ECE 3110: Introduction to Digital Systems

1. ECE 3110: Introduction to Digital Systems Combinational Logic Design Principles

2. Other codes • Character codes (nonnumeric) • ASCII (7-bit string) • Codes for action/condition/states • Codes for Detecting and Correcting Errors • Codes for Serial Data Transmission Dr. Xubin He ECE 3110: Introduction to Digital systems

3. Codes for Actions/Conditions/States • If there are n different actions, conditions, or states, we can represent them with a b-bit binary code with • Ceiling function: the smallest integer greater than or equal to the bracketed quantity. Dr. Xubin He ECE 3110: Introduction to Digital systems

4. Dr. Xubin He ECE 3110: Introduction to Digital systems

5. Codes for serial data transmission and storage • Parallel data: disk storage • Serial data: telephone network • Bit rates: bps, numerically equals to the clock frequency(Hz) • Bit time: reciprocal of bit rate • Bit cell: time occupied by each bit. • Line code: format of actual signal on the line, NRZ (Non-Return-to-Zero) • Synchronization signal: identify the significane of each bit in the stream.

7. Chapter Summary • Positional Number Systems, 2, 8, 10, 16 • Conversions • Representation of Negative Numbers • Addition/Subtraction for unsigned and signed numbers • Binary multiplication/division • BCD, Gray…codes Dr. Xubin He ECE 3110: Introduction to Digital systems

8. Chapter 4 • Combinational Logic Design Principles • Analyze • Synthesis • Fundamental Theory: Switching Algebra Dr. Xubin He ECE 3110: Introduction to Digital systems

9. Combinational logic circuit • Outputs depend only on the current inputs (Not on history) • Contain an arbitrary number of logic gates and inverters, but NO feedback loops. Dr. Xubin He ECE 3110: Introduction to Digital systems

10. Analysis vs. Synthesis • Analysis: • Start with a logic diagram and proceed to a formal description of the function performed by that circuit. • Synthesis: • Do the reverse, starting with a formal description and proceeding to a logic diagram. Dr. Xubin He ECE 3110: Introduction to Digital systems

11. Combinational-Circuit Analysis • Kinds of combinational analysis: • exhaustive (truth table) • algebraic (expressions) • simulation / test bench • Write functional description in HDL • Define test conditions / test vectors, including corner cases • Compare circuit output with functional description (or known-good realization) • Repeat for “random” test vectors Dr. Xubin He ECE 3110: Introduction to Digital systems

12. Switching algebra • a.k.a. “Boolean algebra” • deals with boolean values -- 0, 1 • Positive-logic convention • analog voltages LOW, HIGH --> 0, 1 • Negative logic -- seldom used • Signal values denoted by variables(X, Y, FRED, etc.) Dr. Xubin He ECE 3110: Introduction to Digital systems

13. Complement: X¢ (opposite of X) AND: X × Y OR: X + Y Boolean operators binary operators, describedfunctionally by truth table. Dr. Xubin He ECE 3110: Introduction to Digital systems

14. Logic symbols Dr. Xubin He ECE 3110: Introduction to Digital systems

15. Some definitions • Literal: a variable or its complement • X, X¢, FRED¢, CS_L • Expression: literals combined by AND, OR, parentheses, complementation • X+Y • P × Q × R • A + B × C • ((FRED × Z¢) + CS_L × A × B¢× C + Q5) × RESET¢ • Equation: Variable = expression • P = ((FRED × Z¢) + CS_L × A × B¢× C + Q5) × RESET¢ Dr. Xubin He ECE 3110: Introduction to Digital systems

16. Axioms (postulates) • A1) X=0 if X‡1 A1’ ) X=1 if X‡0 • A2) if X=0, then X’=1A2’ ) if X=1, then X’=0 • A3) 0 • 0=0 A3’ ) 1+1=1 • A4) 1 • 1=1 A4’ ) 0+0=0 • A5) 0 • 1= 1 • 0 =0 A5’ ) 1+0=0+1=1 Logic multiplication and addition precedence

17. Theorems (Single variable) • Proofs by perfect induction Dr. Xubin He ECE 3110: Introduction to Digital systems

18. Two- and three- variable Theorems In all of the theorems, it is possible to replace each variable with an arbitrary logic expression.

19. Duality • Swap 0 & 1, AND & OR • Result: Theorems still true • Principle of Duality (Metatheorem) • Any theorem or identity in switching algebra remains true if 0 and 1 are swapped and • and + are swapped throughout. • Why? • Each axiom (A1-A5) has a dual (A1¢-A5¢) Dr. Xubin He ECE 3110: Introduction to Digital systems

20. Duality • Counterexample:X + X × Y = X (T9)X × X + Y = X (dual)X + Y = X (T3¢)???????????? X + (X×Y) = X (T9)X× (X + Y) = X (dual)(X× X) + (X× Y) = X (T8)X+ (X× Y) = X (T3¢) parentheses,operator precedence! Dr. Xubin He ECE 3110: Introduction to Digital systems

21. Dual of a logic expression • If F(X1, X2, X3,… Xn,, +, ‘) is a fully parenthesized logic expression involving variables X1, X2, X3,… Xn and the operators +,, and ‘, then the dual of F, written FD, is the same expression with + and  swapped. • FD(X1, X2, X3,… Xn, +,, ‘)=F(X1, X2, X3,… Xn,, +, ‘) Dr. Xubin He ECE 3110: Introduction to Digital systems

22. Sumamry • Variables, expressions, equations • Axioms (A1-A5 pairs) • Theorems (T1-T15 pairs) • Single variable • 2- or 3- variable • Prime, complement, logic multiplication/addition, precedence • Duality Dr. Xubin He ECE 3110: Introduction to Digital systems

23. Next… • N-variables theorems • Representations of logic fucntions • Read Chapter 4.2 and take notes • Combinational circuit analysis Dr. Xubin He ECE 3110: Introduction to Digital systems