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Unformatted text preview: x x y y x!y x x x y y Homework 1 Solutions
x!y ECE 152A
Summer 2011
H.O. #4 y x y x!y A similar proof is constructed for 15b.
2.6. Proof using Venn diagrams: 2.9. Timing diagram of the waveforms that can be observed on all wires of the circuit:
x
x1 1
x3 x2 C x3 x2 x1
x3 x1 + x2 + x3 A f x1 + x2 D
x2
x1 x2 B x3 x1 x1 + x2 + x3 x2
x3 x1 A x2
x3 B ( x1 + x2 + x3 ) ! ( x1 + x2 + x3 )
C
D
f 22
2.10. Starting with the canonical sumofproducts for f get = x1 x2 x3 + x1 x2 x3 + x1 x2 x3 + x1 x2 x3 + x1 x2 x3 + x1 x2 x3 + x1 x2 x3 = x1 (x2 x3 + x2 x3 + x2 x3 + x2 x3 ) + x2 (x1 x3 + x1 x3 + x1 x3 + x1 x3 )
+x3 (x1 x2 + x1 x2 + x1 x2 + x1 x2 ) = x1 (x2 (x3 + x3 ) + x2 (x3 + x3 )) + x2 (x1 (x3 + x3 ) + x1 (x3 + x3 ))
+x3 (x1 (x2 + x2 ) + x1 (x2 + x2 )) =
= x1 (x2 · 1 + x2 · 1) + x2 (x1 · 1 + x1 · 1) + x3 (x1 · 1 + x1 · 1)
x1 (x2 + x2 ) + x2 (x1 + x1 ) + x3 (x1 + x1 ) =
= f x1 · 1 + x2 · 1 + x3 · 1
x1 + x2 + x3 2.11. Starting with the canonical productofsums for f can derive:
= (x1 + x2 + x3 )(x1 + x2 + x3 )(x1 + x2 + x3 )(x1 + x2 + x3 ) ·
(x1 + x2 + x3 )(x1 + x2 + x3 )(x1 + x2 + x3 ) = ((x1 + x2 + x3 )(x1 + x2 + x3 ))((x1 + x2 + x3 )(x1 + x2 + x3 )) ·
((x1 + x2 + x3 )(x1 + x2 + x3 ))((x1 + x2 + x3 )(x1 + x2 + x3 )) = (x1 + x2 + x3 x3 )(x1 + x2 + x3 x3 ) ·
(x1 + x2 + x3 x3 )(x1 + x2 x2 + x3 ) = f (x1 + x2 )(x1 + x2 )(x1 + x2 )(x1 + x3 )
24 = (x1 + x2 x2 )(x1 + x2 x3 )
= x1 (x1 + x2 x3 )
= x1 x1 + x1 x2 x3
= x1 x2 x3
2.12. Derivation of the minimum sumofproducts expression:
f = x1 x3 + x1 x2 + x1 x2 x3 + x1 x2 x3
= x1 (x2 + x2 )x3 + x1 x2 (x3 + x3 ) + x1 x2 x3 + x1 x2 x3
= x1 x2 x3 + x1 x2 x3 + x1 x2 x3 + x1 x2 x3 + x1 x2 x3
= x1 x3 + (x1 + x1 )x2 x3 + (x1 + x1 )x2 x3
= x1 x3 + x2 x3 + x2 x3 2.13. Derivation of the minimum sumofproducts expression:
f = x1 x2 x3 + x1 x2 x4 + x1 x2 x3 x4
= x1 x2 x3 (x4 + x4 ) + x1 x2 x4 + x1 x2 x3 x4
= x1 x2 x3 x4 + x1 x2 x3 x4 + x1 x2 x4 + x1 x2 x3 x4
= x1 x2 x3 + x1 x2 (x3 + x3 )x4 + x1 x2 x4
= x1 x2 x3 + x1 x2 x4 + x1 x2 x4 2.14. The simplest POS expression is derived as
f = (x1 + x3 + x4 )(x1 + x2 + x3 )(x1 + x2 + x3 + x4 )
2.20. The simplest SOP implementation of the function is
= (x1 + x3 + x4 )(x1 + x2 + x3 )(x1 + x2 + x3 + x4 )(x1 + x2 + x3 + x4 )
= x1 + 2
= (x1 + x3f+ x4 )(x1 x2 x3 + x1 x((x3 + x1 x+x34+ x3 x2 x3 ))
3 ) 2 x1 + x2 2 x )( 1 + x3
= (1 + x2 + x x ( + + x2 + 4 ) 1
= (x1 + x3 + x4 )(xx1 + x1 )x23 )3 x1 x1 (x2+ xx2 )·x3
= x2 + + +
= (x1 + x3 + x4 )(x1 x3 x2 x1 x3 )(x1 + x2 + x4 ) 2.15. Derivation of the minimum productofsums expression:
f im l x1 + at + of the + x i + x
2.21. The simplest SOP= p(ementx2ion x3 )(x1funct2on is3 )(x1 + x2 + x3 )(x1 + x2 + x3 )
= ((x1 + x2 ) + x3 )((x1 + x2 ) + x3 )(x1 + (x2 + x3 ))(x1 + (x2 + x3 ))
f = x x2 x 3
= (x1 + x2 )(1 x2+3x+)x1 x2 x3 + x1 x2 x3 + x1 x2 x3 + x1 x2 x3 = x1 (x2 + x2 )x3 + x1 (x2 + x2 )x3 + (x1 + x1 )x2 x3
=in a 3x3a+ax1e Venn 2 x3gram:
2.16. (a) Location of all minterms x1 v ri bl x3 + xdia
Another possibility is
m0 f 2
= x 1 x1 x3 +xx1 x3 + x1 x2 m6 m4
m5 m7 m2
m3 2.22. The simplest POS implementation of the functx 3 n is
io
m1 f = (x1 + x2 + x3 )(x1 + x2 + x3 )(x1 + x2 + x3 )
= ((x1 + x3 ) + x2 )((x1 + x3 ) + x2 )(x1 + x2 + x3 )
= (x1 + x3 )(x1 + x2 + x3 ) 25
2.23. The simplest POS implementation of the function is
f = (x1 + x2 + x3 )(x1 + x2 + x3 )(x1 + x2 + x3 )(x1 + x2 + x3 )
= ((x1 + x2 ) + x3 )((x1 + x2 ) + x3 )((x1 + x3 ) + x2 )((x1 + x3 ) + x2 )
= (x1 + x2 )(x1 + x3 ) 2.24. The simplest SOP expression for the function is
f = x1 x3 x4 + x2 x3 x4 + x1 x2 x3
= x1 x3 x4 + x2 x3 x4 + x1 x2 x3 + x1 x2 x3
= x1 x3 x4 + x2 x3 x4 + x1 x3
= x2 x3 x4 + x1 x3 27 ...
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 Spring '12
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