Logic Design

Department of Communication Engineering, NCTU 1
Unit 1 Introduction
1. Digital Systems
2. Number System and Conversion
3. Binary Arithmetic

1.1 Digital Systems

Department of Electrical Engineering, NCTU 3
Logic Design Unit 1 Introduction Sau-Hsuan Wu
 Digital Systems:
 A system that can perform data computation, storage and data
input and output in digital formats
 Why digital systems instead of analog systems?
 Easy and reliable control of data precision (number of bits)
 Easy and reliable circuit implementations (on/off switch)
 Applications of Digital Systems:
 Computation, data processing, signal acquisition and processing,
automatic control, communications and measurements, etc
Input/Output
Interface
Arithmetic/Logic Unit
Data Storage
Control Unit

 Design of digital systems includes 3 phases:
 System design: design functions to meet specific applications
 Logic design : implement functions in logic gates
 Circuit design : design and implement logic gates
 System design involves functional definitions and system
partitions. The aspect of system design varies a lot
 Logic design means coming up with methods to
implement the defined functions, using basic logic units
 Circuit design here refers to the implementations of basic
logic units such as AND, OR, Flip-Flops, etc. with
resistors, diodes and transistors
 We will not cover circuit design in this course

 A simple digital system : Y = max { C, A+B}
 If C > A+B
Y=C
else
Y=A+B
 A simple logic implementation of the above system
Yes, C1=1
Start
A+B > C
Save Y
No, C1=0
C2
A B C
Adder
Comp
Mux
Y
C1
C2

 Logic circuits can be categorized into two classes:
 Combinational logic
 Sequential Logic
 Combinational logic’s outputs depend only on the
present inputs:
 E.g. Adder, multiplier, comparator, multiplexer
 Sequential logic’s output depend on both the present
and the past outputs. In general, a sequential circuit is
composed of a combinational circuit with added
memory elements.
 E.g. C = A+B
E = C+D
F = CD

1.2 Number Systems and
Conversion

 Base R representation of a rational number
 Any positive integer R > 1 can be chosen as the base of a
number
 E.g. : converting an octal number to decimal
3 82 + 2 81 + 1 80 + 1 8-1
= 192 + 16 + 1 + 0.125
= 208.125
4 3 2 1 0 1 2 3
4 3 2 1 0
4 3 2 1 0
1 2 3
1 2 3
( . )
0 1i
N a a a a a a a a
a R a R a R a R a R
a R a R a R
a R
  
  
  

          
      
  
 


其中where

 For bases greater than 10, more than 10 symbols are
needed to represent the digits
 Letters are usually used to represent digits greater than 9
 E.g., for a hexadecimal number, we have digits
0,1,2,3,…9,A,B,C,D,E,F
 E.g. : converting a hexadecimal to decimal
A 162 + 2 161 + B 160 + F 16-1
= 10 256 + 2 16 + 11 + 15/16
= 2603.9375

 Conversion of a decimal number to base R
10 2
For integer,
53 110101
53 2 26 1
26 2 13 0
13 2 6 1
6 2 3 0
3 2 1 1

 
 
 
 
 





使用除法
10 2
For fraction,
.625 .101
.625 2 1.25 1
.25 2 0.5 0
.5 2 1.0 1

 
 
 
使用乘法use division use multiplication

 Conversion between two bases R1 & R2 other than
decimal
 Convert from base R1 to decimal first, then convert the
decimal number to base R2
4 10 7231.3 45.75 63.5151
45 7 6 3
.75 7 5.25 5
.25 7 1.75 1
.75 7 5.25 5
 
 
 
 
 



1.3 Binary Arithmetic

 Arithmetic operations in digital systems are usually done
in binary because design of logic circuits to perform
binary arithmetic is much simpler than decimal arithmetic
 We only need switches: On  1, OFF  0
 Binary addition
0 0 0
0 1 1
1 0 1
1 1 0 carry 1 to the next column
 
 
 
 
10
10
10
Ex.
1111 carries
13 = 1101
11 = 1011
24 11000


 Binary subtraction
0 0 0
0 1 1 borrow 1 from the next column
1 0 1
1 1 0
 
 
 
 
Ex.
1 borrow
11101
10011
1010



 Binary multiplication
0 0 0
0 1 0
1 0 0
1 1 1
 
 
 
 
Ex.
1101
1011
1101
1101
0000
1101
10001111


 Binary division is similar to decimal division
Ex.
1101
1011 10010001
1011
1110
1011
1101
1011
10

1.4 Representation of Negative
Numbers

 For signed integer (both positive and negative integers), the first bit
in a word is used as a sign bit, with 0 for plus and 1 for minus
 There are 3 types of representations of negative numbers:
sign and magnitude, 2’s complement and 1’s complement

 Sign and magnitude：
For an n-bit word, the first bit, the most significant bit (MSB), is the
sign and the remaining n-1 bits represent the magnitude
Ex. 0011=+3, 1011=-3
 2’s complement：
 A positive number is represented by a 0 followed by the magnitude
 A negative number is represented by its 2’s complement N*：
N* = 2n –N
 1’s complement：
 A negative number, -N, is represented by its 1’s complement of N
defined as
 Notice
 Therefore, if
 N* = 2n –N
(2 1)n
N N  
2 1 111 11 (n bits)n
  
00110001100111  NN
00110011*
 NN

 Addition of 2’s complement numbers:
 E.g. 4-bit word, n = 4
5 –6 = –1
 5 + (2n –6) = 2n –1  2’s complement of 1
 So the addition is carried out just as if the numbers were positive
 The addition of two 4-bit words becomes a 5-bit word in general
 Do sign extension before addition
E.g. 5 + 6 = 11
By sign extension 5 = 00101, 6 = 00110
 5+6 = 01011
E.g. –5 –6 = –11
By sign extension, –5  24 –5 = 11010+1 = 11011
–6  24 –6 = 11001+1 = 11010
–11  25 –6 –5 = 10101  2’s complement of 11

 Addition of 1’s complement numbers:
 Similar to 2’s complement except that instead of discarding the
last carry, it is added to the n-bit sum in the position furthest to
the right, least significant bit (LSB).
This is referred to as an end-around carry
 –A+B (B > A)  (2n 1 A) + B
= B A + (2n 1)  removing (2n 1)
 –A –B (A+B < 2n-1) 
(2n 1 A) + (2n 1 B) = [2n 1 (A+B)] + (2n 1)
 Similarly, do sign extension to protect from overflow
( 5) ( 6)
1010
0110
10000
1
0001
 
End-around carry

Logic Design

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