ECE354 S'09 hw9 - sol

ECE354 S'09 hw9 - sol - NEW JERSEY INSTITUTE OF TECHNOLOGY...

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NEW JERSEY INSTITUTE OF TECHNOLOGY DEPT. OF ELECTRICAL & COMPUTER ENGINEERING ACADEMIC YEAR 2008-2009 SEMESTER 2 ECE354 DIGITAL TEST S OLUTION HW9 1. We compute the signal probability expressions for all lines in the circuit. Since inputs A and D have single paths to the output, they need not be assigned a symbol. We can assign to each their numeric value 2 1 . All the other inputs need to be assigned a symbolic value. A B C D F 1 g 2 g 3 g 4 g 5 g 6 g 7 g 8 g 9 g 10 g 11 g C p B p 2 1 2 1 Fig. 1. Signal probability assignment to the input lines Let i E be the signal probability expression for line i g . B p E E = = 2 1 , C p E E = = 4 3 2 1 2 1 1 1 5 B p E E - = × - = , C B p p E E E E E = = = = 3 2 9 8 6 ) 1 ( 2 1 2 1 1 ) 1 ( 4 7 C p E E - = - - = C B B C B C B C B B C B B C B B p p p p p p p p p p p p p p p p E E E E E 2 2 8 5 8 5 10 2 1 2 1 2 1 2 1 2 1 2 1 + - = + - + - = - - + - = - + = After exponent suppression: ) 1 ( 2 1 1 2 1 2 1 ' 10 C B C B B p p p p p E - - = + - = 1

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( 29 2 9 7 9 7 11 2 1 2 1 2 2 1 ) 1 ( 2 1 1 2 1 C B C B C B C C C B C B C p p p p p p p p p p p p p E E E E E + - + - = - - + - = - + = After suppressing high-order exponents of C p , we get: C B C C B C B C B C p p p p p p p p p p E + - = + - + - = 2 1 2 1 2 1 2 2 1 ' 11 C B C B C B C B C B C B C C B C C B B F p p p p p p p p p p p p p p p p p p p E E E 2 2 2 11 10 2 1 4 1 4 1 2 1 ) 1 ( 4 1 2 1 2 1 2 1 2 1 ' ' + - + - - - + - = + - + - = = After exponent suppression: 4 5 2 2 4 5 4 2 2 1 2 1 4 1 4 1 2 1 4 1 4 1 2 2 1 ' C B B C C B B C C B C B C B C B C B B C B C F p p p p p p p p p p p p p p p p p p p p p p E + - - = + - - = + - + - + - + - = By substituting 2 1 = = C B p p we get: 2 1 4 3 2 1 = = = = = = = = g g g g D C B A p p p p p p p p 4 3 5 = g p , 4 1 9 8 7 6 = = = = g g g g p p p p , 8 7 10 = g p 2 1 11 = g p , 16 7 = F p 2. There are 3 fanout stems with two fanout branches each. The cutting algorithm requires turning the circuit into a tree by cutting branches. Therefore, there are 8 cutting patterns of 3 branches each: { } 8 3 1 , , g g g , { } 9
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