, w 2, w 3 Also has two select lines: s 1 and s 0 If s 1 =0 and s 0 =0, then the output f is equal to w 0 If s 1 =0 and s 0 =1, then the output f is equal to w 1 If s 1 =1 and s 0 =0, then the output f is equal to w 2 If s 1 =1 and s 0 =1, then the output f is equal to w 3

49 4-1 Multiplexer (Definition) Has four inputs: w 0, w 1, w 2, w 3 Also has two select lines: s 1 and s 0 If s 1 =0 and s 0 =0, then the output f is equal to w 0 If s 1 =0 and s 0 =1, then the output f is equal to w 1 If s 1 =1 and s 0 =0, then the output f is equal to w 2 If s 1 =1 and s 0 =1, then the output f is equal to w 3 We ll talk more about this when we get to chapter 4, but here is a quick preview.

50 Graphical Symbol and Truth Table [ Figure 4.2a-b from the textbook ]

51 The long-form truth table

52 The long-form truth table [

53 The long-form truth table [

54 The long-form truth table [

55 The long-form truth table [

56 4-1 Multiplexer (SOP circuit) [ Figure 4.2c from the textbook ]

57 Using three 2-to-1 multiplexers to build one 4-to-1 multiplexer s 1 s 0 w 0 w f w 2 w [ Figure 4.3 from the textbook ]

58 Analogy: Railroad Switches

59 Analogy: Railroad Switches w 0 w 1 w 2 w 3 s 1 f

60 Analogy: Railroad Switches w 0 w 1 w 2 w 3 s 0 these two switches are controlled together s 1 f

61 Using three 2-to-1 multiplexers to build one 4-to-1 multiplexer

62 Using three 2-to-1 multiplexers to build one 4-to-1 multiplexer

63 Using three 2-to-1 multiplexers to build one 4-to-1 multiplexer

64 Using three 2-to-1 multiplexers to build one 4-to-1 multiplexer w 0 s 1 s 0 w 1 w 2 f w 3

65 That is different from the SOP form of the 4-1 multiplexer shown below, which uses less gates

66 16-1 Multiplexer s 0 s 1 w 0 w 3 w 4 s 2 s 3 w 7 f w 8 w 11 w 12 w 15 [ Figure 4.4 from the textbook ]

67 [

68 Questions?

69 THE END