CE1111 :Digital Logic Design lecture 01 Introduction Dr. Atef Ali Ibrahim

Course Administration Instructor:Dr. Atef Ali Ibrahim E_mail: Office Place: (Room CS-25 ) Office Hrs: TA: Tutorial: Text Book: M. Morris Mano; “Digital Design”, 4thEdition, prentice hall. Prerequisite: Recommended Text: Notes:Lecture slides and Assignments 2

Course Objectives Design of Digital logic systems. Building large systems from small components (logic gates ) EEG 481 Lec. 1 Introduction3

Chapter 1 4

Course Outlines Number Systems (conversion and coding) Boolean Algebra Gate Level minimization Combinational Logic Design Synchronous sequential logic Registers and counters Memories and Programmable logic 5

What is the Meaning of Digital Logic Design? Digital means: 6

7

Why Digital? Digital systems are easier to design and implement than analog systems. EEG 481 Lec. 1 Introduction8

9

Digital Logic circuit 10

11

12

13

N UMBER S YSTEMS – Representation A number with radix r is represented by a string of digits: A n - 1 A n - 2 … A 1 A 0. A - 1 A - 2 … A - m  1 A - m in which 0  A i < r and. is the radix point. The string of digits represents the power series:  (Number)r =   j = - m j j i i = 0 i rArA (Integer Portion) + (Fraction Portion) i = n - 1 j = - 1

Number Systems – Examples GeneralDecimalBinaryOctalHex. Radix (Base)r Digits0 => r - 10 => 90 => 10 => 70 => F Powers of 4 Radix r 0 r 1 r 2 r 3 r 4 r 5 r -1 r -2 r -3 r , , …. … ….

Decimal Number System 16

Binary Number system EEG 481 Lec. 1 Introduction17

Octal Number system 18

Hexadecimal Number System 19

Numbers with different bases 20 The first 17 numbers in decimal, Binary, octal, and Hexadecimal

CONVERSION OF BASES Decimal to Base R number Base R to Decimal Conversion (736.4) 8 = 7 x x x x 8 -1 = 7 x x x 1 + 4/8 = (478.5) 10 (110110) 2 =... = (54) 10 ( ) 2 =... = (6.785) 10 (F3) 16 =... = (243) 10 (0.325) 6 =... = ( ) 10 - Separate the number into its integer and fraction parts and convert each part separately. - Convert integer part into the base R number → successive divisions by R and accumulation of the remainders. - Convert fraction part into the base R number → successive multiplications by R and accumulation of integer digits Data Types

EXAMPLE Convert to base 2. Integer = Fraction = x x x x (41) 10 = (101001) 2 (0.6875) 10 = (0.1011) 2 ( ) 10 = ( ) 2 Convert (63) 10 to base 5: (223) 5 Convert (1863) 10 to base 8: (3507) 8 Convert ( ) 10 to hexadecimal: (0.A3) 16 Exercise Data Types

Decimal to Octal Conversion 23

Decimal to Hex. Conversion 24

Binary to Hex. Conversion 25

Binary to Octal Conversion 26

Signed Numbers 27

COMPLEMENT OF NUMBERS Two types of complements for base R number system: - R's complement and (R-1)'s complement The (R-1)'s Complement Subtract each digit of a number from (R-1) Example - 9's complement of is 's complement of is (bit by bit complement operation) The R's Complement Add 1 to the low-order digit of its (R-1)'s complement Example - 10's complement of is = 's complement of is = Complements

SIGNED NUMBERS Representation Signed magnitude representation Signed 1's complement representation Signed 2's complement representation Example: Represent +9 and -9 in 7 bit-binary number Only one way to represent +9 ==> Three different ways to represent -9: In signed-magnitude: In signed-1's complement: In signed-2's complement: Need to be able to represent both positive and negative numbers - Following 3 representations

SIGNED NUMBERS Representation Signed Magnitude Signed-1’s complement Signed-2’s complement Decimal Range of Signed 2’s complement : -2 n-1 to 2 n Range of Signed 1’s complement: -2 n-1 +1 to 2 n Range of Signed Magnitude: -2 n-1 +1 to 2 n-1 -1

ARITHMETIC ADDITION: SIGNED MAGNITUDE [1] Compare their signs [2] If two signs are the same, ADD the two magnitudes - Look out for an overflow [3] If not the same, compare the relative magnitudes of the numbers and then SUBTRACT the smaller from the larger --> need a subtractor to add [4] Determine the sign of the result ) > ) > ) > ) > (- 9) -6 + (-9) Overflow or (-9) + (-9) ) (1)0010 overflow

ARITHMETIC ADDITION: SIGNED 2’s COMPLEMENT Example (1) overflow ) (c n-1  c n ) Add the two numbers, including their sign bit, and discard any carry out of leftmost (sign) bit - Look out for an overflow

ARITHMETIC ADDITION: SIGNED 1’s COMPLEMENT Add the two numbers, including their sign bits. - If there is a carry out of the most significant (sign) bit, the result is incremented by 1 and the carry is discarded (1) 0(1) ) end-around carry (1) ) (1)0010 +) overflow Example not overflow (c n-1  c n ) = 0 (c n-1  c n ) Fixed Point Representations

COMPARISON OF REPRESENTATIONS * Easiness of negative conversion S + M > 1’s Complement > 2’s Complement * Hardware - S+M: Needs an adder and a subtractor for Addition - 1’s and 2’s Complement: Need only an adder * Speed of Arithmetic 2’s Complement > 1’s Complement(end-around C)

ARITHMETIC SUBTRACTION Take the complement of the subtrahend (including the sign bit) and add it to the minuend including the sign bits. (  A ) - ( - B ) = (  A ) + B (  A ) - B = (  A ) + ( - B ) Example : (-6) – (-13) = +7 in 8-bit representation – = = (+7) Fixed Point Representations Arithmetic Subtraction in 2’s complement

Binary codes Binary codes are used to represent discrete elements of information The codes must be in binary that the computer can only hold 0’s or 1’s. 36

Cont. Binary Codes Given n binary digits (called bits), there are 2 n distinct elements that can be represented. But, you can represent m elements, m < 2 n Examples: You can represent 4 elements with n = 2 bits: (00, 01, 10, 11). You can represent 4 elements with n = 4 digits: (0001, 0010, 0100, 1000). This second code is called a "one hot" code.

Example: A binary code for the seven colors of the rainbow Code 100 is not used Non-numeric Binary Codes Binary Code Color Red Orange Yellow Green Blue Indigo Violet

Decimal BCD(8421) Excess d 3 d 2 d 1 d 0 : symbol in the codes BCD: d 3 x 8 + d 2 x 4 + d 1 x 2 + d 0 x 1  8421 code. 2421: d 3 x 2 + d 2 x 4 + d 1 x 2 + d 0 x : d 3 x 8 + d 2 x 4 + d 1 x (-2) + d 0 x (-1) Excess-3: BCD + 3 Note: 8,4,2,-2,1,-1 in this table is the weight associated with each bit position. External Representations D ECIMAL C ODES - Binary Codes for Decimal Digits

Chapter 1 40 Binary Coded Decimal (BCD) The BCD code is the 8,4,2,1 code. 8, 4, 2, and 1 are weights BCD is a weighted 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.

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 = (This is conversion) 13  0001|0011 (This is coding)

BCD Addition BCD addition of two numbers involve following rules:- 1) Maximum value of the sum for two digits = 9 (max digit 1) + 9 (max digit 2) + 1 (previous addition carry) = 19 2) If sum of two BCD digits is less than or equal to 9 (1001) without carry then the result is a correct BCD number. 3) If sum of two BCD digits is greater than or equal to 10 (1010) the result is in-correct BCD number. Perform steps 4 for correct BCD sum. 4) Add 6 (0110) to the result. 42

Example -BCD addition Add 599 and 984 using BCD numbers Binary Sum 1, 2 and 3 are greater than So, from Step 3 and 4 add ‘6’ to the sum. carry 1 1 Result End Carry Answer. Result of BCD addition is 1583 Exercise: add using BCD addition 43

G RAY C ODE – Decimal

A LPHANUMERIC C ODES - ASCII Character Codes ASCII means “American Standard Code for Information Interchange” The ASCII code assigns binary patterns for - Numbers 0 to 9 - All the letters of English alphabet, uppercase and lowercase - Many control functions (E.g. BS = Backspace, CR = Carriage Return) and punctuation marks. The ASCII system uses 7 bits to represent each code 45

ASCII Code

P ARITY B IT Error-Detection Codes parity, an extra bit appended onto the code word to make the number of 1’s odd or even. 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 The codeword "1111" has even parity and the codeword "1110" has odd parity. Both can be used to represent 3-bit data. Even Parity Odd Parity Message - Parity Message - Parity

Parity Bit Generation For b 6 b 5... b 0 (7-bit information); even parity bit b even b even = b 6  b 5 ...  b 0 For odd parity bit b odd = b even  1 = b even PARITY BIT GENERATION

PARITY GENERATOR AND PARITY CHECKER Parity Generator Circuit (even parity) b6b5b4b3b2b1b0b6b5b4b3b2b1b0 b even Parity Checker b6b5b4b3b2b1b0b6b5b4b3b2b1b0 b even Even Parity error indicator Error Detecting codes