6. Combinational Circuits. Building Blocks. Digital Circuits. Wires. Q. What is a digital system? A. Digital: signals are 0 or 1.
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1 Digital Circuits 6 Combinational Circuits Q What is a digital system? A Digital: signals are or analog: signals vary continuously Q Why digital systems? A Accurate, reliable, fast, cheap Basic abstractions On, off Wire: propagates on/off value Switch: controls propagation of on/off values through wires George Boole (85 864) Claude Shannon (96 2) Applications Cell phone, ipod, antilock brakes, microprocessors, 2 Wires Building Blocks Wires ON (): connected to power OFF (): not connected to power If a wire is connected to a wire that is on, that wire is also on Typical drawing convention: "flow" from top, left to bottom, right thick wires are ON power connection thin wires are OFF 4
2 Controlled Switch Controlled Switch Controlled switch 3 connections: input, output, control Controlled switch 3 connections: input, output, control control OFF: output is connected to input control OFF input OFF output OFF control OFF input ON output ON 5 Controlled Switch Controlled Switch Controlled switch 3 connections: input, output, control control ON: output is disconnected from input Controlled switch 3 connections: input, output, control control OFF: output is connected to input control ON: output is disconnected from input control ON control OFF control ON output OFF input OFF output OFF output OFF control ON control OFF control ON output OFF input ON output ON output OFF idealized model of pass transistors found in real integrated circuits 7 8
3 Implementing a Controlled Switch Relay implementation 3 connections: input, output, control Magnetic force pulls on a contact that cuts electrical flow First Level of Abstraction Separates physical world from logical world we assume that switches operate as specified that is the only assumption physical realization of switch is irrelevant to design Physical realization dictates performance size speed power New technology immediately gives new computer Better switch? Better computer 9 Controlled Switches: A First Level of Abstraction Controlled Switches: A First Level of Abstraction Some amusing attempts to prove the point: Real-world examples that prove the point: technology switch Technology Information Switch relay pneumatic air pressure vacuum tube fluid relay water pressure electric potential transistor pass transistor in integrated circuit atom-thick transistor 2
4 Controlled Switches: A First Level of Abstraction? Circuit Anatomy VLSI = Very Large Scale Integration Technology: Deposit materials on substrate drawing Key property: Crossing lines are controlled switches drawing Key challenge in physical world: Fabricating physical circuits with billions of controlled switches Key challenge in abstract world: Understanding behavior of circuits with billions of controlled switches circuit Bottom line: Circuit = Drawing (!) need more levels of abstraction to understand circuit behavior 3 4 Second Level of Abstraction: Logic Gates Second Level of Abstraction: Logic Gates 5 6
5 Multiway Gates Multiway gates OR: if any input is ; otherwise AND: if all inputs are ; otherwise Generalized: negate some inputs Multiway Gates Multiway gates OR: if any input is ; otherwise AND: if all inputs are ; otherwise Generalized: negate some inputs 7 8 Building blocks (summary) Wires Boolean Algebra Controlled switches Gates Generalized multiway gates u v w x y z u v w x y z interpretation: value is iff variables with inverters are (and others ) AND u vwx y z abstract AND gate implementation underlying circuit simpler version 9
6 Boolean Algebra Boole Orders Lunch History Developed by Boole to solve mathematical logic problems (847) Shannon master's thesis applied it to digital circuits (937) Boolean algebra Boolean variable: value is or Boolean function: function whose inputs and outputs are, Relationship to circuits Boolean variable: signal Boolean function: circuit possibly the most important, and also the most famous, master's thesis of the [2th] century Howard Gardner Copyright 24, Sidney Harris Truth Table Truth Table for Functions of 2 Variables Truth table Systematic method to describe Boolean function One row for each possible input combination n inputs! 2 n rows Truth table 6 Boolean functions of 2 variables every 4-bit value represents one x y ZERO AND x y XOR OR x y x y truth table for all Boolean functions of 2 variables x y NOR EQ y' x' NAND ONE AND truth table truth table for all Boolean functions of 2 variables 23 24
7 Truth Table for Functions of 3 Variables Universality of AND, OR, NOT Truth table 6 Boolean functions of 2 variables 256 Boolean functions of 3 variables 2^(2^n) Boolean functions of n variables! every 4-bit value represents one every 8-bit value represents one every 2 n -bit value represents one Fact Any Boolean function can be expressed using AND, OR, NOT { AND, OR, NOT } are universal Ex: XOR(x, y) = xy' + x'y notation x' meaning NOT x x y x AND y x + y x OR y x y z AND OR MAJ ODD x y x' Expressing XOR Using AND, OR, NOT y' x'y xy' x'y + xy' x XOR y Exercise Show {AND, NOT}, {OR, NOT}, {NAND} are universal Hint DeMorgan's law: (x' y')' = x + y some functions of 3 variables Sum-of-Products Translate Boolean Formula to Boolean Circuit Sum-of-products Systematic procedure for representing a Boolean function using AND, OR, NOT Form AND term for each in Boolean function OR terms together proves that { AND, OR, NOT } are universal Sum-of-products XOR x y z MAJ x'yz xy'z xyz' xyz x'yz + xy'z + xyz' + xyz expressing MAJ using sum-of-products 27 28
8 Translate Boolean Formula to Boolean Circuit Translate Boolean Formula to Boolean Circuit Sum-of-products XOR Key transformation from abstract to real circuit Example XOR Key transformation from abstract to real circuit AND AND AND AND 29 3 Translate Boolean Formula to Boolean Circuit Translate Boolean Formula to Boolean Circuit Example 2 Majority Example 2 Majority 3 32
9 Translate Boolean Formula to Boolean Circuit Translate Boolean Formula to Boolean Circuit Example 2 Majority Example 2 Majority Simplification Using Boolean Algebra Combinational Circuit Design: Summary Many possible circuits for each Boolean function Sum-of-products not necessarily optimal in: number of switches (space) depth of circuit (time) Ex MAJ(x, y, z) = x'yz + xy'z + xyz' + xyz = xy + yz + xz Problem: Compute the value of a boolean function Ingredients AND gates OR gates NOT gates Wire Instructions Step : represent input and output signals with Boolean variables Step 2: construct truth table to carry out computation Step 3: derive (simplified) Boolean expression using sum-of products Step 4: transform Boolean expression into circuit Bottom line (profound idea): It is easy to design a circuit to compute ANY boolean function size =, depth = 2 size = 7, depth = 2 35 Caveat (stay tuned): Circuit might be huge 36
10 Translate Boolean Formula to Boolean Circuit Example 3 Odd parity if odd number of inputs are otherwise Translate Boolean Formula to Boolean Circuit Example 3 Odd parity if odd number of inputs are otherwise x y z ODD x'y'z x'yz' xy'z' xyz x'y'z + x'yz' + xy'z' + xyz Expressing ODD using sum-of-products Translate Boolean Formula to Boolean Circuit Example 3 Odd parity if odd number of inputs are otherwise Adder Circuit 39
11 Let's Make an Adder Circuit Let's Make an Adder Circuit Goal x + y = z for 4-bit integers We build 4-bit adder: 9 inputs, 4 outputs Each output bit is a boolean function of the inputs Standard method applies same idea scales to 64-bit adder in your computer Goal x + y = z for 4-bit integers Step 2 [first attempt] Build truth table c out + c in x 3 x 2 x x y 3 y 2 y y z 3 z 2 z z Step Represent input and output in binary bit adder truth table + c x 3 x 2 x x y 3 y 2 y y z 3 z 2 z z 2 8+ = 52 rows! x 3 x 2 x x + y 3 y 2 y y z 3 z 2 z z Q Why is this a bad idea? A 28-bit adder: rows >> # electrons in universe! 4 42 Let's Make an Adder Circuit Let's Make an Adder Circuit Goal x + y = z for 4-bit integers c out c 3 c = c 2 c Goal x + y = z for 4-bit integers c out c 3 c = c 2 c Step 2 Do one bit at a time! Build truth table for carry bit Build truth table for summand bit + x 3 x 2 x x y 3 y 2 y y z 3 z 2 z z Step 3 A surprise! carry bit is majority function summand bit is odd parity function + x 3 x 2 x x y 3 y 2 y y z 3 z 2 z z carry bit summand bit carry bit summand bit x i y i c i c i+ x i y i c i z i x i y i c i c i+ MAJ x i y i c i z i ODD 43 44
12 Let's Make an Adder Circuit Adder: Interface Goal x + y = z for 4-bit integers Step 4 Transform Boolean expression into circuit (use known components!) Chain together -bit adders That s it! x x x2 x3 y y y2 y3 input buses output bus z z z2 z3 A bus is a group of wires that connect (carry data values) to other components Adder: Component Level View Adder: Switch Level View x x x2 x3 y y y2 y3 input buses x x x2 x3 y y y2 y3 input buses output bus output bus z z z2 z3 z z z2 z
13 Useful Combinational Circuits Decoder Adder Decoder [n-bit] n address inputs, 2 n data outputs Addressed output bit is ; others are Compact implementation of n Boolean functions Incrementer (easy, add ) x x x2 z z z2 z3 z4 z5 z6 z7 Bitwise AND, XOR (easy) Decoder [next slide] Shifter (clever, but we ll skip details) Multiplexer [next lecture] 3-bit Decoder 49 5 Decoder Decoder application: Your computer s ALU! Decoder [n-bit] n address inputs, 2 n data outputs Addressed output bit is ; others are Compact implementation of n Boolean functions ALU: Arithmetic and Logic Unit implements instructions input, output connects to registers via buses -bit input buses TOY opcode bits sub x x x2 z z z2 z3 z4 z5 z6 z7 Ex: TOY-Lite ( bit words) : add 2: subtract 3: and 4: xor 5: shift left 6: shift right add and xor shift right 3-bit Decoder Details: All circuits compute their result Decoder lines AND all results one-hot OR collects answer -bit output bus 5 52
14 Summary Lessons for software design apply to hardware design! Interface describes behavior of circuit Implementation gives details of how to build it Layers of abstraction apply with a vengeance! On/off Controlled switch [relay, transistor] Gates [AND, OR, NOT] Boolean circuit [MAJ, ODD] Adder Shifter Arithmetic logic unit TOY machine (stay tuned) Your computer 53
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