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COMP541 Combinational Logic - 3

COMP541 Combinational Logic - 3. Montek Singh Jan 21, 2015. Today’s Topics. Synthesis: from t ruth t able to logic implementation Schematic d rawing conventions Non-Boolean values “Don’t Cares”, or X values “Floating values”, or Z values. Mechanically Go From Truth Table to Function.

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COMP541 Combinational Logic - 3

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  1. COMP541Combinational Logic - 3 Montek Singh Jan 21, 2015

  2. Today’s Topics • Synthesis: • from truth table to logic implementation • Schematic drawing conventions • Non-Boolean values • “Don’t Cares”, or Xvalues • “Floating values”, or Zvalues

  3. Mechanically Go From Truth Table to Function

  4. From Truth Table to Logic Equation • Consider a truth table • Standard sum-of-products implementation • OR of all product terms that are 1 • For each row where output is 1 • write the minterm • called “ON-set minterm” • OR all of these minterms

  5. Standard Forms • Not necessarily simplest F • But it is a systematic way to go from truth table to function • Definitions: • “Literal”: a single variable, complemented or not  Ā • “Product terms”: AND of literals  ĀBZ • “Sum terms”: OR of product terms  X + Ā • This is logical product and sum, not arithmetic

  6. Definition: Minterm • Product term in which all variables appear once (complemented or not) • each minterm is 1 in exactly one row, 0 elsewhere

  7. Number of Minterms • For n variables, there will be 2nminterms • Like binary numbers from 0 to 2n-1 • Often numbered same way (often in decimal)

  8. Maxterms • Sum term in which all variables appear once (complemented or not) • each maxtermis 0 in exactly one row, 1 elsewhere

  9. Minterm related to Maxterm • Minterm and maxterm with same subscripts are complements • Example

  10. Implementation: Sum of Minterms • OR all of the minterms of truth table row with a 1 • “ON-set minterms” • F = m0 + m2 + m5 + m7

  11. More General: Sum of Products • Simplifying sum-of-minterms can yield a sum of products • difference is that each term need not be a minterm • i.e., terms do not need to have all variables • Ex: • Implementation is still AND-OR • but products may contain fewer literals simplifies to:

  12. Two-Level Implementation • Sum of products has 2 levels of gates • ANDs followed by an OR • equivalently: NANDs followed by a NAND

  13. More Levels of Gates? • What’s best? • Hard to answer • More gate delays (more on this later) • But maybe we only have 2-input gates • So multi-input ANDs and ORs have to be decomposed

  14. Complement of a Function • Definition: 1s & 0s swapped in truth table • Mechanical way to derive algebraic form • Take the dual • Recall: Interchange AND and OR, and 1s & 0s • Complement each literal

  15. Complement of F • Not surprisingly, just sum of the other minterms • sum of “OFF-set minterms” • Example: • F = m0 + m2 + m5 + m7 • F’ = m1+ m3 + m4 + m6 simplifies to:

  16. Product of Maxterms • Recall that maxterm is true except for its own case • So M1 is only false for 001

  17. Product of Maxterms • Can express F as AND of all rows that should evaluate to 0 • i.e., product of OFF-setMaxterms! • why? • a row in which F=0 (OFF-set)… • … has a Maxterm that is 0 • which makes the product 0 or

  18. Complement of F • Can express F’s complement similarly: • product of ON-setMaxterms! • why? • a row in which F=1 (ON-set)… • … has a Maxterm that is 0 • which makes F’ zero or

  19. More General: Product of Sums • Simplifying product-of-Maxtermscan yield a product of sums • difference is that each term need not be a Maxterm • i.e., terms do not need to have all variables • Ex: • Implementation is still OR-AND • but each sum may contain fewer literals simplifies to: HOW??  homework problem (hint: distributive property)

  20. From Equations to Gates • Simply parse the Boolean equation and replace each operator with a gate • AND, OR, NOT gates • parentheses indicate hierarchy • Example:

  21. Recap • Working (so far) with AND, OR, and NOT • Algebraic identities • Algebraic simplification • Minterms and maxterms • Can now synthesize gate-level implementation from truth table

  22. Drawing Style • Indicate inputs and outputs using arrows • or: inputs at left/top, outputs at right/bottom • If possible, gates should flow from left to right • or: top to bottom • Straight wires best • or: keep bends at a minimum (preferably 90 deg) • Connections: • wires always connect at a “T” junction • a dot at a wire crossing indicates connection • wire crossing without a dot means no connection

  23. Circuit Schematic Rules (cont.) Wire connections • A dot where wires cross indicates a connection • Wires crossing without a dot make no connection • Wires always connect at a T junction

  24. Multiple Output Circuits: Example Output asserted corresponding to most significant TRUE input Example: Priority Encoder Hardware

  25. Example: Priority Encoder Hardware (contd.)

  26. Values that are not 0’s and 1’s Don’t Cares (X) Floating values (Z)

  27. X values • X is neither 1 nor 0 • typically used to represent “unknown” or “illegal” values • Unknown • e.g., an uninitialized value in a simulator • in hardware most flipflops will wake up to a 1 or a 0 value • but could be different each time it wakes up • Don’t Care • an output specified as X means “don’t care” • i.e., left unspecified: whatever comes out is okay • Illegal • e.g., contention at output • two gates fighting

  28. Actually: Several Meanings of X • When used to specify an input value • Means: “Don’t Care”: this particular input variable’s value does not matter when determining the output • Example: Output F is 1 when the inputs A, B, C are 1X1 • Means F = AC // B is a Don’t Care • Unknown/uninitialized signal • If a simulator cannot determine the value of a signal, it will display it as X • Other values that depend on this signal may also become X • Contention (illegal input value) • Sometimes a simulator will use X to denote the value of a node that is being pulled both to 0 and to 1 • Example: Outputs of two gates are shorted; or a gate has p-transistor and n-transistor network simultaneously on!

  29. Don’t Cares (X) More compact representation! Example: Priority Encoder Hardware

  30. Z values • Also neither 1 nor 0 • but actually “floating” • i.e., the output is neither connected to 0 (ground) nor to 1 (power supply) • Could be undesirable: • actual voltage is highly susceptible to noise • e.g., neighboring wires/gates could easily influence value • Could be by design: • useful in buses, memories, multiplexers, etc. • usually one gate drives a wire to a 1 or 0 • all others “float” their outputs • example: tristate buffers/inverters • cover in next lecture

  31. Next • Mon next week: • Combinational building blocks

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