DIGITAL DESIGN
What is System : System is the interconnection of devices or component that convert one form of signal into the another form. It design to preform a specific function.
Types of System: System can be divided into the two categories
1. Analog System
2. Digital System
1. Analog System: Analog system is a system which process continuous signal. The system manipulates physical quantities that are represented in analog form.
For example : The output of Amplifier.
2. Digital System: Digital System is a system which process digital signal. The system manipulates discrete quantities of information that are represented in binary form.
For Example: Digital Computer, Calculate.
Difference between Analog and Digital System: one of the most common example of the difference is a clock. The analog clock represent the time by middle that spin around the dial and points to location on the dial that represent the approximate time. On the other side the digital clock a numeric display indicate the exact time.
Advantage of Digital System:
1. Digital information storage is easy.
2. digital system are generally easier to design.
3. Accurate and precision are greater as compared to Analog system.
4. Digital system are less affected by noise.
Limitation of digital system: only analog signal is available in real world,so an extra hardware is required to convert this analog signal to digital signal by using analog to digital converter.
Number System
Number system is a language of digital system. A number can be represented in various ways using different number systems. The total number of symbols used to represent the number is called a Base (or Radix) for particular number system.
In digital system,following number systems are frequently used.
1. digital number system
2. Binary number system
3. Octal number system
4. Hexadecimal number system
Decimal number system :There are 10 digit symbols are used to represent any number. So its Base is 10 and the symbols are 0,1,2,3,4,5,6,7,8,9.
Binary number system: There are 2 digit symbols (called Bit) are used to represent any number. There are 0 and 1. the base of binary number system is 2.
Octal number system: There are 8 digit symbols are used to represent any numberer are 0,1,2,3,4,5,6,7. because of 8 digits its base is 8.
Hexadecimal number system: there are 16 symbols are used to represent any number. So that its base is 16. the 16 symbols in hexadecimal number system are 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F.
Number representation: Any number has two parts
1. Integer
2. Fraction
These parts are separated by a point ( . )
Most Significant Digit (MSD): the leftmost digit having the highest weight is called as the most significant digit of the number.
Least Significant Digit (LSD): the rightmost digit having the lowest weight is called as the least significant digit of the number.
Binary Number Formats: Binary numbers is a sequence of bits (bits is short for binary digits). We have defined boundaries of these boundaries are:
Binary number formats
Name  Size(bits)  Example 
Bit  1  1 
Nibble  4  0101 
Byte  8  0000 0101 
Word  16  0000 0000 0000 0101 
Double Word  32  0000 0000 0000 0000 0000 0000 0000 0101 
Bit: The smallest unit of data is defined as bit. We can represent any two distinct item with a single bit like true or false, ON or OFF, right or wrong etc.
Nibble: A nibble is a combination of 4 bits. With a nibble, we can represent up to 16 distinct values. The structure of nibble as shown in figure.
MSB LSB

b3 b2 b1 b0
Byte: Byte is a combination of 8 bits. The number of distinct values represent by a byte is 256. ranging from 00000000 to 11111111.
it means
number of distinct value = 2^{N }=2^{4 }= 16
where N= Number of bits used to represent a numberer
Word: word is the combination of 16 bits. Hence it consists of two byte.
Double Word: Double Word is a combination of 32 bits.
Relation between Binary,Decimal,Octal and Hexadecimal numbers:
Decimal
(base 10) 
Binary
(base 2) 
Octal
(base 8) 
Hexadecimal
(base 16) 
0  0000  0  0 
1  0001  1  1 
2  0010  2  2 
3  0011  3  3 
4  0100  4  4 
5  0101  5  5 
6  0110  6  6 
7  0111  7  7 
8  1000  10  8 
9  1001  11  9 
10  1010  12  A 
11  1011  13  B 
12  1100  14  C 
13  1101  15  D 
14  1110  16  E 
15  1111  17  F 
Number base conversion: The conversion can be made from decimal to any radix and viceversa.
Conversion from Decimal to any Radix (Base): A number from decimal to its equivalent representation in the Base r is carried out by separating its integer and fraction parts. These two parts are converted separately.
Example 1: Convert the decimal number 31.6875 into an Binary Number.
Step 1 Separate the Integer and fractional parts.
31 and 0.6875
Step 2 Integer Conversion
(31)_{10} = (11111)_{2}
Step 3 Fraction conversion
(.6875)_{10 } = (.1011)_{2}
(31.6875)_{10 } = (11111.1011)_{2 } Answer
Example 2 : Convert the decimal 3315.3 into an equivalent octal number.
Example 3 : Convert the decimal number ( 3315.3 )_{10 }into an equivalent hexadecimal number.
( 3315.3 )_{10} = ( CF3.4CCC )_{16 }Answer
From any Base to Decimal: the general procedure for conversion from any radix to decimal write down the weights for different positions, multiply each digit in the corresponding weight to obtain product numbers. And in the last add all the product number to get the decimal equivalent.
Example 1: Convert the Binary number ( 11011.101 )_{2} into its equivalent Decimal number.
11011 = 1 x 2^{0 }+ 1 x 2^{1} + 0 x 2^{2} + 1 x 2^{3} + 1 x 2^{4}
= 1 +2 + 8 + 16
= 27
.101 = 1 x 2^{1} + 0 x 2^{2}+ 1 x 2^{3}
= .5 + .125 = .625
( 11011.101) 2 = ( 27.625 )10 Answer
Example 2: Convert the octal number ( 532.125)_{8} into its equivalent decimal number.
532 = 2 x 8^{0 }+ 3 x 8^{1} + 5 x 8^{2 }
= 2 + 24 + 320
= 346
.125 = 1 x 8 ^{1}+ 2 x 8 ^{2} + 5 x 8^{3}
= .125 + .03125 + .0097
= .165
( 532.125)_{8} = ( 346.165)10 Answer
Example 3: convert Hexadecimal number ( F9A.B)_{16} into its equivalent Decimal number.
F9A.B = 10 X 16^{0 } + 9 X 16^{1} + 15 X 16^{2} + 11 X 16^{1}
= 10 + 144 + 3840 + .6875
= ( 3994.6875)10 Answer
Example : Convert 1110101.101001 into Octal number system.
Solve: For integer part starting grouping from LSB to MSB and for fraction part start from MSB to LSB
1 110 101 . 101 001
1 6 5 5 1
( 165.51)_{8}^{ }Answer
Example : Convert the Binary number 1110101.101001 in to the Hexadecimal number.
0111 0101 . 1010 0100
7 5 . A 4
( 75.A4)_{16}^{ }Answer
Example : Convert octal ( 576)8 into Binary number system.
(576)8= (101 111 110)2
Example : Convert the Hexadecimal number ( F9A )16 into binary number system.
Solve : F9A = ( 1111 1001 1010 )2 Answer
Example: Convert the following numbers with the indicated Bases to Decimal
1> ( 4310)5
2> (198 )12
Solve: 1> ( 4310)^{5 = }0 x 5^{0 } + 1 x 5^{1} + 3 x 5^{2} +4 x 5^{3}
= 0 + 5 + 75 + 500
= ( 580 )10^{ }Answer
2> (198)^{1}^{2 }= 8 x 12^{0} + 9 x 12^{1} + 1 x 12^{2}
= 8 + 108 + 144
= ( 260 )10^{ }Answer
Example: Determine the Base of the numbers in each case for the following operation to be correct.
1> 14/2 = 5
Solve Let b= Base
14/2 = 5
(1 x b + 4 ) /2 = 5
b + 4 = 10
b = 6 Answer