Course contents Chapter 1 - section 1.6 Chapter 2 - all sections Chapter – 4.7, and 4.12 Chapter , Chapter 6 - all sections Chapter 7 - all sections Chapter

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 = d n-1 d n-2 ∙ ∙ ∙ d 1 d 0 which represents the value V(D) = d n-1 ×10 n-1 + d n-2 ×10 n-2 + ∙ ∙ ∙ + d 1 × d 0 ×10 0 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 = b n-1 b n-2 ∙ ∙ ∙ b 1 b 0 b n-1 is the most significant bit (MSB), b 0 is the least significant bit (LSB), Every bit b i can only have two values: 0 or 1. 4

Numbers in decimal and binary Decimal representation Binary representation Decimal representation Binary representation

Conversion from binary to decimal Compute a weighted sum of every binary digit contained in the binary number B = b n-1 b n-2 ∙ ∙ ∙ b 1 b 0 V(B) = b n-1 ×2 n-1 + b n-2 ×2 n-2 + ∙ ∙ ∙ + b 1 ×2 1 + b 0 ×2 0 E.g., (1101) 2 = 1× × ×2 1 +1×2 0 =(13) 10 6

Conversion from decimal to binary Perform the successive division by 2 until the quotient becomes 0. Remainder 857 / 2 = 4281LSB 428 / 2 = / 2 = / 2 = / 2 = / 2 = / 2 = 61 6 / 2 = 30 3 / 2 = 11 1 / 2 = 01MSB 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

Figure 2.1. A binary switch. x1=x0= (a) Two states of a switch S x (b) Symbol for a switch 2.1 Variables 10

Figure 2.2. A light controlled by a switch. (a) Simple connection to a battery S (b) Using a ground connection as the return path Battery Light Power supply S Light x x An application 11

Figure 2.3. Two basic functions. (a) The logical AND function (series connection) S Power supply S S Power supplyS (b) The logical OR function (parallel connection) Light x1x1 x2x2 x1x1 x2x2 Functions 12

S Power supplyS Light S X1X1 X2X2 X3X3 L(x 1, x 2, x 3 ) = (x 1 + x 2 ) x 3 A series-parallel connection 13

Figure 2.5. An inverting circuit. S Light Power supply R x 2.2 Inversion (complement, not) 14