MOHD. YAMANI IDRIS/ NOORZAILY MOHAMED NOOR Karnaugh Map Introduction Venn Diagram 2 variable K-map 3 variable K-map 4 variable K-map K-map simplification MOHD. YAMANI IDRIS/ NOORZAILY MOHAMED NOOR

MOHD. YAMANI IDRIS/ NOORZAILY MOHAMED NOOR Karnaugh Map Convert to Minterm Form Simplest SOP expression Produce POS expression Don’t care condition MOHD. YAMANI IDRIS/ NOORZAILY MOHAMED NOOR

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 MOHD. YAMANI IDRIS/ NOORZAILY MOHAMED NOOR

Karnaugh Map – Venn Diagram Venn Diagram represent minterm space Example: two variables (4 minterm) MOHD. YAMANI IDRIS/ NOORZAILY MOHAMED NOOR

Karnaugh Map – Venn Diagram Each minterm set represent Boolean function Example: MOHD. YAMANI IDRIS/ NOORZAILY MOHAMED NOOR

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 MOHD. YAMANI IDRIS/ NOORZAILY MOHAMED NOOR

Karnaugh Map – 2 variable Map form that can be drawn for 2 variable (a,b) MOHD. YAMANI IDRIS/ NOORZAILY MOHAMED NOOR

Karnaugh Map – 2 variable Map form that can be drawn for 2 variable (a,b) – cont.. MOHD. YAMANI IDRIS/ NOORZAILY MOHAMED NOOR

Karnaugh Map – 2 variable Equivalent labeling method MOHD. YAMANI IDRIS/ NOORZAILY MOHAMED NOOR

Karnaugh Map – 2 variable 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 MOHD. YAMANI IDRIS/ NOORZAILY MOHAMED NOOR

Karnaugh Map – 3 variable 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 MOHD. YAMANI IDRIS/ NOORZAILY MOHAMED NOOR

Karnaugh Map – 3 variable 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. MOHD. YAMANI IDRIS/ NOORZAILY MOHAMED NOOR

Karnaugh Map – 4 variable There are 16 cell in 4 variable K-map (w,x,y,z) MOHD. YAMANI IDRIS/ NOORZAILY MOHAMED NOOR

Karnaugh Map – 4 variable 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 MOHD. YAMANI IDRIS/ NOORZAILY MOHAMED NOOR

Karnaugh Map – 5 variable 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 MOHD. YAMANI IDRIS/ NOORZAILY MOHAMED NOOR

Karnaugh Map – 5 variable It is arrange similar to 4 variable K-map Similar square on each adjacent map MOHD. YAMANI IDRIS/ NOORZAILY MOHAMED NOOR

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, MOHD. YAMANI IDRIS/ NOORZAILY MOHAMED NOOR

Simplification using Karnaugh Map 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 MOHD. YAMANI IDRIS/ NOORZAILY MOHAMED NOOR

Simplification using Karnaugh Map Example: MOHD. YAMANI IDRIS/ NOORZAILY MOHAMED NOOR

Simplification using Karnaugh Map Get the ‘product’ for each cluster for adjacent minterm (cluster with power of two size) for given function. Example: MOHD. YAMANI IDRIS/ NOORZAILY MOHAMED NOOR

Simplification using Karnaugh Map There are two minterm cluster A and B where (for previous example) MOHD. YAMANI IDRIS/ NOORZAILY MOHAMED NOOR

Simplification using Karnaugh Map 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 MOHD. YAMANI IDRIS/ NOORZAILY MOHAMED NOOR

Simplification using Karnaugh Map 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 MOHD. YAMANI IDRIS/ NOORZAILY MOHAMED NOOR

Simplification using Karnaugh Map Other types of cluster in 4 variable K-map case are: MOHD. YAMANI IDRIS/ NOORZAILY MOHAMED NOOR

Simplification using Karnaugh Map Minterm cluster must be Square, and Power of two size If not, it is not a certified cluster. Examples of non certified cluster are MOHD. YAMANI IDRIS/ NOORZAILY MOHAMED NOOR

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 MOHD. YAMANI IDRIS/ NOORZAILY MOHAMED NOOR

Conversion to Minterm Form Example: MOHD. YAMANI IDRIS/ NOORZAILY MOHAMED NOOR

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