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Combinational Logic Part 2:. Karnaugh maps (quick). Sum of Minterms Implementation. OR all of the minterms of truth table for which the function value is 1 F = m 0 + m 2 + m 5 + m 7 F = X’Y’Z’ + X’YZ’+ XY’Z + XYZ. F = Y’ + X’YZ’ + XY. Sum of Products Implementation.

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## Combinational LogicPart 2:

Karnaugh maps (quick)

### Sum of Minterms Implementation

• OR all of the minterms of truth table for which the function

value is 1

F = m0 + m2 + m5 + m7

F = X’Y’Z’ + X’YZ’+

XY’Z + XYZ

F = Y’ + X’YZ’ + XY

### Sum of Products Implementation

• Simplifying sum-of-minterms can yield a sum of products

• Difference is that each term need not have all variables

• Resulting gates 

• ANDs and one OR

### Two-Level Implementation

• Sum of products has 2 levels of gates

Fig 2-6

### More Levels of Gates?

• What’s best?

• Hard to answer

• More gate delays (more on this later)

• But maybe we only have 2-input gates

### Product of Maxterms Implementation

• Can express F as AND of Maxterms for all rows that should evaluate to 0

• or

This makes one Maxterm fail each time F should be 0

### Product of Sums Implementation

• ORs followed by AND

### Karnaugh Map

• Graphical depiction of truth table

• A box for each minterm

• So 2 variables, 4 boxes

• 3 variable, 8 boxes

• And so on

• Useful for simplification

• by inspection

• Algebraic manipulation harder

### K-Map from Truth TableExamples

• There are implied 0s in empty boxes

### Function from K-Map

• Can generate function from K-map

Simplifies to X + Y (in a moment)

### In Practice:

• Karnaugh maps were mildly useful when people did simplification

• Computers now do it!

• We’ll cover Karnaugh maps as a way for you to gain insight,

• not as real tool

### Three-Variable Map

• Eight minterms

• Look at encoding of columns and rows

### Simplification

• Adjacent squares (horizontally or vertically) are minterms that vary by single variable

• Draw rectangles on map to simplify function

• Illustration next

instead of

### Adjacency is cylindrical

• Note that wraps from left edge to right edge.

is

### Another Example

• Help me solve this one

### In General

• One box -> 3 literals

• Rectangle of 2 boxes -> 2 literals

• Rectangle of 4 boxes -> 1 literal

• Rectangle of 8 boxes -> Logic 1 (on 3-variable map)

• Covers all minterms

### Slight Variation

• Overlap is OK.

• No need to use full m5-- waste of input

### 4-variable map

• At limit of K-map

### Systematic Simplification

• A Prime Implicant is a product term obtained by combining the maximum possible number of adjacent squares in the map into a rectangle with the number of squares a power of 2.

• A prime implicant is called an Essential Prime Implicant if it is the only prime implicant that covers (includes) one or more minterms.

• Prime Implicants and Essential Prime Implicants can be determined by inspection of a K-Map.

• A set of prime implicants "covers all minterms" if, for each minterm of the function, at least one prime implicant in the set of prime implicants includes the minterm.

### Example of Prime Implicants

CD

C

B

B

D

D

B

C

1

1

1

1

1

1

1

1

BD

BD

1

1

B

B

1

1

1

1

A

A

1

1

1

1

1

1

1

1

A

B

D

D

AD

Minterms covered by single prime implicant

• Find ALL Prime Implicants

B’D’ and BD are ESSENTIAL Prime Implicants

C

### Prime Implicant Practice

• Find all prime implicants for:

### Prime Implicant Practice

B

D

B

C

C

1

1

1

B

1

1

A

1

1

1

1

1

1

D

A

• Find all prime implicants for:

### Algorithm to Find An Optimal Expression for A Function

• Find all prime implicants.

• Include all essential prime implicants in the solution

• Select a minimum cost set of non-essential prime implicants to cover all minterms not yet covered.

• The solution consists of all essential prime and the selected minimum cost set of non-essential prime implicants

minimum cost

selected minimum cost

### The Selection Rule

• Obtaining a good simplified solution: Use the Selection Rule

### Prime Implicant Selection Rule

• Minimize the overlap among prime implicants as much as possible.

• In the solution, make sure that each prime implicant selected includes at least one minterm not included in any other prime implicant selected.

### Selection Rule Example

Essential

Selected

C

C

1

1

1

1

1

1

1

1

1

1

1

1

B

B

1

1

A

A

1

1

1

1

D

D

Minterms covered by essential prime implicants

• Simplify F(A, B, C, D) given on the K-map.

### Don’t Care

• So far have dealt with functions that were always either 0 or 1

• Sometimes we have some conditions where we don’t care what result is

• Example: dealing with BCD

• Only care about first 10

### Mark With an X

• In a K-map, mark don’t care with X

• Simpler implementations

• Can select an X either as 1 or 0

### Example

or

What would we have if Xs were 0?

### Selection Rule Example with Don't Cares

Essential

C

C

1

1

x

x

x

x

1

1

x

x

1

1

B

B

x

x

A

A

x

x

1

1

1

1

D

D

Minterms covered by essential prime implicants

• Simplify F(A, B, C, D) given on the K-map.

Selected

### Product of Sums Example

F

• Find the optimum POS solution:

• Hint: Use and complement it to get the result.