1. Logic Gates II Informatics INFO I101 February 5, 2003 John C. Paolillo, Instructor

2. Items for Today Last time –Logic Circuits –Applications of Boolean Logic This time –More Circuits –Addition and subtraction –Two’s complement representation –Color

3. Quiz Answers and01 000 101 (1) or01 001 111 (2) C A B Exclusive OR (XOR) xor01 001 110 (3)

4. Quiz: 10 minutes Take out a sheet of paper Write you name on it and date it Answer the following questions (next slide)

5. Quiz Questions (1)Give the truth table for Boolean AND (2)Give the truth table for Boolean OR (3)What does the following circuit do? Give a truth table for it, and name the logic gate it corresponds to. C A B

6. Graphic Paint/Copy Modes COPYORXOR

7. Arithmetic Addition and Subtraction

8. Addition: Half Adder +0001 00 01 10 S A B xor01 0 0 1 1 1 0 and01 000 101 C The half adder sends a carry, but can’t accept one +

9. Addition: Full Adder A B S CO CI

10. Addition: Truth Tables CI 0 1 A01010101A01010101 B00110011B00110011 SCO S01101001S01101001 0 1 0 1

11. More Digits Full Adder ci co s abab a0 s0 b0 Full Adder ci co s abab a1 s1 b1 Full Adder ci co s abab a2 s2 b2 Full Adder ci co s abab a3 s3 b3 Full adders can be cascaded

12. Subtraction –0001 00 01 –0100 Subtraction is asymmetrical That makes it harder We have to borrow sometimes

13. Solution: “Easy Subtraction” 456 –123 333 999 –123 876 Subtraction is easy if you don’t have to borrow i.e. if all the digits of the minuend are greater than (or equal to) all those of the subtrahend This will always be true if the minuend is all 9’s: 999, or 999999, or 9999999999 etc.

14. How can we use easy subtraction? Subtract the subtrahend from 999 (or whatever we need) (easy) Add the result to the minuend (easy enough) Add 1 (easy) Subtract 1000 (not too hard) Difference = Minuend + 999 – Subtrahend + 1 – 1000 This works for binary as well as decimal

15. Subtraction Example 10010101 –01101110 ????????? 11111111 –01101110 10010001 This is the same as inverting each bit + 10010101 100100110 +1 100100111 –100000000 00100111 Regular addition Add one Now drop the highest bit (easy: it’s out of range) 00100111

16. Subtraction Procedure Invert each bit Regular addition Add one Now drop the highest bit (easy: it’s out of range) Each of these steps is a simple operation we can perform using our logic circuits Bitwise XOR Cascaded Adders Add carry bit Drop the highest bit (overflows)

17. Negative Numbers Invert each bit Add one These steps make the negative of a number in twos-complement notation Twos complements can be added to other numbers normally Positive numbers cannot use the highest bit (the sign bit) This is the normal representation of negative numbers in binary

18. Counting 000000000 000000011 000000102 000000113 000001004 000001015 000001106 000001117 000010008 000010019 etc. 11111111–1 11111110–2 11111101–3 11111100–4 11111011–5 11111010–6 11111001–7 11111000–8 11110111–9 11110110–10 etc.

19. Representations The number representation you use (encoding) affects the way you need to do arithmetic (procedure) This is true of all codes: encoding (representation) affects procedure (algorithm) Good binary codes make use of properties of binary numbers and digital logic

20. Color (review)

21. The Eight-Color Computer Eight colors: black, yellow, magenta, red, cyan, green, blue, white Three color tubes on a TV monitor: Red, Green, Blue 2 3 =8 Additive color relations: red+green+blue=white

22. Color Perception 3 Electron guns, aimed at 3 different colors of phosphor dots 3 types of retinal sensor cells, sensitive to 3 different bands of light

23. Color: Response Patterns red cones green cones blue cones Wavelength 

24. Color: Neural Encoding Wavelength  110 RGB 101100110010011001101 000111