1. EE 231 Digital Electronics Fall 01 Week 4-1 Multi-Level Logic: Conversion of Forms NAND-NAND and NOR-NOR Networks DeMorgan's Law: A + B = A B; A B = A + B Written differently: A + B = A B; A B = A + B A A BB A B A B   In other words, OR =NAND with complemented inputs AND =NOR with complemented inputs NAND =OR with complemented inputs NOR = AND with complemented inputs A B A B  AA BB 

2. EE 231 Digital Electronics Fall 01 Week 4-2 Multi-Level Logic: Conversion of Forms Example: Map AND/OR network to NAND/NAND network Verify equivalence of the two forms NAND A A B B C C D D Z Z

3. EE 231 Digital Electronics Fall 01 Week 4-3 Multi-Level Logic: Mapping Between Forms Example: Map AND/OR network to NOR/NOR network Step 1 Step 2 NOR Conserve "Bubbles" NOR Conserve "Bubbles" A A’ B B’ C C’ D D’ Z Z

4. EE 231 Digital Electronics Fall 01 Week 4-4 Multi-Level Logic: CAD Tools for Simplification Decomposition: Take a single Boolean expression and replace with collection of new expressions: F = A B C + A B D + A' C' D' + B' C' D' F rewritten as: F = X Y + X' Y' X = A B Y = C + D (12 literals) (4 literals) Before Decomposition After Decomposition F F A A A A B B B B C C C C D D D D X Y

5. EE 231 Digital Electronics Fall 01 Week 4-5 Multi-Level Logic: CAD Tools for Simplification Extraction: common intermediate subfunctions are factored out F = (A + B) C D + E G = (A + B) E' H = C D E can be re-written as: F = X Y + E G = X E' H = Y E X = A + B Y = C D (11 literals) (7 literals) Before Extraction After Extraction X E E E GG Y HH A A AB B B C C CD D D FF

6. EE 231 Digital Electronics Fall 01 Week 4-6 Multi-Level Logic: CAD Tools for Simplification Factoring: expression in two level form re-expressed in multi-level form F = A C + A D + B C + B D + E can be rewritten as: F = (A + B) (C + D) + E (9 literals) (5 literals) Before FactoringAfter Factoring A A A B B B C C C D D D E E F F

7. EE 231 Digital Electronics Fall 01 Week 4-7 Number Systems Sign and Magnitude Representation High order bit is sign: 0 = positive (or zero), 1 = negative Three low order bits is the magnitude: 0 (000) to 7 (111) Two representations for 0 are: 0000 and 1000

8. EE 231 Digital Electronics Fall 01 Week 4-8 Number Systems Ones Complement To negate a number simply flip all the bits. Still two representations of 0!

9. EE 231 Digital Electronics Fall 01 Week 4-9 Number Representations Twos Complement To negate: Twos complement = Ones complement + 1 Only one representation for 0 Easier to implement addition and subtraction

10. EE 231 Digital Electronics Fall 01 Week 4-10 Number Systems Overflow Conditions Add two positive numbers to get a negative number or two negative numbers to get a positive number 5 + 3 = -8 -7 - 2 = +7 0000 0001 0010 0011 1000 0101 0110 0100 1001 1010 1011 1100 1101 0111 1110 1111 +0 +1 +2 +3 +4 +5 +6 +7 -8 -7 -6 -5 -4 -3 -2 0000 0001 0010 0011 1000 0101 0110 0100 1001 1010 1011 1100 1101 0111 1110 1111 +0 +1 +2 +3 +4 +5 +6 +7 -8 -7 -6 -5 -4 -3 -2