ELEC Digital Logic Circuits Fall 2014 Logic Minimization (Chapter 3)
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1 ELE Digital Logic ircuits Fall 204 Logic Minimization (hapter 3) Vishwani D. grawal James J. Danaher Professor Department of Electrical and omputer Engineering uburn University, uburn, L vagrawal@eng.auburn.edu Fall 204, Oct 3... ELE Lecture 5
2 Understanding Minimization... Logic function: a b NOT F = ab + ab + bd + ND ND cd ND OR F c d ND Fall 204, Oct 3... ELE Lecture 5 2
3 ... Understanding Minimization Reducing products: F = ab + ab + bd + cd = b( a + a) + bd + cd = b + bd + cd = b( c + c) + bd + cd = bd + bc + cd + bc = bd + bc + bc = bd + b( c + c) = bd + b Distributivity omplementation Identity omplementation Distribitivity onsensus theorem Distributivity omplement, identity Fall 204, Oct 3... ELE Lecture 5 3
4 ... Understanding Minimization Reduced SOP: F = bd + b b NOT d ND OR F Fall 204, Oct 3... ELE Lecture 5 4
5 ... Understanding Minimization Reducing literals: F = bd + b = d + b b bd d bsorption theorem b d NOT OR Exercise: This circuit uses 8 transistors in MOS technology. an you redesign it with 6 transistors? (Hint: Use de Morgan s Theorem.) Fall 204, Oct 3... ELE Lecture 5 5 F
6 Logic Minimization Generally means In SOP form: Minimize number of products (reduce gates) and Minimize literals (reduce gate inputs) In POS form: Minimize number of sums (reduce gates) and Minimize literals (reduce gate inputs) Fall 204, Oct 3... ELE Lecture 5 6
7 Product or Implicant or ube ny set of literals NDed together. Minterm is a special case where all variables are present. It is the largest product. minterm is also called a 0-implicant of 0- cube. -implicant or -cube is a product with one variable eliminated: Obtained by combining two adjacent 0-cubes D + D = (D + D) = Fall 204, Oct 3... ELE Lecture 5 7
8 ubes (Implicants) of 4 Variables What is this? D Minterm or 0-implicant or 0-cube D implicant or -cube D Fall 204, Oct 3... ELE Lecture 5 8
9 Growing ubes, Reducing Products -implicant or -cube D implicant or 2-cube D D What is this? implicant or -cube D Fall 204, Oct 3... ELE Lecture 5 9
10 Largest ubes or Smallest Products What is this? D implicant or 3-cube Fall 204, Oct 3... ELE Lecture 5 0
11 Implication and overing larger cube covers a smaller cube if all minterms of the smaller cube are included in the larger cube. smaller cube implies (or subsumes) a larger cube if all minterms of the smaller cube are included in the larger cube. Fall 204, Oct 3... ELE Lecture 5
12 Implicants of a Function Minterms, products, cubes that imply the function. F = + D + D + D D Fall 204, Oct 3... ELE Lecture 5 2
13 Prime Implicant (PI) cube or implicant of a function that cannot grow larger by expanding into other cubes PI D D Fall 204, Oct 3... ELE Lecture 5 3
14 Essential Prime Implicant (EPI) If among the minterms subsuming a prime implicant (PI), there is at least one minterm that is covered by this and only this PI, then the PI is called an essential prime implicant (EPI). lso called essential prime cube (EP). EPI Why not this? Fall 204, Oct 3... ELE Lecture 5 4
15 Redundant Prime Implicant (RPI) If each minterm subsuming a prime implicant (PI) is also covered by other essential prime implicants, then that PI is called a redundant prime implicant (RPI). lso called redundant prime cube (RP). EPI RPI Fall 204, Oct 3... ELE Lecture 5 5
16 Selective Prime Implicant (SPI) prime implicant (PI) that is neither EPI nor RPI is called a selective prime implicant (SPI). lso called selective prime cube (SP). SPIs occur in pairs. EPI SPI Fall 204, Oct 3... ELE Lecture 5 6
17 Minimum Sum of Products (MSOP) Identify all prime implicants (PI) by letting minterms and implicants grow. onstruct MSOP with PI only : over all minterms Use only essential prime implicants (EPI) Use no redundant prime implicant (RPI) Use cheaper selective prime implicants (SPI) good heuristic hoose EPI in ascending order, starting from 0-implicant, then -implicant, 2- implicant,... Fall 204, Oct 3... ELE Lecture 5 7
18 Example: F= m(,3,4,5,8,9,3,5) D MSOP: F = D + + D + RPI Fall 204, Oct 3... ELE Lecture 5 8
19 Example: F= m(,3,5,7,8,0,2,3,4) SPI MSOP: F = D + D D Fall 204, Oct 3... ELE Lecture 5 9
20 Functions with Don t are Minterms F(,,) = m(0,3,7) + d(4,5) Include don t care minterms when beneficial. Φ Φ Φ Φ F = + F = + Fall 204, Oct 3... ELE Lecture 5 20
21 Five-Variable Function F(,,,D,E) = m(0,,4,5,6,3,4,5,22,24,25,28,29,30,3) = 0 = D E D E F = D + D + E + D E Fall 204, Oct 3... ELE Lecture 5 2
22 Multiple-Output Minimization Inputs Outputs D F F Fall 204, Oct 3... ELE Lecture 5 22
23 Individual Output Minimization Need five products. F F D D Fall 204, Oct 3... ELE Lecture 5 23
24 Global Minimization Need four products. F F D D Fall 204, Oct 3... ELE Lecture 5 24
25 Minimized SOP and POS F(,,,D) = m(,3,4,7,) + d(5,2,3,4,5) = Π M(0,2,6,8,9,0) D(5,2,3,4,5) Φ Φ Φ 3 Φ 5 Φ Φ 9 0 D 5 Φ 3 Φ Φ Φ D F = + D + D F = D + D + F = ( + D)( + D)( + ) Fall 204, Oct 3... ELE Lecture 5 25
26 SOP and POS ircuits F(,,,D) = m(,3,4,7,)+d(5,2,3,4,5) = Π M(0,2,6,8,9,0) D(5,2,3,4,5) F = + D + D F = ( + D)( + D)( + ) F F D D re two circuits functionally Identical? Fall 204, Oct 3... ELE Lecture 5 26
27 How Don t ares Occur onsider two roads crossing: Highway with traffic signal, red (R), yellow (Y) or green (G). Rural road with red (r) or green (g) signal. Here R, Y, G, r and g are oolean variables; a implies light is on, 0 means light is off. Highway signals R, Y and G are controlled by a computer; only one light can be on. We only need to devise a digital circuit to obtain g, because r = g. Fall 204, Oct 3... ELE Lecture 5 27
28 Traffic Signals Fall 204, Oct 3... ELE Lecture 5 28
29 ompletely Specified Function Truth Table minterm R Y G g G R Y G Y g = R Y G R g Fall 204, Oct 3... ELE Lecture 5 29
30 Incompletely Specified Function Truth Table minterm R Y G g Φ R Φ G Φ Φ Y g = R Φ Φ 6 0 Φ R g 7 Φ Fall 204, Oct 3... ELE Lecture 5 30
31 bsorption Theorem For two oolean variables:, + = Proof: Fall 204, Oct 3... ELE Lecture 5 3
32 onsensus Theorem For three oolean variables:,, + + = + Proof: Fall 204, Oct 3... ELE Lecture 5 32
33 Growing Implicants to PI F = + D + D initial implicants = + D + D + D consensus th. = + D + D absorption th. = + D + D + D consensus th. = + D + D absorption th. D D D Fall 204, Oct 3... ELE Lecture 5 33
34 Identifying EPI Find all prime implicants. From prime implicant SOP, remove a PI. pply consensus theorem to the remaining SOP. If the removed PI is generated, then it is either an RPI or an SPI. If the removed PI is not generated, then it is an EPI Fall 204, Oct 3... ELE Lecture 5 34
35 Example PI SOP: F = D + + D Is D an EPI? F {D} = + D, no new PI can be generated Hence, D is an EPI. D D Fall 204, Oct 3... ELE Lecture 5 35
36 Example (ont.) PI SOP: F = D + + D Is D an EPI? F { D } = D + = D + + D (onsensus theorem) Hence D is not an EPI (it is an RPI) D D Fall 204, Oct 3... ELE Lecture 5 36
37 Finding MSOP. Start with minterm or cube SOP representation of oolean function. 2. Find all prime implicants (PI). 3. Include all EPI s in MSOP. 4. Find the set of uncovered minterms, {U}. 5. MSOP is minimum if {U} is empty. DONE. 6. For a minterm in {U}, include the largest PI from remaining PI s (non-epi s) in MSOP. 7. Go to step 4. Fall 204, Oct 3... ELE Lecture 5 37
38 Finding Uncovered Minterms, {U} {U} = ({PI} # {PMSOP}) # {D} Where: {PI} is set of all prime implicants of the function. {PMSOP} is any partial SOP. {D} is set of don t care minterms. Sharp (#) operation between oolean expressions X and Y, X # Y, is the set of minterms covered by X that are not covered by Y. Fall 204, Oct 3... ELE Lecture 5 38
39 Example: # (Sharp) Operation D # D = { D, D} D # D = { D, D} D D D Fall 204, Oct 3... ELE Lecture 5 39
40 Minterms overed by a Product product from which k variables have been eliminated, covers 2 k minterms. Example: For four variables,,,, D Product covers 2 2 = 4 minterms: ) D 2) D 3) D 4) D Obtained by inserting the eliminated variables in all possible ways. (4) (2) (3) () D Fall 204, Oct 3... ELE Lecture 5 40
41 Quine-Mcluskey Willard V. O. Quine Edward J. Mcluskey b. 929 Fall 204, Oct 3... ELE Lecture 5 4
42 Quine-Mcluskey Tabular Minimization Method W. V. Quine, The Problem of Simplifying Truth Functions, merican Mathematical Monthly, vol. 59, no. 0, pp , October 952. E. J. Mcluskey, Minimization of oolean Functions, ell System Technical Journal, vol. 35, no., pp , November 956. Textbook, Section 3.9, pp Fall 204, Oct 3... ELE Lecture 5 42
43 Q-M Tabular Minimization Minimizes functions with many variables. egin with minterms: Step : Tabulate minterms in groups of increasing number of true variables. Step 2: onduct linear searches to identify all prime implicants (PI). Step 3: Tabulate PI s vs. minterms to identify EPI s. Step 4: Tabulate non-essential PI s vs. minterms not covered by EPI s. Select minimum number of PI s to cover all minterms. MSOP contains all EPI s and selected non-epi s. Fall 204, Oct 3... ELE Lecture 5 43
44 F(,,,D) = m(2,4,6,8,9,0,2,3,5) Q-M Step : Group minterms with true variable, 2 true variables, etc. Minterm D Groups : single 2: two s 3 0 3: three s 5 4: four s Fall 204, Oct 3... ELE Lecture 5 44
45 Q-M Step 2 Find all implicants by combining minterms, and then combining products that differ in a single variable: For example, 2 and 6, or D and D D, written as 0 0. Try combining a minterm (or product) with all minterms (or products) listed below in the table. Include resulting products in the next list. If minterm (or product) does not combine with any other, mark it as PI. heck the minterm (or product) and repeat for all other minterms (or products). Fall 204, Oct 3... ELE Lecture 5 45
46 Step 2 Executed on Example List List 2 List 3 Minterm D PI? Minterms D PI? Minterms D PI? X 2, PI2 8,9,2,3-0- PI X 2,0-00 PI X 4,6 0-0 PI X 4,2-00 PI X 8,9 00- X 0 00 X 8,0 0-0 PI X 8,2-00 X 3 0 X 9,3-0 X 5 X 2,3 0- X 3,5 - PI7 Fall 204, Oct 3... ELE Lecture 5 46
47 Step 3: Identify EPI s overed by EPI x x x x x Minterms PI is EPI x x x x PI2 x x PI3 x x PI4 x x PI5 x x PI6 x x PI7 is EPI x x Fall 204, Oct 3... ELE Lecture 5 47
48 Step 4: over Remaining Minterms Remaining minterms PI2 x x PI3 x x PI4 x x PI5 PI6 Integer linear program (ILP), available from Matlab and other sources: Define integer {0,} variables, xk =, select PIk; xk = 0, do not select PIk. Minimize k xk, subject to constraints: x2 + x3 x4 + x5 x2 + x4 x3 + x6 solution is x3 = x4 =, x2 = x5 = x6 = 0 Fall 204, Oct 3... ELE Lecture 5 48 x x
49 Linear Programming (LP) mathematical optimization method for problems where some cost depends on a large number of variables. n easy to understand introduction is: S. I. Gass, n Illustrated Guide to Linear Programming, New York: Dover Publications, 970. Very useful tool for a variety of engineering design problems. vailable in software packages like Matlab. ourses on linear programming are available in Math, usiness and Engineering departments. Fall 204, Oct 3... ELE Lecture 5 49
50 Q-M MSOP Solution and Verification F(,,,D) = PI + PI3 + PI4 + PI7 = = + D + D + D See Karnaugh map. EPI s in MSOP D Non-EPI s in MSOP Non-EPI s not in MSOP Fall 204, Oct 3... ELE Lecture 5 50
51 Minimized ircuit D D PI PI3 PI4 F PI7 Fall 204, Oct 3... ELE Lecture 5 5
52 QM Minimizer on the Web cretemathwithproof/quinemcluskey.html Note: This minimizer may not always select the best from the selective prime implicants. Fall 204, Oct 3... ELE Lecture 5 52
53 Further Reading Incompletely specified functions: See Example 3.25, pages Multiple output functions: See Example 3.26, pages Fall 204, Oct 3... ELE Lecture 5 53
ELEC Digital Logic Circuits Fall 2015 Logic Minimization (Chapter 3)
ELE 2200-002 igital Logic ircuits Fall 205 Logic Minimization (hapter 3) Vishwani. grawal James J. anaher Professor epartment of Electrical and omputer Engineering uburn University, uburn, L 36849 http://www.eng.auburn.edu/~vagrawal
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