# Design and Implementation of Radix4 Booth Algorithm Using Vhdl

**Topics:**Addition, Elementary arithmetic, Multiplication

**Pages:**3 (380 words)

**Published:**May 26, 2011

• Notice the following equality (Booth did)

• 2J + 2J–1 + 2J–2 + … + 2K = 2J+1 – 2K • Example: 0111 = 1000 - 0001 • We can exploit this to create a faster multiplier

• How?

• Sequence of N 1s in the multiplier yields sequence of N additions • Replace with one addition and one subtraction

© 2009 Daniel J. Sorin from Roth and Lebeck

ECE 152

44

Booth In Action

• For each multiplier bit, also examine bit to its right

• • • • 00: 10: 11: 01: middle of a run of 0s, do nothing beginning of a run of 1s, subtract multiplicand middle of a run of 1s, do nothing end of a run of 1s, add multiplicand

* + + +

43 12 0 0 172 0 688 516

= = = = = = = =

00000101011 00000001100 00000000000 00000000000 11101010100 00000000000 01010110000 01000000100

// // // // //

multiplier multiplier multiplier multiplier multiplier

bits bits bits bits bits

0_ (implicit 0) 00 10 11 01

ICQ: so why is Booth better?

© 2009 Daniel J. Sorin from Roth and Lebeck ECE 152 45

Booth Hardware

Multiplicand

16 16 + or – 16± Multiplier 16 16 Product 32>>1 control 2

16

• Control algorithm: repeat 16 times

• Multiplier LSBs == 10? Subtract multiplicand from product • Multiplier LSBs == 01? Add multiplicand to product • Shift product/multiplier right by 1 (not by 2!) © 2009 Daniel J. Sorin from Roth and Lebeck ECE 152 46

Booth in Summary

• Performance/efficiency

+ Good for sequences of 3 or more 1s • Replaces 3 (or more) adds with 1 add and 1 subtract • Doesn’t matter for sequences of 2 1s • Replaces 2 adds with 1 add and 1 subtract (add = subtract) – Actually bad for singleton 1s • Replaces 1 add with 1 add and 1 subtract

• Bottom line

• Worst case multiplier (101010) requires N/2 adds + N/2 subs • What is the worst case multiplier for straight multiplication? • How is this better than normal multiplication?

© 2009 Daniel J. Sorin from Roth and Lebeck

ECE 152

47

Modified Booth’s Algorithm

• What if we detect...

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