Announcements Project components starting to arrive No assignment due this week assignment 7 will be posted on Thursday. Final exam date. Lets leave it where it is (Dec 13 th ). Project reports due to me by, Sunday Dec 11 th. Individual reports. NO EXTENSIONS Lecture 17 Overview Karnaugh map examples Applications of Combinational logic Karnaugh Maps: Example 2: ABCDO Draw the table Karnaugh Maps: Example 2: ABCDO AB CD Draw the table Karnaugh Maps: Example 2: ABCDO AB CD Draw the table Find any isolated cells - 1's with no neighbours Karnaugh Maps: Example 2: ABCDO AB CD Draw the table Find any isolated cells Find any 2-cell subcubes which are not adjacent to other 2-cell subcubes. Remember wrapping AB CD Karnaugh Maps: Example 2: ABCDO AB CD Draw the table Find any isolated cells Find 2-cell subcubes. Find 4-cell subcubes which are not adjacent to other four cell subcubes Find 8-cell subcubes etc etc. Minimal expression is formed by the smallest number of maximal subcubes Karnaugh Maps: Example 2: ABCDO AB CD A'B'D' Karnaugh Maps: Example 2: ABCDO B'C' AB CD Karnaugh Maps: Example 2: ABCDO C'D AB CD Karnaugh Maps: Example 2: ABCDO AD AB CD Karnaugh Maps: Example 2: ABCDO So sum-of products realization is: O=A'B'D'+B'C'+C'D+AD Requires 9 gates AB CD Karnaugh Maps: Example 2: ABCDO So sum-of products realization is: O=A'B'D'+B'C'+C'D+AD Requires 9 gates A'B'D' B'C' C'D AD Karnaugh Maps: Your turn ABCDO ABCDO AB CD Solve the Karnaugh map and draw the circuit Karnaugh Maps: Fill the map ABCDO AB CD Karnaugh Maps: Find the subcubes ABCDO AB CD Karnaugh Maps: Evaluate ABCDO AB CD A' B'C' sum of products realization is A'+B'C' Karnaugh Maps: Evaluate ABCDO sum of products realization is A'+B'C' Karnaugh Maps: Setting up the Maps DeMorgan's Theorem shows that any logic function can be implemented by using just OR and NOT gates, or by just AND and NOT gates A consequence of this is that any logical expression can be reduced to either a "Sum-of-Products" form or a "Product-of-Sums" form Karnaugh Maps: Boxing zeros for sum of products So far, we have been "boxing the ones" for a sum-of-products It is also possible to realize a sum-of-products by "boxing the zeros" Truth table Karnaugh Map A'B' AC So, realization is O=(A'B'+AC)' (all inverted as we've boxed zeros) Can convert to a product-of-sums with DeMorgan's Theorem: (A'B'+AC)' =(A'B')'(AC)'= (A+B)(AC)'=(A+B)(A'+C') Karnaugh Maps: Boxing zeros for product of sums Knowing this, we can get the same product of sums more directly Truth table Karnaugh Map A+B A'+C' So, realization is O=(A+B)(A'+C') We have to NOT the inputs, then: Karnaugh Maps So O=(A+B)(A'+C') This requires 5 gates: Note: this realization, part sum of products, part product of sums: O=(A+B)(AC)' only requires 3 gates: Sum of Products (XY+WZ)Product of sums (X+Y)(W+Z) Box 1'sBox 0's AND operation among inputs which are not changing OR operation among NOTted inputs which are not changing OR operation between boxes AND operation between boxes Result must be NOTted for final answer Choose whether to box 1's or 0's by looking at the map - if you have more 0's, boxing 0's will lead to greater simplification Another Karnough Example: 7 segment displays Truth Table Karnaugh Map for "a" "x" represents a "don't care" condition - the value can be either 0 or 1 Control segments with 4 binary inputs Need a logic circuit for each segment Box the ones for sum-of-products This subcube: B This subcube: A'C' This subcube: D This subcube: AC Realization (sum-of-products) is B+D+ AC+ A'C' Karnaugh Maps More generally, Karnaugh maps can be used to simplify any logic function (doesnt have to be electronics!) Final comment on Karnaugh Maps AB CD