CS211 Computer Architecture Digital Logic
Class Checkpoint
l What have discussed up until
l  C Programming language
l 
l 
More low-level then Java.
Better idea about what’s really going on.
l  Covered
l Topics
l  Transistors
(Design & Types)
l  Logic Gates
l  Combinational Circuits
l  K-Maps
l 
now & why:
data representation
Computers manipulate data. How is this data stored and
manipulated?
l  Covered
Machine-Level representation of programs
(assembly lanaguage - x86)
l 
Computers don’t work on C code (or Java). Need
instructions closer to hardware.
l  What’s
next? Processor Design… how does machine
instructions actually make the processor work?
Figures & Tables borrowed from:!
•  http://www.allaboutcircuits.com/vol_4/index.html!
But first…
l Before
we go into processor design, we’re going to
cover some topics in Digital Logic.
l Book
covers this only sparingly, we’re going to go
into a bit more detail.
l Specifically:
l  Transistors
l  Logic
gates
l  Combinational & Sequential Circuits
l  Flip-Flops
l  Memory
Transistor: Building Block of
Computers
l Microprocessors
contain millions of transistors
l  Intel
Pentium 4 (2000): 48 million
l  IBM PowerPC 750FX (2002): 38 million
l  IBM/Apple PowerPC G5 (2003): 58 million
l Logically,
each transistor acts as a switch
to implement logic functions
l Combined
l  AND,
OR, NOT
l Combined
l  Adder,
to build higher-level structures
multiplexer, decoder, register, …
l Combined
to build processor
1
Simple Switch Circuit
l Switch
n-type MOS Transistor
open:
No current through
circuit
l  Light is off
l  Vout is +2.9V
l 
l Switch
closed:
Short circuit across
switch
l  Current flows
l  Light is on
l  Vout is 0V
l 
Switch-based circuits can easily represent two states:
on/off, open/closed, voltage/no voltage. p-type MOS Transistor
l p-type is complementary
l  when Gate has positive voltage,
open circuit between #1 and #2
(switch open)
l  when Gate has zero voltage,
short circuit between #1 and #2
(switch closed)
l MOS
l 
= Metal Oxide Semiconductor
two types: n-type and p-type
l n-type
l  when Gate has positive voltage,
short circuit between #1 and #2
(switch closed)
l  when Gate has zero voltage,
open circuit between #1 and #2
(switch open)
Terminal #2 must be
connected to GND (0V).
Gate = 1
Gate = 0
Inverter (NOT Gate)
to n-type
Gate = 1
Truth table
In Gate = 0
Out In Out 0 V 2.9 V 0 1 2.9 V 0 V 1 0 Terminal #1 must be
connected to +2.9V.
2
NOR Gate
OR Gate
A B C 0 0 1 0 1 0 1 0 0 1 1 0 A B C 0 0 0 0 1 1 1 0 1 1 1 1 Add inverter to NOR.
Note: Serial structure on top, parallel on bottom.
NAND Gate (NOT(AND))
AND Gate
A B C 0 0 1 0 1 1 1 0 1 1 1 0 A B C 0 0 0 0 1 0 1 0 0 1 1 1 Add inverter to NAND.
Note: Parallel structure on top, serial on bottom.
3
Basic Logic Gates Symbols
XOR: truth table A^B
A B 0 0 0 0 1 1 1 0 1 1 1 0 XOR B A
+
-
-
A^B= A & B + A&B
XOR
+
B
AO
DeMorgan's Law
DeMorgan's Law
l NOT
l Converting
(A and B) = NOT (A) OR NOT (B)
l NOT (A OR B) = NOT(A) AND NOT (B)
AND to OR (with some help from
NOT)
l Consider the following gate:
To convert AND to OR (or vice versa),
invert inputs and output.
A
BB
NOT(A NOT
AND B) (A)
NOT
(B)
NOT(A)
OR
NOT(B)
0
0
1
1
1
1
0
1
1
1
0
1
A B
1
0
1
0
1
1
0 0 A B
1 1 1
1
0
0
0
0
0 1 1 1 0 0 1 1 0 A ⋅B
1 A ⋅B
0 0 0 1 1 0 1 0 0 1 Same as A OR B 4
Building Functions from Logic
Gates
More than 2 Inputs?
l AND/OR
l 
l 
l 
can take any number of inputs.
l Combinational
AND = 1 if all inputs are 1.
OR = 1 if any input is 1.
Similar for NAND/NOR.
Logic Circuit
l  output
depends only on the current inputs
l  stateless
l Can
implement with multiple two-input gates,
or with single CMOS circuit.
l Sequential
Logic Circuit
l  output
depends on the sequence of inputs (past and
present)
l  stores information (state) from past inputs
l We'll
first look at some useful combinational
circuits,
then show how to use sequential circuits to store
information.
Half adder
Full Adder
A half adder is used to add just two bits.
l  The result consists of two bits: a sum (the
right bit) and a carry out (the left bit)
S
l  Here is the circuit and its block symbol
l Add
l 
X
0
0
1
1
Y
+0
+1
+0
+1
CS
= 00
= 01
= 01
= 10
C
two bits and carry-in,
produce one-bit sum and carry-out.
A B Cin S Cout
0 0 0 0 0 0 0 1 1 0 0 1 0 1 0 0 1 1 0 1 1 0 0 1 0 1 0 1 0 1 1 1 0 0 1 1 1 1 1 1 5
Four-bit Adder (carry-ripple
adder)
Carry Save adder
l Compute
l 101
l 101
l -----l 000
l Then
Disadvantage: Delay through N-1 Stages
Sum and Carry independently
101
101
---------101-
add S + C
l 
l 
l 
Carry Save Adder Design
Carry Save adder
l Delay
reduced compared to Carry ripple adder
l Add 3 Numbers and Produce two numbers S
and C
X:
10011
X:
10011
l Final result is S + shifted carry
Y: + 1 1 0 0 1
Y: + 1 1 0 0 1
Z:
C:
+ 01011
11011
X:
Y:
Z:
10011
+ 11001
+ 01011
S:
00001
C:
Sum:
11011
110111
000
101---------------1010
Z:
S:
+ 01011
00001
X
Y
Z
Xn-1
Yn-1
Zn-1
FA
CSA
n+1
n
C=carry
Xn-2
Yn-2
Z n-2
FA
Cn
Sn-1
Cn-1
Sn-2
X0
Y0
Z0
FA
C1
S0
C =0
0
S=sum
+
N Bit Carry Save Adder Block
6
Decoder
l n
inputs,
l  exactly
2n
Decoder
outputs
one output is 1 for each possible input
pattern
l n
inputs, 2n outputs
l  exactly
one output is 1 for each
0
possible input
pattern
1
2-bit
decoder
2-bit
decoder
2
3
Selecting Memory
Multiplexer (MUX)
A0
.
.
A7
l n-bit
selector and 2n inputs, one output
l  output
256 x 8
RAM
256 x 8
RAM
equals one of the inputs, depending on
selector
256 x 8
RAM
256 x 8
RAM
A8
A9
2 to 4
decoder
4-to-1 MUX
7
Circuit Design
l 
1. 
2. 
3. 
4. 
Converting Truth Table to
Boolean Expression
Designing circuits is a process…
Have a good idea. What kind of circuit might be useful?
Derive a truth table for this circuit.
Derive a Boolean expression for the truth table.
Build a circuit given the Boolean expression
l 
Building the circuit involves mapping the Boolean
expression to actual gates. This part is easy.
l 
Deriving the Boolean expression is easy. Deriving a good
one is tricky.
Converting Truth Table to
Boolean Expression
l Given
a circuit, isolate that rows in which the output of the circuit should
be true.
l A product term that contains exactly one instance of every variable is
called a minterm.
l Given
a circuit, isolate that rows in which the
output of the circuit should be true.
Converting Truth Table to
Boolean Expression
l Given
the expressions for each row, build a
larger Boolean expression for the entire table.
l  This
is a sum-of-products (SOP) form.
8
Converting Truth Table to
Boolean Expression
First Approach: Algebraic
l Simply
l Finally
use the rules of Boolean logic
build the circuit.
l  Problem:
l  Solution:
SOP forms are often not minimal.
Make it minimal. We’ll go over two ways.
The Result
Karnaugh Maps or K-Maps
l K-maps
are a graphical technique to view
minterms and how they relate.
l The
“map” is a diagram made up of squares,
with each square representing a single
minterm.
l Minterms
resulting in a “1” are marked as “1”,
all others are marked “0”
9
2 Variable K-Map
2 Variable K-Map
2 Variable K-Map
Finding Commonality
0
1
0
1
10
Finding the “best” solution
Simplify Example
l Grouping
become simplified products.
l Both are “correct”. “A+B” is preferred.
Simplify Example
3 Variable K-Maps
C
B
l  Note
in higher maps, several variables occupy a given
axis
l  The
sequence of 1s and 0s follow a Gray Code
Sequence.
11
3 Variable K-Maps
3 Variable K-Maps
C
B
3 Variable K-Maps
3 Variable K-Maps
12
Back to our earlier
example…..
3 Variable K-Maps
l The
K-map and the algebraic produce the
same result.
D
Up… up… and let’s
keep going
Few more
examples
D
B
A
B
A
C
C
13
Few more examplesD
B
A
C
14