1. ECE 2110: Introduction to Digital Systems Number Systems

2. 2 Previous class Summary Electronics aspects of digital design Integrated Circuits (wafer, die, SSI, MSI, LSI, VLSI) PLDs: PLAs, PALs, CPLD, FPGA ASIC

3. 3 Digital Design Levels Many representations of digital logic Device Physics and IC manufacturing  Moore’s Law [1965, Gordon Moore]: Transistor level --->Logic design, functional building blocks The number of transistors per square inch in an IC doubles every year [18months].

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

5. 5 Truth tables Gate-level Logic diagrams

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

7. 7 Various hardware description languages  ABEL  VHDL

8. 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. 9 Number Systems To talk about binary data, we must first talk about number systems The decimal number system (base 10) you should be familiar with! Positional number system

10. 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. 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’.

12. 12 953.78 10 = 9 x 10 2 + 5 x 10 1 + 3 x 10 0 + 7 x 10 -1 + 8 x 10 -2 = 900 + 50 + 3 +.7 +.08 = 953.78 1011.11 2 = 1x2 3 + 0x2 2 + 1x2 1 + 1x2 0 + 1x2 -1 + 1x2 -2 = 8 + 0 + 2 + 1 + 0.5 + 0.25 = 11.75 A2F 16 = 10x16 2 + 2x16 1 + 15x16 0 = 10 x 256 + 2 x 16 + 15 x 1 = 2560 + 32 + 15 = 2607 Base 10, Base 2, Base 16

13. 13 Common Powers 2 -3 = 0.125 2 -2 = 0.25 2 -1 = 0.5 2 0 = 1 2 1 = 2 2 2 = 4 2 3 = 8 2 4 = 16 2 5 =32 2 6 = 64 2 7 = 128 2 8 = 256 2 9 = 512 2 10 = 1024 2 11 = 2048 2 12 = 4096 16 0 = 1 = 2 0 16 1 = 16 = 2 4 16 2 = 256 = 2 8 16 3 = 4096 = 2 12 2 10 = 1024 = 1 K 2 20 = 1048576 = 1 M (1 Megabits) = 1024 K = 2 10 x 2 10 2 30 = 1073741824 = 1 G (1 Gigabits)

14. 14 Least Significant Digit Most Significant Digit 53 10 = 110101 2 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. 15 Hex (base 16) to Binary Conversion Each Hex digit represents 4 bits. To convert a Hex number to Binary, simply convert each Hex digit to its four bit value. Hex Digits to binary (cont): 9 16 = 1001 2 A 16 = 1010 2 B 16 = 1011 2 C 16 = 1100 2 D 16 = 1101 2 E 16 = 1110 2 F 16 = 1111 2 Hex Digits to binary: 0 16 = 0000 2 1 16 = 0001 2 2 16 = 0010 2 3 16 = 0011 2 4 16 = 0100 2 5 16 = 0101 2 6 16 = 0110 2 7 16 = 0111 2 8 16 = 1000 2

16. 16 Hex to Binary, Binary to Hex A2F 16 = 1010 0010 1111 2 345 16 = 0011 0100 0101 2 Binary to Hex is just the opposite, create groups of 4 bits starting with least significant bits. If last group does not have 4 bits, then pad with zeros for unsigned numbers. 1010001 2 = 0101 0001 2 = 51 16 Padded with a zero

17. 17 Next… More conversions Addition/Subtraction HW #2