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Topics. Multipliers. Elementary school algorithm. 0 1 1 0 multiplicand x 1 0 0 1 multiplier 0 1 1 0 + 0 0 0 0 0 0 1 1 0 + 0 0 0 0 0 0 0 1 1 0 + 0 1 1 0 0 1 1 0 1 1 0. partial product. Word serial multiplier. +. register.

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  1. Topics • Multipliers.

  2. Elementary school algorithm 0 1 1 0 multiplicand x 1 0 0 1 multiplier 0 1 1 0 + 0 0 0 0 0 0 1 1 0 + 0 0 0 0 0 0 0 1 1 0 + 0 1 1 0 0 1 1 0 1 1 0 partial product

  3. Word serial multiplier + register

  4. Combinational multiplier Uses n-1 adders, eliminates registers:

  5. Array multiplier • Array multiplier is an efficient layout of a combinational multiplier. • Array multipliers may be pipelined to decrease clock period at the expense of latency.

  6. Array multiplier organization 0 1 1 0 x 1 0 0 1 0 1 1 0 + 0 0 0 0 0 0 1 1 0 + 0 0 0 0 0 0 0 1 1 0 0 1 1 0 0 1 1 0 1 1 0 multiplicand multiplier skew array for rectangular layout product

  7. Unsigned array multiplier x2y0 x1y0 x0y0 0 0 x1y1 x0y1 + + x1y2 x0y2 + + xn-1yn-1 0 + + P(2n-1) P0 P(2n-2)

  8. Unsigned array multiplier, cont’d

  9. Array multiplier critical path

  10. Verilog for multiplier row module multrow(part,x,ym,yo,cin,s,cout); /* A row of one-bit multiplies */ input [2:0] part; input [3:0] x; input ym, yo; input [2:0] cin; output [2:0] s; output [2:0] cout; assign {cout[0],s[0]} = part[1] + (x[0] & ym) + cin[0]; assign {cout[1],s[1]} = part[2] + (x[1] & ym) + cin[1]; assign {cout[2],s[2]} = (x[3] & yo) + (x[2] & ym) + cin[2]; endmodule

  11. Verilog for last multiplier row module lastrow(part,cin,s,cout); /* Last row of adders with full carry chain. */ input [2:0] part; input [2:0] cin; output [2:0] s; output cout; wire [1:0] carry; assign {carry[0],s[0]} = part[0] + cin[0]; assign {carry[1],s[1]} = part[1] + cin[1] + carry[0]; assign {cout,s[2]} = part[2] + cin[2] + carry[1]; endmodule

  12. Verilog for multiplier module array_mult(x,y,p); input [3:0] x; input [3:0] y; output [7:0] p; wire [2:0] row0, row1, row2, row3, c0, c1, c2, c3; /* generate first row of products */ assign row0[2] = x[2] & y[0]; assign row0[1] = x[1] & y[0]; assign row0[0] = x[0] & y[0]; assign p[0] = row0[0]; assign c0 = 3’b000; multrow p0(row0,x,y[1],y[0],c0,row1,c1); assign p[1] = row1[0]; multrow p1(row1,x,y[2],y[1],c1,row2,c2); assign p[2] = row2[0]; multrow p2(row2,x,y[3],y[2],c2,row3,c3); assign p[3] = row3[0]; lastrow l({x[3] & y[3],row3[2:1]},c3,p[6:4],p[7]); endmodule

  13. Baugh-Wooley multiplier • Algorithm for two’s-complement multiplication. • Adjusts partial products to maximize regularity of multiplication array. • Moves partial products with negative signs to the last steps; also adds negation of partial products rather than subtracts.

  14. Booth multiplier • Encoding scheme to reduce number of stages in multiplication. • Performs two bits of multiplication at once—requires half the stages. • Each stage is slightly more complex than simple multiplier, but adder/subtracter is almost as small/fast as adder.

  15. Booth encoding • Two’s-complement form of multiplier: • y = -2nyn + 2n-1yn-2 + 2n-2yn-2 + ... • Rewrite using 2a = 2a+1 - 2a: • y = -2n(yn-1-yn) + 2n-1(yn-2 -yn-1) + 2n-2(yn-3 -yn-2) + ... • Consider first two terms: by looking at three bits of y, we can determine whether to add/subtract x, 2x to partial product.

  16. Booth actions yi yi-1 yi-2 increment 0 0 0 0 0 0 1 x 0 1 0 x 0 1 1 2x 1 0 0 -2x 1 0 1 -x 1 1 0 -x 1 1 1 0

  17. Booth example • x = 011001 (2510), y = 101110 (-1810). • y1y0y-1 = 100, P1 = P0 - (10  011001) = 11111001110. • y3y2y1= 111, P2 = P1 0 = 11111001110. • y5y4y3= 101, P3 = P2 - 0110010000 = 11000111110.

  18. Booth structure

  19. Wallace tree • Reduces depth of adder chain. • Built from carry-save adders: • three inputs a, b, c • produces two outputs y, z such that y + z = a + b + c • Carry-save equations: • yi = parity(ai,bi,ci) • zi = majority(ai,bi,ci)

  20. Wallace tree structure

  21. Wallace tree operation • At each stage, i numbers are combined to form ceil(2i/3) sums. • Final adder completes the summation. • Wiring is more complex. • Can build a Booth-encoded Wallace tree multiplier.

  22. Serial-parallel multiplier • Used in serial-arithmetic operations. • Multiplicand can be held in place by register. • Multiplier is shfited into array.

  23. Serial-parallel multiplier structure

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