1. CHAPTER 1 BINARY SYSTEMS
Digital Logic Design
CHAPTER 1
BINARY SYSTEMS

2. Digital Computers & Digital Systems
the most striking property of a digital computer is its generality.
It can follow a sequence of instructions, called a program, that operates on given data.
The user can specify and change programs and/or data according to the specify need.
general-purpose digital computers can perform a wide variety of information-processing tasks.
The general purpose digital computer is the best known example of a digital system.

3. Digital Computers & Digital Systems
Early digital computers were used mostly for numerical computations.
In this case the discrete elements used are the digits. From this application, the term digital computer has emerged
A more appropriate name for a digital computer would be a “discrete information processing system”.

4. Digital Computer Processor or arithmetic unit Control unit
Storage or memory unit
Input devices and control
Output devices and control
Figure 1-1 BLOCK DIAGRAM OF A DIGITAL COMPUTER

5. Digital vs. Analog An analog system has continuous range of values
A mercury thermometer
Human eye
A digital system has a set of discrete values
Digital Thermometer
Digital camera

6. Digital vs. Analog Cheap electronic circuits
Analog signal
Digital signal
Cheap electronic circuits
Easier to calibrate and adjust
Resistance to noise: Clearer picture and sound

7. Binary System
The binary numbers system have two possible values:0 and 1
Discrete elements of information are represented with bits called binary codes.
Example:
(09)10 = (1001)2
(15)10 = (1111)2

8. Decimal Number Review the decimal number system
Base (Radix) is 10 - symbols (0,1, . . 9) Digits
Each position carries a weight:
If we were to write using a power series expansion and base 10 arithmetic:
Weights:
MSD
LSD

9. Binary number system The binary number system
Base is 2 - symbols (0,1) - Binary Digits (Bits)
Each position carries a weight:
If we write using a decimal power series we convert from binary to decimal:
Weights:
MSD
LSD

10. Octal/Hex number systems
The octal number system:
Its base is 8  eight digits 0, 1, 2, 3, 4, 5, 6, 7
For example: (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
E.g. (D63FA)16 = (877562)10

11. Conversion from Decimal to Binary
It is necessary to separate the number into an integer part and a fraction:
Repeatedly divide the decimal number by 2
e.g. Find the binary equivalent of 37. = = = = = = 1

12. Conversion from Decimal to Binary
Conversion from decimal fraction to binary:
Same method used for integers except multiplication is used instead of division.
e.g. convert (0.8542)10 to binary (6 digits):
x 2 = a-1 = 1
x 2 = a-2 = 1
x 2 = a-3 = 0
x 2 = a-4 = 1
x 2 = a-5 = 1
x 2 = a-6 = 0
(0.8542)10 = ( )2

13. Conversion from Decimal to Octal
The conversion from decimal integers to any base –r system:
Similar to the binary, except that division is done by r instead of 2
Convert decimal 153 to octal: = = = 1 3 2

14. Conversion from Decimal to Octal
Conversion from decimal fraction to octal:
Same method used for integers except multiplication is used instead of division.
e.g. convert (0.3152)10 to octal (4 digits):
x 8 = a-1 = 2
x 8 = a-2 = 4
x 8 = a-3 = 1
x 8 = a-4 = 3
(0.3152)10 = (0.2413)8
The conversion of decimal numbers with both integer and fraction parts is done separately and then combing the two answers together .

15. Numbers with different bases
Octal 00 01 02 03 04 05 06 07 10 11 12 13 14 15 16 17

16. Conversion using Table
Conversion from and to binary, octal, and hexadecimal plays and important part in digital computers.
Since 23 = 8 and 24 = 16
Each octal digit corresponds to 3 binary digits
( )2 = ( )8
Each hexa digit corresponds to 4 binary digits
( )2 = (6FD.134)16

17. Complements
Used in digital computers for subtraction operation and for logic manipulation.
There are two types of complements for each base-r system:
The r’s complement & (r-1)’s complement
2’s and 1’s complement for binary numbers
10’s and 9’s complement for decimal numbers

18. The r’s Complement The r’s complement of N=rn – N (N is a decimal #).
The 10’s complement of (52520)10 is
=47480
The 10’s complement of (0.3267)10 is
=0.6733
No integer part , so 10n=100=1
The 10’s complement of (25.639)10 is =
The 2’s complement of (101100)2 is
(26)10 – (101100)2=( )2=
The 2’s complement of (0.0110)2 is
(1 – )2 =0.1010

19. The r’s Complement
The 10’s complement of a decimal number can be formed by leaving all least significant zeros unchanged, subtracting the first nonzero least significant digits from 10, and then subtracting all other higher significant digits from 9.
The 2’s complement can be formed by leaving all least significant zeros and the first nonzero digit unchanged, and then replacing 1’s by 0’s and 0’s by 1’s in all other higher significant digits.

20. The (r - 1)’s complement The (r-1)’s complement of N= rn - r-m- N .
The 9’s complement of (52520)10 is
( )= – = 47479
No fraction part , so 10-m=100=1
The 9’s complement of (0.3267)10 is
( )= – =
No integer part , so 10n=100=1
The 9’s complement of (25.639)10 is
(102 – )= – =
The 1’s complement of (101100)2 is
(26 -1) – (101100)2=( )2=010011
The 1’s complement of (0.0110)2 is
(1 – 2-4) - (0.0110)2 = ( – ) =

21. The (r - 1)’s complement
The 9’s complement of a decimal number is formed simply by subtracting every digit from 9.
The 1’s complement of a binary number is even simpler to form : the 1’s are changed to 0’s and the 0,s to 1’s.
The r’s complement can be obtained from the (r-1)’s complement after the addition 1 to the least significant digit.

22. Subtraction with r’s Complements
The subtraction of two positive numbers (M-N)r, may be done as follows:
Add the minuend M to the r’s complement of the subtrahend N.
Inspect the result obtained for an end carry:
If an end carry occurs, discard it.
If an end carry does not occur, take the r’s complement of the number obtained in step 1 and place a negative sign in front.

23. Decimal Subtraction Subtract 7188 – 3049 using 10’s complement:
10’s complement of N =
Sum =
Discard end carry
Answer =
Subtract 150 – 2100 using 10’s complement:
M = 0150
10’s complement of N =
Sum =
There’s no end carry  negative Answer = (10’s complement of 8050) = – 1950

24. Binary Subtraction Subtract 1010100 – 1000100 using 2’s comp.:
2’s complement of N =
Sum =
Discard end carry
Answer =
Subtract – using 2’s comp.:
M =
2’s complement of N =
Sum =
There’s no end carry  negative Answer = (2’s complement of ) = – 10000

25. Subtraction with (r-1)’s Complement
The subtraction with the (r-1)’s compl. is exactly the same as the one used with the r’s complement.
Add the minuend M to the (r-1)’s complement of the subtrahend N.
Inspect the result obtained for an end carry.
If an end carry occurs, add 1 to the least significant digit (end-around carry).
If an end carry does not occur, take the (r-1)’s complement of the number obtained in step 1 and place a negative sign in front

26. Decimal Subtraction Subtract 72532 – 03250 using 9’s compl.: M = 72532
9’s complement of N =
Sum =
end-around carry
Answer =
Subtract – using 9’s compl.:
M = 03250
9’s complement of N =
Sum =
There’s no end carry  negative Answer = - (9’s complement of 30717)=

27. Binary Subtraction Subtract 1010100 – 1000100 using 1’s compl.:
1’s complement of N =
Sum =
end-around carry
Answer =
Subtract – using 1’s compl.:
M =
1’s complement of N =
Sum =
There’s no end carry  negative Answer = - (1’s complement of )=

28. Signed binary numbers
To represent a negative binary number, the convention is to make the sign bit 1. [sign bit 0 is for positive]
01001
9 (unsigned binary)
+9 (signed binary)
11001
25 (unsigned binary)
– 9 (signed binary)

29. Binary Codes
Digital systems represent and manipulate not only binary numbers, but also many other discrete elements of information.
Any discrete element of information distinct among a group of quantities can be represented by a binary code.
Red is one distinct color of the spectrum.
The latter A is one distinct latter of the alphabet.
To represent a group of 2n distinct elements in a binary code requires a minimum of n bits. This is because it is possible to arrange n bit in 2n distinct way.

30. Binary Codes For example:
A group of four distinct quantities can be represented by a two-bit code, with each quantity assigned one of the following bit combinations: 00, 01, 10, 11.
A group of eight elements requires a three-bit code, with each element assigned to one of the following bit combinations: 000, 001, 010, 011, 100, 101, 110,111.
The examples show that the distinct bit combinations of an n-bit code can be found by counting in binary from 0 to (2n-1).
Some bit combinations are unassigned when the number of elements is not a multiple of the power of 2. e.g. the ten decimal digits 0,1,2,…..,9

31. Binary Codes
A binary code that distinguishes among ten elements must contain at least four bits:
Three bits can distinguish a maximum of eight elements.
four bits can form 16 distinct combinations, but since only ten digits are coded , the remaining six combinations are unassigned and not used

32. Decimal Codes
Decimal numbers are represented in a digital system using a string of bits.
This can be done in various ways.
A set of n-bit strings where different strings represent different numbers or other things is called a code.
A particular combination of n-bit values is called a code word.
A common and convenient code for decimal numbers is Binary-coded decimal (BCD).

33. Decimal Codes
BCD is a binary representation of decimal numbers in a form that can easily be interpreted by a person.
The weights in the BCD code are 8, 4, 2, 1.
The bit assignment 0110, for example, can be interpreted by the weights to represent the decimal digit 6 because 0x8+1x4+1x2+0x1=6
A decimal code that has been used in some old computers is the excess-3 code.
Its code assignment is obtained from the corresponding value of BCD after the addition 3

34. Decimal Codes
Numbers are represented in digital computers either in binary or in decimal through a binary code.
When specifying data, the user likes to give data in decimal form.
The input decimal numbers are stored internally in the computer by means of a decimal code.
Each decimal digit requires at least four binary storage elements.
The decimal numbers are converted to binary when arithmetic operations are done internally with numbers represented in binary.

35. Decimal Codes
It is also possible to perform the arithmetic operations directly in decimal codes.
The decimal number 395, when converted to binary, is equal to and consists of nine binary digits.
The same number, when represented internally in the BCD code, occupies four bits for each decimal digit, for a total of 12 bits:

36. Decimal Codes
It is very important to understand the difference between conversion of a decimal number to binary and the binary coding of a decimal number.
The bits obtained from conversion are binary digits.
Bits obtained from coding are combinations of 1’s and 0’s arranged according to the rules of of the code used.
The binary conversion of decimal 13 is 1101 ;
The coding of decimal 13 with BCD is

37. Decimal Codes
The exess-3, the 2, 4, 2, 1 and the 8, 4, -2, -1 are self complementary codes.
That is, the 9’s complement of the of the decimal number is obtained by changing 1’s to 0’s and 0’s to 1’s.
For example, 395 is represented in 2, 4, 2, 1 code by its 9’s complement 604 is represented by

38. The Reflected Code
Used to represent the digital data converted from the analog data.
The advantage of the reflected code over pure binary numbers is that a number in the reflected code changes by only one bit as it proceeds from one number to the next.
The reflected code is also known as the Gray Code.

39. Alphanumeric Codes
Many application of digital computers require the handling of data that consist not only of numbers, but also of letters.
That is, it is necessary to have a binary code for the alphabet.
an alphanumeric code is a binary code of a group of elements consisting of the ten decimal digits, the 26 letters of the alphabet, and a certain number of special symbols such as $.

40. Alphanumeric Codes
The total number of the elements in an alphanumeric group is greater than 36.
It must be coded with a minimum of six bits.
26 = 64, but 25 = 32 is insufficient.
The internal code is a six-bit Alphanumeric Codes.
ASCII most common code for representing alphanumeric characters.
7-bit code. Typically stored in a byte 8 bit can be used for parity.

41. Error-Detecting Code
An Error-Detecting Code can be used to detect errors during transmission.
Parity bit: an extra bit included with a message to make the total number of 1’s either even or odd.
For example,
with even parity with odd parity
ASCII A =
ASCII T =

42. Binary Storage and Registers
A binary cell is a device that possesses two stable states and is capable of storing one bit of information.
A register is a group of binary cells.
Since a cell stores one bit of information, it follows that a register with n cells can store any discrete quantity of information that contains n bits.
Register Transfer
A transfer of the information stored in one register to another.

43. Register Transfer

44. Register Transfer

45. Binary Logic Boolean algebra
Consists of binary variables and logical operations
Binary variables
The variable are designated by letters of the alphabet such as A,B,C,x,y,z, etc.
Two discrete values (true/false; yes/no; 1/0)
Logical operations
Three basic operations: AND, OR, NOT

46. Definition of Binary Logic
AND: this operation is represented by a dot or by the absence of an operator.
x . y =z or xy=z is read “x and y is equal to z.”
The logical operation AND is interpreted to mean that z=1 if and only if x=1 and y=1; otherwise z=0.
OR: this operation is represented by a plus sign.
x+y=z is read “x OR y is equal to z”
Meaning that z=1 if x=1 or if y=1 0r if both x=1 and y=1 . If both x=0 and y=0, then z=0

47. Definition of Binary Logic
NOT: this operation is represented by a prime (sometimes by a bar).
For example, x’=z (or x=z) is read “x not is equal to z,”
Meaning that z is what x is not . In other words, if x=1, then z=0; but if x=0, then z=1
A truth table is a table of all possible combinations of the variables showing the relation between the values that the variables may take and the result of the operation

48. Truth tables of logical operations

49. Logic Gates
Logic circuits that perform the logical operations of AND, OR , and NOT
Symbols for digital logic circuits
Gates with multiple inputs

50. Integrated Circuits
An integrated circuit (IC) is a small silicon semiconductor crystal, called a chip.
It is contain an electrical components such as transistor, diodes, resistors, and capacitors.
The various components are interconnected inside the chip to form an electronic circuit.
The chip is mounted on a metal or plastic package, and connections are welded to external pins to form the IC.

51. Integrated Circuits
integrated circuits come in two types of packages, the flat package and the dual-in-line (DIP) package.
The dual-in-line package is the most widely used type because of the low price and easy installation on circuit boards.
Most packages have standard sizes, and the number of pins ranges from 8 to 64.
Each IC has a numeric designation printed on the surface of the package for identification.

52. Integrated Circuits
The advantages of constructing digital systems with IC’s.
Size of IC packages is very small
The cost of IC’s is very low.
Their reduced power consumption.
Suitable for high speed operations.
The use of IC’s reduces the number of external wiring connections because many of the connections are internal to the package.
IC’s are classified in two general categories, linear and digital

53. Integrated Circuits
Several logic gates in a single package make it a small –scale integration (SSI) device.
To qualify as a medium-scale integration (MSI) device, the IC must perform a complete logic function and have a complexity of 10 to 100 gates.
A large-scale integration (LSI) device performs a logic function with more than 100 gates.
There are also very-large-scale integration (VLSI) devices that contain thousands of gates in a single chip