Week 3 Digital Logic PDF

Week 3 Logic will get you from A to B. Imagination will take you everywhere. Albert Einstein 0  Digital Logic Represents Binary outcomes statement TRUE FALSE answer YES NO light...

Week 3 Logic will get you from A to B. Imagination will take you everywhere. Albert Einstein 0  Digital Logic Represents Binary outcomes statement TRUE FALSE answer YES NO light OFF On switch CLOSED OPEN one bit 1 0 0  Basic Rules of Boolean Algebra 1. A + 0 =A 10. A + AB = A 2. A + 1 =1 11. A + A’B = A + B 3. A 0 =0 12. (A + B)(A + C) = 4. A 1 =A A + BC 5. A + A = ALoading… 6. A + A’ =1 Note: Symbols can 7. A A =A represent a single 8. A A’ =0 variable or a 9. A’’ =A combination of variables 0  DeMorgan’s Rules A + B = (A’B’)’ AB = (A’+B’)’ 0  Canonical Form Canonical means all variables are represented in each term. X = a’b + ac is a minimum representation Change to Canonical Form Loading… = a’b(c+c’) + a(b+b’)c = a’bc + a’bc’ + abc + ab’c This implies that some variables are redundant. Sum of products 0  Gray Code unsigned decimal gray 000 0 000 001 1 001 010 2 011 011 3 010 100 4 110 101 5 111 110 6 101 111 7 100 0  Karnaugh Maps 2 & 3 Variables B B' B A 0 1 A' 0 A 1 BC B' C' B' C BC B C' A 00 01 11 10 A' 0 A 1 0  Karnaugh Maps 4 Variables CD C' D' C' D CD C D' AB 00 01 11 10 A' B' 00 A' B 01 AB 11 A B' 10 0  Karnaugh Map Example A’B’C’ + AB’C’ + A’BC’ + ABC’ BC B' C' B' C BC B C' A 00 01 11 10 A' 1 1 0 A 1 1 1 0 Karnaugh Map Grouping 1 1 1 1 1 1 1 1 1 1 0 Karnaugh Map Grouping 1 1 1 1 Loading… 1 1 1 1 1 1 0  Karnaugh Map Example Cont. A’B’C’ + AB’C’ + A’BC’ + ABC’ B is covered, B and B’ cancel, So we ignore B C’ is common to the entire grouping, So it is included BC B' C' B' C BC B C' A is covered A 00 01 11 10 over the full A' 1 1 Range so we 0 ignore A A 1 1 1 Final Result : X = C’ 0  Don’t Care 0 0000 1 BCD to 7 segment Segment 1 display Logic 1 0001 0 Each segment is 2 0010 1 controlled by it’s own logic 3 0011 1 To reduce the boolean 4 0100 0 equation in a Karnaugh 5 0101 1 Map we plot the don’t care 6 0110 1 states. 7 0111 1 If appropriate we can use 8 1000 1 these to form larger 9 1001 1 groupings, thus simplifying 10 1010 x the logic. One equation for each 15 1111 x segment. 0 0 0 0 0 End 0