1. 60-265: Computer Architecture 1: Digital Design SIMPLIFICATION and IMPLEMENTATION
F = ∑m(1,4,5,6,7)
F = A’B’C+ (AB’C’+AB’C) + (ABC’+ABC) Use X’ + X = 1.
= A’B’C + (AB’ + AB) Again use X’ + X = 1.
= A’B’C + A Use X + X’.Y = X + Y.
= A + B’C

2. Implementation of Inverter/AND/OR using NAND gates only
Inverter using NAND
AND using NAND
OR using NAND

3. IMPLEMENTATION of F = A + B’C using NAND gates only
To implement the previous simplified function by using NAND gates only:
X NAND Y = (X.Y)’
F = (A + B’.C)’’
= ( (A + B’.C)’ )’
= (A’.(B’.C)’)’

4. Implementation of Inverter/AND/OR using NOR gates only
Inverter using NOR
AND using NOR
OR using NOR

5. IMPLEMENTATION of F = A + B’C using NOR gates only
To implement the same function by using NOR gates only: X NOR Y = (X + Y)’
F = (A + B’C)’’
= ((A + (B’C)’’)’)’
= ((A + (B+C’)’ )’)’ = ((A + P)’)’

6. Variables, Literals and minterms
Variables: A, B or C; # of Variables = n
Literals: A, A’, B, B’, C or C’
# of literals = 2n
minterm: ANDing of n literals such that if a literal x is present, its complement will not be present
A minterm is equal to 1 for only one set of values of the variables. For all other sets of values of the variables, the minterm is equal to 0.

7. Maxterms
Maxterm: ORing of n literals such that if a literal x is present, its complement will not be present
A Maxterm is equal to 0 for only one set of values of the variables. For all other sets of values of the variables, the Maxterm is equal to 1.

8. Examples: minterm and Maxterms for a Function of 3 variables:
A B C minterm F Maxterm
A’.B’.C’ A + B + C
A’.B’.C A + B + C’
A’.B.C’ A + B’ + C
A’.B.C A + B’ + C’
A.B’.C’ A’ + B + C
A.B’.C A’ + B + C’
A.B.C’ A’ + B’ + C
A.B.C A’ + B’ + C’

9. Defining a Function in terms of minterms and Maxterms
Definition: F = ∑ m(1,4,5,6,7) =  M(0, 2, 3)
F represents an output function, which is equal to 1 for the rows with equivalent decimal value of 1, 4, 5, 6 and 7 and which is equal to 0 for the rows with equivalent decimal value of 0, 2 and 3.
F =A’.B’.C +A.B’.C’ +A.B’.C +A.B.C’ +A.B.C
= (A + B + C).(A + B’ + C).(A + B’ + C’)

10. Ex. 2: SIMPLIFICATION and IMPLEMENTATION
F = X.Y’.Z + X.Y.Z + X’.Y’.Z
= (X+X’).Y’.Z +X.Y.Z Use X’ + X = 1.
= Y’.Z + X.Y.Z = (Y’ +Y.X).Z = (Y’+X).Z Use X + X’.Y = X + Y.
To implement using NAND gates only: F = (Y’.Z+X.Z)’’
F =( (Y’.Z)’ . (X.Z)’ )’ = (A.B)’

11. Combinatorial Circuits
Combinatorial Circuit: A circuit, the outputs of which depend upon only the present values of the inputs only ( and not on the history of the past values of the inputs or outputs.)
Characteristic Table: shows the binary relationship between the n input variables and m output variables.
For n variables, the Characteristic Table has 2n entries. 1 1 m n
Half-adder A S HA C B
Full-adder X S Y FA C Z

12. Combinatorial circuit: Half-Adder
Half Adder: It performs the arithmetic addition of two bits.
Symbols: A and B for two input variables , S for sum and C for carry.
A B S C
C = A .B
S= A’B + AB’ = A  B

13. Combinatorial circuit: Full-Adderx
Cout y S
Cin
X Y Cin S Cout
Truth Table for Full-Adder
It performs the arithmetic addition of three input bits.
Cout =x’.y.Cin + x.y’.Cin + x.y.Cin’ + x.y.Cin
=(x’.y + x.y’).Cin + (Cin’+Cin)x.y
=( x’.y + x.y’ ). Cin + x.y
= (x  y).Cin + x.y

14. Combinatorial circuit: Full-Adder
S =x’.y’.Cin + x’.y. Cin’ + x.y’. Cin’ + x.y. Cin
= (x’.y’ + x.y). Cin + (x’.y +x.y’). Cin’
= (x’.y+x.y)’. Cin + (x  y). Cin’
= Q.Cin + P.Cin’
If Q were equal to P’
S = P’.Cin + P.Cin’
= P  Cin
= (x  y)  Cin
To see if it is so :
P = x’.y + x.y’
P’ = (x +y’).(x’ + y) Use De’ Morgan’s Theorem
= x.x’ + x.y + x’.y’ + y.y’
=x.y + x’.y’
= Q

15. Combinatorial circuit: Full-Adder circuit

16. Combinatorial circuit: continued. F1
Boolean inputs
outputs
N-1 F2
The input set has 2n distinct terms.
For each output, find F1 = fn1(I0 ....In-1.)
F2 = fn2(I0 ....In-1.)
For implementing the logic functions, the basic issue is of simplification.
No specific rules for determining the sequence of steps or the specific theorems that may be used.
Karnaugh Map graphical method
The number of cells = 2n Each cell corresponds to a minterm.
Adjacent cells: differ by only one bit.