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l8_arithmetic

l8_arithmetic - L8 Arithmetic Structures Structures...

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L8: Arithmetic Structures L8: Arithmetic Structures Acknowledgements: - J. Rabaey, A. Chandrakasan, B. Nikolic. Digital Integrated Circuits: A Design Perspective . Prentice Hall, 2003. - Kevin Atkinson, Rex Min L8: 6.111 Spring 2004 Introductory Digital Systems Laboratory 1 ., Materials in this lecture are courtesy of the following people and used with permission. Computer Science) - Gaetano Borriello (University of Washington, Department of Computer Science & Engineering, http://www.cs.washington.edu/370)
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Number Systems Basics Number Systems Basics How to represent negative numbers? ± Three common schemes: sign-magnitude, ones complement, twos complement ± Sign-magnitude: MSB = 0 for positive, 1 for negative ² Range: -(2 N-1 – 1) to +(2 N-1 –1) ² Two representations for zero: 0000… & 1000… ² Simple multiplication but complicated addition/subtraction _ ± Ones complement: if N is positive then its negative is N ² Example: 0111 = 7, 1000 = -7 ² Range: -(2 N-1 – 1) to +(2 N-1 –1) ² Two representations ² Subtraction implemented as addition followed by ones complement L8: 6.111 Spring 2004 Introductory Digital Systems Laboratory 2
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Twos Complement Representation Twos Complement Representation Twos complement = bitwise complement + 1 0000 0001 0010 0011 1000 0101 0110 0100 1001 1010 1011 1100 1101 0111 1110 1111 -1 +0 -2 +1 0111 1000 + 1 = 1001 = -7 -3 +2 1001 0110 + 1 = 0111 = 7 -4 +3 -5 +4 ± Asymmetric range: -2 N-1 to +2 N-1 -1 ± Only one representation for zero -6 +5 ± Simple addition and subtraction -7 +6 ± Most common representation -8 +7 4 0100 -4 1100 4 0100 -4 1100 + 3 0011 + (-3) 1101 -3 1101 + 3 0011 7 0111 -7 11001 1 10001 -1 1111 [Katz93, chapter 5] L8: 6.111 Spring 2004 Introductory Digital Systems Laboratory 3
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Overflow Conditions Overflow Conditions Add two positive numbers to get a negative number or two negative numbers to get a positive number 0000 0001 0010 0011 1000 0101 0110 0100 1001 1010 1011 1100 1101 0111 1110 1111 -1 +0 -2 +1 -3 +2 -4 0000 0001 0010 0011 1000 0101 0110 0100 1001 1010 1011 1100 1101 0111 1110 1111 +0 -1 -2 +1 -3 +2 -4 +3 +3 -5 -5 +4 +4 -6 -6 +5 +5 -7 +6 -7 +6 -8 +7 -8 +7 5 + 3 = -8! -7 - 2 = +7! 0 1 1 1 10 0 0 5 0 1 0 1 -7 1 0 0 1 3 0 0 1 1 -2 1 1 0 0 0 1 1 1 -8 0 1 0 0 0 7 1 If carry in to sign equals carry out then can ignore carry out, otherwise have overflow L8: 6.111 Spring 2004 Introductory Digital Systems Laboratory 4
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Binary Full Adder Binary Full Adder A B Full Adder S = A B C i C o = ABC i + ABC i + ABC i + ABC i C i C o = AB + C i (A+B) S A B CI 0 0 0 0
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