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Techniques for Combinational Logic Optimization

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Minimizing Circuits Karnaugh Maps

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Goals of Circuit Minimization (1) Minimize the number of primitive Boolean logic gates needed to implement the circuit. Ultimately, this also roughly minimizes the number of transistors, the chip area, and the cost. Also roughly minimizes the energy expenditure among traditional irreversible circuits. This will be our focus. (2) It is also often useful to minimize the number of combinational stages or logical depth of the circuit. This roughly minimizes the delay or latency through the circuit, the time between input and output. Note: Goals (1) and (2) are often conflicting! In the real world, a designer may have to analyze and optimize some complex trade-off between logic complexity and latency.

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Minimizing Expressions We would like to find the smallest sum-of- products expression that is equivalent to a given function. This will yield a fairly small circuit.

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Simplification of Switching Functions

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Karnaugh Maps (K-Map) A K-Map is a graphical representation of a logic functions truth table

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Relationship to Venn Diagrams b a

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b a

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Two-Variable K-Map

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Three-Variable K-Map Note: The bit sequences must always be ordered using a Gray code!

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Three-Variable K-Map Note: The bit sequences must always be ordered using a Gray code!

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Four-variable K-Map Note: The bit sequences must be ordered using a Gray code!

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Four-variable K-Map

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Plotting Functions on the K-map SOP Form

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Canonical SOP Form Three Variable Example using shorthand notation

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Three-Variable K-Map Example Plot 1s (minterms) of switching function

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Three-Variable K-Map Example Plot 1s (minterms) of switching function

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Four-variable K-Map Example

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Karnaugh Maps (K-Map) Simplification of Switching Functions using K-MAPS

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Or, logically adjacent terms can be combined Terminology/Definition Literal A variable or its complement Logically adjacent terms Two minterms are logically adjacent if they differ in only one variable position Ex: and m6 and m2 are logically adjacent Note:

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Terminology/Definition Implicant Product term that could be used to cover minterms of a function Prime Implicant An implicant that is not part of another implicant Essential Prime Implicant A prime implicant that covers at least one minterm that is not contained in another prime implicant Cover A minterm that has been used in at least one group

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Guidelines for Simplifying Functions Each square on a K-map of n variables has n logically adjacent squares. (i.e. differing in exactly one variable) When combing squares, always group in powers of 2 m, where m=0,1,2,…. In general, grouping 2 m variables eliminates m variables.

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Guidelines for Simplifying Functions Group as many squares as possible. This eliminates the most variables. Make as few groups as possible. Each group represents a separate product term. You must cover each minterm at least once. However, it may be covered more than once.

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K-map Simplification Procedure Plot the K-map Circle all prime implicants on the K- map Identify and select all essential prime implicants for the cover. Select a minimum subset of the remaining prime implicants to complete the cover. Read the K-map

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Example Use a K-Map to simplify the following Boolean expression

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Three-Variable K-Map Example Step 1: Plot the K-map

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Three-Variable K-Map Example Step 2: Circle ALL Prime Implicants

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Three-Variable K-Map Example Step 3: Identify Essential Prime Implicants EPI PI

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Three-Variable K-Map Example Step 4: Select minimum subset of remaining Prime Implicants to complete the cover EPI PI EPI

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Three-Variable K-Map Example Step 5: Read the map

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Solution

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Example Use a K-Map to simplify the following Boolean expression

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Three-Variable K-Map Example Step 1: Plot the K-map 11 11

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Three-Variable K-Map Example Step 2: Circle Prime Implicants Wrong!! We really should draw A circle around all four 1s

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Three-Variable K-Map Example Step 3: Identify Essential Prime Implicants EPI Wrong!! We really should draw A circle around all four 1s

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Three-Variable K-Map Example Step 4: Select Remaining Prime Implicants to complete the cover. EPI

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Three-Variable K-Map Example Step 5: Read the map

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Solution Since we can still simplify the function this means we did not use the largest possible groupings.

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Three-Variable K-Map Example Step 2: Circle Prime Implicants Right!

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Three-Variable K-Map Example Step 3: Identify Essential Prime Implicants EPI

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Three-Variable K-Map Example Step 5: Read the map

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Solution

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Special Cases

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Three-Variable K-Map Example

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Four Variable Examples

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Example Use a K-Map to simplify the following Boolean expression

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Four-variable K-Map

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