mohd. yamani idris/ noorzaily mohamed noor 1 karnaugh map introduction venn diagram 2 variable k-map...

1MOHD. YAMANI IDRIS/ NOORZAILY MOHAMED NOOR

Karnaugh Map

• Introduction

• Venn Diagram

• 2 variable K-map

• K-map simplification

MOHD. YAMANI IDRIS/ NOORZAILY MOHAMED NOOR

Karnaugh Map

• Convert to Minterm Form

• Simplest SOP expression

• Produce POS expression

• Don’t care condition

Karnaugh Map -Introduction

• Systematic method to get simplest Boolean SOP expression

• Objective: Minimum number of literal• Dramatic technique based on special Venn

Diagram Form• Advantage: Easy with visual aid• Disadvantage: Limited to five or six

variables

Karnaugh Map – Venn Diagram• Venn Diagram represent minterm space• Example: two variables (4 minterm)

Karnaugh Map – Venn Diagram• Each minterm set represent Boolean function

Example:

Karnaugh Map – 2 variable

• K-map is abstract form, Venn diagram is arranged as square matrix, where– Each square represent one minterm– Adjacent square is always differentiated with

one literal (therefore theorem a+a’ can be used)

• For two variable case (e.g. variable a, b), map can be drawn as

• Map form that can be drawn for 2 variable (a,b)

• Map form that can be drawn for 2 variable (a,b) – cont..

• Equivalent labeling method

• K-map for a function is determined by placing sign– 1 on equivalent square with minterm

– 0 vice versa

• For example: Carry & Sum for half adder

• There are 8 minterm for three variable (a,b,c). Therefore, 8 cell in three variable K-map

The above array ensure that minterm in adjacent cell only has one literal difference

• There are wrap-around in 3 variable K-map– a’b’c’(m0) is as neighbor next to a’bc’(m2)– ab’c’(m4) is as neighbor next to abc’(m6)

– Each cell in 3 variable K-map contains three adjacent neighbor. Generally, each cell in K-map n variable contain n adjacent neighbor.

• There are 16 cell in 4 variable K-map (w,x,y,z)

• There are 2 wrap-around: vertical & horizontal

• Each cell has 4 adjacent neighbor. For example: cell m0 is a neighbor of m1, m2, m4 and m8

• Map for more than 4 variable are more complicated since the geometry (hypercube configuration) for adjacent neighbor is greater.

• For five variable; e.g. vwxyz there is 25=32 squares

• It is arrange similar to 4 variable K-map

• Similar square on each adjacent map

Simplification using Karnaugh Map

• Based on Theorem:

A+A’=1

• In K-map, each cell contains ‘1’ representing minterm for the given function

• Each adjacent cell cluster contains ‘1’ (the cluster must have a power of two size which is 1,2,4,8….) then get the simplified value for each cluster– grouped 2 adjacent squares will eliminate 1 variable, grouped 4 adjacent

squares will eliminate 2 variable, grouped 8 adjacent squares will eliminate 3 variable, grouped 2n adjacent squares will eliminate n variable,

• Grouped largest possible squares (minterm) in a cluster– The grater the cluster, the lesser the number of literals in your answer

• Get the number of small cluster to gather all squares (minterm) for the function– The lesser the cluster, the lesser the number of ‘product’ and the

minimize function

• Example:

• Get the ‘product’ for each cluster for adjacent minterm (cluster with power of two size) for given function.

• Example:

• There are two minterm cluster A and B where (for previous example)

• Each ‘product’ for cluster, w’xy’ and wy represent sum-of-minterm in that cluster

• Boolean Function is sum-of-product which represent all minterm cluster for that function

F(w,x,y,z)=A+B=w’xy’+wy

• Greater cluster produce ‘product’ with small number of literal. In 4 variable K-map case:

1 cell = 4 literal, example: wxyz, w’xy’z

2 cell = 3 literal, example: wxy, w’y’z’

4 cell = 2 literal, example: wx, x’y

8 cell = 1 literal, example: w, y’, z’

16 cell = no literal example: 1

• Other types of cluster in 4 variable K-map case are:

• Minterm cluster must be– Square, and

– Power of two size

If not, it is not a certified cluster. Examples of non certified cluster are

Conversion to Minterm Form

• Function is easy to draw in K-map when function is given in SOP or SOM canonical form

• How if it is not in sum-of-minterm form?– Convert it to sum-of-product

– Elaborate SOP expression to SOM expression, or fill SOP expression directly to K-map

Conversion to Minterm Form

• Example:

SOP Expression Simplification• To get the simplest SOP expression from K-map, you will

need– Minimum number of literal for each ‘product’, and – Minimum number of ‘product’

• This can be achieved in K-map by using– Largest possible number of minterm in one cluster (i.e. prime implicant

(PI))– No extra cluster (i.e. essential prime implicant (epi))

Implicant: is a "covering" (sum term or product term) of one or more in a sum of product (or a maxterm in a product of sum) of a boolean function

mohd. yamani idris/ noorzaily mohamed noor 1 karnaugh map introduction venn diagram 2 variable k-map...

variable map form

map n variable

adjacent map slide

map abcm0

map simplification slide

variable case

map similar square

minterm slide

Documents

rangkaian logika optimal: peta karnaugh dan … karnaugh...

k map minimization

k-map teknik digital

further improved variable-entered karnaugh map procedures

elct 201: digital logic...

k-map simplification

variable dépendante à k (k > 2) modalités

validating a prescription map used in variable rate ... ·...

بهينه سازي با نقشة کارنو karnaugh map. 2...

k map research

k map week 9

contoh k map - rudist.files.wordpress.com · karnaugh map...

k-map solver circuit

5c digital k-map

digital circuit logic unit 5: karnaugh maps (k … ·...

karnaugh map (k-map) - … · 2 karnaugh map (k-map)...

k grassy creek k map legend - california

karnaugh map(k-map)

k-map using logisim

reviewkuliah … · contohgroupingfungsi2variabel peta...