6.1 Combinational Circuits

Transcription

1 6. Combinational Circuits Digital signals Binar (or logical ) values: or, on or off, high or low voltage Wires. Propagate logical values from place to place. ignals "flow" from left to right. ignals and Wires A drawing convention, sometimes violated Actuall: flow from producer to consumer(s) of signal George Boole (85 864) Claude hannon (96 2) Input Output 2 Logic Gates Multiwa AND Gates Logical gates. Fundamental building blocks. AND(,,,, 4, 5, 6, 7 ). if all inputs are. otherwise. ' NOT AND O 3 4

2 Multiwa O Gates Boolean Algebra O(,,,, 4, 5, 6, 7 ). if at least one input is. otherwise. Histor. Developed b Boole to solve mathematical logic problems (847). hannon master's thesis applied it to digital circuits (937). Basics. Boolean variable: value is or. Boolean function: function whose inputs and outputs are,. elationship to circuits. Boolean variables: signals. Boolean functions: circuits. "possibl the most important, and also the most famous, master's thesis of the [2th] centur" --Howard Gardner 5 6 Truth Table Truth Table for Functions of 2 Variables Truth table. stematic 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. ever 4-bit value represents one AND Truth Table AND(, ) AND Truth Table for All Boolean Functions of 2 Variables ZEO AND XO Truth Table for All Boolean Functions of 2 Variables NO E ' ' NAND O ONE 8 9

3 Truth Table for Functions of 3 Variables Universalit of AND, O, NOT Truth table. 6 Boolean functions of 2 variables. ever 4-bit value represents one 256 Boolean functions of 3 variables. ever 8-bit value represents one 2^(2^N) Boolean functions of N variables ome Functions of 3 Variables z AND O MAJ ODD An Boolean function can be epressed using AND, O, NOT. "Universal." XO(,) = ' ' Epressing XO Using AND, O, NOT ' ' ' ' ' ' XO Eercise. how {AND, NOT}, {O, NOT}, {NAND}, {AND, XO} are universal. Hint. Use DeMorgan s Law: () = ( ) and ( ) = ( ) Notation ' Meaning NOT AND O um-of-products Translate Boolean Formula to Boolean Circuit An Boolean function can be epressed using AND, O, NOT. um-of-products is sstematic procedure. form AND term for each in truth table of Boolean function O terms together Use sum-of-products form. XO(, ) = ' '. Epressing MAJ Using um-of-products z MAJ 'z 'z z' z 'z 'z z' z 2 3

4 Translate Boolean Formula to Boolean Circuit implification Using Boolean Algebra Use sum-of-products form. MAJ(,, z) = 'z 'z z' z. Man possible circuits for each Boolean function. um-of-products not necessaril optimal in: number of gates (space) depth of circuit (time) MAJ(,, z) = 'z 'z z' z = z z. size = 8, depth = 3 size = 4, depth = Epressing a Boolean Function Using AND, O, NOT ODD Parit Circuit Ingredients. AND gates. O gates. NOT gates. Wire. Instructions. tep : represent input and output signals with Boolean variables. tep 2: construct truth table to carr out computation. tep 3: derive (simplified) Boolean epression using sum-of products. tep 4: transform Boolean epression into circuit. ODD(,, z). if odd number of inputs are. otherwise. Epressing ODD Using um-of-products z ODD ''z 'z' 'z' z ''z 'z' 'z' z 6 8

5 ODD Parit Circuit Let's Make an Adder Circuit ODD(,, z). if odd number of inputs are. otherwise. Goal: = z for 4-bit integers. We build 4-bit adder: 9 inputs, 4 outputs. ame idea scales to 28-bit adder. Ke computer component. tep. epresent input and output in binar c z z c 3 c 2 c c 3 2 z z 9 2 Let's Make an Adder Circuit Let's Make an Adder Circuit Goal: = z for 4-bit integers. tep 2. (first attempt) Build truth table. c 3 2 Goal: = z for 4-bit integers. tep 2. (do one bit at a time) Build truth table for carr bit. c 3 c = c 2 c 3 2 Wh is this a bad idea? z z Build truth table for summand bit. z z 28-bit adder: rows > # electrons in universe c. 4-Bit Adder Truth Table 3 2 z z 2 8 = 52 rows Carr Bit i i c i c i ummand Bit i i c i z i 2 22

6 Let's Make an Adder Circuit Let's Make an Adder Circuit Goal: = z for 4-bit integers. tep 3. Derive (simplified) Boolean epression. c 3 c = c 2 c 3 2 Goal: = z for 4-bit integers. tep 4. Transform Boolean epression into circuit. z z Chain together -bit adders. Carr Bit ummand Bit i i c i c i MAJ i i c i z i ODD Let's Make an Adder Circuit ubtractor Goal: = z for 4-bit integers. tep 4. Transform Boolean epression into circuit. Chain together -bit adders. ubtractor circuit: z = -. One approach: new design, like adder circuit. Better idea: reuse adder circuit. 2's complement: to negate an integer, flip bits, then add carr z z 4-Bit ubtractor Interface 4-Bit ubtractor Implementation 25 26

7 TOY Arithmetic Logic Unit: Interface n = 8 for main memor 2 n -to- Multipleer ALU Interface. Add, subtract, bitwise and, bitwise or, shift left, shift right, cop. Associate 3-bit integer with 5 primar ALU operations. ALU performs operations in parallel control wires select which result ALU outputs 2 n -to- multipleer. n select inputs, 2 n data inputs, output. Copies "selected" data input bit to output. op 2, - & ^ <<, >> input 2 Input Input ALU to MUX select shift direction subtract ALU select 27 7 s s s 2 8-to- Mu Interface s 2 s s 8-to- Mu Implementation 28 TOY Arithmetic Logic Unit: Implementation Input Input ~ carr in 6.2: equential Circuits & MUX 6 ^^ op 2, - & ^ <<, >> input 2 subtract << >> shift direction 3 ALU select 29 CO 26: General Computer cience

8 equential vs. Combinational Circuits Flip-Flop Combinational circuits. Output determined solel b inputs. Flip-flop. A small and useful sequential circuit. Abstraction that "remembers" one bit. Can draw solel with left-to-right Basis of important computer components: signal paths. equential circuits. Output determined b inputs memor counter We will consider several flavors. AND previous outputs. Feedback loop Flip-Flop Flip-Flop What is the value of if: = and =? is surel What is the value of if: = and =? is surel = and =? is surel 33 34

9 Flip-Flop Flip-Flop What is the value of if: = and =? is surel = and =? is surel " What is the value of if: = and =? is surel = and =? is surel " = and =? is possibl = and =? is possibl. or possibl Flip-Flop Flip-Flop What is the value of if: = and =? is surel = and =? is surel " = and =? is possibl. or possibl Flip-Flop. =, = (et) Flips bit on. =, = (eset) Flops bit off. = = tatus quo. = = Not allowed. old flip flop While = =, remembers what it was the last time or was. Implementation Interface 37 38

10 Clock How much does it Hert? Clock. Fundamental abstraction. regular on-off pulse From some oscillating device, possibl eternal. nchronizes operations of different circuit elements. GHz clock means billion pulses per second. ccle time Frequenc is inverse of ccle time. Epressed in hertz. Frequenc of Hz means that there is ccle per second. Hence: kilohertz (khz) means ccles/sec. megahertz (MHz) means million ccles/sec. gigahertz (GHz) means billion ccles/sec. terahertz (THz) means trillion ccles/sec. B the wa, no such thing as hert Clock Heinrich udolf Hertz ( ) 39 4 Clocked Flip-Flop Clocked D Flip-Flop Clocked Flip-Flop. ame as flip-flop ecept and onl active when clock is. Implementation flip flop Clocked flip flop Interface Clocked D Flip-Flop. Output follows D input while clock is. Output is remembered while clock is. D Cl Clocked flip flop Implementation Clocked D flip flop D Interface 4 D 42

11 ummar Combinational circuits implement Boolean functions Gates and wires Fundamental building blocks. Truth tables. Describe Boolean functions. um-of-products. stematic method to implement functions. equential circuits add "state" to digital hardware. Flip-flop. epresents bit. TOY register. 6 D flip-flops. TOY main memor. 256 registers. Net time: we build a complete TOY computer (oh es). George Boole (85 864) Claude hannon (96 2) 43 44