6.1 Combinational Circuits. George Boole ( ) Claude Shannon ( )


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1 6. Combinational Circuits George Boole (85 864) Claude Shannon (96 2)
2 Signals and Wires Digital signals Binary (or logical ) values: or, on or off, high or low voltage Wires. Propagate digital signals from place to place. Signals "flow" from left to right. A drawing convention, sometimes violated Actually: flow from producer to consumer(s) of signal 2
3 Logic Gates Logical gates. Fundamental building blocks. x x' x y xy x y x + y NOT AND OR 3
4 Multiway AND Gates AND(x, x, x 2, x 3, x 4, x 5, x 6, x 7 ). if all inputs are. otherwise. 4
5 Multiway OR Gates OR(x, x, x 2, x 3, x 4, x 5, x 6, x 7 ). if at least one input is. otherwise. 5
6 Boolean Algebra History. Developed by Boole to solve mathematical logic problems (847). Shannon master's thesis applied it to digital circuits (937). Basics. Boolean variable: value is or. Boolean function: function whose inputs and outputs are,. Relationship to circuits. Boolean variables: signals. Boolean functions: circuits. "possibly the most important, and also the most famous, master's thesis of the [2th] century" Howard Gardner 6
7 Truth Table Truth table. Systematic method to describe Boolean function. One row for each possible input combination. N inputs 2 N rows. AND Truth Table x y AND(x, y) AND 7
8 Truth Table for Functions of 2 Variables Truth table. 6 Boolean functions of 2 variables. every 4bit value represents one Truth Table for All Boolean Functions of 2 Variables x y ZERO AND x y XOR OR Truth Table for All Boolean Functions of 2 Variables x y NOR EQ y' x' NAND ONE 8
9 Truth Table for Functions of 3 Variables Truth table. 6 Boolean functions of 2 variables. every 4bit value represents one 256 Boolean functions of 3 variables. every 8bit value represents one 2^(2^N) Boolean functions of N variables! x Some Functions of 3 Variables y z AND OR MAJ ODD 9
10 Universality of AND, OR, NOT Any Boolean function can be expressed using AND, OR, NOT. "Universal" Example: XOR(x,y) = xy' + x'y Expressing XOR Using AND, OR, NOT Notation x' x y x + y Meaning NOT x x AND y x OR y x y x' y' x'y xy' x'y + xy' XOR Exercise. Show {AND, NOT}, {OR, NOT}, {NAND}, {AND, XOR} are universal. Hint. Use DeMorgan s Law: (xy) = (x + y ) and (x + y) = (x y )
11 SumofProducts Any Boolean function can be expressed using AND, OR, NOT. Sumofproducts is systematic procedure. form AND term for each in truth table of Boolean function OR terms together x'yz Expressing MAJ Using SumofProducts z xyz' xyz xy'z MAJ y x x'yz + xy'z + xyz' + xyz
12 Translate Boolean Formula to Boolean Circuit Use sumofproducts form. XOR(x, y) = xy' + x'y. 2
13 Translate Boolean Formula to Boolean Circuit Use sumofproducts form. MAJ(x, y, z) = x'yz + xy'z + xyz' + xyz. 3
14 Simplification Using Boolean Algebra Many possible circuits for each Boolean function. Sumofproducts not necessarily optimal in: number of gates (space) depth of circuit (time) MAJ(x, y, z) = x'yz + xy'z + xyz' + xyz = xy + yz + xz. size = 8, depth = 3 size = 4, depth = 2 4
15 Expressing a Boolean Function Using AND, OR, NOT 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 sumof products. Step 4: transform Boolean expression into circuit. 5
16 6 ODD Parity Circuit ODD(x, y, z). if odd number of inputs are. otherwise. x'y'z Expressing ODD Using SumofProducts z xy'z' xyz x'yz' ODD y x x'y'z + x'yz' + xy'z' + xyz
17 ODD Parity Circuit ODD(x, y, z). if odd number of inputs are. otherwise. 7
18 Let's Make an Adder Circuit Goal: x + y = z for 4bit integers. We build 4bit adder: 9 inputs, 4 outputs. Same idea scales to 28bit adder. Key computer component. Step. Represent input and output in binary x 3 x 2 x x y 3 y 2 y y + z 3 z 2 z z + + x 3 x 2 x x y 3 y 2 y y z 3 z 2 z z 8
19 Let's Make an Adder Circuit Goal: x + y = z for 4bit integers. c Step 2. (first attempt) Build truth table. Why is this a bad idea? + x 3 y 3 z 3 x 2 y 2 z 2 x y z x y z 28bit adder: rows > # electrons in universe! 4Bit Adder Truth Table c x 3 x 2 x x y 3 y 2 y y z 3 z 2 z z = 52 rows! 9
20 Let's Make an Adder Circuit Goal: x + y = z for 4bit integers. Step 2. (do one bit at a time) Build truth table for carry bit. Build truth table for summand bit. + c 3 c = c 2 c x 3 x 2 x x y 3 y 2 y y z 3 z 2 z z Carry Bit x i y i c i c i+ Summand Bit x i y i c i z i 2
21 Let's Make an Adder Circuit Goal: x + y = z for 4bit integers. Step 3. Derive (simplified) Boolean expression. + c 3 c = c 2 c x 3 x 2 x x y 3 y 2 y y z 3 z 2 z z Carry Bit Summand Bit x i y i c i c i+ MAJ x i y i c i z i ODD 2
22 Let's Make an Adder Circuit Goal: x + y = z for 4bit integers. Step 4. Transform Boolean expression into circuit. Chain together bit adders. 22
23 Let's Make an Adder Circuit Goal: x + y = z for 4bit integers. Step 4. Transform Boolean expression into circuit. Chain together bit adders. 23
24 Subtractor Subtractor circuit: z = x  y. One approach: design like adder circuit. Better idea: reuse adder circuit. 2's complement: to negate an integer, flip bits, then add x 3 x 2 x x z 3 x + z 2 z  x  y y 3 z y 2 y y y 4Bit Subtractor Interface 4Bit Subtractor Implementation 24
25 Arithmetic Logic Unit: Interface ALU Interface. Add, subtract, bitwise and, bitwise xor, shift left, shift right, copy. Associate 3bit integer with 5 primary ALU operations. ALU performs operations in parallel control wires select which result ALU outputs op 2 Input 6 +,  & ALU 6 ^ <<, >> input 2 Input ALU select shift direction subtract 25
26 Arithmetic Logic Unit: Implementation Input Input ~ + carry in & 6 MUX ^ << >> op 2 +,  & 3 ^ <<, >> input 2 subtract shift direction ALU control 26
27 2 n to Multiplexer n = 8 for main memory 2 n to multiplexer. n select inputs, 2 n data inputs, output. Copies "selected" data input bit to output. x x x 2 x 3 x x 4 y x x 2 x 3 x 4 8 to MUX y x 5 x 6 x 5 x 7 x 6 x 7 s s s 2 s 2 s s 8to Mux Interface 8to Mux Implementation 27
28 6.2: Sequential Circuits S Q R COS 26: General Computer Science
29 Sequential vs. Combinational Circuits Combinational circuits. Output determined solely by inputs. Can draw solely with lefttoright signal paths. Sequential circuits. Output determined by inputs AND previous outputs. Feedback loop. S R Q 29
30 SR FlipFlop What is the value of Q if: S = and R =? Q is surely. S = and R =? Q is surely. S = and R =? Q is possibly... or possibly. S old Q Q R While S = R =, Q remembers what it was the last time S or R was. 3
31 SR FlipFlop SR FlipFlop. S =, R = (set) Flips bit on. S =, R = (reset) Flops bit off. S = R = Status quo. S = R = Not allowed. S Q SR flip flop R S R Q Implementation Interface 3
32 Clock Clock. Fundamental abstraction. regular onoff pulse External analog device. Synchronizes operations of different circuit elements. GHz clock means billion pulses per second. cycle time Clock 32
33 Clocked SR FlipFlop Clocked SR FlipFlop. Same as SR flipflop except S and R only active when clock is. S Cl R SR flip flop S Q R Implementation Clocked SR flip flop S Cl R Q Interface 33
34 Clocked D FlipFlop Clocked D FlipFlop. Output follows D input while clock is. Output is remembered while clock is. D Cl Clocked SR flip flop S Cl R Implementation Q Clocked D flip flop D Cl Q Interface 34
35 StandAlone Register kbit register. Stores k bits. Register contents always available on output. If write enable is asserted, k input bits get copied into register. Ex: Program Counter, 6 TOY registers, 256 TOY memory locations. x D Cl Q y x D Cl Q y 6 reg 6 write data write enable read data x 5 Write D Cl Q y 5 6bit Register Interface 6bit Register Implementation 35
36 Registers TOY registers: fancy 6 x 6bit register file. Want to be able to read two registers, and write to a third in the same instructions: R R2 + R3. 3 address inputs, data input, 2 data outputs. Add decoders and muxes for additional ports. Write Data Write Address A Address R R R2 R3 R4 R5 R8 R9 RA RB RC RD 6 6 A Data B Data B Address 4 R6 R7 RE RF Cl Write 36
37 37 Main Memory TOY main memory: 256 x 6bit register file. 6 Cl Write Address 8 Read Data 6 Write Data F E D B A 9 C F E D B A 9 C F 2E 2D 2B 2A 29 2C F 3E 3D 3B 3A 39 3C F 4E 4D 4B 4A 49 4C F 5E 5D 5B 5A 59 5C F 6E 6D 6B 6A 69 6C F 7E 7D 7B 7A 79 7C F 8E 8D 8B 8A 89 8C F 9E 9D 9B 9A 99 9C 98 A7 A6 A5 A3 A2 A A4 A AF AE AD AB AA A9 AC A8 B7 B6 B5 B3 B2 B B4 B BF BE BD BB BA B9 BC B8 C7 C6 C5 C3 C2 C C4 C CF CE CD CB CA C9 CC C8 D7 D6 D5 D3 D2 D D4 D DF DE DD DB DA D9 DC D8 E7 E6 E5 E3 E2 E E4 E EF EE ED EB EA E9 EC E8 F7 F6 F5 F3 F2 F F4 F FF FE FD FB FA F9 FC F8
38 6.3: Machine Architecture
39 The TOY Datapath 8 6 pc for branch, jump PC + addr for loads, stores pc for jal Memory Addr W Data R Data W 8 load IR op d s t result of arithmetic, logic, or addr for load addr 2 addr Registers W Data B Data W Addr A Addr B Addr A Data W Cond Eval A L U = > pc + 8 store data 39
40 Control Control: controls components, enables connections. Input: opcode, clock, conditional evaluation. (green) Output: control wires. (orange) Registers Cond Eval = > W Data PC Memory Addr R Data IR op d s 4 2 A Data B Data W Addr A Addr A L U 5 + W Data W t B Addr W bit counter Opcode Execute Fetch Clock Control 4
41 Summary Combinational circuits: how to compute things And, Or, Not primitives sufficient for any Boolean function Systematic method: truth tables and sumofproducts Examples Majority Binary adder Multiplexor Sequential circuits: where to put things Flip flop primitive holds one bit Many flip flops make a register (6 for TOY) Many registers make a register file Lots and lot of registers make a memory (256 for TOY) A whole computer Uses combinational circuits to perform computations Uses sequential circuits to store results Uses a little of each for control 4
42 The final secret A" B" OUT" A" B" OUT" A" OUT" All three of our logic primitives can be made using a single* type of electronic primitive: the transistor! *not counting the passive resistors 42
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