Homework
• Reading
– Tokheim, Chapter 3, 4, and 6.1 - 6.3
– Logisim Website
• Machine Projects
– Continue on mp3
• Labs
– Continue in labs with your assigned section
1
Combining Basic Logic Gates
•
•
•
•
•
•
•
Decoders
Encoders
Selectors - Multiplexers
ALUs
Control Units
Buses
Simple computers
2
Binary Decoder
• Logic with n input lines and 2n output lines
• Only one output is a 1 for any given input
n
inputs
Binary
Decoder
2n outputs
3
Building a Binary Decoder
• Start with a 2-bit decoder:
X1
X0
AND
Y0
AND
Y1
AND
Y2
AND
Y3
Enable
4
Then Add Two to Make Three...
X0
X1
X2
2-bit
Decoder
Y0
Y1
Y2
Y3
NOT
2-bit
Decoder
Y4
Y5
Y6
Y7
5
Developing an Encoder
• If we can decode, then we need to encode
• Encode from 1 out of n into a binary weighted form
• A keyboard encoder does this
X0
X1
X2
X3
X4
...
X7
NC
OR
Y0
OR
Y1
OR
Y2
6
Next Comes a Selector
• Like a switch; also called a multiplexer or MUX
S1
S2
a
b
c
MUX
or
Selector
y
d
• Again, build it up from simple basic logic gates
7
A 1-bit Selector
S1
S2
Decoder
a
b
c
d
AND
AND
O
R
AND
y
AND
8
A 4-bit Selector
S
S0 S1
A
a0
b0
c0
d0
B
Y0
MUX
D
S
A
a3
b3
c3
d3
Y
C
B
Y3
C
D
2
4
4
4
MUX
4
Y
4
9
The ALU Is Next
• Logical and arithmetic operations
• Variations in
– Base
• Binary
• Decimal
• BCD
– Implementation
• Serial
• Parallel
• Pipelined
A
Control Signals
B
ALU
CCs
f(A,B)
10
Simple Example - Binary Adder
• Develop a half-adder (HA)
• Use two HA’s to build a full-adder (FA)
11
The Half-Adder
Sum  (a  b)  (a  b )  (a  b )  (a  b)
Carry  a  b
a
b
a
b
HA
Sum
Cout
XOR
Sum
AND
Cout
a
b
0
0
1
1
0
1
0
1
Sum Carry
0
1
1
0
0
0
0
1
12
From HA to FA
cin
a
b
cin
a
b
Sum
Full
Adder
cout
a
b
cin Sum
0
0
0
0
1
1
1
1
0
0
1
1
0
0
1
1
0
1
0
1
0
1
0
1
0
0
0
1
0
1
1
1
Sum
Half
Adder
Half
Adder
0
1
1
0
1
0
0
1
Cout
OR
Cout
13
Using Full Adders for Addition
0
a0
b0
C0
a1
b1
00
+ 11
(0) 1 1
Full
Adder
Full
Adder
00
+01
(0) 0 1
Sum0
C0
Sum1
C1
11
11
+01
+1 1
(1) 0 0 (1) 1 0
Note: The carry flag value after the addition
represents the N+1 bit value in the result
a b
Cin
Sum0
C0
LSB
0 0
0 1
1 0
1 1
0
0
0
0
0
1
1
0
0
0
0
1
MSBs
0 0
0 0
0 1
0 1
1 0
1 0
1 1
1 1
Cn-1 Sumn Cn
0
0
0
1
1
0
0
1
0
1
0
1
0
1
0
1
0
1
0
0
1
1
1
1
14
Using Full Adders for Subtraction
Difference = a
=a
=a
=a
1
a0
b0#
C0
a1
b1#
(1) 0 0
- 11
01
Full
Adder
Full
Adder
00
+01
(0) 0 1
- b
+ (-b)
+ (~b + 1)
+ ~b + 1
Sum0
C0
Sum1
C1
(0)1 1
-01
10
11
+1 1
(1) 1 0
Note: The carry flag value after the ~ and addition
is the opposite of the subtract borrow condition
a b#
C0
Sum
C1
LSB
0 0
0 1
1 0
1 1
1
1
1
1
1
0
0
1
0
1
1
1
MSBs
0 0
0 0
0 1
0 1
1 0
1 0
1 1
1 1
Cn-1
0
1
0
1
0
1
0
1
0
1
1
0
1
0
0
1
Cn
0
0
0
1
0
1
1
1
15
Configurable Add/Subtract ALU
Subtract/Add#
From
Instruction
Decoding
b0
Logic
.
.
.
bn-1
a0
Full
Adder
XOR
.
.
.
XOR
C0
Cn-2
an-1
.
.
.
Full
Adder
EFlags
Register
Sum0
.
.
.
NOR
Sumn-1
Cn-1
Zero
Flag
Sign
Flag
Overflow Flag
= Cn-2*Cn-1#
Carry
Flag
16