Essay Example on Radix 4 Booth Multiplication To improve the Performance

Radix 4 Booth Multiplication To improve the performance of multiplication process encoding is used Booth encoder plays an important role in Booth's multiplier as it reduces the number of partial product stages Consider the multiplication of two N bit numbers multiplicand X and multiplier Y in two's complement form as N 2 X xN 12N 1 ai 2i 1 i 0 N 2 Y yN 12N 1 bi 2i 2 i In this structure each group is encoded and decoded by selecting by 1 2 or 0 instead of shifting and adding for every column of the multiplier term and multiplying to obtain the same result which is shown in table 2 1 Table 2 1 Function table of Booth Multiplier Radix 4 booth encoder performs the process of encoding the multiplicand based on multiplier bits Using overlapping technique 3 bits at a time is compared Grouping of the multiplier bit to obtain the encoding value starts from the LSB of the multiplier and the first block only uses two bits of the multiplier and assumes a zero for the third bit as shown in figure 2 2 111000110 Figure 2 2 3 bit pairing as per booth recoding The number of partial products can be reduced to n 2 if two n bits numbers are multiplied if n is even number or n 1 2 if n is an odd number by using modified radix 4 booth multiplier 

This is useful to perform operation using higher order bits Signed number operation takes place when s_u bit 1 and when s_u 0 it indicates unsigned operation which is shown in table 2 2 From this it is to be noted that when unsigned operation takes place for both multiplier and multiplication the sign extended bit must be extended to zero i e x16 x17 y16 y17 0 to avoid false product generation Similarly when signed multiplication operation takes place the sign bit depends on the nature of the bits used in multiplication multiplicand or multiplier is negative or both the operands are negative For signed bit s_u 1 x15 y16 y16 1 y16 x17 x16 0 Table 2 2 Sign conversion operation Signed_unsigned s_u Operation 0 Unsigned multiplication 1 Signed multiplication Figure 2 3 Logic diagram of sign converter Signed unsigned modified booth multiplication process is shown in figure 2 3 The partial products are generated by using the circuit in figure 2 2 All the partial products are generated in parallel x15x14 x2x1x0 y15y14 y2y1y0 P016P016 P002P001P000 X1 1P116P115 P114 P103P102P001N0 X2 P816P815 P803P802P801N X9 P31P30P29 P3P2P1P0 Figure 2 3 16 x16 multiplier for signed unsigned numbers 3 WALLACE TREE STRUCTURE Wallace tree structure is used to generate the partial products from the given multiplier and multiplicand bit Due to the use of Wallace tree structure the organisation and perfomance of booth multipliers is enhanced Thid section explains about the structure of Wallace tree multiplier from which the partial product generation is used in Modified Radix 4 Booth Multiplier structure 3 1 Wallace Tree Multiplier Like Booth multipliers Wallace Tree Multipliers are also used in high speed multiplication process Since it reduces the partial product stages by employing CSA Carry Save Adder structure it has better performance It can operate only on unsigned numbers since it does not use any encoding technique to encode the multiplier bit as applied in Booth Multiplier By using column compression technique numbers of partial product stages are reduced and the total delay is proportional to the logarithm of word length of multiplier bit The basic process of Wallace Tree Multiplier is shown in figure 3 1