ECE 2110: Introduction to Digital Systems 2. Number Systems & Codes

2 Previous class Summary Electronics aspects of digital design Integrated Circuits (wafer, die, SSI, MSI, LSI, VLSI)

Digital Design Levels Many representations of digital logic Device Physics and IC manufacturing  Moore’s Law [1965, Gordon E. Moore]:  Check Transistor level --->Logic design, functional building blocks The number of transistors per square inch in an IC doubles every 2 years.

4 Digital Design Levels Transistor-level circuit diagrams Example: Multiplexor

5 Truth tables Gate-level Logic diagrams: Test it at Logisim circuit

6 Prepackaged building blocks, e.g. multiplexer Equations: Z = S  A  + S  B

7 Various hardware description languages  ABEL  VHDL

8 Binary Representation The basis of all digital data is binary representation. Binary - means ‘two’  1, 0  True, False  Hot, Cold  On, Off We must be able to handle more than just values for real world problems  1, 0, 56  True, False, Maybe  Hot, Cold, Warm, Cool  On, Off, Leaky

9 2. Number Systems & Codes To talk about binary data, we must first talk about number systems The decimal number system (base 10) we are all familiar with! What if??? Decimal is a Positional number system.  Which system is not?

10 Positional Notation Value of number is determined by multiplying each digit by a weight and then summing. The weight of each digit is a POWER of the BASE and is determined by position.

11 The decimal number system (base 10) you should be familiar with!  A digit in base 10 ranges from 0 to 9.  A digit in base 2 ranges from 0 to 1 (binary number system). A digit in base 2 is also called a ‘bit’.  A digit in base R can range from 0 to R-1  A digit in Base 16 can range from 0 to 16-1 (0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F). Use letters A-F to represent values 10 to 15. Base 16 is also called Hexadecimal or just ‘Hex’.

= 9 x x x x x = = = 1x x x x x x2 -2 = = A2F 16 = 10x x x16 0 = 10 x x x 1 = = 2607 Base 10, Base 2, Base 16

13 Common Powers 2 -3 = = = = = = = = = = = = = = = = = 1 = = 16 = = 256 = = 4096 = = 1024 = 1 K 2 20 = = 1 M (1 Megabits) = 1024 K = 2 10 x = = 1 G (1 Gigabits)

14 Least Significant Digit Most Significant Digit = Most Significant Digit (has weight of 2 5 or 32). For base 2, also called Most Significant Bit (MSB). Always LEFTMOST digit. Least Significant Digit (has weight of 2 0 or 1). For base 2, also called Least Significant Bit (LSB). Always RIGHTMOST digit.

15 Next… Number system conversions Addition/Subtraction Check HW #1, HW#2 due dates on the web:  15/homework.htm 15/homework.htm