hw1_solutions

# hw1_solutions - x x y y x!y x x x y y Homework 1 Solutions...

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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 2-2 2.10. Starting with the canonical sum-of-products 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 product-of-sums 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 ) 2-4 = (x1 + x2 x2 )(x1 + x2 x3 ) = x1 (x1 + x2 x3 ) = x1 x1 + x1 x2 x3 = x1 x2 x3 2.12. Derivation of the minimum sum-of-products 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 sum-of-products 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 product-of-sums 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 ) 2-5 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 2-7 ...
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