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Course contents • • • • • • • Chapter 1 - section 1.6 Chapter 2 - all sections Chapter 4 - 4.1 – 4.7, and 4.12 Chapter 5 - 5.1-5.3, 5.6-5.7 Chapter 6 - all sections Chapter 7 - all sections Chapter 8 - 8.1-8.9 1 1.6 Binary numbers • An electronic signal in logic circuits carries one digit of information. – Each digit is allowed to take on only two possible values, usually denoted as 0 and 1. – -> Information in logic circuits is represented as combinations of 0 and 1 digits. • Q: How to represent numbers (E.g., positive integers) using only binary digits 0 and 1? 2 Decimal (base-10) number system • A decimal integer is expressed by an n-tuple comprising n decimal digits D = dn-1dn-2 ∙ ∙ ∙ d1d0 which represents the value V(D) = dn-1×10n-1 + dn-2×10n-2 + ∙ ∙ ∙ + d1×101 + d0×100 • This is referred to as the positional number representation. 3 Binary (base-2) number system • Logic circuits use the binary system whose positional number representation is B = bn-1bn-2 ∙ ∙ ∙ b1b0 bn-1 is the most significant bit (MSB), b0 is the least significant bit (LSB), Every bit bi can only have two values: 0 or 1. 4 Numbers in decimal and binary Decimal representation Binary representation Decimal representation Binary representation 00 0000 09 1001 01 0001 10 1010 02 0010 11 1011 03 0011 12 1100 04 0100 13 1101 05 0101 14 1110 06 0110 15 1111 07 0111 08 1000 5 Conversion from binary to decimal • Compute a weighted sum of every binary digit contained in the binary number B = bn-1bn-2 ∙ ∙ ∙ b1b0 V(B) = bn-1×2n-1 + bn-2×2n-2 + ∙ ∙ ∙ + b1×21 + b0×20 E.g., (1101)2 = 1×23 + 1×22 + 0×21+1×20=(13)10 6 Conversion from decimal to binary • Perform the successive division by 2 until the quotient becomes 0. Remainder 857 / 2 = 428 1 428 / 2 = 214 0 214 / 2 = 107 0 107 / 2 = 53 1 53 / 2 = 26 1 26 / 2 = 13 0 13 / 2 = 6 1 6/ 2 =3 0 3/ 2 =1 1 1/ 2 =0 1 LSB MSB 7 Chapter 2 Introduction to Logic Circuits Outline 2.1 Variables and Functions 2.2 Inversion 2.3 Truth tables 2.4 Logic gates and networks 2.5 Boolean algebra 2.6 Synthesis using AND, OR and NOT gates 2.7 NAND and NOR logic networks 2.8 Design examples 9 2.1 Variables x=0 x=1 (a) Two states of a switch S x (b) Symbol for a switch Figure 2.1. A binary switch. 10 An application S Battery x Light (a) Simple connection to a battery S Power supply x Light (b) Using a ground connection as the return path Figure 2.2. A light controlled by a switch. 11 Functions Power supply S S x1 x2 Light (a) The logical AND function (series connection) S x1 Power supply S Light x2 (b) The logical OR function (parallel connection) Figure 2.3. Two basic functions. 12 A series-parallel connection S X1 Power supply S S X3 Light X2 L(x1, x2, x3) = (x1 + x2) x3 13 2.2 Inversion (complement, not) R Power supply x S Light Figure 2.5. An inverting circuit. 14