1 COMP541 Combinational Logic - II Montek Singh Aug 27, 2014.
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Transcript of 1 COMP541 Combinational Logic - II Montek Singh Aug 27, 2014.
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Today Digital Circuits (review)
Basics of Boolean Algebra (review) Identities and Simplification
Basics of Logic ImplementationMinterms and maxtermsGoing from truth table to logic implementation
Digital Circuits Digital Circuit = network that processes binary
variablesone or more binary inputsone or more binary outputs
inputs and outputs are called “terminals”a functional specification
relationship between inputs and outputsa timing specification
describes delay from inputs changing to outputs responding
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Digital Circuits Inside the black box
subcircuits or components or elements
connected by wireswires and terminals often
called “nodes”each node has a binary
valueeach node is an input, an
output, or “internal”
Example:E1, E2, E3 are elementsA, B, C are input nodesY, Z are output nodesn1 is an internal node 4
Types of circuits Two types: with memory and without
Combinational Circuitoutput depends only on the current values of the inputs
– provided enough time is given for output to respondoutput does not depend on past inputs or outputscalled “memoryless”example: AND gate
Sequential Circuitanything not combinational is sequentialoutput depends on not only current inputs, but also past
behavior– previous inputs and/or outputs affect behavior
has “memory”, or is “stateful”example: counter
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Combinational Circuits Theorem: A circuit is combinational if:
every element is itself combinationalevery node is either designated as an input, or
connects to exactly one output terminal of an elementoutputs of two elements are never “shorted together”ensures that each node has a unique/unambiguous value
contains no cyclic pathsevery path through the circuit visits each node at most
onceno “feedback”
Conditions above ensure that output is only a function of inputsProof: By induction
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Identities in Boolean Algebra Use identities to manipulate functions You can use distributive law …
… to transform
ZY X F
))(( ZXY X F
))(( ZX YX YZ X
to
Substitution Can substitute arbitrarily large algebraic
expressions for the variablesDistribute an operation over the entire expressionExample:
X + YZ = (X+Y)(X+Z)
Substitute ABC for X
ABC + YZ = (ABC + Y)(ABC + Z)
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Algebraic/Boolean Manipulation Apply algebraic and Boolean identities to
simplify expressionexample: XZZYX YZ X F
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Consensus Theorem
The third term is redundantCan just drop third term (consensus term)
Proof summary (for first version):For third term to be true, Y & Z both must be 1Then one of the first two terms is already 1!
Exercise: Provide a similar proof for the 2nd version
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Complement of a Function Definition: 1s & 0s swapped in truth table Mechanical way to derive algebraic form
Take the dualRecall: Interchange AND and OR, and 1s & 0s
Complement each literalx becomes x’ (x’ means complement of x)