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1 Slides for Lecture 5 ENEL 353: Digital Circuits Fall 203 Term Steve Norman, PhD, PEng Electrical & Computer Engineering Schulich School of Engineering University of Calgary October, 203
2 ENEL 353 F3 Section 02 Slides for Lecture 5 slide 2/20 Previous Lecture Continued presentation of Kmap preliminaries: more about Kmap layout relationships between minterms and Kmap cells adjacency and using adjacency for simplification introduction to grouping cells into 2cell, 4cell, or 8cell rectangles
3 ENEL 353 F3 Section 02 Slides for Lecture 5 slide 3/20 Today s Lecture Kmap terminology: implicant, prime implicant, distinguished cell, essential prime implicant. Using Kmaps to find minimal SOP expressions for functions. Related reading in Harris & Harris: Section 2.7, up to the end of Section
4 ENEL 353 F3 Section 02 Slides for Lecture 5 slide 4/20 Rectangles of three, six, or nine cells are not helpful A rectangle of 2, 4, or 8 cells corresponds to a single product in an SOP expression. A rectangle of 3, 6, or 9 cells does not correspond to a single product in an SOP expression, and is therefore not helpful in minimization of logic functions.
5 ENEL 353 F3 Section 02 Slides for Lecture 5 slide 5/20 An imprecise description of Kmap minimization methods Cover all the cells, and only the cells. Use as few singlecell, 2cell, 4cell, and 8cell rectangles as possible. Make the rectangles as large as possible. canonical SOP not quite minimal minimal SOP (far from minimal!)
6 ENEL 353 F3 Section 02 Slides for Lecture 5 slide 6/20 Leadup to an important definition: Implicant X implies Y, means, Whenever X is true, then Y must also be true. Example: The fact that Joe is in the ICT Building implies that Joe is on the U of C campus. Let s look at a typical SOP expression: F = Ā BC + ĀB C + AC Because of the way OR is defined, Ā BC = implies F =. Similarly, ĀB C = implies F =, and AC = implies F =. The products Ā BC, ĀB C, and AC are said to be implicants of F.
7 ENEL 353 F3 Section 02 Slides for Lecture 5 slide 7/20 Important definition: Implicant For a given logic function F, the implicants of F are all of the products from all of the valid SOP expressions for F. For example, here is an exhaustive list of valid SOP expressions for the twoinput NAND function: F = Ā B + ĀB + A B = Ā + S B = ĀB + B = Ā + B So what are all the implicants of this particular F?
8 ENEL 353 F3 Section 02 Slides for Lecture 5 slide 8/20 Implicants and Kmaps From a Kmap for a function F, the implicants of F correspond to all of the individual cells; all rectangles composed of 2, 4, or 8 cells. (That s for functions of up to four variables things get more complicated with functions of five or six variables.) Example : Let s use a Kmap to find all the implicants of NAND (A,B). Example 2: Let s work with an example 4variable Kmap, and find a few implicants and SOP expressions.
9 ENEL 353 F3 Section 02 Slides for Lecture 5 slide 9/20 Important definition: Prime implicant A prime implicant is an implicant that is not fully contained by any other implicant. Example: If ABC and AB C are both implicants of F, they cannot be prime implicants, because they are fully contained by AB. In a Kmap, a prime implicant is a rectangle that cannot be doubled in size without collecting one or more 0cells.
10 ENEL 353 F3 Section 02 Slides for Lecture 5 slide 0/20 Illustration of prime and nonprime implicants Examples, for a 3variable function and a 4variable function... nonprime implicants prime implicants
11 ENEL 353 F3 Section 02 Slides for Lecture 5 slide /20 Illustration of prime and nonprime implicants, continued A B C D C 00 0 A D The circled implicant is a prime implicant because it can t be doubled in size without collecting a 0cell. What is the product for the circled implicant? 0 B What are all the other prime implicants?
12 ENEL 353 F3 Section 02 Slides for Lecture 5 slide 2/20 The prime implicant theorem The theorem says: The products in a minimal SOP expression must all be prime implicants. Sketch of proof: Suppose an SOP expression for F includes nonprime implicant X. Replace X with a prime implicant that fully contains X. The resulting new SOP expression is valid for F and is simpler than the original SOP expression, so the original expression could not have been minimal. Practical consequence: When you use a Kmap to look for a minimal SOP expression, you should totally ignore nonprime implicants.
13 ENEL 353 F3 Section 02 Slides for Lecture 5 slide 3/20 What the prime implicant theorem does not say This is not generally true: An SOP expression for F in which the products are prime implicants is a minimal SOP expression for F. Example: Let s study this Kmap... A B C 00 0 A 0 0 C B
14 ENEL 353 F3 Section 02 Slides for Lecture 5 slide 4/20 Important definitions: Distinguished cell and essential prime implicant A distinguished cell of function F is a cell that is covered by exactly one prime implicant of F. An essential prime implicant of F is a prime implicant that covers at least one distinguished cell of F. A B C 00 0 A 0 Let s study this Kmap some more... 0 C B
15 ENEL 353 F3 Section 02 Slides for Lecture 5 slide 5/20 Distinguished cells, essential prime implicants, and this year s textbook Harris and Harris do not provide definitions of distinguished cell or essential prime implicant. However, both concepts are useful for efficient discovery of minimal SOP expressions from Kmaps, so in ENEL 353 you must know exactly what these terms mean. (Many other textbooks define distinguished cell and essential prime implicant exactly as done on the previous slide.)
16 ENEL 353 F3 Section 02 Slides for Lecture 5 slide 6/20 Essential prime implicants and minimal SOP expressions Fact: If the essential prime implicants of F cover all of its cells, the OR of the essential prime implicants is a unique minimal SOP expression for F. Sketch of proof: The minimal SOP expression must be a sum of prime implicants. If an essential prime implicant is not used in a sum of prime implicants, at least one cell is not covered, so no essential prime implicant can be left out. Important consequence: Finally, for at least some functions, we have a way to be certain that an SOP expression is a minimal SOP expression!
17 ENEL 353 F3 Section 02 Slides for Lecture 5 slide 7/20 Review of an earlier example A B C D C A D Which prime implicants are essential prime implicants? Can we use the essential prime implicants to make a minimal SOP expression? B
18 ENEL 353 F3 Section 02 Slides for Lecture 5 slide 8/20 Using only essential prime implicants may fail to cover all the cells Unfortunately, when looking for minimal SOP expressions, we can t always declare victory after we find all the essential prime implicants. Let s look at this example... A B C D 00 A D C 0 B
19 ENEL 353 F3 Section 02 Slides for Lecture 5 slide 9/20 A note about nonessential prime implicants We ve just seen that prime implicants in a Kmap can be divided into those that are essential prime implicants and those that are not essential prime implicants. To call a prime implicant nonessential does not mean that it is useless or unimportant! Here is the correct distinction... essential PI: contains a distinguished cell, must appear in a minimal SOP expression nonessential PI: does not contain a distinguished cell, might or might not be needed in a minimal SOP expression
20 ENEL 353 F3 Section 02 Slides for Lecture 5 slide 20/20 Upcoming topics Finding minimal SOP expressions when essential prime implicants do not cover all the cells. (Not covered in detail in Harris & Harris.) Don tcares, Xcells, and SOP minimization. (Related reading in Harris & Harris: Section )
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