1. ECE 3110: Introduction to Digital Systems
Combinational-Circuit Synthesis

2. Previous… Representations of Logic Functions
Truth table
Canonical sum
Minterm list: on-set
Canonical product
Maxterm list: off-set
Combinational Circuit Analysis/Synthesis

3. Combinational circuit synthesis
Description--->combinational logic circuit.
Description:
Word description of a problem using English-language connectives
Write corresponding logic expression/truth table
Manipulate the expression if necessary.
Build a circuit from the expression.
A circuit realizes an expression if its output function equals that expression, and the circuit is called a realization of the function.

4. Prime-number detector
Word description:
Given a 4-bit input combination N= N3N2N1N0, This function produces a 1 output for N=1,2,3,5,7,11,13, and produces a 0 otherwise.

5. Brute-force design Truth table --> canonical sum (sum of minterms)
row N3 N2 N1 N0 F
Brute-force design
Truth table --> canonical sum (sum of minterms)

6. Minterm list --> canonical sum

7. Combinational-Circuit Design
Sometimes you can write an equation or equations directly using “logic” (the kind in your brain).
Another example: Alarm

8. Corresponding Circuit
Description:The ALARM output is 1 if the PANIC input is 1, or if the ENABLE input is 1, the EXITING input is 0, and the house is not secure; the house is secure if the WINDOW,DOOR, and GARAGE inputs are all 1.
Algebraic expression
Corresponding Circuit

9. Alarm-circuit transformation
Sum-of-products form
Useful for programmable logic devices
“Multiply out”:

10. Circuit manipulation Standard gate types
AND, OR, Inverter, NAND,NOR
In most logic technologies, inverting gates like NAND/NOR are faster than noninverting gates like AND/OR
Translate any logic expression into an equivalent sum-of-products expression by multiplying it out. (T8)
Translate any logic expression into an equivalent product-of-sums expression by adding it out.(T8’)

11. Sum-of-products form
AND-OR
NAND-NAND

12. DeMorgan Symbols

13. Any sum-of-products expression can be realized in either as an AND-OR circuit or as a NAND-NAND circuit.
Dual statement is also true:
Any product-of-sums expression can be realized in either as an OR-AND circuit or as a NOR-NOR circuit.

14. Product-of-sums form OR-AND NOR-NOR
P-of-S preferred in CMOS, TTL (NAND-NAND)

15. Prime detector

16. Minimization
Logic Function minimization : Simplifying the logic function to reduce the number and size of gates.
Minimization methods: 1- Using theorems T9,T9’, T10,T10’ 2- Karnaugh map

17. Algebraic simplification
Starting from a truth table, or a minterm (maxterm) list
Reduce the cost of a 2-level circuit:
Minimize # of first-level gates
Minimize # of inputs of gates
Do not consider the cost of input inverters: assuming that both true and complemented versions of all input variables are available.

18. Algebraic simplification
T10, X.Y+X.Y’=X T10’: (X+Y).(X+Y’)=X
Generalization of the theorems:
(Given product term).Y+(Given product term).Y’= Given product term
(Given product term+Y).(Given product term+Y’)= Given product term
Reduce number of gates and gate inputs

19. Algebraic simplification
Theorem T10,
Reduce number of gates and gate inputs

20. Resulting circuit

21. Next:
K-map
Simplifying SOP