Systems Multipliers and Other Circuits Shankar Balachandran* - PowerPoint PPT Presentation

Spring 2015 Week 9 Module 52 Digital Circuits and Systems Multipliers and Other Circuits Shankar Balachandran* Associate Professor, CSE Department Indian Institute of Technology Madras *Currently a Visiting Professor at IIT Bombay

Arithmetic and Logic Unit (ALU)  An ALU is a combinational circuit that can perform any of a number of different arithmetic and logical operation on a pair of n -bit operands.  Operation to be performed is specified by a set of control (function-select) inputs.  ALUs are designed in a bit-slice fashion and slices can be cascaded to form larger ALUs. Some ALUs are designed as an m -bit group instead of a single bit slice.  m -bit group enables us to improve carry propagation within a group using carry lookahead or other similar enhancements. Arithmetic Circuits 2

Unsigned Binary Multiplication  Consider unsigned m - and n -bit integers, X and Y   m 1 n 1 i j     X x 2 Y y 2 i j   i 0 j 0             m n 1 m 1 n 1 m 1 n 1    k i j i j         P X Y  p 2  x 2  y 2   x y 2   k i j i j          k 0 i 0 j 0 i 0 j 0 Arithmetic Circuits 3

Array Multiplier x 3 x 2 x 1 x 0 y 0 y 1 HA FA FA HA y 2 FA FA FA HA y 3 FA FA FA HA p 7 p 5 p 6 p 4 p 3 p 2 p 0 p 1 Arithmetic Circuits 4

Performance of Array Multiplier m x 3 x 2 x 1 x 0 y 0 y 1 HA FA FA HA n y 2 Critical path 1 FA FA FA HA Critical path 2 y 3 FA FA FA HA p 7 p 5 p 6 p 4 p 3 p 2 p 0 p 1         sum        t t m 1 n 2 t n 1 t mult and carry Arithmetic Circuits 5

Signed Multiplication Positive Multiplicand Negative Multiplicand 0 1 1 1 0 1 0 0 1 0 (-14) (+14) (+11) (+11) x 0 1 0 1 1 x 0 1 0 1 1 0 0 0 0 1 0 0 1 0 1 1 1 1 0 1 1 1 0 0 0 0 1 0 0 1 0 1 1 1 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 1 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 + + 0 1 0 0 1 1 0 1 0 1 0 1 1 0 0 1 1 0 (-154) (+154) Negative Multiplier: Negate multiplier and multiplicand and use the above scheme for positive multiplier. Arithmetic Circuits 7

Modular Multiplication Using small multipliers to build larger multipliers  A ≡ a 7 a 6 a 5 a 4 a 3 a 2 a 1 a 0 ≡ A 7-4 A 3-0 B ≡ b 7 b 6 b 5 b 4 b 3 b 2 b 1 b 0 ≡ B 7-4 B 3-0 A B A 7-4 A 3-0 x B 7-4 B 3-0 x A 7-4 *B 3-0 A 3-0 *B 3-0 A 3-0 * B 3-0 4 × 4 multiplier A 7-4 *B 7-4 A 7-4 * B 3-0 A 3-0 *B 7-4 4 × 4 multiplier A 3-0 * B 7-4 4 × 4 multiplier A 7-4 * B 7-4 4 × 4 multiplier A 3-0 * B 7-4 P 15-12 P 11-8 P 7-4 P 3-0 A 7-4 * B 7-4 A 3-0 * B 3-0 A 7-4 * B 3-0 a more convenient arrangement Arithmetic Circuits 8

Adding Three 4-Bit Numbers ( X, Y, Z ) x 3 y 3 z 3 x 2 y 2 z 2 x 1 y 1 z 1 x 0 y 0 z 0 FA FA FA FA FA FA FA 0 s 0 s 3 s 2 s 1 Arithmetic Circuits 9

Comparator – Equality Check      x ... x x x y ... y y y ?  Iterative algorithm: n  1 2 1 0 n  1 2 1 0   1. i 0; EQ 1; x i y i 0        2. if x y EQ 1 i i i  then EQ 1;  i 1 EQ i+1  else EQ 0; EQ i  i 1   3. i i 1;    4. i n if 2; then go to step else end; x n-1 y n-1 y 2 x 2 y 1 x 0 y 0 x 1 x y x x y x y y EQUIV EQUIV EQUIV EQUIV EQ n EQ n-1 EQ 3 EQ 0 EQ 2 EQ 1 EQO EQI EQO EQI EQO EQI EQO EQI Arithmetic Circuits 10

4-Bit Comparator  Magnitude comparator for n -bit unsigned binary numbers  Inputs: a 3 a 2 a 1 a 0 4 4 b 3 b 2 b 1 b 0 b 3- 0 a 3- 0 altb in aeqb in agtb in altb in altb in  Outputs: COMP4 aeqb in aeqb in altb out aeqb out agtb out agtb in agtb in  Logic Equations:        agtb a > b a = b ag tb     out 3 0 3 0 3 0 3 0 in     a eqb a = b aeqb out 3 0  3 0  in        altb a b a b altb < =     out 3 0 3 0 3 0 3 0 in                       a b a b a b a b a b a b a b a b a b a b a b   3 0 3 0 3 3 3 3 2 2 3 3 2 2 1 1 3 3 2 2 1 1 0 0              a b a b a b a b a b   3 0 3 0 3 3 2 2 1 1 0 0           a b a b a b 3 0  3 0  3 0  3 0  3 0  3 0  Arithmetic Circuits 11

12-Bit Comparator (unsigned) x 11…0 y 11…0 y 3…0 x 11…8 y 11…8 y 7…4 x 3…0 x 7…4 4 4 4 4 4 4 b 3- 0 a 3- 0 b 3- 0 a 3- 0 b 3- 0 a 3- 0 XLTY8 XLTY XLTY4 0 altb in altb in altb in altb in altb in altb in XEQY8 XEQY4 XEQY COMP4 COMP4 COMP4 aeqb in aeqb in aeqb in aeqb in aeqb in 1 aeqb in XGTY8 XGTY XGTY4 agtb in agtb in agtb in agtb in agtb in agtb in 0 Arithmetic Circuits 12

MIN/MAX Circuit  Use a 4-bit comparator to design an 8-bit MIN/MAX circuit.  Inputs: unsigned inputs, X ≡ x 7..0 , Y ≡ y 7..0 and a control signal MINMAX    Output:   min X Y , MINMAX 1 if    Z    max X Y , MINMAX 0 if   MINMAX X>Y MINMAX Z S 0 0 1 Y X COMP 12 0 1 X 0 X>Y Y 1 0 X 0 12 S 1 1 1 Y 0 Z MUX 1 12 Arithmetic Circuits 13

End of Week 9: Module 52 Thank You Multipliers+Others 14