IKI10201 04b-Simplification of Boolean Functions Bobby Nazief Semester-I 2005 - 2006 The materials on these slides are adopted from Prof. Daniel Gajski’s.

IKI10201 04b-Simplification of Boolean

Functions

Bobby NaziefSemester-I 2005 -

2006

The materials on these slides are adopted from Prof. Daniel Gajski’s transparency for Principles of Digital Design.

2

Tabulation Method

• Map method is a trial-and-error procedure

• Tabulation method performs thorough search

• It starts with SOM and consists of 2 steps:

– PIs generation

• group minterms by number of 1s

• compare minterms & find pairs that differ in 1 variable

• generate subcubes

• repeat the above 3 steps to generate subcubes until no more subcubes can be generated

– Minimal cover generation

• find EPIs through a selection table

• find minimal cover through the POS of PIs

3

• K-map representation:

• PIs generation:

– 0-subcubes

Example: simplify w’y’z’ + wz + xyz + w’y

00 01 11 10

00 1 0 1 1

01 1 0 1 1

11 0 1 1 0

10 0 1 1 0

yzwx

4

– 1-subcubes

– 2-subcubes

Example: simplify w’y’z’ + wz + xyz + w’y (cont.)

5

• Minimal cover generation:– EPIs selection

• PI list: w’z’, w’y, yz, wz• EPI list: w’z’, wz• POS: (P2 + P3)(P2 + P3) = P2 + P3

– Minimal cover expressions:• F1 = w’z’ + wz + w’y• F2 = w’z’ + wz + yz

Example: simplify w’y’z’ + wz + xyz + w’y (cont.)

6

• K-map representation:

• PIs generation:

– 0-subcubes, 1-subcubes

Another example

00 01 11 10

00 0 0 0 1

01 0 0 1 1

11 0 1 1 0

10 1 1 0 0

yzwx

7

• Minimal cover generation:– EPIs selection

• PI list: w’yz’, x’y’z, w’xy, wx’z, xyz, wyz• EPI list: w’yz’, x’y’z• POS: (P3 + P5)(P4 + P6)(P5 + P6) =

(P3 + P5)(P4P5 + P5P6 + P4P6 + P6) = P3P4P5 + P4P5 + P3P6 + P5P6

– Minimal cover expressions:• F1 = w’yz’ + x’y’z + wx’z + xyz• F2 = w’yz’ + x’y’z + w’xy + wyz• F3 = w’yz’ + x’y’z + xyz + wyz

Another example (cont.)

8

Technology Mapping for Gate Arrays

• Gate arrays contain only one type of m-input gate (such as 3-input NAND, 3-input NOR)

• Technology mapping is a transformation of Boolean expressions into a logic schematic containing only this type (NAND or NOR) of gate

– SOP/POS NAND/NOR gate implementation

9

Conversion & Optimization

• Conversion:

• Optimization:

• Conversion procedure: replace AND & OR gates with NAND (NOR) gates by using Rules 1 & 2 (3 & 4), and eliminate double inverters whenever possible

10

Translation standard forms to NAND/NOR schematics

11

Conversion to NAND (NOR) gates

12

Technology Mapping for Custom Libraries

• Libraries contain gates with different functions and different delays

• Technology mapping means covering schematic with library gates

• Minimize delay on critical paths

• Minimize cost on non-critical paths

13

Example design with custom libraries

• F = w’z’ + z(w + y)

• AND-OR implementation (delay = 7.2ns, cost = 28)

• NAND implementation (delay = 5.2ns, cost = 22)

14

Example design with custom libraries (cont.)

• Alternatif A (delay = 5.4ns, cost = 20)

• Alternatif B (delay = 3.8ns, cost = 20)

• Alternatif B-optimized (delay = 3.8ns, cost = 18)

15

Design with static 1-hazard

Timing Diagram

16

Hazard-free design

Timing Diagram