K-map ( karnaugh map ) 8,4,-2,-1 to binary code conversion
I'm taking computer science courses and some digital design knowledge is required, so I'm taking digital design 101.
Image above is representing the conversion process of 8,4,-2,-1 to binary using K-map (Karnaugh map).
I have no idea why 0001, 0011, 0010, 1100, 1101, 1110 are marked as 'X'.
For 0001, 0011, 0010, they could be expressed as 8,4,-2,-1 as 0111, 0110, 0101. And for 1100, 1101, 1110, 1110 can still be expressed as 1100 in 8,4,-2,-1 form as 1100. rests cannot be expressed in 8,4,-2,-1 since 1100 is the biggest amount of number in 8,4,-2,-1 binary form (I think).
Is there something I'm missing?
I understand the excess-3 to binary code conversion provided from my textbook example ( m10-m15 are marked as 'X' since excess-3 were used to express only 0-9.)
Answer
According to the definition of BCD, 1 decimal digit (NOT one number) is represented by 4 bits.
The 4 given inputs can therefore represent only values from interval from 0 to 9.
The corresponding and complete truth-table looks like this:
decimal | 8 4 -2 -1 | decimal || BCD
/index | A B C D | result || W X Y Z
----------------------------------||---------
0 | 0 0 0 0 | 0 || 0 0 0 0 ~ 0
1 | 0 0 0 1 | -1 || X X X X
2 | 0 0 1 0 | -2 || X X X X
3 | 0 0 1 1 | -2-1=-3 || X X X X
4 | 0 1 0 0 | 4 || 0 1 0 0 ~ 4
5 | 0 1 0 1 | 4-1=3 || 0 0 1 1 ~ 3
6 | 0 1 1 0 | 4-2=2 || 0 0 1 0 ~ 2
7 | 0 1 1 1 | 4-2-1=1 || 0 0 0 1 ~ 1
8 | 1 0 0 0 | 8 || 1 0 0 0 ~ 8
9 | 1 0 0 1 | 8-1=7 || 0 1 1 1 ~ 7
10 | 1 0 1 0 | 8-2=6 || 0 1 1 0 ~ 6
11 | 1 0 1 1 | 8-2-1=5 || 0 1 0 1 ~ 5
12 | 1 1 0 0 | 8+4=12 || X X X X
13 | 1 1 0 1 | 8+4-1=11 || X X X X
14 | 1 1 1 0 | 8+4-2=10 || X X X X
15 | 1 1 1 1 | 8+4-2-1=9 || 1 0 0 1 ~ 9
The K-maps then match the truth-table by its indexes:
Using the K-maps, it can be indeed simplified to these boolean expressions:
W = A·B + A·¬C·¬D
X = ¬B·C + ¬B·D + B·¬C·¬D
Y = ¬C·D + C·¬D
Z = D