Lec-09-Arith - Motivation ArithmeticCircuits .logicdesign...

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Contemporary Logic Design Arithmetic Circuits © R.H. Katz   Transparency No. 5-1 Motivation Arithmetic circuits are excellent examples of comb. logic design •   Time vs. Space Trade-offs     Doing things fast requires more logic and thus more space     Example: carry lookahead logic  •   Arithmetic Logic Units      Critical component of processor datapath        Inner-most "loop" of most computer instructions
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Contemporary Logic Design Arithmetic Circuits © R.H. Katz   Transparency No. 5-2 Chapter Overview Binary Number Representation Binary Addition    Full Adder Revisted ALU Design BCD Circuits Combinational Multiplier Circuit Design Case Study: 8 Bit Multiplier
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Contemporary Logic Design Arithmetic Circuits © R.H. Katz   Transparency No. 5-3 Number Systems Representation of Negative Numbers Representation of positive numbers same in most systems Major differences are in how negative numbers are represented Three major schemes: sign and magnitude ones complement twos complement Assumptions: we'll assume a 4 bit machine word 16 different values can be represented roughly half are positive, half are negative
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Contemporary Logic Design Arithmetic Circuits © R.H. Katz   Transparency No. 5-4 Number Systems Sign and Magnitude Representation 0000 0111 0011 1011 1111 1110 1101 1100 1010 1001 1000 0110 0101 0100 0010 0001 +0 +1 +2 +3 +4 +5 +6 +7 -0 -1 -2 -3 -4 -5 -6 -7 0 100 = + 4 1 100 = - 4 + - High order bit is sign: 0 = positive (or zero), 1 = negative Three low order bits is the magnitude: 0 (000) thru 7 (111) Number range for n bits = +/-2      -1 Representations for 0 n-1
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Contemporary Logic Design Arithmetic Circuits © R.H. Katz   Transparency No. 5-5 Number Systems Sign and Magnitude Cumbersome addition/subtraction Must compare magnitudes to determine sign of result Ones Complement N is positive number, then N is its negative 1's complement N = (2   - 1) - N n Example: 1's complement of 7 2     =  10000 -1    =  00001              1111 -7    =    0111              1000 = -7 in 1's comp. Shortcut method:       simply compute bit wise complement       0111 -> 1000 4
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Contemporary Logic Design Arithmetic Circuits © R.H. Katz   Transparency No. 5-6 Number Systems Ones Complement Still two representations of 0!  This causes some problems Some complexities in addition 0000 0111 0011 1011 1111 1110 1101 1100 1010 1001 1000 0110 0101 0100 0010 0001 +0 +1 +2 +3 +4 +5 +6 +7 -7 -6 -5 -4 -3 -2 -1 -0 0 100 = + 4 1 011 = - 4 + -
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Contemporary Logic Design Arithmetic Circuits © R.H. Katz   Transparency No. 5-7
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  • Spring '10
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  • Pallavolo Modena, Sisley Volley Treviso, Associazione Sportiva Volley Lube, R.H. Katz   Transparency, Katz   Transparency No.

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Lec-09-Arith - Motivation ArithmeticCircuits .logicdesign...

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