iki10201 04b-simplification of boolean functions
DESCRIPTION
IKI10201 04b-Simplification of Boolean Functions. Bobby Nazief Semester-I 2005 - 2006. The materials on these slides are adopted from Prof. Daniel Gajski’s transparency for Principles of Digital Design. Tabulation Method. Map method is a trial-and-error procedure - PowerPoint PPT PresentationTRANSCRIPT
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