Karnaugh Maps. Kiril Solovey. TelAviv University, Israel. April 8, Kiril Solovey (TAU) Karnaugh Maps April 8, / 22


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1 Karnaugh Maps Kiril Solovey TelAviv University, Israel April 8, 2013 Kiril Solovey (TAU) Karnaugh Maps April 8, / 22
2 Reminder: Canonical Representation Sum of Products Function described for the inputs for which it returns 1 f (x, y, z) Kiril Solovey (TAU) Karnaugh Maps April 8, / 22
3 Reminder: Canonical Representation Sum of Products Function described for the inputs for which it returns 1 f (x, y, z) = (2, 3, 4, 6) = x yz + x yz + xy z + xyz Kiril Solovey (TAU) Karnaugh Maps April 8, / 22
4 Reminder: Canonical Representation Product of Sums Function described for the inputs for which it returns 0 f (x, y, z) = (0, 1, 5, 7) = (x +y +z) (x +y +z ) (x +y +z ) (x +y +z ) Kiril Solovey (TAU) Karnaugh Maps April 8, / 22
5 Canonical Representation is Wasteful The function will be written in SOP as f (x, y) = 1 f (x, y) = x y + x y + xy + xy Kiril Solovey (TAU) Karnaugh Maps April 8, / 22
6 Karnaugh Maps Provide a geometric visualisation of boolean functions Simplify SOP/POS representation Reduce number of variables in boolean expression Reduce number of minterms in expression Kiril Solovey (TAU) Karnaugh Maps April 8, / 22
7 Kmaps for 3variable functions Kiril Solovey (TAU) Karnaugh Maps April 8, / 22
8 Kmaps for 4variable functions Kiril Solovey (TAU) Karnaugh Maps April 8, / 22
9 Properties of Kmaps SOP Every two neighbors (horizontal or vertical) differ in one bit Elements from first and last column (or row) are neighbors too If the value of two neighbors is 1 then their minterms can be merged xy + xy = x(y + y) = x 1 = x Kiril Solovey (TAU) Karnaugh Maps April 8, / 22
10 Properties of Kmaps SOP Every two neighbors (horizontal or vertical) differ in one bit Elements from first and last column (or row) are neighbors too If the value of two neighbors is 1 then their minterms can be merged xy + xy = x(y + y) = x 1 = x x 1 x 2 x 3 x 4 x 1 x 2x 3 x 4 Kiril Solovey (TAU) Karnaugh Maps April 8, / 22
11 Properties of Kmaps SOP Every two neighbors (horizontal or vertical) differ in one bit Elements from first and last column (or row) are neighbors too If the value of two neighbors is 1 then their minterms can be merged xy + xy = x(y + y) = x 1 = x x 1 x 3 x 4 Kiril Solovey (TAU) Karnaugh Maps April 8, / 22
12 Properties of Kmaps POS Similarly, for neighbors with value 0: (x + y)(x + y ) = xx + yx + y x + yy = x + x(y + y ) + 0 = x (x 1 + x 2 + x 3 + x 4 )(x 1 + x 2 + x 3 + x 4 ) Kiril Solovey (TAU) Karnaugh Maps April 8, / 22
13 Properties of Kmaps POS Similarly, for neighbors with value 0: (x + y)(x + y ) = xx + yx + y x + yy = x + x(y + y ) + 0 = x (x 1 + x 3 + x 4 ) Kiril Solovey (TAU) Karnaugh Maps April 8, / 22
14 Properties of Kmaps Similarly, clusters of 4,8,16 can be merged Cluster: Collection of cells that form a rectangle All cells in cluster have the same number of neighbors (> 0) Notice that variables whose value changes in the cluster vanish x 1 x 4 Kiril Solovey (TAU) Karnaugh Maps April 8, / 22
15 Kmaps Goal: cover all 1 s using a minimal number of clusters or all 0 s This results in the simplified representation of function (SOP/POS) Example: Kiril Solovey (TAU) Karnaugh Maps April 8, / 22
16 Clusterfinding algorithm Put isolated 1 s (without neighbors) in separate clusters of size 1 Find 1 s for which there is only one cluster of size 2 that contains them Add the respective clusters of size 2 Find 1 s for which there is only one cluster of size 4 that contains them Add the respective clusters of size 4 Repeat for clusters of 8 and 16 * Use minimal number of clusters to cover leftover 1 s Kiril Solovey (TAU) Karnaugh Maps April 8, / 22
17 Example Kiril Solovey (TAU) Karnaugh Maps April 8, / 22
18 Example Kiril Solovey (TAU) Karnaugh Maps April 8, / 22
19 Example Kiril Solovey (TAU) Karnaugh Maps April 8, / 22
20 Example Kiril Solovey (TAU) Karnaugh Maps April 8, / 22
21 Example Kiril Solovey (TAU) Karnaugh Maps April 8, / 22
22 Unnecessary clusters Kiril Solovey (TAU) Karnaugh Maps April 8, / 22
23 Examples (valid solution) Kiril Solovey (TAU) Karnaugh Maps April 8, / 22
24 Don t cares The output of some functions is not defined for certain inputs For instance: circuit that implements a digital decimal digit Input is 4 bits (required to represent the number 9) Output of circuit for input > 9 is not defined Simplifying a function with don t cares ( ) Assign to every a value (0,1) Goal: Increase size of existing clusters with real 1 s Never add new clusters Kiril Solovey (TAU) Karnaugh Maps April 8, / 22
25 Don t cares Kiril Solovey (TAU) Karnaugh Maps April 8, / 22
26 Implementing a FullAdder FA computes the addition of 3 input bits Has two output bits: S i for result, C i+1 for carry Task: implement FA with logical circuit Steps for solution: Find truth tables of Si and C i+1 Find minimal expression for Si, C i+1 using Kmaps Draw circuits for minimal expressions Kiril Solovey (TAU) Karnaugh Maps April 8, / 22
27 Implementing a FullAdder Kiril Solovey (TAU) Karnaugh Maps April 8, / 22
28 Implementing a FullAdder Kiril Solovey (TAU) Karnaugh Maps April 8, / 22
29 Beyond Kmaps Kmaps applicable to functions with few variables (35) How to implement more complex functions? Example: Implement adder of two 4digit numbers 8 inputs 5 outputs Use more complex building blocks Kiril Solovey (TAU) Karnaugh Maps April 8, / 22
30 Exercise: number of 1 s Input: 6 bits of input ABCDEF Output: number of 1 s in input Inventory: 3 FA, 1 HA Solution: Solve separately for ABC, DEF Combine the two results Result is represented by S 2 S 1 S 0 Kiril Solovey (TAU) Karnaugh Maps April 8, / 22
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