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Basic Gates - Design Techniques for Digital Systems - Lecture Slides, Slides of Digital Systems Design
In the course of the Design Techniques for Digital Systems, we study the key concept regarding the digital system. The major points in these lecture slides are:Basic Gates, Boolean Algebra, Boolean Function Representations, Canonical Form, Two-Level Function Minimization, Shannon Expansion, Universal Gates, Universality Check, Logic Minimization, Essential Prime Implicates
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2012/2013
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Midterm 1 Review
Course materials covered so far ^ Boolean algebra^ Axioms & theorems ^ Basic gates ^ Boolean function representations^ Canonical form, SoP, PoS ^ Two-level function minimization^ Kmap, Quine-McCluskey Approach ^ Other issues^ Shannon expansion
Boolean Algebra (2)^ ( a^ +^ c )( a ’ +^ c ’)( b ’ +
c^ +^ d ’)( a^ +^ b ’ +^ d
’) = ( a^ +^ c )( a ’ +^ c ’)(
b ’ +^ d ’)
Proof: ( a^ +^ c )( a ’ +
c ’)( b ’ +^ c^ +^ d ’)( a^ +^ b ’ +^ d ’) = (a + c) (a’ + c’) ((ac) + (b’ + d’))
distributivity = (a + c) (a’ + c’) ac + (a + c) (a’ + c’) (b’ + d’)
distributivity = (a + c) (a’ac + c’ac) + (a + c) (a’ + c’) (b’ + d’) distributivity= (a + c) (0 + 0) + (a + c) (a’ + c’) (b’ + d’)
complement = 0 + (a + c) (a’ + c’) (b’ + d’)
nullity = (a + c) (a’ + c’) (b’ + d’)
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Universal Gates (1)^ Universality check: {
f ( x , y )}, where^ f^ ( x ,^ y ) =^ x + y ’, assuming 0 & 1 are available as inputsStrategy: Construct AND, OR & NOT using {
f ( x , y )} x 0 f(x,y)y^ x 0 f(x,y)y x f(x,y)y^
x 0 f(x,y)y^ x xy^ 0 f(x,y)y f(x,y)^ What if 0 & 1 arenot available?
Two-level Logic Minimization: SoP^0
ab^00 01 11 10^ Essential prime implicants: cd a’c’d, ad’, ab x^100^ Non-essential prime implicants:bc’, bd’, cd,^1 1 1 001110 0 1 0 Minimum SoP cover:a’c’d+ad’+ab+bc’ x^ x^ x^110^ a’c’d+ad’+ab+bd’ Essential prime implicants must be included in the cover!
Two-level Logic Minimization: PoS^0
ab^00 01 11 10^ Essential prime implicates: cd b+d, b+c’, b’+c+d’ x^000^ Non-essential prime implicates:a’+c+d^1 0 x^101110 1 1 0 Minimum PoS cover:(b+d)(b+c’)(b’+c+d’) x^1 1 010 Essential prime implicates must be included in the cover!