1. Introduction to Digital Systems
Multiple-outputs minimization
Timing Hazards

2. Previous…
PoS minimization
Don’t care values

3. Multiple-Output Minimization
Most digital applications require multiple outputs derived from the same input variables.
Example: 3 inputs and 2 outputs
Each output function could be minimized using K-map and realized independently.
The output functions could share one or more product terms (prime implicant) which reduces the total number of gates. X F1 Y
Logic Circuit F2 Z

4. Example F1= XZ+YZ’ F2=XY’+YZ’
To find the common terms multiply the two functions (F1.F2)
The common terms are : YZ’, XY’Z XY X XY X 00 01 11 10 Z 00 01 11 10 Z 2 6 4 2 6 4 1 1 1 1 1 1 3 7 5 1 3 7 5 1 1 Z 1 1 Z 1 Y XY Y X Z 00 01 11 10 2 6 4 1 1 1 3 7 5 1 Z 1 Y

5. Example - Logic Diagram
Independent realization Minimal realization F1= XZ+YZ’ F2= XY’+YZ’ F1 F1 X Y X F2 Z Y F2 Z

6. Real-World Logic Design
More than 6 inputs -- can’t use Karnaugh maps
Design correctness more important than gate minimization
Use “higher-level language” to specify logic operations
Use programs to manipulate logic expressions and minimize logic.
PALASM, ABEL, CUPL -- developed for PLDs
VHDL, Verilog -- developed for ASICs

7. Timing Hazards
The Truth Table determines the Steady State behavior of a Combinational Logic Circuit
Transient behavior: - Output could produce glitches (a short pulse) when input variables change. - Glitches occur when the paths between inputs and output have different delays. - Timing Hazards refer to the possibility of having glitches during input transitions.
Hazards : - Definitions. - Finding hazards. - Eliminating hazards.

8. Definitions
Static Hazards: * Static-1 Hazard : Two input combinations that : differ in only one variable both produce logic possibly produce Logic 0 glitch during input variable transition * Static-0 Hazard : Two input combinations that differ in only one variable Both produce logic Possibly produce Logic 1 glitch during input variable transition
Dynamic hazards: - The output could change more than once during input transitions - Caused by multiple paths with different delays from input to the output 1 1

9. Example F= YZ+XZ’ Delay in each gate is T .
Input changes from XYZ=111 to 110 X Y Z YZ Z’
XZ’ F T
glitch Y F Z X

10. Finding Timing hazards using K-map
Two-level AND-OR Circuits :
Static 0 hazards do not exist in the sum-of products (AND-OR) implementation.
Static 1 hazards are possible
The K-map of the function F in the previous example : - Cell 6 ( 110 ) and cell 7 ( 111 ) are covered in two product terms XY X
Faster 00 01 11 10 Z Y 1 2 6 4 F Z 1 1 1 3 7 5 1 X 1 1 1 Z
Static 1 hazard Y

11. Timing hazards in OR-AND circuits
Static 1 hazards do not exist in the Product-of sums (Two-level OR- AND) implementation.
Static 0 hazards are possible
The minimal product of F = ( X+Z)(Y+Z’) - Cell 0 ( 000 ) and cell 1 ( 001 ) are covered in two sum terms - Static 0 hazard occurs when inputs switched between 000 to 001
Faster XY X 00 01 11 10 X Z F Z 2 6 4 1 1 Z Y 1 3 7 5 1 1 1
Static 0 hazard Y

12. Eliminating Timing Hazards
AND-OR Circuit - Add a prime implicant that combines the two inputs that cause static 1 hazard.---->consensus - Cells 6 & 7 are combined : XY
The hazard-free circuit is : XY X 00 01 11 10 Z Y 1 2 6 4 F 1 1 1 Z 1 1 3 7 5 1 1 1 1 Z X 1 1 Y

13. Eliminating Timing Hazards
OR-AND Circuit - Add a prime implicant that combines the two inputs that cause static 0 hazard.---->consensus - Cells 0 & 1 are combined : X+Y
The hazard-free circuit is : XY X 00 01 11 10 Z 2 6 4 X 1 1 F Z Z 1 3 7 5 1 1 1 Y Y

14. Summary
A properly designed two-level SOP (AND-OR) circuit has no static-0 hazards. It may have static-1 hazards.
A properly designed two-level POS (OR-AND) circuit has no static-1 hazards. It may have static-0 hazards.
Dynamic hazards do not occur in a properly designed two- level AND-OR or OR-AND circuit. It may occur in multilevel circuits.
A brute-force method of obtaining a hazard-free realization is to use the complete sum or complete product.
Hazard analysis and elimination are typically needed in the design of asynchronous sequential circuits.