1. Digital Systems Design
Review
D: n bit binary number
D = (dn-1 ∙ ∙ ∙ d1 d0)2
If D is an unsigned binary number
D = (2n-1 dn-1+∙ ∙ ∙ 21 d d0)10
If D is a sign-magnitude binary number
D = + (2n-2 dn-2+∙ ∙ ∙ 21 d d0 ) if dn-1=0
= – (2n-2 dn-2+∙ ∙ ∙ 21 d d0 ) if dn-1=1
(–D) = (d’n-1 dn-2 ∙ ∙ ∙ d1 d0)2
If D is in two`s complement system
D = (-2n-1 dn-1+ 2n-2 dn-2 + ∙ ∙ ∙ 21 d d0)10
(–D) = 2n – D = (2n-1) – D + 1 = (d’n-1 d’n-2 ∙ ∙ ∙ d’1 d’0)2 + 1
Digital Systems Design

2. Digital Systems Design
Review
Two’s complement multiplication
Shift and two’s complement addition except for the last step. Remember MSB represent (-2n-1)
x –
0000 initial partial product, which is zero.
1011
11011 partial product
0000
partial product
shifted-and-negated
Digital Systems Design

3. BCD: Binary-Coded Decimal
Review
BCD: Binary-Coded Decimal
0-9 encoded with their 4-bit unsigned binary representation (0000 – 1001). The codewords (1010 – 1111) are not used.
8-bit byte represent values from 0 to 99.
BCD Addition:
Carry
937 Sum
Add
BCD sum
BCD result
Digital Systems Design

4. 2. Combinational Logic Circuits
Boolean Algebra
switching algebra
deals with Boolean values --- 0, 1
Positive-logic convention
analog voltages LOW, HIGH  0, 1
Negative logic --- seldom used
Signal values denoted by variables (X, Y, FRED, etc.)
Digital Systems Design

5. Digital Systems Design
Boolean operators
Complement: X¢ (opposite of X)
AND: X × Y
OR: X + Y
Digital Systems Design

6. Digital Systems Design
Literal: a variable or its complement
X, X¢, FRED¢, CS_L
Expression: literals combined by AND, OR, parentheses, complementation
X+Y
P × Q × R
A + B × C
((FRED × Z¢) + CS_L × A × B¢ × C + Q5) × RESET¢
Equation: Variable = expression
P = ((FRED × Z¢) + CS_L × A × B¢ × C + Q5) × RESET¢
Digital Systems Design

7. Digital Systems Design
Basic Logic Gates
Digital Systems Design

8. Digital Systems Design
Theorems
Digital Systems Design

9. Digital Systems Design
More Theorems
Digital Systems Design

10. Digital Systems Design
Duality
Swap 0 & 1, AND & OR
Result: Theorems still true
Why?
Each axiom (T1-T5) has a dual (T1¢-T5¢)
Counterexample: X + X × Y = X (T9) X × X + Y = X (dual)
X + (X × Y) = X (T9) X × (X + Y) = X (dual) (X × X) + (X × Y) = X (T8)
Digital Systems Design

11. Digital Systems Design
N-variable Theorems
Digital Systems Design

12. DeMorgan Symbol Equivalence
Digital Systems Design

13. Digital Systems Design
Similar for OR
Digital Systems Design

14. Complement of a function
F1 = XYZ’ + X’Y’Z
F1’ = (XYZ’ + X’Y’Z)’ = (XYZ’)’ × (X’Y’Z)’
= (X’+Y’+Z) × (X+Y+Z’)
Complement = take dual +complement each literal
Dual of F1 = (X+Y+Z’) × (X’+Y’+Z)
F1’ = (X’+Y’+Z) × (X+Y+Z’)
Digital Systems Design

15. Digital Systems Design
Standard Forms:
Product and sum terms
Minterm: A product term in which all variables appear exactly once, either complemented or not (2n minterms)
For a two variable function, minterms are
X’Y’, X’Y, XY’, XY
m0 , m1 , m2 , m3
Maxterms: A sum term that contains all variables in complemented or uncomplemented form
X+Y, X+Y’, X’+Y, X’+Y’
M0 , M1 , M2 , M3
Digital Systems Design

16. Digital Systems Design

17. Digital Systems Design
Alternative representations
F(X,Y,Z) = X’Y’Z’ + X’YZ’ +XY’Z + XYZ
= m0 + m2 + m5 + m7 =
F’(X,Y,Z) = X’Y’Z + X’YZ + XY’Z’ + XYZ’
= m1 + m3 + m4 + m6
F(X,Y,Z) = (m1 + m3 + m4 + m6)’ = m1’ m3’ m4’ m6’
= M1 M3 M4 M6
Digital Systems Design

18. Digital Systems Design
Maxterms are seldom used, we’ll use minterms rather.
Properties of minterms:
There are 2n minterms. 1-1 with binary numbers 0-(2n-1)
Every Boolean function can be expressed as sum of minterms.
Absent minterms belong to complement function
A function that include all minterms is equal to logic 1.
Digital Systems Design