1. Chapter 3 Digital Logic Structures

2. Basic Logic Gates

3. Basic Relations of Boolean Algebra
+ = OR
= AND
‘ = NOT
x + 0 = x x + 1 = 1 x + x = x x + x’ = 1 x + y = y + x (Commutative) x + (y+z) = (x+y)+z (Associative) x(y+z’) = xy + xz (Distributive) (x+y)’ = x’ y’ (DeMorgan) (x’)’ = x
x0 = 0 x1 = x xx = x xx’ = 0 xy = yx (Commutative) x(yz) = (xy)z (Associative) x+yz = (x+y)(x+z) (Distributive) (xy)’ = x’+y’ (DeMorgan)

4. DeMorgan’s Law not(A and B) = (not A) or (not B) =
not(A or B) = (not A) and (not B) = =

5. More than 2 Inputs? AND/OR can take any number of inputs.
AND = 1 if all inputs are 1.
OR = 1 if any input is 1.
Similar for NAND/NOR.
Can implement with multiple two-input gates, or with single CMOS circuit.
NAND and NOR are not associative.
Jim Conrad’s example:
NAND(NAND(0,0), 1) = NAND(1, 1) = 0
NAND(0, NAND(0,1)) = NAND(0, 0) = 1

6. Summary
MOS transistors are used as switches to implement logic functions.
n-type: connect to GND, turn on (with 1) to pull down to 0
p-type: connect to +2.9V, turn on (with 0) to pull up to 1
Basic gates: NOT, NOR, NAND
Logic functions are usually expressed with AND, OR, and NOT
DeMorgan's Law
Convert AND to OR (and vice versa) by inverting inputs and output

7. Building Functions from Logic Gates
Combinational Logic Circuit
output depends only on the current inputs
stateless
Sequential Logic Circuit
output depends on the sequence of inputs (past and present)
stores information (state) from past inputs
We'll first look at some useful combinational circuits, then show how to use sequential circuits to store information.

8. Decoder 2-bit decoder n inputs, 2n outputs
exactly one output is 1 for each possible input pattern
2-bit
decoder
Uses of decoder:
convert memory/register address to a control line that selects that location
convert an opcode to one of n control lines

9. Multiplexer (MUX) 4-to-1 MUX n-bit selector and 2n inputs, one output
output equals one of the inputs, depending on selector
Another view: decode S, and AND each output with one of the MUX inputs.
Also explain multi-bit inputs.
Uses of multiplexer:
select which input to use for function
select which computed value to pass to next stage (or to place on bus)
4-to-1 MUX

10. Mux (cont.) A B C D Out S0 S1 In general, a MUX has 2n data inputs
n select (or control) lines
and 1 output.
It behaves like a channel selector.
S[1:0] A B C D
Out
A 4-to-1 MUX:
Out takes the value of A,B, C or D
depending on the value of S (00, 01, 10, 11)

11. Adder Half Adder Full Adder Half-adder truth table 2 inputs
2 outputs: sum and carry
Full Adder
performs the addition in column i
3 inputs: ai, bi and ci
2 outputs: si and ci+1
ci is the carry in from bit position i-1
ci+1 is the carry out to bit position i+1
Half-adder truth table

12. Full Adder
Add two bits and carry-in, produce one-bit sum and carry-out. A B
Cin S
Cout 1
A half-adder is one that doesn't take a carry-in.
Sum is one when 1 or 3 inputs are one. Carry-out is one when 2 or 3 inputs are one.

13. Full Adder (cont) where
- verify that this corresponds to the gate-level implementation.

14. Four-bit Adder 1010 Cin 0101 A + 1101 B 10010 S
This is called a "ripple-carry" adder. The sum becomes valid as the carry ripples its way from the low bit to the high bit. How many gate delays until the output is settled?

15. Logical Completeness Can implement ANY truth table with AND, OR, NOT.1
1. AND combinations that yield a "1" in the truth table.
2. OR the results of the AND gates.
Note the use of the bubbles (NOT) in the input.