ECE 2110: Introduction to Digital Systems Combinational-Circuit Synthesis

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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.

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.

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

Minterm list --> canonical sum

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

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

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

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

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

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.

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

Next… Minimization using Karnaugh maps