Schedule. ECEN 301 Discussion #25 Final Review 1. Date Day Class No. 1 Dec Mon 25 Final Review. Title Chapters HW Due date. Lab Due date.

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1 Schedule Date Day Class No. Dec Mon 25 Final Review 2 Dec Tue 3 Dec Wed 26 Final Review Title Chapters HW Due date Lab Due date LAB 8 Exam 4 Dec Thu 5 Dec Fri Recitation HW 6 Dec Sat 7 Dec Sun 8 Dec Mon 27 Final Review 9 Dec Tue LAB 9 ECEN 3 Discussion #25 Final Review

2 Summary Jacob 6:2 2 O be wise; what can I say more? ECEN 3 Discussion #25 Final Review 2

3 Lecture 25 Final Review ECEN 3 Discussion #25 Final Review 3

4 Final Exam 7 2 December Chapters 2 6, 8, questions 3 multiple choice (answer on bubble sheet!) point each long answer (show your work!) 4 or 5 points each Closed book! Three 3x5 cards allowed Calculators allowed No time limit Study lecture slides and homework ECEN 3 Discussion #25 Final Review 4

5 Final Exam Review Overview. Exam Review 2. Exam 2 Review 3. Binary Numbers Signed & Unsigned Conversions 4. Logic Functions Conversions among 3 representations 5. Boolean Algebra 6. Combinational Logic 7. Sequential Logic 8. Digital to Analog Converters (DACs) ECEN 3 Discussion #25 Final Review 5

6 Binary Numbers Unsigned Binary word (B): a sequence of n s and s B = b n- b n-2 b 2 b b.b - b -2 b -(m-) b -m Binary point E: B =. (n = 8, m=4) Converting from binary (B) to decimal (D) B = b n- b n-2 b 2 b b.b - b -2 b -(m-) b -m D = b n- 2 n- + b n-2 2 n b 2 + b 2 + b b b -(m-) 2 -(m-) + b -m 2 -m ECEN 3 Discussion #25 Final Review 6

7 Binary Numbers Unsigned Example: What is. 2 in decimal? ECEN 3 Discussion #25 Final Review 7

8 Binary Numbers Unsigned Example: What is. 2 in decimal? ECEN 3 Discussion #25 Final Review 8

9 Binary Numbers Signed 3 common representations for signed integers:. Sign magnitude 2. s compliment 3. 2 s compliment Most common for computers ECEN 3 Discussion #25 Final Review 9

10 Binary Numbers Sign-Magnitude Range: 2 n Representations 2 n binary => 5 decimal The MSB encodes the sign: => -5 = + => = => - Problem Difficult addition/subtraction check signs convert to positive use adder or subtractor as required How to add two sign-magnitude numbers? Ex: + (-4) ECEN 3 Discussion #25 Final Review

11 Binary Numbers s Complement Range: 2 n Representations binary => 6 decimal => -6 => => - Problem Difficult addition/subtraction no need to check signs as before cumbersome logic circuits end-around-carry How to add to one s complement numbers? Ex: 4 + (-3) 2 n To negate a number, Invert it, bit-by-bit. MSB still encodes the sign: = + = ECEN 3 Discussion #25 Final Review

12 Binary Numbers Two s Complement Range: 2 n 2 n Representation: If number is positive or zero, normal binary representation, zeroes in upper bit(s) If number is negative, start with positive number flip every bit (i.e., take the one s complement) then add one MSB still encodes the sign: = + = (5) (9) + + ( s comp) ( s comp) (-5) (-9) ECEN 3 Discussion #25 Final Review 2

13 Binary Numbers Signed Example2: What is the 2 s complement of 2? ECEN 3 Discussion #25 Final Review 3

14 Binary Numbers Signed Example2: What is the 2 s complement of 2? (53) ( s comp) + (add ) (-53) ECEN 3 Discussion #25 Final Review 4

15 Binary Numbers Decimal to Binary Positive numbers start with empty result if decimal number is odd, prepend to result else prepend divide number by 2, throw away fractional part (INTEGER divide) if number is non-zero, go back to else you are done Negative numbers do above for positive version of number and negate result. ECEN 3 Discussion #25 Final Review 5

16 Binary Numbers Decimal to Binary Example3: convert -95 to 2 s complement binary notation ECEN 3 Discussion #25 Final Review 6

17 Binary Numbers Decimal to Binary Example3: convert -95 to 2 s complement binary notation. First find positive version (95) in binary then take 2 s complement Is it odd? 95 ECEN 3 Discussion #25 Final Review 7

18 Binary Numbers Decimal to Binary Example3: convert -95 to 2 s complement binary notation. First find positive version (95) in binary then take 2 s complement Is it odd? 95 Integer division by 2: 95/2 = ECEN 3 Discussion #25 Final Review 8

19 Binary Numbers Decimal to Binary Example3: convert -95 to 2 s complement binary notation. First find positive version (95) in binary then take 2 s complement Is it odd? 95 Integer division by 2: 97/2 = ECEN 3 Discussion #25 Final Review 9

20 Binary Numbers Decimal to Binary Example3: convert -95 to 2 s complement binary notation. First find positive version (95) in binary then take 2 s complement Is it odd? 95 Integer division by 2: 48/2 = ECEN 3 Discussion #25 Final Review 2

21 Binary Numbers Decimal to Binary Example3: convert -95 to 2 s complement binary notation. First find positive version (95) in binary then take 2 s complement Is it odd? 95 Integer division by 2: 24/2 = ECEN 3 Discussion #25 Final Review 2

22 Binary Numbers Decimal to Binary Example3: convert -95 to 2 s complement binary notation. First find positive version (95) in binary then take 2 s complement Integer division by 2: 2/2 = 6 Is it odd? ECEN 3 Discussion #25 Final Review 22

23 Binary Numbers Decimal to Binary Example3: convert -95 to 2 s complement binary notation. First find positive version (95) in binary then take 2 s complement Integer division by 2: 6/2 = 3 Is it odd? ECEN 3 Discussion #25 Final Review 23

24 Binary Numbers Decimal to Binary Example3: convert -95 to 2 s complement binary notation. First find positive version (95) in binary then take 2 s complement Integer division by 2: 3/2 =.5 Is it odd? ECEN 3 Discussion #25 Final Review 24

25 Binary Numbers Decimal to Binary Example3: convert -95 to 2 s complement binary notation. First find positive version (95) in binary then take 2 s complement Integer division by 2: /2 =.5 Is it odd? ECEN 3 Discussion #25 Final Review 25

26 Binary Numbers Decimal to Binary Example3: convert -95 to 2 s complement binary notation. First find positive version (95) in binary then take 2 s complement Is it odd? s complement ECEN 3 Discussion #25 Final Review 26

27 Hexadecimal Notation Binary is hard to read and write by hand Hexadecimal is a common alternative 6 digits are ABCDEF = x478f = xdead = xbeef = xa5a5. Separate binary code into groups of 4 bits (starting from the right) 2. Translate each group into a single hex digit x is a common prefix for writing numbers which means hexadecimal Binary Hex Dec A B C 2 D 3 E 4 F 5 ECEN 3 Discussion #2 Boolean Algebra 27

28 Binary Numbers Hexadecimal Example4: convert 2 to hexadecimal notation ECEN 3 Discussion #25 Final Review 28

29 Binary Numbers Hexadecimal Example4: convert 2 to hexadecimal notation 7 ECEN 3 Discussion #25 Final Review 29

30 Binary Numbers Hexadecimal Example4: convert 2 to hexadecimal notation D 7 ECEN 3 Discussion #25 Final Review 3

31 Binary Numbers Hexadecimal Example4: convert 2 to hexadecimal notation 4 D 7 ECEN 3 Discussion #25 Final Review 3

32 Binary Numbers Hexadecimal Example4: convert 2 to hexadecimal notation F 4 D 7 ECEN 3 Discussion #25 Final Review 32

33 Binary Numbers Hexadecimal Example4: convert 2 to hexadecimal notation 8 F 4 D 7 ECEN 3 Discussion #25 Final Review 33

34 Binary Numbers Hexadecimal Example4: convert 2 to hexadecimal notation A 8 F 4 D 7 ECEN 3 Discussion #25 Final Review 34

35 Binary Numbers Hexadecimal Example4: convert 2 to hexadecimal notation 3 A 8 F 4 D 7 NB: add a leading zero to complete 4 bits ECEN 3 Discussion #25 Final Review 35

36 Binary Numbers Hexadecimal Example4: convert 2 to hexadecimal notation x3a8f4d7 ECEN 3 Discussion #25 Final Review 36

37 Binary Numbers Example5: Complete the following table Unsigned Sign-magitude s complement 2 s complement HE ECEN 3 Discussion #25 Final Review 37

38 Binary Numbers Example5: Complete the following table Unsigned Sign-magitude s complement 2 s complement HE x xeb 93 x5d -93 NA xa3 ECEN 3 Discussion #25 Final Review 38

39 Logic Functions 3 different ways to represent logic functions:. Equation: a mathematical representation of a logic function A bar over a variable represent an inverting or a NOT operation out sab sab sab sab Final logic output Each letter variable represents a top-level input to the logic function Mathematical operations (i.e. addition and multiplication) are boolean algebra operations ECEN 3 Discussion #25 Final Review 39

40 Logic Functions 3 different ways to represent logic functions: 2. Gates: a visual block representation of the function Top-level inputs Final output Four 3-input AND gates feeding into one 4-input OR gate ECEN 3 Discussion #25 Final Review 4

41 Logic Functions 3 different ways to represent logic functions: 3. Truth Table: indicates what the output will be for every possible input combination A B C Z If there are n inputs (left-hand columns) there will be 2 n entries (rows) in the table E: 3 inputs require 2 3 = 8 rows There will always be at least one output (right-hand columns) For each input combination (row) output(s) will be either or ECEN 3 Discussion #25 Final Review 4

42 Logic Functions Gates Type Symbol Equation Truth Table NOT OUT IN IN OUT A B OUT AND OUT A B OR OUT A B A B OUT ECEN 3 Discussion #25 Final Review 42

43 Logic Functions Gates Type Symbol Equation Truth Table A B OUT NAND OUT A B A B OUT NOR OUT A B ECEN 3 Discussion #25 Final Review 43

44 Logic Functions Translation These are three different ways of representing logical information A B Logic Equations You can convert any one of them to any other Logic Gates Truth Tables A B OUT ECEN 3 Discussion #25 Final Review 44

45 Equations to Gates y NOT( s) AND a AND NOT( b) OR y s a b s s a b y a y OR b s a y b ECEN 3 Discussion #25 Final Review 45

46 Logic Functions Gates to Equations out out s a b s a b s a b s a b out ( s a b) ( s a b) ( s a b) ( s a b) OR sab sab sab sab ECEN 3 Discussion #25 Final Review 46

47 Logic Functions Truth Tables to Gates Each row of truth table is an AND gate Each output column is an OR gate S A B OUT s a b s a b s a b s a b out ECEN 3 Discussion #25 Final Review 47

48 Logic Functions Truth Table to Equations Write out truth table a combination of AND s and OR s equivalent to gates easily converted to gates S A B OUT out sab sab sab sab ECEN 3 Discussion #25 Final Review 48

49 Logic Functions Equations to Truth Tables For each AND term fill in the proper row on the truth table out sab sab sab sab S A B OUT ECEN 3 Discussion #25 Final Review 49

50 ECEN 3 Discussion #25 Final Review 5 Boolean Algebra Rules Z Y Z Z Y Y Y Y Z Y Z Y Y Z ) ( ) ( 7. ) ( Z Y Z Y Z Y Z Y Z Y Z Y Y Y Y Y ) ( 4. ) ( ) ( 3. ) ( ) ( 2...

51 Boolean Algebra DeMorgan s Law To distribute the bar, change the operation. A B A B NOR Symbols A B A B NAND Symbols ECEN 3 Discussion #25 Final Review 5

52 Boolean Algebra Example6: simplify the following function OUT ABD ABD BCD ACD ECEN 3 Discussion #25 Final Review 52

53 Boolean Algebra Example6: simplify the following function OUT ABD ABD BCD ACD Rule 4 OUT AD B B BCD ACD Rule 4 OUT AD BCD ACD Rule 4 OUT D A AC BCD Rule 8 OUT D A C BCD Rule 4 OUT AD CD BCD Rule 4 OUT CD( B) AD Rule 2 OUT CD AD Rule 4 OUT D A C ECEN 3 Discussion #25 Final Review 53

54 Boolean Algebra Example7: Simplify the equation created by the following truth table A B C Z ECEN 3 Discussion #25 Final Review 54

55 Boolean Algebra Example7: Simplify the equation created by the following truth table A B C Z Z ABC ABC ABC ABC ABC ABC ECEN 3 Discussion #25 Final Review 55

56 Boolean Algebra Example7: Simplify the equation created by the following truth table A B C Z Z AC Z AC Z AC Z A C Z ABC ABC ABC Z AC B B AB C AB AB A B B A ABC ABC C AB C C ABC ECEN 3 Discussion #25 Final Review 56

57 Boolean Algebra A Example8: Determine the truth table B Z C A B C Z???????? ECEN 3 Discussion #25 Final Review 57

58 Boolean Algebra A Example8: Determine the truth table x A B B Z C A B C x Z???????? ECEN 3 Discussion #25 Final Review 58

59 Boolean Algebra A Example8: Determine the truth table x A B B Z C A B C x x 2 Z???????? x 2 BC B C ECEN 3 Discussion #25 Final Review 59

60 Boolean Algebra A Example8: Determine the truth table x A B B Z C A B C x x 2 x 3 Z???????? x 3 A C x 2 BC B C ECEN 3 Discussion #25 Final Review 6

61 Boolean Algebra A Example8: Determine the truth table x A B B Z C A B C x x 2 x 3 Z x 3 A C x 2 BC B C Z A B BC A C ECEN 3 Discussion #25 Final Review 6

62 Boolean Algebra Example8: Determine the truth table Z A B BC A C ABC ABC ABC AC B B ABC AC ABC A C BC A C B A B C x x 2 x 3 Z ECEN 3 Discussion #25 Final Review 62

63 Combinational Logic Decoders Decode the input and signify its value by raising just one of its outputs. A decoder with n inputs has 2 n outputs W Y 2-to-4 Decoder W Y Z Z A B DECODER Symbol ECEN 3 Discussion #25 Final Review 63

64 Combinational Logic Decoders Write the truth table W Y A B W Y Z Z ECEN 3 Discussion #25 Final Review 64

65 Combinational Logic Multiplexors Connect one of its inputs to its output according to select signals A B S Useful for selecting one from a collection of data inputs. Usually has 2 n inputs and n select lines. ECEN 3 Discussion #25 Final Review 65 C MULTIPLEOR Symbol

66 Combinational Logic Multiplexors Write the truth table A B S C A B S C MULTIPLEOR Symbol ECEN 3 Discussion #25 Final Review 66

67 Sequential Logic SR Latch SR Latch with additional inputs: Enable (E) S and R can only change Q when E is Preset (PRE) regardless of S, R, or E, put Q to when PRE is Clear (CLR) regardless of S, R, E, or PRE, put Q to when CLR is CLR S Q E R PRE Precedence:. If CLR =, Q = 2. If PRE =, Q = 3. If E =, Q is set based on SR: If S = and R =, Q = held If S = and R =, Q = If S = and R =, Q = If S = and R =, Q = unstable 4. Else Q is held Q CLR PRE R S E SR can only change Q only in blue regions (where E = ) BUT CLR and PRE will change Q ANYTIME ECEN 3 Discussion #25 Final Review 67

68 Sequential Logic D Latch D Latch has only 2 states: Set (set Q to ): D = Reset (reset Q to ): D = D Latch with enable (E): Q can only change when E = D E S E R Q D E Q E D Q new Q old Q old Q D E D can only change Q only in blue regions (where E = ) ECEN 3 Discussion #25 Final Review 68

69 Sequential Logic D Flip-Flop D FF: 2 SR latches in master/slave configuration. The output (Q) changes on the rising clock edge Master Slave D S Q S Q Q D Q R Q R E E Q D D Q new D can only change Q only on rising clock edge (arrows) ECEN 3 Discussion #25 Final Review 69

70 Sequential Logic JK Flip-Flop JK FF: 2 SR latches in master/slave configuration. The output (Q) changes on the falling clock edge JK FF has 4 allowed states: Present state (keep Q as is): J =, K = Reset (reset Q to ): J =, K = Set (set Q to ): J =, K = Toggle (set Q to Q): J =, K = J K Q new Q old Q old J S E Q S E Q Q J Q K K R Q R Q Indicates falling clock edge ECEN 3 Discussion #25 Final Review 7

71 Sequential Logic T Flip-Flop T FF: JK FF with J and K inputs connected T FF has 2 allowed states: Present state (keep Q as is): T = Toggle (set Q to Q): T = T Q new Q old Q old T J Q K Q T Q ECEN 3 Discussion #25 Final Review 7

72 Sequential Logic Digital Counters Ripple counter: with N bits, cycles through the numbers from to 2 N N JK FFs cascaded together to produce an N-bit up counter J Q J Q J Q NB: for 3-bit counter we need 3 FFs K K B 2 B B K B 2 B B ECEN 3 Discussion #25 Final Review 72

73 Sequential Logic Digital Counters Synchronous counter: with N bits, cycles through the numbers from to 2 N Input clock drives all FFs simultaneously B B B 2 B 2 T Q T Q T Q B B ECEN 3 Discussion #25 Final Review 73

74 Sequential Logic Example9: Assuming the outputs of the following circuit start in a state, determine the outputs for 4 clock cycles Q 2 Q Q J Q K T Q D Q ECEN 3 Discussion #25 Final Review 74

75 Sequential Logic Example9: Assuming the outputs of the following circuit start in a state, determine the outputs for 4 clock cycles. Set outputs to 2. Based on output values change FF inputs 3. On each clock cycle: a) change FF outputs based on inputs b) Change FF inputs based on new outputs J Q K J K Q new Q 2 Q Q T Q D Q Q old Q old T Q new Q old Q old D Q new ECEN 3 Discussion #25 Final Review 75

76 Sequential Logic Example9: Assuming the outputs of the following circuit start in a state, determine the outputs for 4 clock cycles. Set outputs to 2. Based on output values change FF inputs 3. On each clock cycle: a) change ALL FF outputs based on inputs J Q K J K Q new Q 2 Q Q T Q D Q Q old Q old T Q new Q old Q old D Q new ECEN 3 Discussion #25 Final Review 76

77 Sequential Logic Example9: Assuming the outputs of the following circuit start in a state, determine the outputs for 4 clock cycles. Set outputs to 2. Based on output values change FF inputs 3. On each clock cycle: a) change ALL FF outputs based on inputs b) Change ALL FF inputs based on new outputs J Q K J K Q new Q 2 Q Q T Q D Q Q old Q old T Q new Q old Q old D Q new ECEN 3 Discussion #25 Final Review 77

78 Sequential Logic Example9: Assuming the outputs of the following circuit start in a state, determine the outputs for 4 clock cycles Cycle Q 2 Q Q start Inputs changed due to outputs J Q K J K Q new Q 2 Q Q T Q D Q Q old Q old T Q new Q old Q old D Q new ECEN 3 Discussion #25 Final Review 78

79 Sequential Logic Example9: Assuming the outputs of the following circuit start in a state, determine the outputs for 4 clock cycles Cycle Q 2 Q Q start Outputs change on new clock cycle J Q K J K Q new Q 2 Q Q T Q D Q Q old Q old T Q new Q old Q old D Q new ECEN 3 Discussion #25 Final Review 79

80 Sequential Logic Example9: Assuming the outputs of the following circuit start in a state, determine the outputs for 4 clock cycles Cycle Q 2 Q Q start J Q K Q 2 Q Q T Q D Q Inputs changed due to outputs J K Q new Q old T Q new Q old Q old D Q new Q old ECEN 3 Discussion #25 Final Review 8

81 Sequential Logic Example9: Assuming the outputs of the following circuit start in a state, determine the outputs for 4 clock cycles Cycle Q 2 Q Q start 2 Outputs change on new clock cycle J Q K J K Q new Q old Q old Q 2 Q Q T Q new Q old Q old D Q new ECEN 3 Discussion #25 Final Review 8 T Q D Q

82 Sequential Logic Example9: Assuming the outputs of the following circuit start in a state, determine the outputs for 4 clock cycles Cycle Q 2 Q Q start 2 J Q K Q 2 Q Q T Q D Q Inputs changed due to outputs J K Q new Q old T Q new Q old Q old D Q new Q old ECEN 3 Discussion #25 Final Review 82

83 Sequential Logic Example9: Assuming the outputs of the following circuit start in a state, determine the outputs for 4 clock cycles Cycle Q 2 Q Q start 2 3 Outputs change on new clock cycle J Q K J K Q new Q old Q old Q 2 Q Q T Q new Q old Q old D Q new ECEN 3 Discussion #25 Final Review 83 T Q D Q

84 Sequential Logic Example9: Assuming the outputs of the following circuit start in a state, determine the outputs for 4 clock cycles Cycle Q 2 Q Q start 2 3 J Q K Q 2 Q Q T Q D Q Inputs changed due to outputs J K Q new Q old T Q new Q old Q old D Q new Q old ECEN 3 Discussion #25 Final Review 84

85 Sequential Logic Example9: Assuming the outputs of the following circuit start in a state, determine the outputs for 4 clock cycles Cycle Q 2 Q Q start J Q K Q 2 Q Q T Q D Q Outputs change on new clock cycle J K Q new Q old T Q new Q old Q old D Q new Q old ECEN 3 Discussion #25 Final Review 85

86 Digital to Analog Converter (DAC) DAC: converts an unsigned binary word to an analog output voltage or current Resolution δv: minimum step size by which the output voltage (or current) can increment Output voltage v a : the analog value represented by the binary word B E: let n=4 v a = (2 3 b b b + 2 b )δv Max output voltage v amax : the maximum analog value E: let n=4 v amax = ( )δv = (2 n ) δv ECEN 3 Discussion #25 Final Review 86

87 Digital to Analog Converter (DAC) Building a DAC: use a summing amplifier v in + + R S v S R S2 v S2 R Sn R F R n- + + v o b n- b n-2 R n-2 R R F + + v a + v Sn b R b Summing amplifier DAC ECEN 3 Discussion #24 DAC 87

88 Digital to Analog Converter (DAC) Building a DAC: the analog output (v a ) is proportional to the binary word B v in b n- b n-2 b R n- R n-2 R R F + + v a Choose v a R i n i R 2 R R i F i b i v in v si in summing amplifier R b THUS v a R R F v in n i 2 i b i ECEN 3 Discussion #25 Final Review 88

89 Digital to Analog Converter (DAC) Example: find the smallest resolution δv of an 8-bit DAC v amax = 2V v in b n- b n-2 R n- R n-2 R F + + v a R b b R ECEN 3 Discussion #25 Final Review 89

90 Digital to Analog Converter (DAC) Example: find the smallest resolution δv of an 8-bit DAC v amax = 2V v in b n- b n-2 b b R n- R n-2 R R R F + + v a v amax v 2 v n amax n v 47. mv v amin ECEN 3 Discussion #25 Final Review 9

91 Digital to Analog Converter (DAC) Example: find the resistor values for a DAC with: range = 5V δv = V v in = 5V R F = 2kΩ v in b n- b n-2 R n- R n-2 R F + + v a R b b R ECEN 3 Discussion #25 Final Review 9

92 Digital to Analog Converter (DAC) Example: find the resistor values for a DAC with: range = 5V δv = V v in = 5V R F = 2kΩ v in b n- b n-2 b R n- R n-2 R R R F + + v a b 4 n log log 4 ( v a max 5 / log 2 v log a min 2 ) / v ECEN 3 Discussion #25 Final Review 92

93 Digital to Analog Converter (DAC) Example: find the resistor values for a DAC with: range = 5V δv = V v in = 5V R F = 2kΩ v in b n- b n-2 b b R n- R n-2 R R R F + + v a v v R a a max R R R R F R v F F a max v 2 5 k v in 3 n in i v 2 in 5 n 2 n i b i ECEN 3 Discussion #25 Final Review 93

94 Digital to Analog Converter (DAC) Example: find the resistor values for a DAC with: range = 5V δv = V v in = 5V R F = 2kΩ v in b n- b n-2 b b R n- R n-2 R R R F + + v a R R 2 2 5k 3 R i R 2 R 2 i R k R 3 R k ECEN 3 Discussion #25 Final Review 94

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