1. Gate-Level Minimization
Mano & Ciletti
Chapter 3
By Suleyman TOSUN Ankara University

2. Outline Intro to Gate-Level Minimization The Map Method
variable map methods
Product-of-Sums Method
Don’t care Conditions
NAND and NOR Implementations

3. Gate-Level Minimization
Finding an optimal gate-level implementation of Boolean functions.
Difficult to perform manually.
Can use computer-based logic synthesis tool
Exp: espresso logic minimization software
Karnough Map (K-map) can be used for manual design of digital circuits.

4. The Map Method The truth table representation of a function is unique.
But, not the algebraic expression
Several versions of an algebraic expression exist.
Difficult to minimize algebraic functions manually.
The map method is a simple proceure to minimize Boolean functions.
Pictorial form of a truth table.
Called Karnough Map or K-Map.

5. Two-Variable Map Four Minterms Two variables
Four squares for four minterms
Figure b shows the relationship between the squares and the variables x and y.

6. Two-Variable Map (Cont.)
May only be useful to represent 16 Boolean functions.
Exp: If m1=m2=m3=1 then
m1+m2+m3=x’y+xy’+xy=x+y (OR function)

7. Three-Variable Map There are 8 minterms for 3 variables.
So, there are 8 squares.
Minterms are arranged not in a binary sequence, but in sequence similar to the Gray code.

8. Three-Variable Map (Cont.)
The square for square m5 corresponds to row 1 and column 01 (101).
Another look to m5 is m5=xy’z.
When variables are 0, they are primed (ex: x’). Otherwise not primed (x).

9. Three-Variable Map (Cont.)
Two adjacent squares differs one variable (one primed other is not).
So, they can be minimized
Ex: m5+m7=xy’z+xyz=xz(y’+y)=xz
So, try to cover as many adjacent squares as possible by the orders of two.
1,2,4,8… squares that has the logical value1.

10. Example 1 Simplfy the Boolean function F(x,y,z)=Σ(2,3,4,5)
F(x,y,z)=x’y+xy’

11. Adjacent squares Some adjacent squares don’t touch each other.
m0 is adjacent to m2 and m4 is adjacent to m6.
m0+m2=x’y’z’+x’yz’=x’z’(y’+y)=x’z’
m4+m6=xy’z’+xyz’=xz’(y’+y)=xz’

12. Example 2 Simplfy the Boolean function F(x,y,z)=Σ(3,4,6,7)
F(x,y,z)=yz+xz’

13. Example 3 Simplfy the Boolean function F(x,y,z)=Σ(0,2,4,5,6)
F(x,y,z)=z’+xy’

14. Example 4 F=A’C+A’B+AB’C+BC Express F as a sum of minterms.
F(A,B,C)=Σ(1,2,3,5,7)
Find the minimal sum-of-products expression
F=C+A’B

15. Four-Variable Map
16 minterms (and squares) for 4 variables.

16. Example 5 Simplfy F(w,x,y,z)=Σ(0,1,2,4,5,6,8,9,12,13,14)
F(w,x,y,z)=y’+w’z’+xz’

17. Example 6
Simplfy F=A’B’C’+B’CD’+A’BCD’+AB’C’
F=B’D’+B’C’+A’CD”

18. Prime Implicant
A product term obtained by combining the max. possible number of adjacent squares.
If a minterm is covered by only one prime implicant, that prime implicant is said to be essential.

19. Example
F(A,B,C,D)=Σ(0,2,3,5,7,8,9,10,11,13,15)

20. Example
F=BD+B’D’+CD+AD=BD+B’D’+CD+AB’
=BD+B’D’+B’C+AD=BD+B’D’+B’C+AB’

21. Five-Variable Map
Not simple, is not usually used. 5 variables, 32 squares.

22. Example 7 Simplfy F(A,B,C,D,E)=Σ(0,2,4,6,9,13,21,23,25,29,31)
F=A’B’E’+BD’E’+ACE

23. Product-of-Sums Simplification
Mark the empty squares by 0’s.
Combine them (as we did for sum-of-products)
We obtain F’ (complement of the function)
Take the complement of F’ ((F’)’) to obtain F.

24. Example 8 Simplfy F(A,B,C,D)=Σ(0,1,2,5,8,9,10) F=B’D’+B’C’+A’C’D
F’=AB+CD+BD’ => F=(A’+B’)(C’+D’)(B’+D)

25. Example 8 gate impl.
Two different implementations of the same function.

26. Don’t Care Conditions
In practice’ some combinations are not specified as 1’s or 0’s.
Four bit binary codes has six unused combinations.
Functions having unspecified outputs are called incompletely specified functions.
We don’t care the unspecified minterms
These minterms are called don’t-care conditions.
They can be used for minimization.
They are indicated as X’s in the map.
They can be assumed as 1’s or 0’s to have best simplification.

27. Example 9
Simplfy F(w,x,y,z)=Σ(1,3,7,11,15) having don’t conditions d(w,x,y,z)= Σ(0,2,5)

28. NAND and NOR Implementation
Generally used for circuit design
Easier to fabricate.
Basic gates
Other functions can be generated from them.
Rules have been developed to convert functions to NAND and NOR only implementations.

29. NAND Circuits NAND gate is universal
Any digital system can be implemented with it.
AND, OR and NOT can be implemented with NANDs.

30. Two graphic symbols of NAND

31. Two-Level Implementation
Have the function in sum-of-product form.
Put bubbles (inverters) to have two different representations of NAND gate
Either AND-invert or Invert-OR

32. Example: F=AB+CD

33. Example 10
Implement F(x,y,z)=Σ(1,2,3,4,5,7) with only NAND gates.

34. Multilevel NAND Circuits
F=A(CD+B)+BC’

35. NOR Implementation
Dual of NAND operation

36. Example: F=(A+B)(C+D)E

37. Example: F=(AB’+A’B)(C+D’)

38. Exclusive-Or (XOR) Function
1 if only x is one or if only y is 1
XNOR
1 if both inputs are 1 or both inputs are 0
Some identities of XOR

39. XOR Implementations

40. Odd Funct’on N variable XOR function is odd function defined as
The logical sum of the 2n/2 minterms whose binary numerical values have an odd number of 1’s. If n=3, 4 minterms have odd number of 1’s.

41. Logic Diagram of odd and even functions

42. Parity Generation
A parity bit is an extra bit included with a binary message
If number of 1’s is odd, parity bit is 1; else 0.
The circuit that generates the parity bit in the transmitter is called parity generator.
Parity bit can be generated using XOR function.

43. 3-Bit Parity Generator

44. Parity Checker Bits are transmitted to the destination with parity.
The circuit that checks the parity in the receiver is called a parity checker.
Parity checker can be implmeneted with XOR gates.

45. 3-Bit Parity Checker