Transcript Chapter # 1: digital circuits
Chapter 1: Digital Systems and Binary Numbers
• A digital system is a system that manipulates discrete elements of information represented internally in binary form.
• Digital computers – general purposes – many scientific, industrial and commercial applications • Digital systems – telephone switching exchanges – digital camera – electronic calculators, PDA's – digital TV
Signal
• An information variable represented by physical quantity • For digital systems, the variable takes on discrete values – Two level, or binary values are the most prevalent values • Binary values are represented abstractly by: – digits 0 and 1 – words (symbols) False (F) and True (T) – words (symbols) Low (L) and High (H) – and words On and Off.
• Binary values are represented by values or ranges of values of physical quantities
Binary Numbers
• Decimal number
Base or radix … a 5
a
4
a
3
a
2
a
1 .
a
1
a
2
a
3 …
a j
Decimal point Power
10 5
a
5 10 4
a
4 10 3
a
3 10 2
a
2 10 1
a
1 10 0
a
0 10 1
a
1 10 2
a
2 10 3
a
3
Example:
7,329 3 2 1 0 • General form of base-r system
a n r a n
1
r n
1
a
2
r a r
1 1
a
0
a
1
r
1
a
2
r
2 Coefficient:
a j
= 0 to
r
1
a
m
r
m
Binary Numbers Example: Base-2 number
(11010.11) 2 (26.75) 10 4 1 2 0 2 2 1 2 1
Example: Base-5 number
(4021.2) 5 4 5 3 0 5 2 2 5 1 1 5 0
Example: Base-8 number
(127.4) 8 1 8 3 2 8 2 1 8 1 7 8 0
Example: Base-16 number
(B65 F) 16 3 0 1 1 (511.5) 10 1 (87.5) 10 2 2 1 0 (46, 687) 10
Binary Numbers Example: Base-2 number
(110101) 2 (53) 10
Special Powers of 2
2
10
(1024) is Kilo, denoted "K"
2
20
(1,048,576) is Mega, denoted "M"
2
30
(1,073, 741,824)is Giga, denoted "G"
Powers of two Table 1.1
Arithmetic operation
Arithmetic operations with numbers in base
r
numbers.
follow the same rules as decimal
Binary Arithmetic
• Single Bit Addition with Carry • Multiple Bit Addition • Single Bit Subtraction with Borrow • Multiple Bit Subtraction • Multiplication • BCD Addition
Binary Arithmetic
•
Addition
•
Subtraction Augend: 101101 Minuend: 101101 Addend: +100111 Subtrahend:
100111 The binary multiplication table is simple: Sum: 1010100 Difference: 000110 0
0 = 0 | 1
0 = 0 | 0
1 = 0 | 1
1 = 1
•
Multiplication Multiplicand Multiplier Partial Products Product 1011
101 1011 0000 - 1011 - - 110111
Number-Base Conversions
Name Binary Octal Radix 2 8 Digits 0,1 0,1,2,3,4,5,6,7 Decimal 10 0,1,2,3,4,5,6,7,8,9 Hexadecimal 16 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F The six letters (in addition to the 10 integers) in hexadecimal represent: 10, 11, 12, 13, 14, and 15, respectively.
Number-Base Conversions
Example1.1
Convert decimal 41 to binary. The process is continued until the integer quotient becomes 0.
Number-Base Conversions
The arithmetic process can be manipulated more conveniently as follows:
Number-Base Conversions
Example 1.2
Convert decimal 153 to octal. The required base
r
is 8.
Example1.3
Convert (0.6875) 10 to binary.
The process is continued until the fraction becomes 0 or until the number of digits has sufficient accuracy.
Number-Base Conversions
Example1.3
To convert a decimal fraction to a number expressed in base
r
, a similar procedure is used. However, multiplication is by
r
instead of 2, and the coefficients found from the integers may range in value from 0 to
r
1 instead of 0 and 1.
Number-Base Conversions
Example1.4
Convert (0.513) 10 to octal.
From Examples 1.1 and 1.3:
From Examples 1.2 and 1.4: (41.6875) 10 = (101001.1011) 2 (153.513) 10 = (231.406517) 8
Octal and Hexadecimal Numbers
Numbers with different bases: Table 1.2
.
Octal and Hexadecimal Numbers
Conversion from binary to octal can be done by positioning the binary number into groups of three digits each, starting from the binary point and proceeding to the left and to the right. (10 110 001 101 011 ． 111 100 000 110) 2 = (26153.7406) 8 2 6 1 5 3 7 4 0 6 Conversion from binary to hexadecimal is similar, except that the binary number is divided into groups of four digits: Conversion from octal or hexadecimal to binary is done by reversing the preceding procedure.
Complements
There are two types of complements for each base-
r
diminished radix complement.
system: the radix complement and the
r
's complement and the second as the (
r
1)'s complement.
■
Diminished Radix Complement Example:
For binary numbers,
r
= 2 and
r
– 1 = 1, so the 1's complement of
N
is (2
n
1) –
N
.
Example:
Complements
■
Radix Complement
The
r
's complement of an
n
-digit number
N
and as 0 for complement is obtained by adding 1 to the ( 1) –
N
] + 1.
N
= 0. Comparing with the (
r
in base r is defined as
r
r n
– 1) 's complement, since
N r n
for –
N N
≠ 0 1) 's complement, we note that the
r
's = [(
r n
Example: Base-10
The 10's complement of 012398 is 987602 The 10's complement of 246700 is 753300
Example: Base-2
The 2's complement of 1101100 is 0010100 The 2's complement of 0110111 is 1001001
Complements
■
Subtraction with Complements
The subtraction of two
n
-digit unsigned numbers
M
–
N
in base
r
can be done as follows:
Complements Example 1.5
Using 10's complement, subtract 72532 – 3250.
Example 1.6
Using 10's complement, subtract 3250 – 72532 There is no end carry.
Therefore, the answer is – (10's complement of 30718) = 69282.
Complements Example 1.7
Given the two binary numbers
X
X – Y and (b)
Y
X
= 1010100 and by using 2's complement.
Y
= 1000011, perform the subtraction (a) There is no end carry. Therefore, the answer is
Y
–
X
= (2's complement of 1101111) = 0010001.
Complements
Subtraction of unsigned numbers can also be done by means of the (
r
complement. Remember that the (
r
1)'s 1) 's complement is one less then the
r
's complement.
Example 1.8
Repeat Example 1.7, but this time using 1's complement.
There is no end carry, Therefore, the answer is
Y
–
X
= (1's complement of 1101110) = 0010001.
Signed Binary Numbers
To represent negative integers, we need a notation for negative values. It is customary to represent the sign with a bit placed in the leftmost position of the number. The convention is to make the sign bit 0 for positive and 1 for negative.
Example:
Table 1.3
lists all possible four-bit signed binary numbers in the three representations.
Signed Binary Numbers
Signed Binary Numbers
■
Arithmetic Addition
The addition of two numbers in the signed-magnitude system follows the rules of ordinary arithmetic. If the signs are the same, we add the two magnitudes and give the sum the common sign. If the signs are different, we subtract the smaller magnitude from the larger and give the difference the sign of the larger magnitude.
The addition of two signed binary numbers with negative numbers represented in signed-2's-complement form is obtained from the addition of the two numbers, including their sign bits. A carry out of the sign-bit position is discarded.
Example:
Binary Codes
■
BCD Code
A number with
k
decimal digits will require 4
k
bits in BCD. Decimal 396 is represented in BCD with 12bits as 0011 1001 0110, with each group of 4 bits representing one decimal digit. A decimal number in BCD is the same as its equivalent binary number only when the number is between 0 and 9. A BCD number greater than 10 looks different from its equivalent binary number, even though both contain 1's and 0's. Moreover, the binary combinations 1010 through 1111 are not used and have no meaning in BCD.
Signed Binary Numbers
■
Arithmetic Subtraction
In 2 ’ s-complement form:
1.
2.
Take the 2 ’ s complement of the subtrahend (including the sign bit) and add it to the minuend (including sign bit). A carry out of sign-bit position is discarded.
(
A
) (
A
)
B B
) )
A A
) )
B B
) )
Example:
( 6) ( 13) (11111010 11110011) (11111010 + 00001101) 00000111 (+ 7)
Binary Codes
Example:
Consider decimal 185 and its corresponding value in BCD and binary: ■
BCD Addition
Binary Codes
Example:
Consider the addition of 184 + 576 = 760 in BCD: ■
Decimal Arithmetic
Binary Codes
■
Other Decimal Codes
Binary Codes
■
Gray Code
Binary Codes
■
ASCII Character Code
Binary Codes
■
ASCII Character Code
ASCII Character Codes
• American Standard Code for Information Interchange
(Refer to Table 1.7)
• A popular code used to represent information sent as character-based data.
• It uses 7-bits to represent: – 94 Graphic printing characters.
– 34 Non-printing characters • Some non-printing characters are used for text format (e.g. BS = Backspace, CR = carriage return) • Other non-printing characters are used for record marking and flow control (e.g. STX and ETX start and end text areas).
ASCII Properties ASCII has some interesting properties:
Digits 0 to 9 span Hexadecimal values 30 16 Upper case A - Z span 41 16 to 5A 16 .
to 39 16 .
Lower case a - z span 61 16
•
to 7A 16 .
Lower to upper case translation (and vice versa) occurs by flipping bit 6.
Delete (DEL) is all bits set, a carryover from when punched paper tape was used to store messages. Punching all holes in a row erased a mistake!
Binary Codes
■
Error-Detecting Code
To detect errors in data communication and processing, an eighth bit is sometimes added to the ASCII character to indicate its parity. A
parity
bit is an extra bit included with a message to make the total number of 1's either even or odd.
Example:
Consider the following two characters and their even and odd parity:
Binary Codes
■
Error-Detecting Code
• • • •
Redundancy (e.g. extra information), in the form of extra bits, can be incorporated into binary code words to detect and correct errors. A simple form of redundancy is parity, an extra bit appended onto the code word to make the number of 1 ’ s odd or even. Parity can detect all single-bit errors and some multiple-bit errors.
A code word has even parity if the number of 1 ’ s in the code word is even.
A code word has odd parity if the number of 1 ’ s in the code word is odd.
Binary Storage and Registers
■
Registers
A
binary cell
is a device that possesses two stable states and is capable of storing one of the two states.
A
register
is a group of binary cells. A register with
n
cells can store any discrete quantity of information that contains
n
bits.
n cells 2
n
possible states
• A binary cell – two stable state – store one bit of information – examples: flip-flop circuits, ferrite cores, capacitor • A register – a group of binary cells – AX in x86 CPU • Register Transfer – a transfer of the information stored in one register to another – one of the major operations in digital system – an example
Transfer of information
•
The other major component of a digital system
–
circuit elements to manipulate individual bits of information
Binary Logic ■
Definition of Binary Logic
Binary logic consists of binary variables and a set of logical operations. The variables are designated by letters of the alphabet, such as A, B, C, x, y, z, etc, with each variable having two and only two distinct possible values: 1 and 0, There are three basic logical operations: AND, OR, and NOT.
Binary Logic ■
The truth tables for AND, OR, and NOT are given in Table 1.8
.
Binary Logic ■
Logic gates
Example of binary signals
Binary Logic ■
Logic gates
Graphic Symbols and Input-Output Signals for Logic gates: Fig. 1.4 Symbols for digital logic circuits Fig. 1.5 Input-Output signals for gates
Binary Logic ■
Logic gates
Graphic Symbols and Input-Output Signals for Logic gates: Fig. 1.6 Gates with multiple inputs
Number-Base Conversions
Complements
Complements
Signal Example – Physical Quantity: Voltage
HIGH LOW OUTPUT 5.0
4.0
3.0
2.0
1.0
0.0
Volts INPUT HIGH
Threshold Region
LOW
Signal Examples Over Time
Time Analog Digital
Asynchronous Synchronous Continuous in value & time Discrete in value & continuous in time Discrete in value & time
A Digital Computer Example
Memory CPU Control unit
Inputs: Keyboard, mouse, modem, microphone
Input/Output
Synchronous or Asynchronous?
Datapath
Outputs: CRT, LCD, modem, speakers
Binary Codes for Decimal Digits
There are over 8,000 ways that you can chose 10 elements from the 16 binary numbers of 4 bits. A few are useful:
Decimal 6 7 8 9 0 1 2 3 4 5 8,4,2,1 0000 0001 0010 0011 0100 0101 0110 0111 1000 1001 Excess3 0011 0100 0101 0110 0111 1000 1001 1010 1011 1100 8,4, -2,-1 0000 0111 0110 0101 0100 1011 1010 1001 1000 1111 Gray 0000 0100 0101 0111 0110 0010 0011 0001 1001 1000
UNICODE
• UNICODE extends ASCII to 65,536 universal characters codes
– For encoding characters in world languages – Available in many modern applications – 2 byte (16-bit) code words – See Reading Supplement – Unicode on the Companion Website http://www.prenhall.com/mano
•
Negative Numbers Complements
–
1's complements
– – – ( 2
n N
2's complements
2
N
Subtraction = addition with the 2's complement Signed binary numbers
»
signed-magnitude, signed 1's complement, and signed 2's complement.
• • •
M - N M + the 2 ’ s complement of N
–
M + (2 n - N) = M - N + 2 n If M
≧
N
–
Produce an end carry, 2 n , which is discarded If M < N
–
We get 2 n - (N - M), which is the 2 ’ s complement of (N-M)
• • •
Binary Storage and Registers A binary cell
– – –
two stable state store one bit of information examples: flip-flop circuits, ferrite cores, capacitor A register
– –
a group of binary cells AX in x86 CPU Register Transfer
– – –
a transfer of the information stored in one register to another one of the major operations in digital system an example
Special Powers of 2
2
10
(1024) is Kilo, denoted "K"
2
20
(1,048,576) is Mega, denoted "M"
2
30
(1,073, 741,824)is Giga, denoted "G"
Converting Binary to Decimal
• To convert to decimal, use decimal arithmetic to form S respective power of 2).
(digit × • Example:Convert 11010 2 to N 10 :
Non-numeric Binary Codes
• Given
n
binary digits (called bits), a binary code is a mapping from a set of represented elements to a subset of the 2
n
binary numbers.
• Example: A binary code for the seven colors of the rainbow • Code 100 is not used
Color Red Orange Yellow Green Blue Indigo Violet Binary Number 000 001 010 011 101 110 111
Commonly Occurring Bases
Name Binary Octal Radix 2 8 Digits 0,1 0,1,2,3,4,5,6,7 Decimal 10 0,1,2,3,4,5,6,7,8,9 Hexadecimal 16 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F The six letters (in addition to the 10 integers) in hexadecimal represent:
Binary Numbers and Binary Coding
• Information Types – Numeric » Must represent range of data needed » Represent data such that simple, straightforward computation for common arithmetic operations » Tight relation to binary numbers – Non-numeric » Greater flexibility since arithmetic operations not applied.
» Not tied to binary numbers
Number of Elements Represented
• Given
n
digits in radix
r,
there are elements that can be represented.
r n
distinct • But, you can represent m elements, m <
r n
• Examples:
– You can represent 4 elements in radix
r
digits: (00, 01, 10, 11). = 2 with
n
= 2 – You can represent 4 elements in radix
r
digits: (0001, 0010, 0100, 1000).
= 2 with
n
= 4 – This second code is called a "one hot" code.
Binary Coded Decimal (BCD)
• The BCD code is the 8,4,2,1 code.
• This code is the simplest, most intuitive binary code for decimal digits and uses the same powers of 2 as a binary number, but only encodes the first ten values from 0 to 9.
• Example: 1001 (9) = 1000 (8) + 0001 (1) • How many “ invalid ” code words are there?
• What are the “ invalid ” code words?
Excess 3 Code and 8, 4, – 2, – 1 Code
Decimal 0 1 2 3 4 5 6 7 8 9 Excess 3 0011 0100 0101 0110 0111 1000 1001 1010 1011 1100 8, 4, – 2, – 1 0000 0111 0110 0101 0100 1011 1010 1001 1000 1111 • What interesting property is common to these two codes?
Gray Code Decimal 8,4,2,1 Gray 0 1 2 3 4 5 6 7 8 9 0000 0001 0010 0011 0100 0101 0110 0111 1000 1001 0000 0100 0101 0111 0110 0010 0011 0001 1001 1000
• What special property does the Gray code have in relation to adjacent decimal digits?
Gray Code
(Continued) • Does this special Gray code property have any value?
• An Example: Optical Shaft Encoder 111 000 100 000 110 B 0 B 1 B 2 001 101 001 111 G 0 G 1 G 2 101 010 011 100 011 (a) Binary Code for Positions 0 through 7 110 010 (b) Gray Code for Positions 0 through 7
Warning: Conversion or Coding?
• Do NOT mix up conversion of a decimal number to a binary number with coding a decimal number with a BINARY CODE.
• 13
10
= 1101
2
• 13
(This is conversion) 0001|0011 (This is coding)
Single Bit Binary Addition with Carry Given two binary digits (X,Y), a carry in (Z) we get the following sum (S) and carry (C): Carry in (Z) of 0: Carry in (Z) of 1: Z X Z X 0 0 1 0 0 0 1 0 0 1 1 1 0 1 + Y + 0 + 1 + 0 + 1 C S 0 0 0 1 0 1 1 0 1 1 + Y + 0 + 1 + 0 + 1 C S 0 1 1 0 1 0 1 1
Multiple Bit Binary Addition
• Extending this to two multiple bit examples:
Carries 0 0 Augend 01100 10110 Addend +10001 +10111 Sum
• Note: The 0 is the default Carry-In to the least significant bit.
Binary Multiplication The binary multiplication table is simple: 0
0 = 0 | 1
0 = 0 | 0
1 = 0 | 1
1 = 1 Extending multiplication to multiple digits: Multiplicand Multiplier Partial Products Product 1011
101 1011 0000 - 1011 - - 110111
BCD Arithmetic
8 +5 Given a BCD code, we use binary arithmetic to add the digits: 1000 +0101 Eight Plus 5
13 1101 is 13 (> 9) Note that the result is MORE THAN 9, so must be represented by two digits!
To correct the digit, subtract 10 by adding 6 modulo 16.
8 1000 Eight +5 13 +0101 1101 +0110 carry = 1 0011 Plus 5 is 13 (> 9) so add 6 leaving 3 + cy 0001 | 0011 Final answer (two digits)
If the digit sum is > 9, add one to the next significant digit
BCD Addition Example
• Add 2905
BCD
to 1897
BCD
showing carries and digit corrections.
0001 1000 1001 0111 + 0010 1001 0000 0101 0
Error-Detection Codes
• Redundancy (e.g. extra information), in the form of extra bits, can be incorporated into binary code words to detect and correct errors. • A simple form of redundancy is parity, an extra bit appended onto the code word to make the number of 1 ’ s odd or even. Parity can detect all single-bit errors and some multiple-bit errors.
• A code word has even parity if the number of 1 ’ s in the code word is even.
• A code word has odd parity if the number of 1 ’ s in the code word is odd.
4-Bit Parity Code Example
• Fill in the even and odd parity bits:
Even Parity Odd Parity Message - Parity Message - Parity 000 001 010 011 000 001 010 011 100 101 110 111 100 101 110 111 -
• The codeword "1111" has even parity and the codeword "1110" has odd parity. Both can be used to represent 3 bit data.