CS 61C L22 Representations of Combinatorial Logic Circuits (1)Garcia, Spring 2004 © UCBLecturer PSOE Dan Garciawww.cs.berkeley.edu/~ddgarciainst.eecs.berkeley.edu/~cs61cCS61C : Machine Structures Lecture 22 – Representations of Combinatorial Logic Circuits 2004-10-20E-voting talk today⇒ At 4pm in 306 Soda SU Prof.David Dill will give a talk aboutimportant issues in electronic voting.This affects us all! Get there early...www.verifiedvoting.orgvotingintegrity.comCS 61C L22 Representations of Combinatorial Logic Circuits (2)Garcia, Spring 2004 © UCBReview…• We use feedback to maintain state• Register files used to build memories• D-FlipFlops used for Register files• Clocks usually tied to D-FlipFlop load• Setup and Hold times important• Pipeline big-delay CL for faster clock• Finite State Machines extremely useful• You’ll see them again in 150, 152 & 164CS 61C L22 Representations of Combinatorial Logic Circuits (3)Garcia, Spring 2004 © UCBRepresentations of CL Circuits…• Truth Tables• Logic Gates• Boolean AlgebraCS 61C L22 Representations of Combinatorial Logic Circuits (4)Garcia, Spring 2004 © UCBTruth Tables0CS 61C L22 Representations of Combinatorial Logic Circuits (5)Garcia, Spring 2004 © UCBTT Example #1: 1 iff one (not both) a,b=1011101110000ybaCS 61C L22 Representations of Combinatorial Logic Circuits (6)Garcia, Spring 2004 © UCBTT Example #2: 2-bit adderHowManyRows?CS 61C L22 Representations of Combinatorial Logic Circuits (7)Garcia, Spring 2004 © UCBTT Example #3: 32-bit unsigned adderHowManyRows?CS 61C L22 Representations of Combinatorial Logic Circuits (8)Garcia, Spring 2004 © UCBTT Example #3: 3-input majority circuitCS 61C L22 Representations of Combinatorial Logic Circuits (9)Garcia, Spring 2004 © UCBLogic Gates (1/2)CS 61C L22 Representations of Combinatorial Logic Circuits (10)Garcia, Spring 2004 © UCBAnd vs. Or review – Dan’s mnemonicAND GateCABSymbolA B C0 0 00 1 01 0 01 1 1DefinitionANDCS 61C L22 Representations of Combinatorial Logic Circuits (11)Garcia, Spring 2004 © UCBLogic Gates (2/2)CS 61C L22 Representations of Combinatorial Logic Circuits (12)Garcia, Spring 2004 © UCB2-input gates extend to n-inputs• N-input XOR is theonly one which isn’tso obvious• It’s simple: XOR is a1 iff the # of 1s at itsinput is odd ⇒CS 61C L22 Representations of Combinatorial Logic Circuits (13)Garcia, Spring 2004 © UCBTruth Table ⇒ Gates (e.g., majority circ.)CS 61C L22 Representations of Combinatorial Logic Circuits (14)Garcia, Spring 2004 © UCBTruth Table ⇒ Gates (e.g., FSM circ.)100110000010010101000001001100000000OutputNSInputPSor equivalently…CS 61C L22 Representations of Combinatorial Logic Circuits (15)Garcia, Spring 2004 © UCBBoolean Algebra• George Boole, 19th Centurymathematician• Developed a mathematicalsystem (algebra) involvinglogic• later known as “Boolean Algebra”• Primitive functions: AND, OR and NOT• The power of BA is there’s a one-to-onecorrespondence between circuits madeup of AND, OR and NOT gates andequations in BA + means OR,• means AND, x means NOTCS 61C L22 Representations of Combinatorial Logic Circuits (16)Garcia, Spring 2004 © UCBBoolean Algebra (e.g., for majority fun.)y = a • b + a • c + b • cy = ab + ac + bcCS 61C L22 Representations of Combinatorial Logic Circuits (17)Garcia, Spring 2004 © UCBBoolean Algebra (e.g., for FSM)100110000010010101000001001100000000OutputNSInputPSor equivalently…y = PS1 • PS0 • INPUTCS 61C L22 Representations of Combinatorial Logic Circuits (18)Garcia, Spring 2004 © UCBBA: Circuit & Algebraic SimplificationBA also great for circuit verificationCirc X = Circ Y?use BA to prove!CS 61C L22 Representations of Combinatorial Logic Circuits (19)Garcia, Spring 2004 © UCBLaws of Boolean AlgebraCS 61C L22 Representations of Combinatorial Logic Circuits (20)Garcia, Spring 2004 © UCBBoolean Algebraic Simplification ExampleCS 61C L22 Representations of Combinatorial Logic Circuits (21)Garcia, Spring 2004 © UCBCanonical forms (1/2)Sum-of-products(ORs of ANDs)CS 61C L22 Representations of Combinatorial Logic Circuits (22)Garcia, Spring 2004 © UCBCanonical forms (2/2)CS 61C L22 Representations of Combinatorial Logic Circuits (23)Garcia, Spring 2004 © UCBPeer InstructionA. (a+b)• (a+b) = bB. N-input gates can be thought ofcascaded 2-input gates. I.e.,(a ∆ bc ∆ d ∆ e) = a ∆ (bc ∆ (d ∆ e))where ∆ is one of AND, OR, XOR, NANDC. You can use NOR(s) with clever wiringto simulate AND, OR, & NOT ABC1: FFF2: FFT3: FTF4: FTT5: TFF6: TFT7: TTF8: TTTCS 61C L22 Representations of Combinatorial Logic Circuits (24)Garcia, Spring 2004 © UCB“And In conclusion…”• Use this table and techniques welearned to transform from 1 to
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