2. Digital Systems Design 1 Signal Expressions Multiply out: F = ((X + Y)  Z) + (X  Y  Z) = (X  Z) + (Y  Z) + (X  Y  Z)

3. Digital Systems Design 2 New circuit, same function Sum of Products

4. Digital Systems Design 3 Product of Sums Form

5. Digital Systems Design 4 Sum-of-products form AND-OR NAND-NAND

6. Digital Systems Design 5 Product-of-sums form OR-AND NOR-NOR

7. Digital Systems Design 6 Shortcut: Symbol substitution

8. Digital Systems Design 7 Different circuit, same function

9. Digital Systems Design 8 Combinational-Circuit Analysis Combinational circuits -- outputs depend only on current inputs (not on history). Kinds of combinational analysis: –exhaustive (truth table) –algebraic (expressions) –simulation / test bench Write functional description in HDL Define test conditions / test vectors Compare circuit output with functional description (or known- good realization) Repeat for “ random ” test vectors

10. Digital Systems Design 9 Combinational-Circuit Design Sometimes you can write an equation or equations directly using “ logic ” (the kind in your brain). Example (alarm circuit): Corresponding circuit:

11. Digital Systems Design 10 Alarm-circuit transformation Sum-of-products form –Useful for programmable logic devices “ Multiply out ” :

12. Digital Systems Design 11 Brute-force design Truth table --> canonical sum (sum of minterms) Example: prime-number & 1 detector –4-bit input, N 3 N 2 N 1 N 0 row N 3 N 2 N 1 N 0 F 0 0 0 0 0 0 1 0 0 0 1 1 2 0 0 1 0 1 3 0 0 1 1 1 4 0 1 0 0 0 5 0 1 0 1 1 6 0 1 1 0 0 7 0 1 1 1 1 8 1 0 0 0 0 9 1 0 0 1 0 10 1 0 1 0 0 11 1 0 1 1 1 12 1 1 0 0 0 13 1 1 0 1 1 14 1 1 1 0 0 15 1 1 1 1 0 F =   (1,2,3,5,7,11,13)

13. Digital Systems Design 12 Minterm list --> canonical sum Recall T10

14. Digital Systems Design 13 Algebraic simplification Theorem T10 Reduce number of gates and gate inputs

15. Digital Systems Design 14 Resulting circuit

16. Systems Design 15 Circuit Optimization Goal: To obtain the simplest implementation for a given function Optimization is a more formal approach to simplification that is performed using a specific procedure or algorithm Optimization requires a cost criterion to measure the simplicity of a circuit Two distinct cost criteria we will use: –Literal cost (L) –Gate input cost (G) and cost with NOTs (GN)

17. Digital Systems Design 16 Literal – a variable or it complement Literal cost – the number of literal appearances in a Boolean expression corresponding to the logic circuit diagram Examples: –F = BD + AB ’ C + AC ’ D ’ L = 8 –F = BD + AB ’ C + AB ’ D ’ + ABC ’ L = –F = (A + B)(A + D)(B + C +D ’ )(B ’ + C ’ + D) L = –Which solution is best? Literal Cost 11 10

18. Digital Systems Design 17 Gate Input Cost Gate input costs - the number of inputs to the gates in the implementation corresponding exactly to the given equation or equations. (G - inverters not counted, GN - inverters counted) For SOP and POS equations, it can be found from the equation(s) by finding the sum of: –all literal appearances –the number of terms excluding terms consisting only of a single literal,(G) and –optionally, the number of distinct complemented single literals (GN). Example: –F = BD + AB ’ C + AC ’ D ’ G = 11, GN = 14 –F = BD + AB ’ C + AB ’ D ’ + ABC ’ G =, GN = –F = (A + B)(A + D)(B + C +D ’ )(B ’ + C ’ + D) G =, GN = –Which solution is best? 1519 14 17

19. Digital Systems Design 18 Example 1: F = A + B C + B ’ C ’ Cost Criteria (continued) A B C F L = 5  L (literal count) counts the AND inputs and the single literal OR input. G = L + 2 = 7  G (gate input count) adds the remaining OR gate inputs GN = G + 2 = 9  GN(gate input count with NOTs) adds the inverter inputs

20. Digital Systems Design 19 Example 2: F = A B C + A ’ B ’ C ’ L = 6 G = 8 GN = 11 F = (A + C ’ )( B ’ + C)( A ’ + B) L = 6 G = 9 GN = 12 Same function and same literal cost But first circuit has better gate input count and better gate input count with NOTs Select it! Cost Criteria (continued) A B C F F A B C

21. Digital Systems Design 20 Boolean Function Optimization Minimizing the gate input (or literal) cost of a (a set of) Boolean equation(s) reduces circuit cost. We choose gate input cost. Boolean Algebra and graphical techniques are tools to minimize cost criteria values. Some important questions: –When do we stop trying to reduce the cost? –Do we know when we have a minimum cost? Treat optimum or near-optimum cost functions for two-level (SOP and POS) circuits first. Introduce a graphical technique using Karnaugh maps (K- maps, for short)

22. Digital Systems Design 21 Karnaugh Maps (K-map) A K-map is a collection of squares – Each square represents a minterm – The collection of squares is a graphical representation of a Boolean function – Adjacent squares differ in the value of one variable – Alternative algebraic expressions for the same function are derived by recognizing patterns of squares The K-map can be viewed as –A reorganized version of the truth table –A topologically-warped Venn diagram as used to visualize sets in algebra of sets

23. Digital Systems Design 22 Some Uses of K-Maps Provide a means for: –Finding optimum or near optimum SOP and POS standard forms, and two-level AND/OR and OR/AND circuit implementations for functions with small numbers of variables –Visualizing concepts related to manipulating Boolean expressions, and –Demonstrating concepts used by computer-aided design programs to simplify large circuits

24. Digital Systems Design 23 Two Variable Maps A 2-variable Karnaugh Map: – Note that minterm m0 and minterm m1 are “ adjacent ” and differ in the value of the variable y – Similarly, minterm m0 and minterm m2 differ in the x variable. – Also, m1 and m3 differ in the x variable as well. – Finally, m2 and m3 differ in the value of the variable y y = 0 y = 1 x = 0 m 0 = m 1 = x = 1 m 2 = m 3 = yx yx yx yx

25. Digital Systems Design 24 K-Map and Truth Tables The K-Map is just a different form of the truth table. Example – Two variable function: – We choose a,b,c and d from the set {0,1} to implement a particular function, F(x,y). Function Table K-Map y = 0 y = 1 x = 0 a b x = 1 c d

26. Digital Systems Design 25 K-Map Function Representation Example: F(x,y) = x For function F(x,y), the two adjacent cells containing 1 ’ s can be combined using the Minimization Theorem: F = x y = 0 y = 1 x = 0 0 0 x = 1 1 1 xyxyx)y,x(F 

27. Digital Systems Design 26 K-Map Function Representation Example: G(x,y) = x + y For G(x,y), two pairs of adjacent cells containing 1 ’ s can be combined using the Minimization Theorem: G = x+y y = 0 y = 1 x = 0 0 1 x = 1 1 1  yxyxxyyxyx)y,x(G  Duplicate x y