arithmatic circuitII

# arithmatic circuitII - MULTIPLIER Consider two unsigned...

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MULTIPLIER - Consider two unsigned numbers X and Y that are of M and N bits . - i =M-1 X = X i 2 i i = 0 j =N-1 Y = Y j 2 j j = 0 Z = XY = X ·· Y × Z k 2 k k0 = MN1 + == X i 2 i i0 = M1 ⎝⎠ ⎜⎟ ⎛⎞ Y j 2 j j0 = N1 = X i Y j 2 ij + = = =

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The Binary Multiplication 1 0 1 0 0 0 0 0 × 0 + Partial Products AND operation
Array Multiplier MxN Multiplication viewed as forming N Partial products of M bits each, Each row in the partial product array is either a copy of the multiplicand or a row of zeroes. then summing the appropriately shifted products to produce result P. Generation of N Partial products require NxM two bit AND gates. Binary multiplication is equivalent to a logical AND operation. Most of the area devoted to the adding of the N Partial products, which require N-1 M bit adders.

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The Array Multiplier HA FA X 0 X 1 X 2 X 3 Y 1 X 0 X 1 X 2 X 3 Y 2 X 0 X 1 X 2 X 3 Y 3 Z 1 Z 2 Z 3 Z 4 Z 5 Z 6 Z 0 Z 7

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2’s compliment array Multiplication M-2 N-2 P = (-y M-1 2 M-1 + y j 2 j ) (-x N-1 2 N-1 + x i 2 i ) j = 0 I = 0 N-2 M-2 = ∑∑ x i y j 2 i+j + x N-1 y M-1 2 M+N-2 i=0 j=0 N-2 M-2 -( x i y M-1 2 i+M-1 + x N-1 y j 2 j+N-1 ) i=0 j=0 Two of the partial products have negative weight and thus should be subtracted .

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Y5 Y4 Y3 Y2 Y1 Y0 X5 X4 X3 X2 X1 X0 X0Y4 X0Y3 X0Y2 X0Y1 X0Y0 X1Y4 X1Y3 X1Y2 X1Y1 X1Y0 X2Y4 X2Y3 X2Y2 X2Y1 X2Y0 X3Y4 X3Y3 X3Y2 X3Y1 X3Y0 X4Y4 X4Y3 X4Y2 X4Y1 X4Y0 X5Y5 1 1 X4Y5 X3Y5 X2Y5 X1Y5 X0Y5 1 1 1 1 1 1 1 1 X5Y4 X5Y3 X5Y2 X5Y1 X5Y0 1 1 1 1 1 1 Subtraction done by taking 2’s compliment of the terms
Y5 Y4 Y3 Y2 Y1 Y0 X5 X4 X3 X2 X1 X0 1 X5Y0 X0Y4 X0Y3 X0Y2 X0Y1 X0Y0 X5Y1 X1Y4 X1Y3 X1Y2 X1Y1 X1Y0 X5Y2 X2Y4 X2Y3 X2Y2 X2Y1 X2Y0 X5Y3 X3Y4 X3Y3 X3Y2 X3Y1 X3Y0 X5Y4 X4Y4 X4Y3 X4Y2 X4Y1 X4Y0 1 X5Y5 X4Y5 X3Y5 X2Y5 X1Y5 X0Y5 P11 P10 P0

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The MxN Array Multiplier — Critical Path HA FA FA HA HA FA FA FA FA FA FA HA Critical Path 1 Critical Path 2 Critical Path 1 & 2 All critical paths have same length speeding up one by a faster adder does not make sense .

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## This note was uploaded on 09/24/2010 for the course EEE MEL G621 taught by Professor Gurunarayanan during the Fall '08 term at Birla Institute of Technology & Science, Pilani - Hyderabad.

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arithmatic circuitII - MULTIPLIER Consider two unsigned...

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