Implement following function G and the function F given belove together using only one decoder and external gates (OR, AND, NOT...)
Implement following function G and the function F given belove together using
only one decoder and external gates (OR, AND, NOT...)
Overlay Version of the K-map
The overlay version of the Karnaugh map, shown above, is simply two (four for a 6-variable map) identical maps except for the most significant bit of the 3-bit address across the top.
If we look at the top of the map, we will see that the numbering is different from the previous Gray code map. If we ignore the most significant digit of the 3-digit numbers, the sequence 00, 01, 11, 10 is at the heading of both sub-maps of the overlay map. The sequence of eight 3-digit numbers is not Gray code. Though the sequence of four of the least significant two bits is.
Let’s put our 5-variable Karnaugh Map to use. Design a circuit that has a 5-bit binary input (A, B, C, D, E), with A being the MSB (Most Significant Bit). It must produce an output logic High for any prime number detected in the input data.
We show the solution above on the older Gray code (reflection) map for reference. The prime numbers are (1,2,3,5,7,11,13,17,19,23,29,31). Plot a 1 in each corresponding cell. Then, proceed with a grouping of the cells. Finish by
Five variable K-map:
Overlay Version of the K-map
The overlay version of the Karnaugh map, shown above, is simply two (four for a 6-variable map) identical maps except for the most significant bit of the 3-bit address across the top.
If we look at the top of the map, we will see that the numbering is different from the previous Gray code map. If we ignore the most significant digit of the 3-digit numbers, the sequence 00, 01, 11, 10 is at the heading of both sub-maps of the overlay map. The sequence of eight 3-digit numbers is not Gray code. Though the sequence of four of the least significant two bits is.
Let’s put our 5-variable Karnaugh Map to use. Design a circuit that has a 5-bit binary input (A, B, C, D, E), with A being the MSB (Most Significant Bit). It must produce an output logic High for any prime number detected in the input data.
We show the solution above on the older Gray code (reflection) map for reference. The prime numbers are (1,2,3,5,7,11,13,17,19,23,29,31). Plot a 1 in each corresponding cell. Then, proceed with a grouping of the cells. Finish by writing the simplified result.
Note that 4-cell group A’B’E consists of two pairs of cells on both sides of the mirror line. The same is true of the 2-cell group AB’DE. It is a group of 2-cells by being reflected about the mirror line. When using this version of the K-map look for mirror images in the other half of the map.
Out = A’B’E + B’C’E + A’C’DE + A’CD’E + ABCE + AB’DE + A’B’C’D
Below we show the more common version of the 5-variable map, the overlay map.
If we compare the patterns in the two maps, some of the cells in the right half of the map are moved around since the addressing across the top of the map is different. We also need to take a different approach to spot commonality between the two halves of the map.
Overlay one half of the map atop the other half. Any overlap from the top map to the lower map is a potential group. The figure below shows that group AB’DE is composed of two stacked cells. Group A’B’E consists of two stacked pairs of cells.
For the A’B’E group of 4-cells ABCDE = 00xx1 for the group. That is A, B, E is the same 001 respectively for the group. And, CD=xx that is it varies, no commonality in CD=xx for the group of 4-cells. Since ABCDE = 00xx1, the group of 4-cells is covered by A’B’XXE = A’B’E.
The above 5-variable overlay map is shown stacked.
An example of a six variable Karnaugh map follows. We have mentally stacked the four sub-maps to see the group of 4-cells corresponding to Out = C’F’
A magnitude comparator (used to illustrate a 6-variable K-map) compares two equal, greater than, or less than each other on three respective outputs. A three-bit magnitude comparator has two inputs A2A1A0 and B2B1B0 An integrated circuit magnitude comparator (7485) would actually have four inputs, But, the Karnaugh map below needs to be kept to a reasonable size. We will only solve for the A>B output.
G(A, B, C, D) = {Odd numbers not included in the function F}
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