Digital Logic Design Chapter 1

• Digital logic design covers digital systems and binary numbers.
• The lectures occur for 2 hours per week.
• The topics include number systems, codes, Boolean algebra, De-Morgan's Theorems, logic gates, canonical forms, combinational logic circuits, minimization techniques, arithmetic and data handling circuits, decoders, encoders, multiplexers, de-multiplexers, combinational circuit design, flip-flops, and counters.

Chapter 1 Overview

• Digital systems and binary numbers are covered.
• Binary numbers, number-base conversions, octal, hexadecimal numbers are covered.
• Complements, signed binary numbers, binary codes, binary storage, registers and logic are included.

Digital Systems

• The digital age correlates to the information age.
• Digital computers are used for general purposes and many scientific, industrial, and commercial applications.
• Digital systems include telephone switching exchanges, digital cameras, electronic calculators, and digital TVs.
• Discrete information-processing systems manipulate discrete elements of information such as {1, 2, 3, ...} and {A, B, C, ...}.

Analog vs Digital Signals

• Physical quantities or signals in analog systems vary continuously over a specified range.
• Physical quantities or signals in digital systems assume only discrete values, which leads to greater accuracy.

Binary Digital Signals

• An information variable in these systems is represented by a physical quantity.
• Digital systems use discrete values for variables, commonly two-level or binary values.
• Binary values can be represented abstractly by digits like 0 and 1.
• Binary logic contains words such as False (F), True (T), Low (L), and High (H), or On and Off.
• Binary values represent values or ranges of values of physical quantities.

Decimal Number System

• The base (radix) of the decimal system is 10.
• The digits include 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9.
• The digit position determines if it represents an integer or a fraction.
• The digit weight equals (Base)position.
• Magnitude represents the Sum of "Digit x Weight".
• Formal notation: d2B2+d1B1+d0B0+d-1B-1+d-2*B-2; example: (512.74)10.

Octal Number System

• The base is 8, with digits including 0, 1, 2, 3, 4, 5, 6, and 7.
• Weight = (Base) Position
• Magnitude equals the sum of "Digit x Weight"
• Formal notation is 582+181+280+78-1+4*8-2 which equals (330.9375)10 or (512.74)8.

Binary Number System

• The base is 2, with digits 0 and 1, called binary digits or "bits".
• Weight = (Base) Position
• Magnitude is the sum of "Bit x Weight".
• Formal Notation is 122+021+120+02-1+1*2-2=(5.25)10 or (101.01)2.

Hexadecimal Number System

• Base 16 uses digits 0-9 and A-F (16 digits total).
• Weight equals (Base) Position.
• Magnitude is the Sum of "Digit x Weight"
• Formal Notation 1162+14161+5160+716-1+10*16-2, which equals (485.4765625)10 or (1E5.7A)16.

Powers of 2

• 20 = 1
• 21 = 2
• 22 = 4
• 23 = 8
• 24 = 16
• 25 = 32
• 26 = 64
• 27 = 128
• 28 = 256
• 29 = 512
• 210 = 1024, Kilo
• 211 = 2048
• 212 = 4096
• 220 = 1M, Mega
• 230 = 1G, Giga
• 240 = 1T, Tera

Decimal Addition

• Add two numbers that are of Decimal base, Subtract a Base if the result is greater than 10

Binary Addition

• The Column Addition of binary digits in binary base

Binary Subtraction

• This requires borrowing a "Base" when needed.

Binary Multiplication

• It's conducted bit by bit

Number Base Conversions

• Evaluate magnitude to convert between number bases like octal, decimal, binary, and hexadecimal.

Decimal to Binary Conversion

• To convert from Decimal(Integer) to Binary, divide the number by the 'Base' (=2), take the remainder (either 0 or 1) as a coefficient and take the quotient and repeat the division
• To convert from Decimal(Fraction) to Binary, multiply the number by the ‘Base’ (=2), take the integer (either 0 or 1) as a coefficient, take the resultant fraction and repeat the division

Decimal to Octal Conversion

• To convert from Decimal to Octal, divide the number by the base 8, take remainder as coefficient and repeat division with quotient
• To convert decimal fraction to octal, multiply by 8, take integer as coefficient and repeat with fraction

Binary to Octal Conversion

• Since 8 = 23, each group of 3 bits represents an octal digit.
• This works both ways: Binary to Octal & Octal to Binary.

Binary to Hexadecimal Conversion

• Since 16 = 24, each group of 4 bits represents a hexadecimal digit.
• This works both ways: Binary to Hex & Hex to Binary.

Octal to Hexadecimal Conversion

• An intermediate step requires a conversion to binary
• This works both ways (Octal to Hex & Hex to Octal)

Complements

• Two types for each base-r system: radix and diminished radix complement.

Diminished Radix Complement

• (r-1)'s Complement.
• Given a number N in base r of n digits, (rn-1) - N is the (r-1)'s complement of N.
• For 6-digit decimal numbers, the 9's complement is (106 - 1) - N = 999999 - N.
• The 9’s complement of 546700 is 999999 – 546700 = 453299.
• For 7-digit binary numbers, the 1's complement is (27 – 1) – N = 1111111 – N.
• The 1’s complement of 1011000 is 1111111–1011000 = 0100111.
• Subtraction from (rn – 1) will never require a borrow, making it possible to compute diminishing radix by digit.
• For binary: 1 - 0 = 1 and 1 - 1 = 0.
• In 1’s complement all 0’s become 1’s and vice versa.

Radix Complement

• The r's complement of an n-digit number N in base r is rn - N for N ≠ 0 and as 0 for N = 0.
• obtained by adding 1 to the (r  1) 's complement, since rn – N = [(rn  1) – N] + 1.
• The 10's complement of 012398 is 987602 and the 10's complement of 246700 is 753300.
• The 2's complement of 1101100 is 0010100 and the 2's complement of 0110111 is 1001001.

Subtraction with Complements

• Subtracting using 1s complement requires that we determine the 1s complement of the smaller number, then add the 1s complement to the larger number, then remove the final carry to add it to the result.
• Subtracting using 2s complement involves determining the 2s complement of the smaller number, then adding the 2s complement to the larger number, before discarding the final carry

Signed Binary Numbers

• Sign is represented with a bit placed in the leftmost position, with 0 positive and 1 negative.

Binary Codes

• A number with k decimal digits requires 4k bits in BCD
• Decimal 396 is 0011 1001 0110, with each group of 4 bits representing one decimal digit.
• A decimal number in BCD matches the equivalent binary number only when in the 0-9 range.
• Only binary combinations (0-9) are used; 1010-1111 have no BCD meaning.

Other Decimal Codes

• There are other decimal codes, such as BCD 8421, 2421, Excess-3, 8,4, -2, -1.

Gray Code

• Advantage is that only one bit in the code group changes when going from one number to the next.
• It is used for error detection, and representation of analog data, lowering power design.

ASCII

• The American Standard Code for Information Interchange is the character code.

Definition of Binary Logic

• Binary logic uses binary variables and a set of logical operations.
• Variables are letters like A, B, C, x, y, z, etc having two distinct values: 1 and 0.
• Three basic operations: AND, OR, and NOT.
• AND: a dot (.) or absence of operator; x · y = z or xy = z means z = 1 only if x = 1 and y = 1; otherwise z = 0.
• OR: represented by a plus sign; x + y = z means z = 1 if x = 1 or y = 1 or both, and z = 0 only if both are 0.
• NOT: prime or overbar; x' = z (or x = z) means z is not x; if x = 1, then z = 0, and if x = 0, then z = 1.

Binary Logic Gates

• The truth tables, boolean expressions, and logic gates apply to AND, OR, and NOT.
• A graphic symbol shows input/output signals for Logic gates.