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Lect # 1 Binary Systems

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Lect # 1 Binary Systems

Digital vs. Analog

- Analog – a continuous range of values
- Digital – a discrete set of values
- Like money
- Can’t get smaller than cents
- Typically also has maximum value

Benefits of using digital

Digital signal

Analog signal

- Cheap electronic circuits
- Easier to calibrate and adjust
- Resistance to noise: Clearer picture and sound

DIGITAL SYSTEMS

- discrete information processing system
- Discrete elements of info. are represented in a digital system by physical quantities called signals.
- Electrical signals such as voltages and currents are the most common.
- Digital systems that are constrained to take discrete values are further constrained to take binary values.

DIGITAL SYSTEMS

Processor

or

Arithmetic Unit

Control

Unit

Block Diagram of a Digital Computer

Storage

or

Memory Unit

Input

Devices

and Control

Output

Devices

and Control

The memory unit stores programs as well as input, output, and intermediate data.

The processor unit performs arithmetic and other data-processing tasks as specified by a program.

The control unit supervises the flow of information between the various units. It retrieves the instructions 1 by 1, from the program which is stored in memory. For each instruction, it informs the processor to execute the operation specified by the instruction.

Both program and data are stored in memory.

The control unit supervises the program instructions, and the processor manipulates the data as specified by the program.

The program and data prepared by the user are transferred into the memory unit by means of an input device. An output device receives the result of the computations and the printed results are presented to the user.

The input and output devices are special digital systems driven by electromechanical parts and controlled by electronic digital circuits.

DIGITAL SYSTEMS

- An electronic calculator is a digital system similar to a digital computer, why?
ANS. The input device being a keyboard; the output device a numerical display;

- *Instructions are entered by means of the function keys such as + and -;
- Results are displayed directly in numeric form
- A digital computer, however, is a more powerful device than a calculator, how?
ANS. A digital computer can accommodate many other input and output devices;

- it can perform not only arithmetic computations but logical operations as well

- it can be programmed to make decisions based on internal
- and external conditions

- A processor when combined with the control unit, forms a component referred to as a central processing unit or CPU
- A CPU enclosed in a small IC package is called a microprocessor
- A CPU combined with memory and interface control to form a small-size computer is called a microcomputer

Binary System

- Discrete elements of information are represented with bits called binary codes.
- Example: (09)10 = (1001)2
- (15)10 = (1111)2
- Question: Why are commercial products made with digital circuits as opposed to analog?
- Most digital devices are programmable: By changing the program in the device, the same underlying hardware can be used for many different applications.

MSD

LSD

Binary Code

Review the decimal number system.

Base (Radix) is 10 - symbols (0,1, . . 9) Digits

For Numbers > 9, add more significant digits in position to the left, e.g. 19>9.

Each position carries a weight.

Weights:

- If we were to write 1936.25 using a power series expansion and base 10 arithmetic:

Octal/Hex number systems

The octal number system [from Greek: OKTW].

- Its base is 8 eight digits 0, 1, 2, 3, 4, 5, 6, 7

- (236.4)8 = (158.5)10

The hexadecimal number system.

- Its base is 16 first 10 digits are borrowed from the
- decimal system and the letters A, B, C, D, E, F are used for
- the digits 10, 11, 12, 13, 14, 15

- (D63FA)16 = (877562)10

Conversion from Decimal to Binary

- Find the binary equivalent of 37.

1

= 18 + 0.5

LSB

0

= 9 + 0

1

= 4 + 0.5

0

= 2 + 0

0

= 1 + 0

MSB

1

= 0 + 0.5

?

ANS:

Conversion from Binary to Decimal

Conversion from Decimal to Octal

Conversion from decimal to octal:

The decimal number is first divided by 8. The remainder is

the LSB. The quotient is then divide by 8 and the remainder is

the next significant bit and so on.

- Convert 1122 to octal.

LSB

MSB

Conversion from Decimal to Octal

- Convert (0.3152)10 to octal (give answer to 4 digits).

0.3152 x 8 = 2 + 0.5216 a-1 = 2

0.5216 x 8 = 4 + 0.1728 a-2 = 4

0.1728 x 8 = 1 + 0.3824 a-3 = 1

0.3824 x 8 = 3 + 0.0592 a-4 = 3

(1122.3152)10 = ( ? )8

00

01

02

03

04

05

06

07

10

11

12

13

14

15

16

17

Octal

Table 1-2

page: 8

Conversion using Table

Conversion from and to binary, octal, and hexadecimal

plays and important part in digital computers.

since

and

each octal digit corresponds to 3 binary digits

and each hexa digit corresponds to 4 binary digits.

- (010 111 100 . 001 011 000)2 = (274.130)8
- (0110 1111 1101 . 0001 0011 0100)2 = (6FD.134)16

from

table

Addition of Binary Numbers

Subtraction of Binary Numbers

Sign Magnitude Representation

Complements

Complements: They are used in digital computers for subtraction operation and for logic manipulation.

Decimal Numbers

10’s complement and 9’s complement

Binary Numbers

2’s complement and 1’s complement

9’s complement of N = (10n-1) – N (N is a decimal #)

10’s complement of N = 10n– N (N is a decimal #)

1’s complement of N = (2n-1) – N (N is a binary #)

1’s complement can be formed by changing 1’s to 0’s and 0’s to 1’s

2’s complement of a number is obtained by leaving all least

significant 0’s and the first 1 unchanged, and replacing 1’s with

0’s and 0’s with 1 in all higher significant digits.

Complements

9’s complement of N = (10n-1) – N (N is a decimal #)

- The 9’s complement of 12345 = (105 – 1) – 12345 = 87654
- The 9’s complement of 012345 = (106 – 1) – 012345 = 987654

10’s complement of N = [(10n – 1) – N] + 1(N is a decimal #)

- The 10’s complement of 739821 = 106– 739821 = 260179
- The 10’s complement of 2500 = 104 – 2500 = 7500

Find the 9’s and 10’s-complement of 00000000

ANS: 99999999 and 00000000

1’s and 2’s Complements

1’s complement of N = (2n-1) – N (N is a binary #)

1’s complement can be formed by changing 1’s to 0’s and 0’s to 1’s

2’s complement of a number is obtained by leaving all least

significant 0’s and the first 1 unchanged, and replacing 1’s with

0’s and 0’s with 1 in all higher significant digits.

- The 1’s complement of 1101011 = 0010100
- The 2’s complement of 0110111 = 1001001

Find the 1’s and 2’s-complement of 10000000

Answer: 01111111 and 10000000

COMPLEMENTS

Comparison between 1’s and 2’s Complements

- 1’s complement
- easier to implement by digital components since the only thing that must be done is to change 0’s to 1’s and 1’s into 0’s.

- 2’s complement
- may be obtained in two ways:
- (1) By adding 1 to the least significant digit of the 1’s complement

COMPLEMENTS

Comparison between 1’s and 2’s Complements

- 2’s complement
(2) by leaving all leading 0’s in the least significant positions and the first 1 unchanged, and only then changing all 1’s into 0’s and all 0’s into 1’s.

- During subtraction of two numbers by complements, the 2’s complement is advantageous
- only one arithmetic addition operation is required

COMPLEMENTS

Comparison between 1’s and 2’s Complements

- 1’s complement
- requires two arithmetic additions when an end-around carry occurs.

- 1’s complement has the additional disadvantage of possessing two arithmetic zeros: one w/ all 0’s and one w/ all 1’s.

COMPLEMENTS

Comparison between 1’s and 2’s Complements

- Consider the subtraction of two equal binary numbers 1100-1100 = 0
- Using 1’s complement:
1100

+ 0011

+ 1111

- Using 1’s complement:

- complement again to obtain –0000
- Using 2’s complement:
- 1100
- + 0100
- + 0000

COMPLEMENTS

Comparison between 1’s and 2’s Complements

- While the 2’s complement has only one arithmetic zero, the 1’s complement zero can be positive or negative, which may complicate matters.

- 1’s complement is also useful in logical manipulations, since the change of 1’s to 0’s and vice versa is equivalent to a logical inversion operation.

Subtraction Using Complements

Subtraction with digital hardware using complements:

- Subtraction of two n-digit unsigned numbers M – N
- base r:
- Add M to the r’s complement of N: M + (rn – N)
- If MN, the sum will produce an end carry and is equal to rnthat can be discarded. The result is then M – N.
- 3. If MN, the sum will not produce an end carry and is equal to rn – (N – M)

Decimal Subtraction using complements

- Subtract 150 – 2100 using 10’s complement:

M = 150

10’s complement of N = + 7900

Sum = 8050

There’s no end carry negative

Answer: – (10’s complement of 8050) = – 1950

- Subtract 7188 – 3049 using 10’s complement:

M =7188

10’s complement of N = + 6951

Sum = 14139

Discard end carry 104 = – 10000

Answer = 4139

Binary Subtraction using complements

Binary subtraction is done using the same procedure.

- Subtract 1010100 – 1000011 using 2’s complement:

A = 1010100

2’s complement of B = + 0111101

Sum = 10010001

end carry

Discard end carry = – 10000000

Answer = 0010001

Subtract 1000011 – 1010100 using 2’s complement:

Answer = – 0010001

Binary Subtraction using complements

- Subtract 1010100 – 1000011 using 1’s complement:

A = 1010100

1’s complement of B = + 0111100

Sum = 10010000

End-around carry = + 1

Answer = 0010001

Subtract 1000011 – 1010100 using 1’s complement:

Answer = – 0010001

Arithmetic addition

Negative numbers must be initially in 2’s complement form and

if the obtained sum is negative, it is in 2’s complement form.

–611111010

+13 00001101

+7 00000111

+ 600000110

+13 00001101

+19 00010011

Add –6 and –13

Answer = 11101101

BINARY CODES

- Error-Detection Codes
- Binary information, be it pulse-modulated signals or digital computer input or output, may be transmitted through some form of communication medium such as wires or radio waves.
- Any external noise introduced into a physical communication medium changes bit values from 0 to 1 or vice versa.
- An error-detection code can be used to detect errors during transmission.
- The detected error cannot be corrected, but its presence is indicated.

Error-Detecting Code

- To detect the error in data communication, an eighth bit is added to 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
- The 8-bit characters included parity bits (with even parity) are transmitted to their destination. If the parity of received character is not even it means at least one bits has been changed.

BINARY CODES

- A parity bit is an extra bit included w/ a message to make the total number of 1’s either odd or even.
- During transfer of info. from one location to another, the parity bit is handled as follows: In the sending end, the message is applied to a “parity-generation” network where the required P bit is generated.
- The message, including the parity bit, is transferred to its destination.

(a) Message

P (odd)

(b) Message

P (even)

0000

1

0000

0

0001

0

0001

1

0010

0

0010

1

0011

1

0011

0

0100

0

0100

1

0101

1

0101

0

0110

1

0110

0

0111

0

0111

1

1000

0

1000

1

BINARY CODES

PARITY-BIT GENERATION

1001

1

1001

0

1010

1

1010

0

1011

0

1011

1

1100

1

1100

0

1101

0

1101

1

1110

0

1110

1

1111

1

1111

0

BINARY CODES

PARITY-BIT GENERATION

BINARY CODES

- In the receiving end, all incoming bits are applied to a “parity-check” network to check the proper parity adopted.
- An error is detected if the checked parity does not correspond to the adopted one.
- The parity method detects the presence of one, three, or any odd combination of errors.
- An even combination is undetectable.
- Consider odd parity for a bit data 1111.
- Received= 11111 (No error)
- Received= 11110 (1-bit error detected)
- Received= 11100 (2-bit error not detected)
- Received= 11000 (3-bit error detected)

BINARY CODES

- Reflected Code(also known as the Gray code)
Digital systems can be designed to process data in discrete form only. Many physical systems supply continuous output data. These data must be converted into digital or discrete form before they are applied to a digital system.

Advantage:

A number in the reflected code changes by only one bit as it proceeds from one number to the next.

Reflected Code

Decimal Equivalent

0000

0

0001

1

0011

2

0010

3

0110

4

0111

5

0101

6

0100

7

1100

8

BINARY CODES

1101

9

1111

10

1110

11

1010

12

1011

13

1001

14

1000

15

BINARY CODES

- Alphanumeric Codes
Many applications of digital computers require the handling of data that consist not only of numbers, but also of letters.

An alphanumeric code is a binary code of a group of elements consisting of the 10 decimal digits, the 26 letters of the alphabet, and a certain number of special symbols such as $.

BINARY CODES

- The total number of elements in an alphanumeric group is greater than 36. Therefore, it must be coded with a minimum of 6 bits (26 = 64, but 25 = 32 is insufficient).
- One possible arrangement of a 6-bit alphanumeric code is shown in the table under the name “internal code.” The need to represent more than 64 characters gave rise to 7- and 8- bit alphanumeric codes. One such code is known as ASCII. Another is known as EBCDIC.

Character

6-Bit internal code

7-Bit ASCII code

8-Bit EBCDIC code

12-Bit card code

A

010 001

100 0001

1100 0001

12,1

B

010 010

100 0010

1100 0010

12,2

C

010 011

100 0011

1100 0011

12,3

D

010 100

100 0100

1100 0100

12,4

E

010 101

100 0101

1100 0101

12,5

F

010 110

100 0110

1100 0110

12,6

G

010 111

100 0111

1100 0111

12,7

ALPHANUMERIC CHARACTER CODES

H

011 000

100 1000

1100 1000

12,8

I

011 001

100 1001

1100 1001

12,9

J

100 001

100 1010

1101 0001

11,1

K

100 010

100 1011

1101 0010

11,2

L

100 011

100 1100

1101 0011

11,3

M

100 100

100 1101

1101 0100

11,4

N

100 101

100 1110

1101 0101

11,5

O

100 110

100 1111

1101 0110

11,6

P

100 111

101 0000

1101 0111

11,7

ALPHANUMERIC CHARACTER CODES

Q

101 000

101 0001

1101 1000

11,8

R

101 001

101 0010

1101 1001

11,9

S

110 010

101 0011

1110 0010

0,2

T

110 011

101 0100

1110 0011

0,3

U

110 100

101 0101

1110 0100

0,4

V

110 101

101 0110

1110 0101

0,5

W

110 110

101 0111

1110 0110

0,6

X

110 111

101 1000

1110 0111

0,7

Y

111 000

101 1001

1110 1000

0,8

Z

111 001

101 1010

1110 1001

0,9

12-BIT CARD CODE

When discrete info. is transferred through punch cards, the alphanumeric characters use a 12-bit binary code.

- A punch card consists of 80 columns and 12 rows - in each column an alphanumeric character is represented by holes punched in the appropriate rows. A hole is sensed as a 1 and the absence of a hole is sensed as a 0.

- The 12 rows are marked, starting from the top, as the 12, 11, 0, 1, 2,…,9 punch. The first 3 are called the zone punch and the last 9 are called the numeric punch.

The 12-bit card code is inefficient w/ respect to the number of bits used.

BINARY CODES

- Most computers translate the input code into an internal 6-bit code. As an example, the internal code representation of the name “John Doe” is:
100001 100110 011000 100101 110000

J O H N blank

- 010100100110 010101

D O E

Binary Storage and Registers

- Register is a group of binary cells that are responsible for storing and holding the binary information.
- Register transfer operation is transferring binary operation from one set of registers to another set of registers.
- Digital logic circuits process the binary information stored in the registers.

Binary Logic

- Binary logic or Boolean algebra deals with variables that take on two discrete values.
- Binary logic consists of binary variables (e.g. A,B,C, x,y,z and etc.) that can be 1 or 0 and logical operations such as:
AND: x.y=z or xy=z (see AND truth table)

OR: x + y =z (see OR truth table)

NOT: x’ =z (not x is equal to z)

- Note that binary logic is different from binary arithmetic. For example in binary logic 1 + 1 =1 (1OR1) but in binary arithmetic 1 + 1 = 10 (1 PLUS 1)

Logic Gates

- Logic gates are electronic circuits that operate on one or more input signals to produce an output signal.
- Electrical signal can be voltage.
- Voltage-operated circuits respond to two separate levels equal to logic 1 or 0.
- Logical gates can be considered as a block of hardware that produced the equivalent of logic 1 or logic 0 output signals if input logical requirement are satisfied

AND

- Symbol is dot
- C = A · B

- Or no symbol
- C = AB

- Truth table ->
- C is 1 only if
- Both A and B are 1

OR

- Symbol is +
- Not addition
- C = A + B

- Truth table ->
- C is 1 if either 1
- Or both!

NOT

- Unary
- Symbol is bar
- C = Ā

- Truth table ->
- Inversion