Today

1. Today • Continue to take some of the “MAGIC” out of how computers work: • Example of Integrated Circuit Design – adding two numbers. CISC101 - Prof. McLeod

2. Summary: “Truth” or Logic Tables AND OR I1 I2 Output I1 I2 Output NOT I Output 0 0 0 0 0 0 0 1 0 0 1 1 0 1 1 0 0 1 0 1 1 0 1 1 1 1 1 1 NAND NOR XOR I1 I2 Output I1 I2 Output I1 I2 Output 0 0 1 0 0 1 0 0 0 0 1 1 0 1 0 0 1 1 1 0 1 1 0 0 1 0 1 1 1 0 1 1 0 1 1 0 CISC101 - Prof. McLeod

3. Summary, Cont. • Gates: You can use transistors to build a circuit, or “gate”, that provides the logic for each of these operators. • Notation: • ab (you will also see just “ab”) means “a AND b” • a+b means “ a OR b” • a means NOT(a) • ab means “a XOR b” CISC101 - Prof. McLeod

4. Digital Circuits • Generally, a digital circuit can be represented by: • Remember that an input or output can only be either on (1) or off (0). • The gates defined above can be used to design circuits that provide a desired logic. CISC101 - Prof. McLeod

5. For Example: Adding Binary Numbers • Show how to build a circuit to add binary numbers - a “one bit full adder”. • First: How do youadd decimal (base 10) numbers?: 0 1 0 0 0 4 5 7 0 1 2 3 4 7 2 6 9 1 7 3 Carry digits + Sum CISC101 - Prof. McLeod

6. Adding Binary Numbers, for Example • The binary numeric system consists of zeros and ones – only. Also called “base 2”. • Adding binary numbers: Carry bits 1 1 1 0 0 0 1 1 0 1 0 1 1 0 0 1 1 0 1 1 0 1 0 0 1 1 0 0 0 1 1 1 0 0 + Sum CISC101 - Prof. McLeod

7. Half Bit Adder • Look at the first column of the process: carry output 1 1 1 0 0 0 1 1 0 1 0 1 1 0 0 1 1 0 1 1 0 1 0 0 1 1 0 0 0 1 1 1 0 0 input 1 + input 2 sum output CISC101 - Prof. McLeod

8. Half Bit Adder - Cont. • Construct a truth table to describe the operation, considering every possible input: Which gate produces the sum? The carry? CISC101 - Prof. McLeod

9. Half Bit Adder - Cont. • Write the boolean expressions for the sum output and the carry output: • Where “S” is the sum output and “Co” is the carry output. or CISC101 - Prof. McLeod

10. Half Bit Adder - Cont. • Logical diagram for the sum output: • (This is another way to build a “XOR” gate.) not and or CISC101 - Prof. McLeod

11. Test XOR Gate Logic not and 1 0 or 0 0 1 1 1 0 1 0 1 1 1 1 0 0 1 0 CISC101 - Prof. McLeod

12. Test XOR Gate, Cont. not and 0 1 or 0 0 0 0 0 0 1 1 0 0 1 0 1 0 1 0 CISC101 - Prof. McLeod

13. Half Bit Adder - Cont. • Back to the expression for “S”: So, S can be generated by an “XOR” gate and Co by an “AND” gate: I1 S I2 Co CISC101 - Prof. McLeod

14. Full Bit Adder • Look at one column of the addition process that also adds the carry bit: carry output carry input 1 1 1 0 0 0 1 1 0 1 0 1 1 0 0 1 1 0 1 1 0 1 0 0 1 1 0 0 0 1 1 1 0 0 input 1 + input 2 sum output CISC101 - Prof. McLeod

15. Full Bit Adder - Cont. • Construct a logic table to describe what happens - list every input possibility: CISC101 - Prof. McLeod

16. Full Bit Adder - Cont. • Create boolean expressions: • Where “I” is input, “Ci” is carry input, “Co” is carry output and “S” is sum output. • (Don’t panic! You don’t have to know where these expressions come from and you don’t have to remember them!) CISC101 - Prof. McLeod

17. Full Bit Adder - Cont. • You can write the logical diagram from these expressions as they are but it can be simplified using two “half-bit” adders and an OR gate: Ci S I1 Co I2 CISC101 - Prof. McLeod

18. Full Bit Adder - Cont. • How many transistors required? • AND and OR have 2 each, NOT has 1 (and a resistor) • Then XOR must have 8 (2 NOTs, 2 ANDs, 1 OR) • Half-bit adder must have 10 (1 XOR, 1 AND) • Full-bit adder must have 22 (2 half-bits, 1 OR) CISC101 - Prof. McLeod

19. N Bit Full Adder CISC101 - Prof. McLeod

20. N Bit Full Adder – Cont. • How many transistors? • 22 times N • Integers might occupy 4 bytes or 32 bits for example: 704 transistors, minimum! • Just to add two integers! CISC101 - Prof. McLeod

21. Aside - Minecraft Constructions • People with a lot of time on their hands have built adders like these in Minecraft. • For example, see these Youtube videos: • One bit: http://www.youtube.com/watch?v=tTAuD3Y4meQ • 32 bit: http://www.youtube.com/watch?v=R-UdWu60eaI CISC101 - Prof. McLeod

22. Other Operations • So, a full adder can be used to produce a N+1-bit number as the sum of two other N bit numbers. • How about multiplication? • From Grade 4: 11011  1101 11011 000000 1101100 11011000 101011111 Base 2: 27 • 13 81 270 351 Base 10: CISC101 - Prof. McLeod

23. Multiplication, Cont. • So what kind of logic would you build to do multiplication of x times y? • Create a running sum, s. • For each digit, xi: • If xi equals 1 then add 2i y to x. • Use a bitwise shift operation to calculate 2i y (ie. Add a zero to the right of the number). • So, an N-bit adder would be used along with some other simple circuits. CISC101 - Prof. McLeod

24. Summary of Digital Logic • The CPU needs to carry out many other operations than just addition: • subtraction • comparison • multiplication • division, etc. • Some common functions (trig, exp, log, etc) would work faster if they were hard coded into the processor too. • Lots (and lots!) of transistors!! CISC101 - Prof. McLeod

25. Aside – My CPU, for Example • Intel Core i7 2670QM, 2.2 GHz (up to 3.2 GHz): • 32 nm technology • 6MB cache • 4 cores, 8 threads • 995 million transistors, total. • 988 pins. CISC101 - Prof. McLeod

26. “Don’t Worry, Be Happy!” • You do not need to be able to design microprocessor circuits! • Besides, INTEL and AMD have some fancy programs that do all the work. • The purpose of this discussion has been to give you some idea of how an integrated circuit carries out instructions. (Not MAGIC! Just some good technology!) CISC101 - Prof. McLeod