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# Note for DIGITAL LOGIC DESIGN - DESIGN By Ramanjaneya Reddy G

• Digital Logic Design - DLD
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#### Note for DIGITAL LOGIC DESIGN - DESIGN By Ramanjaneya Reddy G

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Page 1 of 22 UNIT – I - BINARY SYSTEMS DIGITAL SYSTEMS  Digital systems have such a prominent role in everyday life that we refer to the present technological period as the digital age.  Digital systems are used in communication, business transactions, traffic control, spacecraft guidance, medical treatment, whether monitoring, the internet and many other commercial, industrial and scientific enterprises.  One characteristic of digital system is their ability to represent and manipulate discrete elements of information. Any set that is restricted to a finite number of elements contains discrete information.  Examples of discrete sets are the 10 decimal digits, the 26 letters of the alphabet, the 52 playing cards and the 64 squares of a chessboard.  The information is represented in a digital system by physical quantities called signals. The signals in electronic digital systems use two discrete values and are therefore said to be binary. A binary digit called bit has two values: 0 and1.  The general purpose digital computer is the best example of a digital system. A digital computer can accommodate many input and output devices. One very useful device is a communication unit that provides interaction with other users through the internet.  Like a digital computer, most digital devices are programmable. By changing the program in a programmable device, the same underlying hardware can be used for many different applications.  Discrete systems can be made to operate with extreme reliability by using error ‐ correction code.  To understand the operation of each digital module, it is necessary to have a basic knowledge of digital circuits and their logical functions. BINARY NUMBERS  A decimal number such as 7392 represents a quantity equal to 7000+300+90+2  7392 is a shorthand notation for what should be written as 7 X 103 + 3 X 102 + 9 X 101 +2 X 100  In general a number with a decimal point is represented by a series of coefficients as a5 a4 a3 a2 a1 a0 . a‐1 a‐2 a‐3  The coefficients aj are any of the 10 digits (0,1,2,3,4,5,6,7,8,9) and the subscript value j gives the place value and hence the power of 10 by which the coefficient must be multiplied.  Thus the preceding decimal number can be expressed as 105a5 +104a4 +103a3 +102a2 +101a1 +100a0 +10‐1a‐1+10‐2 a‐2 10‐3a‐3  The decimal system is said to be of base or radix 10. 10 because it uses 10 digits and the coefficients are multiplied by powers of 10.  A radix point (decimal point) distinguishes positive powers of 10 from negative power of 10.  For example the number 26.75 is expressed as 2 X 101 + 6 X 100 + 7 X 10‐1 + 5 X 10‐2  A binary system has only two possible values: 0 and 1. Each coefficient aj is multiplied by a power of radix 2 and the results are added to obtain the decimal equivalent of the number.  For example (26.75)10 = (11010.11)2 the binary number is expressed as 1 X 24 + 1 X 23 + 0 X 22 +1 X 21 + 0 X 20 + 1 X 2‐1 + 1 X 2‐2  In general a number is expressed in a base r system has coefficients multiplied by powers of r as an . rn + an‐1 . rn‐1 + ‐ ‐ ‐ + a2 . r2 + a1 . r + a0 + a‐1 . r‐1 + a‐2 . r‐2 + ‐ ‐ ‐ + a‐m . r‐m  The coefficients aj range in value from 0 to r‐1  For an example (4021.2)5 = 4 X 53 +0 X 52 + 2 X 51 + 1 X 50 + 2 X 5‐1 = (511.4)10  The decimal i.e base 10 system has 10 digits: 0,1,2,3,4,5,6,7,8,9  The octal i.e base 8 system has 8 digits: 0,1,2,3,4,5,6,7  The Hexadecimal i.e base 16 system has 16 digits: 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F  The letters of alphabet are used to supplement the 10 decimal digits when the base of the number is greater than 10.  In hexadecimal system letters A.B.C,D.E and F are used for the digits 10,11,12,13,14 and 15  For an example (B65F)16 = 11 X 163 + 6 X 162 + 5 X 161 + 15 X 160 = (46687)10

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Page 2 of 22  The digits in a binary number are called bits. When a bit is equal to 0, it does not contribute to sum during the conversion. Therefore the conversion from binary to decimal can be obtained by adding only the numbers with powers of two corresponding to bits that are equal to1.  For an example (110101)2 = 32 + 16 + 4 +1 = (53)10  In computer work 210 is referred as K (Kilo), 220 is referred as M (Mega) 230 is referred as G (Giga) and 240 is referred as T (Tera).  The addition and subtraction and multiplication of binary numbers is as follows NUMBER – BASE CONVERSIONS  Representation of a number in a different radix are said to be equivalent if they have the same decimal representation.  For an example (0011)8 and (1001)2 are equivalent – both have decimal value 9  If the number includes a radix point, it is necessary to separate the number into an integer part and fraction part, since each part must be converted differently. EX – 1 Convert decimal 41 to binary. EX – 2 Convert decimal 153 to octal. EX – 3 Convert (0.6875)10 to binary.  The answer is (0.6875)10 = (0.1011)2 EX – 4 Convert (0.513)10 to octal. EX – 5 Convert (41.6875)10 to decimal.  The conversion of decimal numbers with both integer and fraction parts is done by converting the integer and the fraction separately and then combining the two answers.  Using the results of examples 1 and 3 we obtain (41.687)10 = (101001.1011)2