EEL 3705 / 3705L Digital Logic Design Fall 2006 Instructor: Dr. Michael Frank Module #5: Combinational Logic Optimization (Thanks to Dr. Perry for the slides) 9/17/2018 M. Frank, EEL3705 Digital Logic, Fall 2006

M. Frank, EEL3705 Digital Logic, Fall 2006 Wednesday, October 4, 2006 Administrivia: Midterm #1 grades posted, handing papers back today Averages: 96%, 89%, 96% (98.5 points overall) This week’s lab: “Top secret code display” (w. K-maps) Design project #1: Due this Friday!! Test your designs at end of lab period, or in TA office hours Don’t delay writing the large required amount of documentation! Homework assignment #3: To be posted very soon, watch for it. Plan for today: Go over midterm solutions briefly Start in on next lecture topic, 3.5: Techniques for Combinational Logic Optimization Emphasis on use of Karnaugh maps 9/17/2018 M. Frank, EEL3705 Digital Logic, Fall 2006

Topic 3.5: Techniques for Combinational Logic Optimization 9/17/2018 M. Frank, EEL3705 Digital Logic, Fall 2006

Topic 3.5 – Minimizing Circuits Subtopics: 3.5.1. Simplification using Boolean identities Not covering in depth this semester in lecture 3.5.2. Karnaugh Maps (CIO #5) Focus of this lecture 3.5.3. Don’t care conditions 3.5.4. The Quine-McCluskey Method 9/17/2018 M. Frank, EEL3705 Digital Logic, Fall 2006

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. 9/17/2018 M. Frank, EEL3705 Digital Logic, Fall 2006

Minimizing DNF Expressions Using DNF (or CNF) expressions guarantees that you can always find some circuit that implements any desired Boolean function. However, the resulting circuit may be far larger than is really required! 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. However, circuits of other forms (not either CNF or DNF) might be even smaller for complex functions. 9/17/2018 M. Frank, EEL3705 Digital Logic, Fall 2006

Simplification of Switching Functions Chapter 3 Simplification of Switching Functions 9/17/2018 M. Frank, EEL3705 Digital Logic, Fall 2006

M. Frank, EEL3705 Digital Logic, Fall 2006 Karnaugh Maps (K-Map) A K-Map is a graphical representation of a logic function’s truth table 9/17/2018 M. Frank, EEL3705 Digital Logic, Fall 2006

Relationship to Venn Diagrams 9/17/2018 M. Frank, EEL3705 Digital Logic, Fall 2006

Relationship to Venn Diagrams 9/17/2018 M. Frank, EEL3705 Digital Logic, Fall 2006

Relationship to Venn Diagrams b 9/17/2018 M. Frank, EEL3705 Digital Logic, Fall 2006

Relationship to Venn Diagrams b 9/17/2018 M. Frank, EEL3705 Digital Logic, Fall 2006

Relationship to Venn Diagrams 9/17/2018 M. Frank, EEL3705 Digital Logic, Fall 2006

M. Frank, EEL3705 Digital Logic, Fall 2006 Two-Variable K-Map 9/17/2018 M. Frank, EEL3705 Digital Logic, Fall 2006

Three-Variable K-Map Note: The bit sequences must always be ordered using a Gray code! 9/17/2018 M. Frank, EEL3705 Digital Logic, Fall 2006

Three-Variable K-Map Note: The bit sequences must always be ordered using a Gray code! 9/17/2018 M. Frank, EEL3705 Digital Logic, Fall 2006

Three-Variable K-Map Note: The bit sequences must always be ordered using a Gray code! Edges are adjacent 9/17/2018 M. Frank, EEL3705 Digital Logic, Fall 2006

Four-variable K-Map Note: The bit sequences must be ordered using a Gray code! Note: The bit sequences must be ordered using a Gray code! 9/17/2018 M. Frank, EEL3705 Digital Logic, Fall 2006

M. Frank, EEL3705 Digital Logic, Fall 2006 Four-variable K-Map 9/17/2018 M. Frank, EEL3705 Digital Logic, Fall 2006

M. Frank, EEL3705 Digital Logic, Fall 2006 Four-variable K-Map Edges are adjacent Edges are adjacent 9/17/2018 M. Frank, EEL3705 Digital Logic, Fall 2006

Plotting Functions on the K-map SOP Form 9/17/2018 M. Frank, EEL3705 Digital Logic, Fall 2006

Canonical SOP Form using shorthand notation Three Variable Example 9/17/2018 M. Frank, EEL3705 Digital Logic, Fall 2006

Three-Variable K-Map Example Plot 1’s (minterms) of switching function 1 1 1 1 9/17/2018 M. Frank, EEL3705 Digital Logic, Fall 2006

Three-Variable K-Map Example Plot 1’s (minterms) of switching function 1 1 1 1 9/17/2018 M. Frank, EEL3705 Digital Logic, Fall 2006

Four-variable K-Map Example 1 1 1 1 1 9/17/2018 M. Frank, EEL3705 Digital Logic, Fall 2006

Simplification of Switching Functions using K-MAPS Karnaugh Maps (K-Map) Simplification of Switching Functions using K-MAPS 9/17/2018 M. Frank, EEL3705 Digital Logic, Fall 2006

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: Or, logically adjacent terms can be combined 9/17/2018 M. Frank, EEL3705 Digital Logic, Fall 2006

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 9/17/2018 M. Frank, EEL3705 Digital Logic, Fall 2006

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 2m , where m=0,1,2,…. In general, grouping 2m variables eliminates m variables. 9/17/2018 M. Frank, EEL3705 Digital Logic, Fall 2006

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. 9/17/2018 M. Frank, EEL3705 Digital Logic, Fall 2006

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 9/17/2018 M. Frank, EEL3705 Digital Logic, Fall 2006

M. Frank, EEL3705 Digital Logic, Fall 2006 Example Use a K-Map to simplify the following Boolean expression 9/17/2018 M. Frank, EEL3705 Digital Logic, Fall 2006

Three-Variable K-Map Example Step 1: Plot the K-map 1 1 1 1 1 9/17/2018 M. Frank, EEL3705 Digital Logic, Fall 2006

Three-Variable K-Map Example Step 2: Circle ALL Prime Implicants 1 1 1 1 1 9/17/2018 M. Frank, EEL3705 Digital Logic, Fall 2006

Three-Variable K-Map Example Step 3: Identify Essential Prime Implicants PI EPI PI 1 1 EPI 1 1 1 9/17/2018 M. Frank, EEL3705 Digital Logic, Fall 2006

Three-Variable K-Map Example Step 4: Select minimum subset of remaining Prime Implicants to complete the cover. PI EPI 1 1 EPI 1 1 1 9/17/2018 M. Frank, EEL3705 Digital Logic, Fall 2006

Three-Variable K-Map Example Step 5: Read the map. 1 1 1 1 1 9/17/2018 M. Frank, EEL3705 Digital Logic, Fall 2006

M. Frank, EEL3705 Digital Logic, Fall 2006 Solution 9/17/2018 M. Frank, EEL3705 Digital Logic, Fall 2006

M. Frank, EEL3705 Digital Logic, Fall 2006 Example Use a K-Map to simplify the following Boolean expression 9/17/2018 M. Frank, EEL3705 Digital Logic, Fall 2006

Three-Variable K-Map Example Step 1: Plot the K-map 1 1 1 1 9/17/2018 M. Frank, EEL3705 Digital Logic, Fall 2006

Three-Variable K-Map Example Step 2: Circle Prime Implicants Wrong!! We really should draw A circle around all four 1’s 1 1 1 1 9/17/2018 M. Frank, EEL3705 Digital Logic, Fall 2006

Three-Variable K-Map Example Step 3: Identify Essential Prime Implicants EPI EPI Wrong!! We really should draw A circle around all four 1’s 1 1 1 1 9/17/2018 M. Frank, EEL3705 Digital Logic, Fall 2006

Three-Variable K-Map Example Step 4: Select Remaining Prime Implicants to complete the cover. EPI EPI 1 1 1 1 9/17/2018 M. Frank, EEL3705 Digital Logic, Fall 2006

Three-Variable K-Map Example Step 5: Read the map. 1 1 1 1 9/17/2018 M. Frank, EEL3705 Digital Logic, Fall 2006

M. Frank, EEL3705 Digital Logic, Fall 2006 Solution Since we can still simplify the function this means we did not use the largest possible groupings. 9/17/2018 M. Frank, EEL3705 Digital Logic, Fall 2006

Three-Variable K-Map Example Step 2: Circle Prime Implicants Right! 1 1 1 1 9/17/2018 M. Frank, EEL3705 Digital Logic, Fall 2006

Three-Variable K-Map Example Step 3: Identify Essential Prime Implicants EPI 1 1 1 1 9/17/2018 M. Frank, EEL3705 Digital Logic, Fall 2006

Three-Variable K-Map Example Step 5: Read the map. 1 1 1 1 9/17/2018 M. Frank, EEL3705 Digital Logic, Fall 2006

M. Frank, EEL3705 Digital Logic, Fall 2006 Solution 9/17/2018 M. Frank, EEL3705 Digital Logic, Fall 2006

M. Frank, EEL3705 Digital Logic, Fall 2006 Special Cases 9/17/2018 M. Frank, EEL3705 Digital Logic, Fall 2006

Three-Variable K-Map Example 1 1 1 1 1 1 1 1 9/17/2018 M. Frank, EEL3705 Digital Logic, Fall 2006

Three-Variable K-Map Example 9/17/2018 M. Frank, EEL3705 Digital Logic, Fall 2006

Three-Variable K-Map Example 1 1 1 1 9/17/2018 M. Frank, EEL3705 Digital Logic, Fall 2006

Four Variable Examples 9/17/2018 M. Frank, EEL3705 Digital Logic, Fall 2006

M. Frank, EEL3705 Digital Logic, Fall 2006 Example Use a K-Map to simplify the following Boolean expression 9/17/2018 M. Frank, EEL3705 Digital Logic, Fall 2006

M. Frank, EEL3705 Digital Logic, Fall 2006 Four-variable K-Map 1 1 1 1 1 1 1 1 9/17/2018 M. Frank, EEL3705 Digital Logic, Fall 2006

M. Frank, EEL3705 Digital Logic, Fall 2006 Four-variable K-Map 1 1 1 1 1 1 1 1 9/17/2018 M. Frank, EEL3705 Digital Logic, Fall 2006

M. Frank, EEL3705 Digital Logic, Fall 2006 Four-variable K-Map 1 1 1 1 1 1 1 1 9/17/2018 M. Frank, EEL3705 Digital Logic, Fall 2006

M. Frank, EEL3705 Digital Logic, Fall 2006 Example Use a K-Map to simplify the following Boolean expression D=Don’t care (i.e. either 1 or 0) 9/17/2018 M. Frank, EEL3705 Digital Logic, Fall 2006

M. Frank, EEL3705 Digital Logic, Fall 2006 Four-variable K-Map 1 d 1 1 1 d 1 d 1 1 9/17/2018 M. Frank, EEL3705 Digital Logic, Fall 2006

M. Frank, EEL3705 Digital Logic, Fall 2006 Four-variable K-Map 1 d 1 1 1 d 1 d 1 1 9/17/2018 M. Frank, EEL3705 Digital Logic, Fall 2006

M. Frank, EEL3705 Digital Logic, Fall 2006 Five Variable K-Maps 9/17/2018 M. Frank, EEL3705 Digital Logic, Fall 2006

M. Frank, EEL3705 Digital Logic, Fall 2006 Five variable K-map Use two four variable K-maps A=1 A=0 9/17/2018 M. Frank, EEL3705 Digital Logic, Fall 2006

Use Two Four-variable K-Maps A=0 map A=1 map 9/17/2018 M. Frank, EEL3705 Digital Logic, Fall 2006

M. Frank, EEL3705 Digital Logic, Fall 2006 Five variable example 9/17/2018 M. Frank, EEL3705 Digital Logic, Fall 2006

Use Two Four-variable K-Maps A=0 map A=1 map 1 1 1 1 1 1 1 1 9/17/2018 M. Frank, EEL3705 Digital Logic, Fall 2006

Use Two Four-variable K-Maps A=0 map A=1 map 1 1 1 1 1 1 1 1 9/17/2018 M. Frank, EEL3705 Digital Logic, Fall 2006

M. Frank, EEL3705 Digital Logic, Fall 2006 Five variable example 9/17/2018 M. Frank, EEL3705 Digital Logic, Fall 2006

Plotting POS Functions 9/17/2018 M. Frank, EEL3705 Digital Logic, Fall 2006

K-map Simplification Procedure Plot the K-map for the function F 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 Use DeMorgan’s theorem to convert F to F in POS form 9/17/2018 M. Frank, EEL3705 Digital Logic, Fall 2006

M. Frank, EEL3705 Digital Logic, Fall 2006 Example Use a K-Map to simplify the following Boolean expression 9/17/2018 M. Frank, EEL3705 Digital Logic, Fall 2006

Three-Variable K-Map Example Step 1: Plot the K-map of F 1 1 1 1 1 9/17/2018 M. Frank, EEL3705 Digital Logic, Fall 2006

Three-Variable K-Map Example Step 2: Circle ALL Prime Implicants 1 1 1 1 1 9/17/2018 M. Frank, EEL3705 Digital Logic, Fall 2006

Three-Variable K-Map Example Step 3: Identify Essential Prime Implicants PI EPI PI 1 1 EPI 1 1 1 9/17/2018 M. Frank, EEL3705 Digital Logic, Fall 2006

Three-Variable K-Map Example Step 4: Select minimum subset of remaining Prime Implicants to complete the cover. PI EPI 1 1 EPI 1 1 1 9/17/2018 M. Frank, EEL3705 Digital Logic, Fall 2006

Three-Variable K-Map Example Step 5: Read the map. 1 1 1 1 1 9/17/2018 M. Frank, EEL3705 Digital Logic, Fall 2006

M. Frank, EEL3705 Digital Logic, Fall 2006 Solution 9/17/2018 M. Frank, EEL3705 Digital Logic, Fall 2006

M. Frank, EEL3705 Digital Logic, Fall 2006 TPS Quiz 9/17/2018 M. Frank, EEL3705 Digital Logic, Fall 2006