Saturday, 11 February 2012

Implementation of Boolean functions


AND-OR implementation:
Any Boolean function in SOP form can be implemented in AND-OR form. Product terms can be implemented using AND gates. Sum terms can be implemented using OR gates.
For example f= AB’ + A’B can be implemented as follows
 
OR-AND implementation:
Any Boolean function in POS form can be implemented in OR-AND form.
For example F= (A+B’) (A’+B)
 
NAND implementation:
NAND and NOR are universal gates. So any Boolean function can be implemented with NAND gate.
 
Any circuit can be implemented using NAND gates. The procedure for this implementation is as follows
  1. Obtain the Boolean function in the SOP form
  2. Replace AND gates with NAND gates  
  3. Replace OR gates with invert OR gates.  
  4. Invert OR gate is nothing but NAND gate. So replace invert OR gate with NAND gate.
Ex: F=AB + CD
When open collector TTL NAND gate outputs are connected to a single wire, they implement wired-AND logic. This is equivalent to AND-OR-Invert logic.
NOR implementation:
NOR is also a universal gate. So any Boolean function can be implemented with NOR gate.
 
Any circuit can be implemented using NOR gates. The procedure for this implementation is as follows
  1. Obtain the Boolean function in the POS form
  2. Replace OR gates with NOR gates
  3. Replace AND gates with invert AND gates 
  4. Invert AND gate is nothing but NOR gate. So replace invert AND gate with NOR gate. 
Ex: F = (A+B) (C+D)
When the output of ECL NOR gates is tied together, they perform wired-OR function. This is equivalent to OR-AND-invert function
 
Two level logic implementation:
Generally AND, OR, NAND, NOR gates are used to implement Boolean function. In case of two level implementations, using above four gates total 16 combinations are possible. Out of 16, 8 implementations reduce to a single operation. Those implementations are called degenerate forms. For example AND-NAND implementation reduces to NAND operation, thus it is a degenerate form. Remaining 8 combinations are called non-degenerate forms.
Non-degenerate forms:
AND-OR, AND-NOR, NAND-NAND, NAND-AND, OR-AND, NOR-OR, OR-NOR, OR-NAND.

all remaining combinations are degenerate forms.
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