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ECE 3110: Introduction to Digital Systems
Combinational-Circuit Synthesis
Previous… Representations of Logic Functions
Truth table Canonical sum Minterm list: on-set Canonical product Maxterm list: off-set
Combinational Circuit Analysis/Synthesis
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 1 010 1 0 1 0 011 0 0 1 1 112 1 1 0 0 013 1 1 0 1 114 1 1 1 0 015 1 1 1 1 0
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
Algebraic expression
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.
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 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’)
Sum-of-products form
AND-OR
NAND-NAND
DeMorgan Symbols
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)
Prime detector
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
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.
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
Algebraic simplification
Theorem T10,
Reduce number of gates and gate inputs
Resulting circuit
Next:
K-map Simplifying SOP