1. CEC 220 Digital Circuit Design Number Systems & Conversions Friday, January 9 CEC 220 Digital Circuit Design Slide 1 of 16

2. Number Systems & Conversions Friday, January 9 CEC 220 Digital Circuit Design Number Systems and Conversions  Digital Systems and Switching Circuits  Number Systems and Conversion  Binary, Hexadecimal, and Octal representations  Examples Slide 2 of 16

3. Number Systems & Conversions Digital Systems and Switching Circuits Friday, January 9 CEC 220 Digital Circuit Design What is the difference between analog and digital systems/signals? Analog – Continuous  Natural Phenomena (Pressure, Temperature, Speed … )  Difficult to realize processing using electronics Digital – Discrete  Binary Digit  Signal Processing as Bit unit  Easy to realize processing using electronics  High performance due to Integrated Circuit Technology Slide 3 of 16

4. Number Systems & Conversions Digital Systems and Switching Circuits Friday, January 9 CEC 220 Digital Circuit Design What is a binary signal? Binary = Two values (0, 1) Each digit is referred to as a “bit” Number representation with only two values (0, 1) Can be implemented with simple electronics devices For Example Voltage: High = (1) and Low = (0) Switch: On = (1) and Off = (0) Slide 4 of 16

5. Number Systems & Conversions Digital Systems and Switching Circuits Friday, January 9 CEC 220 Digital Circuit Design What is a switching circuit? Combinational Circuit: Outputs depend on only present inputs, not on past inputs Sequential Circuit: Outputs depend on both present inputs and past inputs Sequential circuits have “memory” !!!!! Slide 5 of 16

6. Number Systems & Conversions Number Systems and Conversion Friday, January 9 CEC 220 Digital Circuit Design Decimal: (base 10) Decimal: (base 10) Binary: (base 2) Binary: (base 2) Base “R” Slide 6 of 16

7. Number Systems & Conversions Number Systems and Conversion Friday, January 9 CEC 220 Digital Circuit Design Examples  Base 8 to Decimal  Base 16 to Decimal Slide 7 of 16

8. Number Systems & Conversions Number Systems and Conversion Friday, January 9 CEC 220 Digital Circuit Design Conversion of Decimal to base R...... Quotient Slide 8 of 16

9. Number Systems & Conversions Number Systems and Conversion Friday, January 9 CEC 220 Digital Circuit Design Example:  Convert Decimal 53 (i.e. base 10) to base 2 (i.e., binary) 53 2 26 2 13 2 6 2 3 2 1 2 rem. = 1 = a 0 rem. = 0 = a 1 rem. = 1 = a 2 rem. = 0 = a 3 rem. = 1 = a 4 0 rem. = 1 = a 5 Slide 9 of 16

10. Number Systems & Conversions Number Systems and Conversion Friday, January 9 CEC 220 Digital Circuit Design Conversion of a Decimal fraction to base R...... Slide 10 of 16

11. Number Systems & Conversions Number Systems and Conversion Friday, January 9 CEC 220 Digital Circuit Design Example:  Convert Decimal 0.625 to base 2 (i.e., binary) Slide 11 of 16

12. Number Systems & Conversions Number Systems and Conversion Friday, January 9 CEC 220 Digital Circuit Design Example:  Convert 231.3 4 to base 7  First convert to decimal  Convert the integer portion (45 10 = ??? 7 )  Convert the decimal portion (.75 10 =.??? 7 ) 45 7 6 7 0rem. = 6 rem. = 3 Slide 12 of 16 …

13. Number Systems & Conversions Number Systems and Conversion Friday, January 9 CEC 220 Digital Circuit Design Binary 000 001 010 011 100 101 110 111 QUESTION: How many binary “bits” do we need to represent a single Octal digit? QUESTION: How many binary “bits” do we need to represent a single Octal digit? Slide 13 of 16 Octal 0 1 2 3 4 5 6 7

14. Number Systems & Conversions Number Systems and Conversion Friday, January 9 CEC 220 Digital Circuit Design BinaryHex Decimal 000000 000111 001022 001133 010044 010155 011066 011177 100088 100199 1010A10 1011B11 1100C12 1101D13 1110E14 1111F15 QUESTION: How many binary “bits” do we need to represent a single Hexadecimal digit? QUESTION: How many binary “bits” do we need to represent a single Hexadecimal digit? Slide 14 of 16

15. C Number Systems & Conversions Number Systems and Conversion Friday, January 9 CEC 220 Digital Circuit Design Converting to/from Binary, Hex, and Octal  An example of converting Binary to Octal  An example of converting Binary to Hexadecimal Slide 15 of 16 51127 1001101.010111 2 = 115.27 8 D45 1001101.010111 2 = 4D.5C 16

16. Next Lesson Friday, January 9 CEC 220 Digital Circuit Design Binary Arithmetic Representation of Negative Numbers Slide 16 of 16