SLIDE 1
Karnaugh Maps
Used to design circuits for up to four inputs. Alternate representation of a truth table:
a b c
- utput
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
Thursday, 19 November 2015 Questions about project and/or labs? - - PowerPoint PPT Presentation
Thursday, 19 November 2015 Questions about project and/or labs? - - PowerPoint PPT Presentation
Thursday, 19 November 2015 Questions about project and/or labs? Last time: Full adders; Karnaugh maps Today: Karnaugh maps again-- in-class exercise DeMorgans Law Karnaugh Maps Used to design circuits for up to four inputs. Alternate
SLIDE 2
SLIDE 3
Karnaugh Maps
1 1 1 1 1
0,0 0,1 1,1 1,0 0 1 a, b: c: This block of 1’s represents the expression b -- these are all the places in the grid where b = 1. 0,0 0,1 1,1 1,0 0 1 a, b: c:
1 1 1 1 1
This block represents ac -- these are all the places in the grid where a = 1 and c = 0.
This is the boolean function b + ac
SLIDE 4
Karnaugh Maps
Cover all the ones with a set of (possibly
- verlapping) rectangles. Each rectangle must
have dimensions that are a power of 2 (e.g., 2x1, 2x2, 4x2, 1x4, etc.). Each rectangle represents one term of the function represented.
SLIDE 5
Karnaugh Maps
1 1 1 1 1 1 1 1 1
0,0 0,1 1,1 1,0 0,0 0,1 1,1 1,0
a,b: c,d:
cd ac bc ad
ac + ad + bc + cd
Note that the box for bc “wraps around” from top to bottom.
SLIDE 6
Karnaugh Maps
1 1 1 1 1 1 1 1
0,0 0,1 1,1 1,0 0,0 0,1 1,1 1,0
a,b: c,d:
ad bd The four corners wrap around to form a 2- by-2 rectangle
ad + bd
SLIDE 7
In-class Exercise: Find formulas for three different Karnaugh maps. (Solutions will be handed out next week.)
Karnaugh Maps
SLIDE 8
Boolean Algebra
So far we have been concerned merely with finding valid expressions for boolean functions. Most of these have been of the following form: (“and-expr”) + (“and-expr”) + … + (“and-expr”) where “and-expr” just indicates an “and” of several variables or their negations. Example: abc + bd + acd
SLIDE 9
Boolean Algebra
Using logic gates, an “or” of “ands” resembles:
0 or more “not” gates Any number of “and” gates A final “or” gate Inputs:
SLIDE 10
Boolean Algebra
But there are many other ways to express a circuit besides an “or” of “ands”. Some of these may require fewer gates (less circuitry). For example, the expression ab + bc (an “or” of “ands”) can be built with two “not” gates, two “and” gates, and an “or” gate. But the equivalent expression b(a+c) requires one fewer “and” gate.
SLIDE 11
Boolean Algebra
Some rules of boolean algebra are listed in Appendix B on page B-6. Some of them look like things from arithmetic, such as “factoring”: a(b+c) = ab + ac The values “0” (false) and “1” (true) behave sort
- f like arithmetic values zero and 1:
0 + a = a 1a = a
SLIDE 12
Boolean Algebra
However, some of them look different:
ab + c = (a+b)(a+c) a + 1 = 1 a + a = 1 aa = 0
One very important law is “DeMorgan’s Law”:
a + b = ab ab = a + b
SLIDE 13
Boolean Algebra
We can use Karnaugh maps and DeMorgan’s law to create circuits in an “and of ors” fashion as follows: 1 1 1 1 1
0,0 0,1 1,1 1,0 0 1 a, b: c:
1 1 1
0,0 0,1 1,1 1,0 0 1 a, b: c: (1) Negate the entries in the Karnaugh map
SLIDE 14
Boolean Algebra
0,0 0,1 1,1 1,0 0 1 a, b: c:
1 1 1
(2) Find the expression for the negated table
ab + bc
SLIDE 15