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Appendix C

Appendix C. Basics of Digital Logic Part I. Modern Computer. Digital electronics operate with only two voltage levels – high and low other levels are temporary and occur while transitioning between these values binary system matches the underlying abstraction inherent in electronics

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Appendix C

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  1. Appendix C Basics of Digital Logic Part I

  2. Modern Computer • Digital electronics • operate with only two voltage levels – high and low • other levels are temporary and occur while transitioning between these values • binary system matches the underlying abstraction inherent in electronics • Use signals instead of voltage levels which are complements or inverses of one another • logically true : 1 (asserted) • logically false : 0 (deasserted) Florida A & M University - Department of Computer and Information Sciences

  3. Logic Block • Each input maybe 0 or 1 • Each output maybe 0 or 1 Output(s) Input(s) Logic block Florida A & M University - Department of Computer and Information Sciences

  4. Logic Blocks • Combinational • without memory • output depends on only on the current input • completed in one machine cycle • Sequential • Retains memory of previous state • output depends on both the inputs and the logic block state Florida A & M University - Department of Computer and Information Sciences

  5. Combinational Logic Block • Defined by the values of the outputs for each possible set of input values (truth table) • n inputs  2n combinations of input values Florida A & M University - Department of Computer and Information Sciences

  6. Truth Table - example • Output y is 1 when inputs w and x have the same value • Output z is 1 when both inputs are 0 Florida A & M University - Department of Computer and Information Sciences

  7. Truth Table • Completely describes combinational logic function • Grow in size quickly • May not be easy to understand Florida A & M University - Department of Computer and Information Sciences

  8. Boolean Algebra • Another approach to express logic function as logic equations • All variables have the values 0 or 1 • Three operators • OR • AND • NOT (unary) Florida A & M University - Department of Computer and Information Sciences

  9. Boolean Algebra • OR operator • result is 1 if either of the operands (variables) is 1 • also known as logical sum • notation: A + B • AND operator • result is 1 only if both operands (variables) is 1 • also known as logical product • notation: A  B • NOT operator • Result is 1 only if operand is 0 • Inversion or negation of the value • Notation: Florida A & M University - Department of Computer and Information Sciences

  10. Boolean Algebra Laws • Identity : A + 0 = A A  1 = A • Zero and One : A  0 = 0 A + 1 = 1 • Inverse: • Commutative: A  B = B  A A + B = B + A • Associative: A + (B + C) = (A + B) + C A  (B  C) = (A  B)  C • Distributive: A  (B + C) = (A  B) + (A  C) A + (B  C) = (A + B)  (A + C) Florida A & M University - Department of Computer and Information Sciences

  11. Boolean Algebra Laws • Idempotent : A + A = A A  A = A • DeMorgan’s : Florida A & M University - Department of Computer and Information Sciences

  12. Boolean Algebra - example • y = wx + • z = Florida A & M University - Department of Computer and Information Sciences

  13. Logic Gates • Used to build logic blocks by implementing basic logic functions • AND gate • May have multiple inputs • Output equal to the AND of all inputs • OR gate • May have multiple inputs • Output equal to the OR of all inputs • NOT gate inverter • Single input Florida A & M University - Department of Computer and Information Sciences

  14. w z x Logic Gates - example • y = wx + • z = What is logic gate implementation for y? Florida A & M University - Department of Computer and Information Sciences

  15. Logic Gate Implementation Algorithm 1. Determine the truth table for problem statement 2. Consider each output independently a. Consider all nonzero entries for the output b. Write the logic equation c. Simplify the logic equation using the laws of Boolean algebra (if possible) 3. Draw the digital logic gate implementation of the simplified logic equation Florida A & M University - Department of Computer and Information Sciences

  16. Logical Function • Find a logical expression for the logical function with three inputs A, B, and C and four outputs W, X, Y and Z such that W is true if at least two of the inputs is true, X is true if exactly one of the inputs is true, Y is true if B is false, and Z is true if all three inputs are true. W X Y Z A B C Digital Logic Gate Implementation Florida A & M University - Department of Computer and Information Sciences

  17. Logical Function – Truth Table Florida A & M University - Department of Computer and Information Sciences

  18. Logic Function –Boolean Algebra • Find a logical expression for the logical function with three inputs A, B, and C and four outputs W, X, Y and Z such that W is true if at least two of the inputs is true, X is true if exactly one of the inputs is true, Y is true if B is false, and Z is true if all three inputs are true. Florida A & M University - Department of Computer and Information Sciences

  19. Function – Simplified Boolean Algebra • Find a logical expression for the logical function with three inputs A, B, and C and four outputs W, X, Y and Z such that W is true if at least two of the inputs is true, X is true if exactly one of the inputs is true, Y is true if B is false, and Z is true if all three inputs are true. Florida A & M University - Department of Computer and Information Sciences

  20. Multiplexor / Selector • n inputs • 1 output – one of the inputs • Control (selector) lines determine which input becomes the output • log2 n = # of required control Florida A & M University - Department of Computer and Information Sciences

  21. Multiplexor (2-1) Florida A & M University - Department of Computer and Information Sciences

  22. Multiplexor (4-1) Florida A & M University - Department of Computer and Information Sciences

  23. Don’t Cares • Situations where one does not care what the value of some output is • Another output is true • Subset of input combinations determines the values of the outputs • Make it easier to optimize implementation of a logic function due to simpler expression • Two types • Output : don’t care about its value for some input combination • Input : output depends on only some of the inputs • Appear as x’s in truth table Florida A & M University - Department of Computer and Information Sciences

  24. Don’t Cares – example • Find a logical expression for the logical function with three inputs A, B, and C and three outputs W, Y and Z such that: • W is true if A or C are true, no matter the value of B, • Y is true if A or B is true, no matter the value of C, and • Z is true if exactly one of the inputs is true and don’t care about its value when both W and Y are true. Florida A & M University - Department of Computer and Information Sciences

  25. Don’t Cares – example • Find a logical expression for the logical function with three inputs A, B, and C and three outputs W, Y and Z such that: • W is true if A or C are true, no matter the value of B, • Y is true if A or B is true, no matter the value of C, and • Z is true if exactly one of the inputs is true and don’t care about its value when both W and Y are true. Florida A & M University - Department of Computer and Information Sciences

  26. Don’t Cares – example • Find a logical expression for the logical function with three inputs A, B, and C and three outputs W, Y and Z such that: • W is true if A or C are true, no matter the value of B, • Y is true if A or B is true, no matter the value of C, and • Z is true if exactly one of the inputs is true and don’t care about its value when both W and Y are true. Florida A & M University - Department of Computer and Information Sciences

  27. Don’t Cares – example • Find a logical expression for the logical function with three inputs A, B, and C and three outputs W, Y and Z such that: • W is true if A or C are true, no matter the value of B, • Y is true if A or B is true, no matter the value of C, and • Z is true if exactly one of the inputs is true and don’t care about its value when both W and Y are true. Florida A & M University - Department of Computer and Information Sciences

  28. Don’t Cares – example • Find a logical expression for the logical function with three inputs A, B, and C and three outputs W, Y and Z such that: • W is true if A or C are true, no matter the value of B, • Y is true if A or B is true, no matter the value of C, and • Z is true if exactly one of the inputs is true and don’t care about its value when both W and Y are true. Florida A & M University - Department of Computer and Information Sciences

  29. Don’t Cares – example • Find a logical expression for the logical function with three inputs A, B, and C and three outputs W, Y and Z such that: • W is true if A or C are true, no matter the value of B, • Y is true if A or B is true, no matter the value of C, and • Z is true if exactly one of the inputs is true and don’t care about its value when both W and Y are true. Florida A & M University - Department of Computer and Information Sciences

  30. Don’t Cares – example • Find a logical expression for the logical function with three inputs A, B, and C and three outputs W, Y and Z such that: • W is true if A or C are true, no matter the value of B, • Y is true if A or B is true, no matter the value of C, and • Z is true if exactly one of the inputs is true and don’t care about its value when both W and Y are true. Florida A & M University - Department of Computer and Information Sciences

  31. Don’t Cares – example • Find a logical expression for the logical function with three inputs A, B, and C and three outputs W, Y and Z such that: • W is true if A or C are true, no matter the value of B, • Y is true if A or B is true, no matter the value of C, and • Z is true if exactly one of the inputs is true and don’t care about its value when both W and Y are true. Florida A & M University - Department of Computer and Information Sciences

  32. Array of Logic Elements • Combinational operations must be performed on an entire word (32 bits) • Represent by showing that a given operation will happen to entire collection of inputs • Example: • A and B are both 32 bits • Output, which is also 32 bits, gets A or B Florida A & M University - Department of Computer and Information Sciences

  33. What is the digital logic implementation? Florida A & M University - Department of Computer and Information Sciences

  34. What is the digital logic implementation? • All multiplexers controlled by the same selector input • 0 selects A • 1 selects B Florida A & M University - Department of Computer and Information Sciences

  35. Arithmetic Logic Unit (ALU) • Brawn of the computer • Performs arithmetic operations • Addition • Subtraction • Performs logical operations • AND • OR Florida A & M University - Department of Computer and Information Sciences

  36. Arithmetic Logic Unit (ALU) • MIPS need 32-bit wide ALU • Connect 32 1-bit ALUs Florida A & M University - Department of Computer and Information Sciences

  37. ALU Logical Operations • Operation value selects logical operation • 0 for a AND b • 1 for a OR b Florida A & M University - Department of Computer and Information Sciences

  38. ALU Addition (Adder) • Three inputs • two operands • carryin : carryout of neighbor adder • Two outputs • sum • carryout Florida A & M University - Department of Computer and Information Sciences

  39. Adder Implementation? Florida A & M University - Department of Computer and Information Sciences

  40. Adder Implementation Florida A & M University - Department of Computer and Information Sciences

  41. AND, OR, Adder implementation 32-bit ALU constructed from 32 1-bit ALUs Florida A & M University - Department of Computer and Information Sciences

  42. AND, OR, Adder implementation 32-bit ALU constructed from 32 1-bit ALUs Florida A & M University - Department of Computer and Information Sciences

  43. Adder Modification for Subtraction • Add a 2:1 multiplexor that chooses between b and • CarryIn of ALU0 set to 1 for subtraction • Remember two’s complement Florida A & M University - Department of Computer and Information Sciences

  44. NOR functionality • Recall DeMorgan’s Florida A & M University - Department of Computer and Information Sciences

  45. slt Functionality • RECALL: All bits except the least significant bit is 0, with the least significant bit set according to comparison • Expand multiplexor to 4:1 with new input Less • Less = 0 on ALU1 – ALU31 • (a - b) < 0  a < b • ALU31 (most significant bit ALU) has extra output line Set • adder output (also used for overflow detection) • used only for slt • input LESS on ALU0 (least significant bit ALU) Florida A & M University - Department of Computer and Information Sciences

  46. ALU Building Blocks ALU0 – ALU30 ALU31 Florida A & M University - Department of Computer and Information Sciences

  47. ALU Florida A & M University - Department of Computer and Information Sciences

  48. Branch Instructions • Branch if either two registers are equal or if they are unequal • Equality Test: subtract b from a and then test to see if result is 0 REMEMBER: a – b = 0  a = b • Implement: Florida A & M University - Department of Computer and Information Sciences

  49. MIPS ALU • Combine CarryIn and Binvert into one single control line Bnegate • both set to 1 for subtraction • both set to 0 for addition and logical operations • 4 bit control lines • Ainvert (1 bit) • Bnegate (1 bit) • Operation (selector) for multiplexor (2 bits) Florida A & M University - Department of Computer and Information Sciences

  50. MIPS ALU Florida A & M University - Department of Computer and Information Sciences

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