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CENG 241 Digital Design 1 Lecture 2 Amirali Baniasadi amirali@ece.uvic.ca

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2 This Lecture zReview of last lecture zBoolean Algebra

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3 Boolean Function: Example zTruth table zx y z F1 F2 z 0 0 0 0 0 z 0 0 1 1 1 z 0 1 0 0 0 z 0 1 1 0 1 z 1 0 0 1 1 z 1 0 1 1 1 z 1 1 0 1 0 z 1 1 1 1 0 A Boolean Function can be represented in only one truth table forms

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4 Canonical & Standard Forms zConsider two binary variables x, y and the AND operation zfour combinations are possible: x.y, x’.y, x.y’, x’.y’ zeach AND term is called a minterm or standard products zfor n variables we have 2 n minterms zConsider two binary variables x, y and the OR operation zfour combinations are possible: x+y, x’+y, x+y’, x’+y’ zeach OR term is called a maxterm or standard sums zfor n variables we have 2 n maxterms

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5 Minterms zx y z Terms Designation z 0 0 0 x’.y’.z’ m0 z 0 0 1 x’.y’.z m1 z 0 1 0 x’.y.z’ m2 z 0 1 1 x’.y.z m3 z 1 0 0 x.y’.z’ m4 z 1 0 1 x.y’.z m5 z 1 1 0 x.y.z’ m6 z 1 1 1 x.y.z m7

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6 Maxterms zx y z Designation Terms z 0 0 0 M0 x+y+z z 0 0 1 M1 x+y+z’ z 0 1 0 M2 x+y’+z z 0 1 1 M3 x+y’+z’ z 1 0 0 M4 x’+y+z z 1 0 1 M5 x’+y+z’ z 1 1 0 M6 x’+y’+z z 1 1 1 M7 x’+y’+z’

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7 Boolean Function: Exampl How to express algebraically z1.Form a minterm for each combination forming a 1 z2.OR all of those terms zTruth table example: zx y z F1 minterm z 0 0 0 0 z 0 0 1 1 x’.y’.z m1 z 0 1 0 0 z 0 1 1 0 z 1 0 0 1 x.y’.z’ m4 z 1 0 1 0 z 1 1 0 0 z 1 1 1 1 x.y.z m7 zF1=m1+m4+m7=x’.y’.z+x.y’.z’+x.y.z=Σ(1,4,7)

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8 Boolean Function: Exampl How to express algebraically z1.Form a maxterm for each combination forming a 0 z2.AND all of those terms zTruth table example: zx y z F1 maxterm z 0 0 0 0 x+y+z M0 z 0 0 1 1 z 0 1 0 0 x+y’+z M2 z 0 1 1 0 x+y’+z’ M3 z 1 0 0 1 z 1 0 1 0 x’+y+z’ M5 z 1 1 0 0 x’+y’+z M6 z 1 1 1 1 zF1=M0.M2.M3.M5.M6 = л(0,2,3,5,6)

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9 Implementations Three-level implementation vs. two-level implementation Two-level implementation normally preferred due to delay importance.

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10 Digital Logic Gates

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11 All gates -except for the inverter and buffer- can be extended to have more than two inputs zA gate can be extended to multiple inputs if the operation represented is commutative & associative zx+y=y+x z(x+y)+z=x+(y+z) Extension to Multiple Inputs

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12 Extension to Multiple Inputs We define multiple input NAND and NOR as:

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13 Extension to Multiple Inputs What about multiple input XOR? ODD function: 1 if the number of 1’s in the input is odd

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14 Positive and Negative Logic Two values of binary signals

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15 Integrated Circuits (ICs) zLevels of Integration zSSI: fewer than 10 gates on chip zMSI:10 to 1000 gates on chip zLSI: thousands of gates on chip zVLSI:Millions of gates on chip z Digital Logic Families zTTL transistor-transistor logic zECL emitter-coupled logic zMOS metal-oxide semiconductor zCMOS complementary metal-oxide semiconductor

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16 Digital Logic Parameters zFan-out: maximum number of output signals zFan-in : number of inputs zPower dissipation zPropagation delay zNoise margin: maximum noise

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17 CAD- Computer-Aided Design zHow do they design VLSI circuits???? zBy CAD tools zMany options for physical realization: FPGA, ASIC… zHardware Description Language (HDL): zRepresents logic design in textual format zResembles a programming language

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18 Gate-Level Minimization zThe Map Method: zA simple method for minimizing Boolean functions zMap: diagram made up of squares zEach square represents a minterm

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19 Two-Variable Map

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20 Two-Variable Map Maps representing x.y and x+y

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21 Three-Variable Map

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22 Three-Variable Map-example 1

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23 Summary zExtension to multiple inputs zPositive & Negative Logic zIntegrated Circuits zGate Level Minimization

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