1. CEC 220 Digital Circuit Design Number Systems & Conversions Wednesday, Aug 26 CEC 220 Digital Circuit Design Slide 1 of 16

2. Number Systems & Conversions Wednesday, Aug 26 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 Wednesday, Aug 26 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 Wednesday, Aug 26 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 Wednesday, Aug 26 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 Wednesday, Aug 26 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 Wednesday, Aug 26 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 Wednesday, Aug 26 CEC 220 Digital Circuit Design Conversion of Decimal to base R...... Quotient Slide 8 of 16

9. Number Systems & Conversions Number Systems and Conversion Wednesday, Aug 26 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 How would I check this result?

10. Number Systems & Conversions Number Systems and Conversion Wednesday, Aug 26 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 Wednesday, Aug 26 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 Wednesday, Aug 26 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 … How would I check this result?

13. Number Systems & Conversions Number Systems and Conversion Wednesday, Aug 26 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 Wednesday, Aug 26 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 Wednesday, Aug 26 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 Wednesday, Aug 26 CEC 220 Digital Circuit Design Binary Arithmetic Representation of Negative Numbers Slide 16 of 16